diff --git a/-dE2T4oBgHgl3EQfQgYx/content/tmp_files/2301.03770v1.pdf.txt b/-dE2T4oBgHgl3EQfQgYx/content/tmp_files/2301.03770v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4b499a161196bf49c152705f9ecd08ac2e18a6a0 --- /dev/null +++ b/-dE2T4oBgHgl3EQfQgYx/content/tmp_files/2301.03770v1.pdf.txt @@ -0,0 +1,3142 @@ +arXiv:2301.03770v1 [cs.DB] 10 Jan 2023 +Scalable Time-Range 푘-Core Qery on Temporal Graphs +Junyong Yang +Wuhan University +Wuhan, China +thomasyang@whu.edu.cn +Ming Zhong∗ +Wuhan University +Wuhan, China +clock@whu.edu.cn +Yuanyuan Zhu +Wuhan University +Wuhan, China +yyzhu@whu.edu.cn +Tieyun Qian +Wuhan University +Wuhan, China +qty@whu.edu.cn +Mengchi Liu +South China Normal University +Guangzhou, China +liumengchi@scnu.edu.cn +Jeffery Xu Yu +The Chinese University of Hong +Kong +Hong Kong, China +yu@se.cuhk.edu.hk +ABSTRACT +Querying cohesive subgraphs on temporal graphs with various +time constraints has attractedintensive research interests recently. +In this paper, we study a novel Temporal 푘-Core Query (TCQ) +problem: given a time interval, find all distinct 푘-cores that exist +within any subintervals from a temporal graph, which general- +izes the previous historical 푘-core query. This problem is chal- +lenging because the number of subintervals increases quadrati- +cally to the span of time interval. For that, we propose a novel +Temporal Core Decomposition (TCD) algorithm that decremen- +tally induces temporal 푘-cores from the previously induced ones +and thus reduces “intra-core” redundant computationsignificantly. +Then, we introduce an intuitive concept named Tightest Time +Interval (TTI) for temporal 푘-core, and design an optimization +technique with theoretical guarantee that leverages TTI as a key +to predict which subintervals will induce duplicated푘-cores and +prunes the subintervals completely in advance, thereby eliminat- +ing “inter-core” redundant computation. The complexity of op- +timized TCD (OTCD) algorithm no longer depends on the span +of query time interval but only the scale of final results, which +means OTCD algorithm is scalable. Moreover, we propose a com- +pact in-memory data structure named Temporal Edge List (TEL) +to implement OTCD algorithm efficiently in physical level with +bounded memory requirement. TEL organizes temporal edges +in a “timeline” and can be updated instantly when new edges ar- +rive, and thus our approach can also deal with dynamic temporal +graphs. We compare OTCD algorithm with the incremental his- +torical 푘-core query on several real-world temporal graphs, and +observe that OTCD algorithm outperforms it by three orders of +magnitude, even though OTCD algorithm needs none precom- +puted index. +1 +INTRODUCTION +1.1 +Motivation +Discovering communities or cohesive subgraphs from temporal +graphs has great values in many application scenarios, thereby +attracting intensive research interests [1, 5, 12, 19, 25, 27, 34, +∗The corresponding author. +This work is licensed under the Creative Commons BY-NC-ND 4.0 International +License. Visit https://creativecommons.org/licenses/by-nc-nd/4.0/ to view a copy +of this license. For any use beyond those covered by this license, obtain +permission by emailing info@vldb.org. Copyright is held by the owner/author(s). +Publication rights licensed to the VLDB Endowment. +Proceedings of the VLDB Endowment, Vol. 14, No. 1 ISSN 2150-8097. +doi:XX.XX/XXX.XX +v10 +v1 +v2 +v3 +v4 +v5 +v6 +v7 +v8 +v9 +1 +1 +6 +6 +6 +6 +5 +5 +2 +2 +2 +2 +7 +7 +2 +6 +5 +5 +4 +3 +5 +5 +5 +5 +3 +2 +2 +8 +4 +1 +2-core of time interval [1,8] +2-core of time interval [5,6] +2-core of time interval [2,4] +2-core of time interval [2,6] +Figure 1: A running example of temporal graph. +36] in recent years. Here, a temporal graph refers to an undi- +rected multigraph in which each edge has a timestamp to indi- +cate when it occurred, as illustrated in Figure 1. For example, +consider a graph consisting of bank accounts as vertices and +fund transfer transactions between accounts as edges with natu- +ral timestamps. For applications such as anti-money-laundering, +we would like to search communities like 푘-cores that contain +a known suspicious account and emerge within a specific time +interval like the World Cup, and investigate the associated ac- +counts. +To address the community query/search problem for a fixed +time interval, the concept of historical 푘-core [36] is proposed +recently, which is the 푘-core induced from the subgraph of a +temporal graph in which all edges occurred out of the time in- +terval have been excluded and the parallel edges between each +pair of vertices have been merged. Also, the PHC-Query method +is proposed to deal with historical 푘-core query/search by using +a precomputed index efficiently. +However, we usually do not know the exact time interval of +targeted historical 푘-core in real-world applications. Actually, if +we can know the exact time interval, a traditional core decom- +position on the projected graph over the given time interval is +efficient enough to address the problem. Thus, it is more reason- +able to assume that we can only offer a flexible time interval +and need to induce cores from all its subintervals. For example, +for detecting money laundering by soccer gambling during the +World Cup, the 푘-cores emerged over a few of hours around one +of the matches are more valuable than a large 푘-core emerging +over the whole month. +Therefore, we aim to generalize historical 푘-core query by al- +lowing the result 푘-cores to be induced by any subinterval of a +given time interval, like “flexible versus fixed”. The historical 푘- +core query can be seen as a special case of our problem that only +evaluates the whole interval. Consider the following example. + +Junyong Yang, Ming Zhong, Yuanyuan Zhu, Tieyun Qian, Mengchi Liu, and Jeffery Xu Yu +Example 1. As illustrated in Figure 1, given a time interval +[1,8], historical 푘-core query only returns the largest core marked +by the grey dashed line. In contrast, our temporal 푘-core query re- +turns four cores marked by dashed lines with different colors. These +cores can reveal various insights unseen by the largest one. For ex- +ample, some cores like red and blue that emerge in bursty periods +may be caused by special events. Also, some persistent or periodic +cores may be found. Further, we can analyze the interaction be- +tween cores and how they evolve over time, such as the small cores +like red and blue are merged to the large cores like yellow. Lastly, +some underlying details may be found. During the merge, the ver- +tex 푣5 may play a vital role because it appears in all the cores. +Overall, our general and flexible query model can support many +interesting temporal community analytics applications. +The general and flexible temporal k-core query we study is +naturally a generalization of existing query models like histori- +cal 푘-core and also potentially a common technique for various +temporal graph mining tasks mentioned in the above example. +1.2 +Contribution +In this paper, we study a novel temporal 푘-core query problem: +given a time interval, find all distinct 푘-cores that exist within +any subintervals from a temporal graph. Although the existing +PHC-Query returns the historical 푘-core of a fixed time inter- +val efficiently, it cannot be trivially applied to deal with the new +problem. Because inducing 푘-cores for each subinterval individ- +ually from scratch is not scalable, since the number of subinter- +vals increases quadratically with the span of time interval. More- +over, PHC-Query suffers from two other intrinsic shortcomings. +Firstly, it relies on a PHC-Index that precomputes the coreness +of all vertices over all time intervals, thereby incurring heavy of- +fline time and space overheads. Secondly, due to its sophisticated +construction, it is unclear if PHC-Index can be updated dynami- +cally. It is against the dynamic nature of temporal graphs. +In order to overcome the above challenges, we present a novel +temporal core decomposition algorithm and auxiliary optimiza- +tion and implementation techniques. Our contributions can be +summarized as follows. +• We formalize a general time-range cohesive subgraph query +problem on ubiquitous temporal graphs, namely, tempo- +ral푘-core query. Many previous typical푘-core query mod- +els on temporal graphs can be equivalently represented +by temporal 푘-core query with particular constraints. +• To address temporal 푘-core query, we propose a simple +and yet efficient algorithm framework based on a novel +temporal core decomposition operation. By using tempo- +ral core decomposition, our algorithm always decremen- +tally induces a temporal k-core from the previous induced +temporal k-core except the initial one, thereby reducing +redundant computation significantly. +• Moreover, we propose an intuitive concept named tight- +est time interval for temporal k-core. According to the +properties of tightest time intervals, we design three prun- +ing rules with theoretical guarantee to directly skip subin- +tervals that will not induce distinct temporal 푘-core. As +a result, the optimized algorithm is scalable in terms of +the span of query time interval, since only the necessary +subintervals are enumerated. +• For physical implementation of our algorithm, we pro- +pose a both space and time efficient data structure named +temporal edge list to represent a temporal graph in mem- +ory. It can be manipulated to perform temporal core de- +composition and tightest time interval based pruning rapidly +with bounded memory. More importantly, temporal edge +list can be incrementally updated with evolving temporal +graphs, so that our approach can support dynamic graph +applications naturally. +• Lastly, we evaluate the efficiency and effectiveness of our +algorithm on real-world datasets. The experimental re- +sults demonstrate that our algorithm outperforms the im- +proved PHC-Query by three orders of magnitude. +The rest of this paper is organized as follows. Section 2 for- +mally introduces the data model and query model, and also gives +a baseline algorithm. Sections 3-5 present our algorithm, opti- +mization and implementation techniques respectively. Section +6 briefly discusses some meaningful extension of our approach. +Section 7 presents the experiments and analyzes the results. Sec- +tion 8 investigates the related work. Section 9 concludes our +work. +2 +PRELIMINARY +In this section, we propose a generalized 푘-core query problem +on temporal graphs, which facilitates various temporal commu- +nity query/search demands. The previous historical푘-core query [36] +can be seen as a special case of the proposed problem. Specifi- +cally, we introduce the data model and query model of the pro- +posedproblem in Section 2.1 and 2.2 respectively, and then present +a nontrivial baseline that addresses the proposed problem based +on the existing PHC-Query. +2.1 +Data Model +A temporal graph is normally an undirected graph G = (V, E) +with parallel temporal edges. Each temporal edge (푢,푣,푡) ∈ E is +associated with a timestamp 푡 that indicates when the interac- +tion happened between the vertices 푢,푣 ∈ V. For example, the +temporal edges could be transfer transactions between bank ac- +counts in a finance graph. Without a loss of generality, we use +continuous integers that start from 1 to denote timestamps. Fig- +ure 1 illustrates a temporal graph as our running example. +There are two useful concepts derived from the temporal graph. +Given a time interval [푡푠,푡푒], we define the projected graph of G +over [푡푠,푡푒] as G[푡푠,푡푒] = (V[푡푠,푡푒], E[푡푠,푡푒]), where V[푡푠,푡푒] = +V and E[푡푠,푡푒] = {(푢,푣,푡)|(푢,푣,푡) ∈ E, 푡 ∈ [푡푠,푡푒]}. Moreover, +we define the detemporalized graph of G[푡푠,푡푒] as a simple graph +퐺[푡푠,푡푒] = (푉[푡푠,푡푒], 퐸[푡푠,푡푒]), where 푉[푡푠,푡푒]=V[푡푠,푡푒] and 퐸[푡푠,푡푒] += {(푢,푣)|(푢,푣,푡) ∈ E[푡푠,푡푒] }. +2.2 +Query Model +For revealing communities in graphs, the 푘-core query is widely +adopted. Given an undirected graph 퐺 and an integer 푘, 푘-core +is the maximal induced subgraph of 퐺 in which all vertices have +degrees at least 푘, which is denoted by C푘 (퐺). The coreness of +a vertex 푣 in a graph 퐺 is the largest value of 푘 such that 푣 ∈ +C푘 (퐺). +For temporal graphs, the Historical 푘-Core Query (HCQ) [36] +is proposed recently. It aims to find a 푘-core that appears during +a specific time interval. Formally, a historical 푘-core H푘 +[푡푠,푡푒] (G) +is a 푘-core in the detemporalized projected graph 퐺[푡푠,푡푒] of G. +Thus, HCQ can be defined as follows. + +Scalable Time-Range 푘-Core Qery on Temporal Graphs +Definition 1 (Historical 푘-Core Qery). For a temporal +graph G, given an integer 푘 and a time interval [푡푠,푡푒], return +H푘 +[푡푠,푡푒] (G) = C푘 (퐺[푡푠,푡푒]). +In this paper, we propose a novel query model called Tempo- +ral 푘-Core Query (TCQ) that generalizes HCQ. The main differ- +ence is that the query time interval [푇푠,푇푒] of TCQ is a range +but not fixed query condition like [푡푠,푡푒] of HCQ. In TCQ,푇푠 and +푇푒 are the minimum start time and maximum end time of query +time interval respectively, and thereby the 푘-cores induced by +each subinterval [푡푠,푡푒] ⊆ [푇푠,푇푒] are all potential results of +TCQ. Moreover, TCQ directly returns the maximal induced sub- +graphs of G in which all vertices have degrees (note that, the +number of neighbor vertices but not neighbor edges) at least 푘 +as results. We call these subgraphs as temporal 푘-cores and de- +note by T 푘 +[푡푠,푡푒] (G) a temporal 푘-core that appears over [푡푠,푡푒] +on G. Obviously, a historical 푘-core H푘 +[푡푠,푡푒] (G) is the detempo- +ralized temporal 푘-core T 푘 +[푡푠,푡푒] (G). Therefore, TCQ can be seen +as a group of HCQ and HCQ can be seen as a special case of +TCQ. +The formal definition of TCQ is as follows. +Definition 2 (Temporal푘-Core Qery). For a temporalgraph +G, given an integer 푘 and a time interval [푇푠,푇푒], return all dis- +tinct T 푘 +[푡푠,푡푒] (G) with [푡푠,푡푒] ⊆ [푇푠,푇푒]. +Note that, TCQ only returns the distinct temporal 푘-cores +that are not identical to each other, since multiple subintervals +of [푇푠,푇푒] may induce an identical subgraph of G. For brevity, +T 푘 +[푡푠,푡푒] (G) is abbreviated as T 푘 +[푡푠,푡푒] if the context is self-evident. +2.3 +Baseline Algorithm +A straightforward solution to TCQ is to enumerate each subin- +terval [푡푠,푡푒] ⊆ [푇푠,푇푒] and induce T 푘 +[푡푠,푡푒] respectively, which +takes 푂(|푇푒 −푇푠|2|E|) time. However, the span of query time in- +terval (namely,푇푒−푇푠) can be extremely large in practice, which +results in enormous time consumption for inducing all temporal +푘-cores from scratch independently. Therefore, we start from a +non-trivial baseline based on the existing PHC-Query. +2.3.1 +A Short Review of PHC-Qery. PHC-Query relies on a heavy- +weight index called PHC-Index that essentially precomputes the +coreness of all vertices in the projected graphs over all possible +time intervals. The index is logically a table that stores a set of +timestamp pairs for each vertex 푣 ∈ V (column) and each rea- +sonable coreness 푘 (row). Given a value of 푘, the coreness of a +vertex 푣 is exactly 푘 in the projected graph over [푡푠,푡푒] for each +timestamp pair 푡푠 and 푡푒 in the cell (푘, 푣). In particular, due to +the monotonicity of coreness of a vertex with respect to 푡푒 when +푡푠 is fixed, PHC-Index can reduce its space cost significantly by +only storing the necessary but not all possible timestamp pairs. +Specifically, for a vertex 푣, a coreness 푘 and a start time 푡푠, only a +discrete set of core time need to be recorded, since the coreness +of the vertex over [푡푠,푡푒] will not change with the increase of +푡푒 until 푡푒 is a core time. Consequently, given an HCQ instance, +PHC-Query leverages PHC-Index to directly determine whether +a vertex has the coreness no less than the required 푘, by compar- +ing the query time interval with the retrieved timestamp pairs, +and then induces historical 푘-cores with qualified vertices. +2.3.2 +Incremental PHC-Qery Algorithm. The main idea of our +baseline algorithm is to induce temporal 푘-cores incrementally, +Algorithm 1: Baseline iPHC-Query algorithm. +Input: G, 푘, 푇푠, 푇푒 +Output: all distinct T 푘 +[푡푠,푡푒] (G) with [푡푠,푡푒] ⊆ [푇푠,푇푒] +1 for 푡푠 ← 푇푠 to 푇푒 do +2 +V ← ∅, E ← ∅, H푣 ← ∅, H푒 ← ∅ +3 +for 푘 and 푡푠, retrieve the core time of each vertex in +G from PHC-Index and push them into H푣 +4 +push the temporal edges with timestamps in [푡푠,푇푒] +in G into H푒 +5 +for 푡푒 ← 푡푠 to 푇푒 do +6 +pop a vertex from H푣 and add it to V, until the +min core time of H푣 exceeds 푡푒 +7 +pop an edge from H푒 and add it to E if both +vertices linked by this edge are in V, until the +min timestamp of H푒 exceeds 푡푒 +8 +push all edges that have been popped from H푒 +and are not added to E back to H푒 +9 +collect T 푘 +[푡푠,푡푒] = (V, E) if it is neither empty nor +identical to other existing results +thereby reducing redundant computation. With a temporal 푘- +core T 푘 +[푡푠,푡푒], we induce T 푘 +[푡푠,푡푒+1] simply by appending new ver- +tices to T 푘 +[푡푠,푡푒], whose coreness has become no less than푘 due to +the expand of time interval. Those vertices can be directly iden- +tified by using core time retrieved from PHC-Index since 푡푠 is +fixed. The correctness of baseline algorithm is guaranteed while +the correctness of PHC-Query holds. +The pseudo code of incremental PHC-Query (iPHC-Query) al- +gorithm is presented in Algorithm 1. It enumerates all subinter- +vals of a given [푇푠,푇푒] in a particular order for fulfilling efficient +incremental temporal 푘-core induction. Specifically, it anchors +the value of 푡푠 (line 1), and increases the value of 푡푒 from 푡푠 to +푇푒 (line 5), so that T 푘 +[푡푠,푡푒+1] can always be incrementally gen- +erated from an existing T 푘 +[푡푠,푡푒]. For each 푡푠 anchored and the +input 푘, the algorithm firstly retrieves the core time of all ver- +tices from PHC-Index, and pushes the vertices into a minimum +heap H푣 ordered by their core time (line 3). Moreover, all tem- +poral edges with timestamps in [푡푠,푇푒] are pushed into another +minimum heap H푒 ordered by their timestamp (line 4). Then, the +algorithm maintains a vertex set V and an edge set E, which rep- +resent the vertices and edges of T 푘 +[푡푠,푡푒] respectively, whenever +푡푒 is increased by the following steps. It pops remaining vertices +with core time no greater than 푡푒 from H푣 and adds them to V +(line 6), since the corenesss of these vertices are no less than +푘 according to PHC-Index. Also, it pops remaining edges with +timestamp no greater than 푡푒 from H푒 and adds them to E if +both vertices linked by the edges are in V (line 7). Then, it puts +back the popped edges that are not in E into H푒 (line 8), because +they could still be contained by other temporal 푘-cores induced +later. Lastly, a temporal푘-core comprised of V and E that are not +empty is collected if it has not been induced before (line 9). +The complexity of baseline mainly depends on the mainte- +nance of both V and E. For the maintenance of V, each ver- +tex in T 푘 +[푡푠,푇푒]is added to V from H푣 at most once in the inner +loop (lines 5-9), which takes logarithmic time for a heap. There- +fore, the total cost is bounded by �푇푒 +푡=푇푠 |V[푡,푇푒]| log |V[푡,푇푒]|. +The case is more complicated for the maintenance of E, since + +Junyong Yang, Ming Zhong, Yuanyuan Zhu, Tieyun Qian, Mengchi Liu, and Jeffery Xu Yu +each edge with a timestamp within [푡,푇푒] is likely to be trans- +ferred between H푒 and E (lines 7-8), until both its endpoints are +contained by V. In the worst case, the total cost is bounded by +�푇푒 +푡=푇푠 |푇푒 − 푡||E[푡,푇 푒]| log |E[푡,푇 푒]|. While, the real cost in prac- +tice can be much lower since the |푇푒 − 푡| part should be a more +reasonable value. +Although the baseline algorithm can achieve incremental in- +duction of temporal k-core for each start time, PHC-Index incurs +a huge amount of extra space and time overheads. Moreover, its +incremental induction only offers a kind of “intra-core” optimiza- +tion that reduces the redundant computation in each temporal +푘-core induction, and lacks of a kind of “inter-core” optimization +that can directly avoids inducing some temporal 푘-cores. In the +following sections, we first propose a novel algorithm that can +outperform baseline algorithm without any precomputation and +index, and then optimize it significantly to further improve the +efficiency by at least three orders of magnitude. +3 +ALGORITHM +In this section, we propose a novel efficient algorithm to address +TCQ. Our algorithm leverages a fundamental operation called +temporal core decomposition to induce T 푘 +[푡푠,푡푒] from T 푘 +[푡푠,푡푒+1] decre- +mentally. More importantly, our algorithm does not require any +precomputation and index space, and can still outperform the +baseline algorithm. Next, Section 3.1 introduces the temporal +core decomposition operation, and Section 3.2 presents our al- +gorithm. +3.1 +Temporal Core Decomposition (TCD) +Firstly, we introduce Temporal Core Decomposition (TCD) as +a basic operation on temporal graphs, which is derived from +the traditional core decomposition [2] on ordinary graphs. TCD +refers to a two-step operation of inducing a temporal 푘-core +T 푘 +[푡푠,푡푒] of a given time interval [푡푠,푡푒] from a given temporal +graph G. The first step is truncation: remove temporal edges with +timestamps not in [푡푠,푡푒] from G, namely, induce the projected +graph G[푡푠,푡푒]. The second step is decomposition: iteratively peel +vertices with degree (the number of neighbor vertices but not +neighbor edges) less than 푘 and the edges linked to them to- +gether. The correctness of TCD is as intuitive as core decom- +position. +An excellent property of TCD operation is that, it can induce +a temporal푘-core T 푘 +[푡푠,푡푒] from another temporal푘-core T 푘 +[푡푠′,푡푒′] +with [푡푠,푡푒] ⊂ [푡푠′,푡푒′], so that we can develop a decremental al- +gorithm based on TCD operation to achieve efficient processing +of TCQ. To prove the correctness of this property, let us consider +the following Theorem 1. +Lemma 1. Given time intervals [푡푠,푡푒] and [푡푠′,푡푒′] such that +[푡푠,푡푒] ⊂ [푡푠′,푡푒′], we have T 푘 +[푡푠,푡푒] is a subgraph of T 푘 +[푡푠′,푡푒′]. +Proof. For each vertex in T 푘 +[푡푠,푡푒], its coreness in G[푡푠′,푡푒′] is +certainly no less than in G[푡푠,푡푒] (namely, ⩾ 푘), because G[푡푠,푡푒] +is a subgraph of G[푡푠′,푡푒′]. Thus, all vertices in T 푘 +[푡푠,푡푒] will be +contained by T 푘 +[푡푠′,푡푒′] that is a temporal 푘-core of G[푡푠′,푡푒′]. +□ +Theorem 1. Given a time interval [푡푠, 푡푒] and a temporal 푘- +core T 푘 +[푡푠′,푡푒′] with [푡푠,푡푒] ⊂ [푡푠′,푡푒′], the subgraph induced by +using TCD operation from T 푘 +[푡푠′,푡푒′] for [푡푠,푡푒] is T 푘 +[푡푠,푡푒]. +v3 +v4 +v5 +v6 +v7 +v8 +6 +6 +6 +6 +5 +5 +2 +2 +2 +6 +5 +5 +5 +5 +5 +5 +3 +2 +4 +v3 +v4 +v5 +v6 +6 +6 +6 +6 +5 +5 +6 +5 +5 +5 +5 +5 +5 +v7 +v8 +v3 +v4 +v5 +v6 +6 +6 +6 +6 +5 +5 +5 +5 +5 +5 +truncation +decomposition +Figure 2: Temporal core decomposition from T 2 +[2,6] to +T 2 +[5,6]. +Proof. Firstly, we prove for any temporal graph G′ satisfy- +ing that T 푘 +[푡푠,푡푒] is a subgraph of G′ and G′ is a subgraph of +G, we can induce T 푘 +[푡푠,푡푒] from G′ by using TCD operation. For +each vertex in T 푘 +[푡푠,푡푒], its coreness is not less than 푘 in G′ over +[푡푠,푡푒], because this temporal 푘-core is a subgraph of G′. Mean- +while, for each vertex in G′ but not in T 푘 +[푡푠,푡푒], its coreness in G′ +is not greater than in G, because G′ is a subgraph of G. Thus, +its coreness in G′ over [푡푠,푡푒] is less than 푘, because it is not +in the temporal 푘-core T 푘 +[푡푠,푡푒] of G. As a result, T 푘 +[푡푠,푡푒] is also +a temporal 푘-core of G′, and thereby can be induced by using +TCD operation from G′. +Then, consider two temporal 푘-cores T 푘 +[푡푠,푡푒] and T 푘 +[푡푠′,푡푒′] with +[푡푠,푡푒] ⊆ [푡푠′,푡푒′]. Due to Lemma 1, we have T 푘 +[푡푠,푡푒] is a sub- +graph of T 푘 +[푡푠′,푡푒′]. Let G′ +[푡푠,푡푒] be the temporal graph induced by +the first step of TCD from T 푘 +[푡푠′,푡푒′], which is certainly a subgraph +of T 푘 +[푡푠′,푡푒′]. Since G′ +[푡푠,푡푒] only removes the temporal edges not +in [푡푠,푡푒], which means these edges are not contained by T 푘 +[푡푠,푡푒], +it is obviously T 푘 +[푡푠,푡푒] is a subgraph of G′ +[푡푠,푡푒]. Thus, the correct- +ness of this theorem holds. +□ +For example, Figure 2 illustrates the procedure of TCD from +T 2 +[2,6] to T 2 +[5,6] on our running example graph in Figure 1. The +edges with timestamps not in [5, 6] (marked by dashed lines) +are firstly removed from T 2 +[2,6] by truncation, which results in +the decrease of degrees of vertices 푣5, 푣7 and 푣8. Then, the ver- +tices with degree less than 2 (marked by dark circles), namely, 푣7 +and 푣8 are further peeled by decomposition, together with their +edges. The remaining temporal graph is T 2 +[5,6]. +3.2 +TCD Algorithm +We propose a TCD algorithm to address TCQ by using temporal +core decomposition. In general, given a TCQ instance, the TCD +algorithm enumerates each subinterval of [푇푠,푇푒] in a particu- +lar order, so that the temporal 푘-cores of each subinterval are in- +duced decrementally from previously induced temporal 푘-cores +except the initial one. +Specifically, we enumerate a subinterval [푡푠,푡푒] of [푇푠,푇푒] as +follows. Initially, let 푡푠 = 푇푠 and 푡푒 = 푇푒. It means we induce +the largest temporal 푘-core T 푘 +[푇푠,푇푒] at the beginning. Then, we +will anchor the start time 푡푠 = 푇푠 and decrease the end time 푡푒 +from 푇푒 until 푡푠 gradually. As a result, we can always leverage +TCD to induce the temporal푘-core of current subinterval [푡푠,푡푒] +from the previously induced temporal 푘-core of [푡푠, 푡푒 + 1] but +not from G[푡푠,푡푒] or even G. Whenever the value of 푡푒 is de- +creased to 푡푠, the value of 푡푠 will be increased to 푡푠 + 1 until +푡푠 = 푇푒, and the value of 푡푒 will be reset to 푇푒. Then, we in- +duce T 푘 +[푡푠+1,푡푒] from T 푘 +[푡푠,푡푒], and start over the decremental TCD + +Scalable Time-Range 푘-Core Qery on Temporal Graphs +Algorithm 2: TCD algorithm. +Input: G, 푘, [푇푠,푇푒] +Output: all distinct T 푘 +[푡푠,푡푒] with [푡푠,푡푒] ⊆ [푇푠,푇푒] +1 for 푡푠 ← 푇푠 to 푇푒 do +// anchor a new start time +2 +푡푒 ← 푇푒 +// reset the end time +3 +if 푡푠 = 푇푠 then +4 +T 푘 +[푡푠,푡푒] ← TCD(G[푇푠,푇푒], 푘, [푡푠,푡푒]) +5 +else +6 +T 푘 +[푡푠,푡푒] ← TCD(T 푘 +[푡푠−1,푡푒], 푘, [푡푠,푡푒]) +7 +collect T 푘 +[푡푠,푡푒] if it is distinct +8 +for 푡푒 ← 푇푒 − 1 to 푡푠 do +// iteratively +decremental induction +9 +T 푘 +[푡푠,푡푒] ← TCD(T 푘 +[푡푠,푡푒+1], 푘, [푡푠,푡푒]) +10 +collect T 푘 +[푡푠,푡푒] if it is distinct +procedure. The pseudo code of TCD algorithm is given in Algo- +rithm 2. Note that, the details of TCD(G, 푘, [푡푠,푡푒]) function is +left to Section 5.2, in which we design a specific data structure +to implement TCD operation efficiently in physical level. +Figure 3 gives a demonstration of TCD algorithm for finding +temporal 2-cores of time interval [1,8] on our running example +graph. The temporal 푘-cores are induced line by line and from +left to right. Each arrow between temporal 푘-cores represents +a TCD operation from tail to head. We can see that, compared +with inducing each temporal 푘-core independently, the TCD al- +gorithm reduces the computational overhead significantly. For +most induced temporal 푘-cores, a number of vertices and edges +have already been excluded while inducing the previous tempo- +ral 푘-cores. Moreover, with the increase of 푡푠 and the decrease of +푡푒 when 푡푠 is fixed, the size of T 푘 +[푡푠,푡푒] will be reduced monotoni- +cally until no temporal푘-core exists over [푡푠,푡푒], so that the time +and space costs of TCD operation will also be reduced gradually. +Lastly, we compare TCD algorithm with Baseline algorithm +abstractly. When 푡푠 is fixed, Baseline algorithm conducts an in- +cremental procedure, in which each vertex is popped once and +each edge may be popped and pushed back many times, and in +contrast, TCD algorithm conducts a decremental procedure, in +which each vertex is peeled once and each edge is also removed +once due to Lemma 1. Therefore, TCD algorithm that is well im- +plemented in physical level (see Section 5.2) can be even more +efficient than Baseline algorithm, though it does not need any +precomputed index. +4 +OPTIMIZATION +In this section, we dive deeply into the procedure of TCD al- +gorithm and optimize it dramatically by introducing an intu- +itive concept called tightest time interval for temporal 푘-cores. +In a nutshell, we directly prune subintervals without inducing +their temporal 푘-cores if we can predict that the temporal 푘- +cores are identical to other induced temporal 푘-cores, and tight- +est time interval is the key to fulfill prediction. In this way, the +optimized TCD algorithm only performs TCD operations that +are necessary for returning all distinct answers to a given TCQ +instance. Conceptually, the new pruning operation of optimized +algorithm eliminates the “inter-core” redundant computation, and +v10 +v3 +v4 +v5 +v6 +v7 +v8 +v9 +6 +6 +6 +6 +5 +5 +2 +2 +2 +2 +7 +7 +6 +5 +5 +5 +5 +5 +5 +3 +2 +2 +8 +4 +1 +v10 +v3 +v4 +v5 +v6 +v7 +v8 +v9 +6 +6 +6 +6 +5 +5 +2 +2 +2 +2 +7 +7 +6 +5 +5 +5 +5 +5 +5 +3 +2 +4 +1 +v3 +v4 +v5 +v6 +v7 +v8 +6 +6 +6 +6 +5 +5 +2 +2 +2 +6 +5 +5 +5 +5 +5 +5 +3 +2 +4 +2 +v3 +v4 +v5 +v6 +v7 +v8 +5 +5 +2 +2 +5 +5 +5 +5 +5 +3 +4 +v5 +v7 +v8 +2 +2 +3 +4 +v5 +v7 +v8 +2 +2 +2 +3 +v5 +v7 +v8 +2 +2 +2 +2 +2 +2 +2 +2 +2 +5 +v3 +v4 +v5 +v6 +6 +6 +6 +6 +5 +5 +5 +5 +5 +5 +v3 +v4 +v5 +v6 +6 +6 +6 +5 +5 +5 +5 +5 +56 +v3 +v4 +v5 +v6 +5 +5 +5 +5 +5 +5 +v3 +v4 +v5 +v6 +6 +6 +6 +6 +v3 +v4 +v5 +v6 +5 +5 +5 +5 +5 +5 +v10 +v3 +v4 +v5 +v6 +v7 +v8 +v9 +6 +6 +6 +6 +5 +5 +2 +2 +2 +7 +6 +5 +5 +5 +5 +5 +5 +3 +2 +2 +8 +4 +2 +v10 +v3 +v4 +v5 +v6 +v7 +v8 +v9 +6 +6 +6 +6 +5 +5 +2 +2 +2 +7 +6 +5 +5 +5 +5 +5 +5 +3 +2 +2 +4 +2 +7 +7 +v3 +v4 +v5 +v6 +v7 +v8 +6 +6 +6 +6 +5 +5 +2 +2 +2 +6 +5 +5 +5 +5 +5 +5 +3 +2 +4 +v3 +v4 +v5 +v6 +v7 +v8 +5 +5 +2 +2 +5 +5 +5 +5 +5 +3 +4 +v5 +v7 +v8 +2 +2 +3 +4 +v5 +v7 +v8 +2 +2 +2 +3 +v5 +v7 +v8 +2 +2 +2 +2 +2 +2 +2 +2 +2 +5 +v3 +v4 +v5 +v6 +v7 +6 +6 +6 +6 +5 +5 +6 +5 +5 +5 +5 +5 +5 +3 +v3 +v4 +v5 +v6 +v7 +5 +5 +5 +5 +5 +5 +5 +5 +3 +v3 +v4 +v5 +v6 +v7 +6 +6 +6 +6 +5 +5 +6 +5 +5 +5 +5 +5 +5 +3 +v3 +v4 +v5 +v6 +v7 +6 +6 +6 +6 +5 +5 +6 +5 +5 +5 +5 +5 +5 +3 +v3 +v4 +v5 +v6 +6 +6 +6 +6 +5 +5 +5 +5 +5 +5 +v3 +v4 +v5 +v6 +6 +6 +6 +6 +5 +5 +5 +5 +5 +5 +v3 +v4 +v5 +v6 +6 +6 +6 +5 +5 +5 +5 +5 +56 +v3 +v4 +v5 +v6 +6 +6 +6 +5 +5 +5 +5 +5 +56 +v3 +v4 +v5 +v6 +6 +6 +6 +6 +v3 +v4 +v5 +v6 +6 +6 +6 +6 +ts +te +1 +2 +3 +4 +5 +6 +1 +2 +3 +4 +5 +6 +8 +7 +7 +8 +Figure 3: A demonstration of TCD algorithm for finding +temporal 2-cores of time interval [1,8]. +the original TCD operation eliminates the “intra-core” redun- +dant computation. Thus, the computational complexity of opti- +mized algorithm no longer depends on the span of query time in- +terval [푇푠,푇푒] like the baseline algorithm and the original TCD +algorithm but only depends on the scale of final results. +Next, we introduce the concept and properties of tightest time +interval in Section 4.1, present three pruning rules based on tight- +est time interval for TCD algorithm in Section 4.2, and briefly +conclude and discuss the optimized TCD algorithm in Section 4.3. +4.1 +Tightest Time Interval (TTI) +We have such an observation, a temporal 푘-core of [푡푠,푡푒] may +only contain edges with timestamps in a subinterval [푡푠′,푡푒′] ⊂ +[푡푠,푡푒], since the edges in [푡푠, 푡푠′) and (푡푒′,푡푒] have been re- +moved by core decomposition. For example, consider a tempo- +ral 푘-core T 2 +[4,8] illustrated in Figure 3. We can see that it does +not contain edges with timestamps 4, 7 and 8. As a result, if +we continue to induce T 2 +[4,7] from T 2 +[4,8] and to induce T 2 +[4,6] +from T 2 +[4,7], the returned temporal 푘-cores remain unchanged. +The sameness of temporal 푘-cores induced by different subinter- +vals inspires us to further optimize TCD algorithm by pruning +subintervals directly. As illustrated in Figure 3, the subintervals +such as [4,7], [4,6], [5,8], [5,7] and [5,6] all induce the identical +temporal 푘-cores to [4,8], so that they can be potentially pruned +in advance. +For that, we propose the concept of Tightest Time Interval +(TTI) for temporal 푘-cores. Given a temporal 푘-core of [푡푠,푡푒], +its TTI refers to the minimal time interval [푡푠′,푡푒′] that can in- +duce an identical temporal 푘-core to T 푘 +[푡푠,푡푒], namely, there is no +subinterval of [푡푠′,푡푒′] that can induce an identical temporal 푘- +core to T 푘 +[푡푠,푡푒]. We formalize the definition of TTI as follows. +Definition 3 (Tightest Time Interval). Given a temporal +푘-core T 푘 +[푡푠,푡푒], its tightest time interval T 푘 +[푡푠,푡푒].TTI is [푡푠′,푡푒′], if +and only if +1) T 푘 +[푡푠′,푡푒′] is an identical temporal 푘-core to T 푘 +[푡푠,푡푒]; +2) there does not exist [푡푠′′,푡푒′′] ⊂ [푡푠′,푡푒′], such that T 푘 +[푡푠′′,푡푒′′] +is an identical temporal 푘-core to T 푘 +[푡푠,푡푒]. + +Junyong Yang, Ming Zhong, Yuanyuan Zhu, Tieyun Qian, Mengchi Liu, and Jeffery Xu Yu +It is easy to prove the TTI of a temporal 푘-core of [푡푠,푡푒] is +surely a subinterval of [푡푠,푡푒]. To evaluate the TTI of a given +T 푘 +[푡푠,푡푒], we have the following theorem. +Theorem 2. Given a temporal 푘-core T 푘 +[푡푠,푡푒], T 푘 +[푡푠,푡푒].TTI = +[푡푚푖푛,푡푚푎푥 ], where 푡푚푖푛 and 푡푚푎푥 are the minimum and maxi- +mum timestamps in T 푘 +[푡푠,푡푒] respectively. +Proof. On one hand, T 푘 +[푡푚푖푛,푡푚푎푥 ] is identical to T 푘 +[푡푠,푡푒]. Be- +cause we can induce T 푘 +[푡푚푖푛,푡푚푎푥 ] by TCD operation from T 푘 +[푡푠,푡푒] +due to [푡푚푖푛,푡푚푎푥 ] ⊆ [푡푠,푡푒]. Meanwhile, during the operation, +none edge is actually removed since there is no edge with times- +tamp outsides [푡푚푖푛,푡푚푎푥] in T 푘 +[푡푠,푡푒], and thus the temporal 푘- +core T 푘 +[푡푠,푡푒] will remain unchanged. On the other hand, any time +interval [푡푠′,푡푒′] ⊂ [푡푚푖푛,푡푚푎푥 ] cannot induce a temporal푘 core +that is identical to T 푘 +[푡푠,푡푒], since the edges with timestamp either +푡푚푖푛 or 푡푚푎푥 in T 푘 +[푡푠,푡푒] are excluded at least. +□ +With Theorem 2, we can evaluate the TTI of a given tempo- +ral 푘-core instantly (by 푂(1) time, see Section 5), which guar- +antees the following optimization based on TTI will not incur +extra overheads. +Moreover, there are the following important properties of TTI +that support our pruning strategies. +Property 1 (Uniqeness). Given a temporal 푘-core T 푘 +[푡푠,푡푒], +there exists no other time interval than T 푘 +[푡푠,푡푒].TTI evaluated by +Theorem 2 that is also a TTI of T 푘 +[푡푠,푡푒]. +Proof. Let T 푘 +[푡푠,푡푒].TTI be [푡푠′,푡푒′], and [푡푠′′,푡푒′′] ≠ [푡푠′,푡푒′] +be any other time interval. There are only two possibilities. Firstly, +[푡푠′,푡푒′] ⊄ [푡푠′′,푡푒′′]. However, the edges with timestamp 푡푠′ +and 푡푒′ are contained by T 푘 +[푡푠,푡푒] according to Theorem 2, and +thereby [푡푠′′,푡푒′′] that does not cover [푡푠′,푡푒′] cannot induce +T 푘 +[푡푠,푡푒]. Thus, the first possibility does not satisfy the first con- +dition in Definition 3. Secondly, [푡푠′,푡푒′] ⊂ [푡푠′′,푡푒′′]. However, +since [푡푠′,푡푒′] can induce T 푘 +[푡푠,푡푒], [푡푠′′,푡푒′′] is certainly not the +tightest even if it can also induce T 푘 +[푡푠,푡푒]. Thus, the second possi- +bility does not satisfy the second condition in Definition 3. Con- +sequently, [푡푠′′,푡푒′′] ≠ [푡푠′,푡푒′] is not a TTI of T 푘 +[푡푠,푡푒]. +□ +Property 2 (Eqivalence). Given two temporal푘-cores T 푘 +[푡푠,푡푒] +and T 푘 +[푡푠′,푡푒′], they are identical temporal graphs if and only if +T 푘 +[푡푠,푡푒].TTI = T 푘 +[푡푠′,푡푒′].TTI. +Proof. If T 푘 +[푡푠,푡푒].TTI = T 푘 +[푡푠′,푡푒′].TTI, T 푘 +[푡푠,푡푒] and T 푘 +[푡푠′,푡푒′] are +both identical to the temporal 푘-core of the TTI according to +Definition 3, and thus are identical to each other. Conversely, if +T 푘 +[푡푠,푡푒] and T 푘 +[푡푠′,푡푒′] are identical, they must have a same unique +TTI according to Theorem 2 and Property 1. +□ +Property 3 (Inclusion). Given two temporal 푘-cores T 푘 +[푡푠,푡푒] +and T 푘 +[푡푠′,푡푒′], we have T 푘 +[푡푠,푡푒].TTI ⊆ T 푘 +[푡푠′,푡푒′].TTI, if [푡푠,푡푒] ⊆ +[푡푠′,푡푒′]. +Proof. Since [푡푠,푡푒] ⊆ [푡푠′,푡푒′], we have T 푘 +[푡푠,푡푒] is a sub- +graph of T 푘 +[푡푠′,푡푒′] according to Lemma 1. Thus, the minimum +timestamp in T 푘 +[푡푠,푡푒] is certainly no earlier than the the min- +imum timestamp in T 푘 +[푡푠′,푡푒′], and the maximum timestamp in +T 푘 +[푡푠,푡푒] is certainly no later than the the maximum timestamp in +T 푘 +[푡푠′,푡푒′]. Then, according to Theorem 2, we have T 푘 +[푡푠,푡푒].TTI ⊆ +T 푘 +[푡푠′,푡푒′].TTI. +□ +Figure 4a abstracts Figure 3 as a schedule table of subinter- +val enumeration, and TCD algorithm will traverse the cells row +by row and from left to right. For example, the cell in row 1 +and column 6 represents a subinterval [1, 6], in which [2, 6] is +the TTI of T 2 +[1,6]. In particular, the grey cells indicate that the +temporal 푘-cores of the corresponding subintervals do not exist. +Figure 4a clearly reveals that TCD algorithm suffers from induc- +ing a number of identical temporal 푘-cores (with the same TTIs). +For example, the TTI [5, 6] repeats six times, which means six +cells will induce identical temporal 푘-cores. +4.2 +Pruning Rules +The main idea of optimizing TCD algorithm is to predict the in- +duction of identical temporal푘-cores by leveraging TTI, thereby +skipping the corresponding subintervals during the enumera- +tion. Specifically, whenever a temporal 푘-core of [푡푠,푡푒] is in- +duced, we evaluate its TTI [푡푠′,푡푒′]. If 푡푠′ > 푡푠 or/and 푡푒′ < 푡푒, +it is triggered that a number of subintervals on the schedule can +be pruned in advance. According to different relations between +[푡푠,푡푒] and [푡푠′,푡푒′], our pruning technique can be categorized +into three rules which are not mutually exclusive. In other words, +the three rules may be triggered at the same time, and prune dif- +ferent subintervals respectively. Next, we present these pruning +rules in Section 4.2.1, Section 4.2.2 and Section 4.2.3, respectively. +4.2.1 +Rule 1: Pruning-on-the-Right. Consider the schedule illus- +trated in Figure 4a. For each row, TCD algorithm traverses the +cells (namely, subintervals) from left to right. If the TTI [푡푠′,푡푒′] +in the current cell [푡푠,푡푒] meets such a condition, namely, 푡푒′ < +푡푒, a pruning operation will be triggered, and the following cells +in this row from [푡푠,푡푒 − 1] until [푡푠,푡푒′] will be skipped be- +cause these subintervals will induce identical temporal 푘-cores +to T 푘 +[푡푠,푡푒]. Since the pruned cells are on the right of trigger cell, +we call this rule Pruning-On-the-Right (PoR). The pseudo code +of PoR is given in lines 2-4 of Algorithm 3. The correctness of +PoR is guaranteed by the following lemma. +Lemma 2. Given a temporal푘-core T 푘 +[푡푠,푡푒] whose TTI is [푡푠′,푡푒′], +for any time interval [푡푠,푡푒′′] with 푡푒′′ ∈ [푡푒′,푡푒], T 푘 +[푡푠,푡푒′′].TTI = +[푡푠′,푡푒′]. +Proof. On one hand, since [푡푠,푡푒′′] ⊆ [푡푠,푡푒], T 푘 +[푡푠,푡푒′′].TTI +⊆ T 푘 +[푡푠,푡푒].TTI = [푡푠′,푡푒′] according to Inclusion (Property 3). +On the other hand, we can prove [푡푠′,푡푒′] ⊆ T 푘 +[푡푠,푡푒′′].TTI. If +we induce T 푘 +[푡푠′,푡푒′] from T 푘 +[푡푠,푡푒] by TCD operation, it is easy +to know T 푘 +[푡푠,푡푒] will remain unchanged, because it only con- +tains the edges with timestamps in [푡푠′,푡푒′] according to Theo- +rem 2. Thus, we have T 푘 +[푡푠′,푡푒′].TTI = [푡푠′,푡푒′] according to Equiv- +alence (Property 2). Also, since [푡푠′,푡푒′] ⊆ [푡푠,푡푒′′], [푡푠′,푡푒′] = +T 푘 +[푡푠′,푡푒′].TTI ⊆ T 푘 +[푡푠,푡푒′′].TTI according to Inclusion (Property 3). +□ +With Lemma 2, we can predict that the TTIs in the cells [푡푠,푡푒− +1], · · · , [푡푠,푡푒′] are the same as the trigger cell [푡푠,푡푒], when the + +Scalable Time-Range 푘-Core Qery on Temporal Graphs +ts te +1 +2 +3 +4 +5 +6 +1 +2 +3 +4 +5 +6 +8 +7 +7 +8 +[6,6] +[5,6] +[5,6] +[3,6] +[2,8] +[1,8] +[6,6] +[5,6] +[5,6] +[3,6] +[2,7] +[1,7] +[6,6] +[5,6] +[5,6] +[3,6] +[2,6] +[2,6] +[5,5] +[5,5] +[3,5] +[2,5] +[2,5] +[2,4] +[2,4] +[2,3] +[2,3] +[2,2] +[2,2] +(a) Without pruning. +ts te +1 +2 +3 +4 +5 +6 +1 +2 +3 +4 +5 +6 +8 +7 +7 +8 +[5,6] +[3,6] +[2,8] +[1,8] +[2,7] +[1,7] +[6,6] +[2,6] +[5,5] +[3,5] +[2,5] +[2,4] +[2,3] +[2,2] +Cell without core induced +Pruning-on-the-Right +Pruning-on-the-Underside +Pruning-on-the-Left +[x,y] +Cell with core induced, TTI = [x,y] +Pruning triggered by cell [1,6] +Pruning triggered by cell [3,8] +Pruning triggered by cell [4,8] +(b) With pruning. +Figure 4: Examples of subinterval pruning based on tightest time interval. +PoR rule is satisfied. Thus, the temporal푘-cores induced by these +subintervals are all identical to the induced T 푘 +[푡푠,푡푒] according to +Equivalence (Property 2). +For example, Figure 4b illustrates two instances of PoR (the +cells in orange and blue colors with left arrow). When T 2 +[3,8] has +been induced, we evaluate its TTI as [3, 6], and thus PoR is trig- +gered. PoR immediately excludes the following two cells [3, 7] +and [3, 6] from the schedule. As a proof, we can see the TTIs in +these two cells are both [3, 6] in Figure 4a. +4.2.2 +Rule 2: Pruning-on-the-Underside. We now consider 푡푠′ > +푡푠, which causes pruning in the following rows but not the cur- +rent row. So we call this rule Pruning-On-the-Underside (PoU). +Specifically, if 푡푠′ > 푡푠, for each row 푟 ∈ [푡푠 + 1,푡푠′], the cells +[푟,푡푒], [푟,푡푒 − 1], · · · , [푟,푟] will be skipped. The pseudo code of +PoU is given in lines 5-8 of Algorithm 3. The correctness of PoU +is guaranteed by the following lemmas. +Lemma 3. Given a temporal푘-core T 푘 +[푡푠,푡푒] whose TTI is [푡푠′,푡푒′], +for any time interval [푡푠′′,푡푒] with푡푠′′ ∈ [푡푠,푡푠′], we have the TTI +of T 푘 +[푡푠′′,푡푒] is [푡푠′,푡푒′]. +Proof. The proof of this lemma is similar to Lemma 2 and +thus is omitted. +□ +Lemma 4. Given a temporal푘-core T 푘 +[푡푠,푡푒] whose TTI is [푡푠′,푡푒′], +for any time interval [푟,푐] with 푟 ∈ [푡푠 + 1,푡푠′] and 푐 ∈ [푡푠,푡푒], +we have T 푘 +[푟,푐] is identical to T 푘 +[푡푠,푐]. +Proof. For 푟 ∈ [푡푠 + 1,푡푠′], we have T 푘 +[푟,푡푒].TTI = [푡푠′,푡푒′] +according to Lemma 3. Thus, T 푘 +[푟,푡푒] is identical to T 푘 +[푡푠,푡푒] ac- +cording to the Equivalence (Property 2). Then, we have T 푘 +[푟,푐] is +identical to T 푘 +[푡푠,푐] when 푐 = 푡푒 − 1 since them are induced by +the same TCD operation from identical temporal graphs, and so +on for the rest [푟,푐] with the decrease of 푐 until 푐 = 푡푠. +□ +Lemma 4 indicates that, PoU safely prunes some cells in the +following rows, since these cells contain the same TTIs as their +upper cells, which even have not been enumerated yet except +the trigger cell. For example, Figure 4b illustrates two PoU in- +stances (the cells in yellow and blue colors with up arrow). On +enumerating the cell [1, 6], since the contained TTI is [2, 6], the +cells [2, 6], · · · , [2, 2] are pruned by PoU, because the TTIs in +these cells are the same as the cells [1, 6], · · · , [1, 2] respectively, +though the TTIs of cells [1, 5], · · · , [1, 2] have not been evalu- +ated. +4.2.3 +Rule 3: Pruning-on-the-Lef. Lastly, if both 푡푠′ > 푡푠 and +푡푒′ < 푡푒, for each row 푟 ∈ [푡푠′+1, 푡푒′], the cells [푟,푡푒], [푟,푡푒 −1], +· · · , [푟,푡푒′ + 1] will also be skipped, besides the cells pruned by +PoR and PoU. Although these cells are in the rows under the +current row 푡푠, the temporal 푘-core of each of them is identical +to the temporal 푘-core of a cell (namely, [푟,푡푒′]) on the right in +the same row but not its upper cell like PoU. So we call this rule +Pruning-On-the-Left (PoL). The pseudo code of PoL is given in +lines 9-12 of Algorithm 3. The correctness of PoL is guaranteed +by the following lemma. +Lemma 5. Given a temporal푘-core T 푘 +[푡푠,푡푒] whose TTI is [푡푠′,푡푒′], +for any time interval [푟,푐] with 푟 ∈ [푡푠′ + 1,푡푒′] and 푐 ∈ [푡푒′ + +1,푡푒], we have T 푘 +[푟,푐] is identical to T 푘 +[푟,푡푒′]. +Proof. Assume T 푘 +[푟,푐].TTI = [푟 ′,푐′]. According to Inclusion +(Property 3), we have [푟 ′,푐′] ⊆ [푡푠′,푡푒′] since [푟,푐] ⊆ [푡푠,푡푒]. +Thus,푐′ ⩽ 푡푒′. Then, according to Lemma 2, we have T 푘 +[푟,푡푒′].TTI += [푟 ′,푐′] since 푡푒′ ∈ [푐′,푐]. Lastly, according to Equivalence +(Property 2), we have T 푘 +[푟,푐] is identical to T 푘 +[푟,푡푒′]. +□ +For example, Figure 4b illustrates a PoL instance (the cells +in blue color with right arrow). On enumerating the cell [4, 8], +PoL is triggered since the contained TTI is [5, 6]. Then, the cells +[6, 8] and [6, 7] are pruned by PoL because the TTIs contained +in them are the same as the cell [6, 6] on the right of them. PoL +is more tricky than PoU because the cells are pruned for contain- +ing the same TTIs as other cells that are scheduled to traverse +after them by TCD algorithm. Note that, the cell [4, 8] triggers +all three kinds of pruning. In fact, a cell may trigger PoL only, +PoU only, or all three rules. +4.3 +Optimized TCD Algorithm +Compared with TCD algorithm, the improvement of Optimized +TCD (OTCD) algorithm is simply to conduct a pruning opera- +tion whenever a temporal 푘-core has been induced. Specifically, +we evaluate the TTI of this temporal 푘-core, check each pruning +rule to determine if it is triggered, and prune the specific subin- +tervals on the schedule in advance. The pseudo code of pruning +operation is given in Algorithm 3. Note that, the “prune” in Al- +gorithm 3 is a logical concept, and can have different physical +implementations. + +Junyong Yang, Ming Zhong, Yuanyuan Zhu, Tieyun Qian, Mengchi Liu, and Jeffery Xu Yu +Algorithm 3: Pruning operation. +Input: [푡푠, 푡푒] and T 푘 +[푡푠,푡푒] +1 [푡푠′,푡푒′] ← T 푘 +[푡푠,푡푒].TTI // Theorem 2 +2 if 푡푒′ < 푡푒 then +// Rule 1: PoR +3 +for 푐 ← 푡푒 - 1 to 푡푒′ do +4 +prune the subinterval [푡푠,푐] +5 if 푡푠′ > 푡푠 then +// Rule 2: PoU +6 +for 푟 ← 푡푠 + 1 to 푡푠′ do +7 +for 푐 ← te to r do +8 +prune the subinterval [푟,푐] +9 if 푡푠′ > 푡푠 and 푡푒′ < 푡푒 then +// Rule 3: PoL +10 +for r ← ts’+1 to te’ do +11 +for c ← te to te’+1 do +12 +prune the subinterval [푟,푐] +As illustrated in Figure 4b, OTCD algorithm completely elim- +inates repeated inducing of identical temporal 푘-cores, namely, +each distinct temporal 푘-core is induced exactly once during the +whole procedure. It means, the real computational complexity +of OTCD algorithm is the summation of complexity for induc- +ing each distinct temporal 푘-core but not the temporal 푘-core +of each subinterval of [푇푠,푇푒]. Therefore, we say OTCD algo- +rithm is scalable with respect to the query time interval [푇푠,푇푒]. +For many real-world datasets, the span of [푇푠,푇푒] could be very +large, while there exist only a limited number of distinct tem- +poral 푘-cores over this period, so that OTCD algorithm can still +process the query efficiently. +5 +IMPLEMENTATION +In this section, we address the physical implementation of pro- +posed algorithm. We first introduce a data structure for temporal +graph representation in Section 5.1, based on which we explain +the details of TCD Operation implementation in Section 5.2. +5.1 +Temporal Edge List (TEL) +We propose a novel data structure called Temporal Edge List +(TEL) for representing an arbitrary temporal graph (including +temporal 푘-cores that are also temporal graphs), which is both +the input and output of TCD operation. Conceptually, TEL(G) +preserves a temporal graph G = (V, E) by organizing its edges +in a 3-dimension space, each dimension of which is a set of bidi- +rectional linked lists, as illustrated in Figure 5. The first dimen- +sion is time, namely, all edges in E are grouped by their times- +tamps. Each group is stored as a bidirectional linked list called +Time List (TL), and TL(푡) denotes the list of edges with a times- +tamp 푡. Then, TEL(G) uses a bidirectional linked list, in which +each node represents a timestamp in G, as a timeline in ascend- +ing order to link all TLs, so that some temporal operations can +be facilitated. Moreover, the other two dimensions are source +vertex and destination vertex respectively. We use a container +to store the Source Lists (SL) or Destination Lists (DL) for each +vertex 푣 ∈ V, where SL(푣) or DL(푣) is a bidirectional linked list +that links all edges whose source or destination vertex is 푣. Ac- +tually, an SL or DL is an adjacency list of the graph, by which +we can retrieve the neighbor vertices and edges of a given vertex +efficiently. Given a temporal graph G, TEL(G) is built in mem- +ory by adding its edges iteratively. For each edge (푢,푣,푡) ∈ E, +it is only stored once, and TL(푡), SL(푢) and DL(푣) will append its +pointer at the tail respectively. +Figure 5 illustrates a partial TEL of our example graph. The +SLs and DLs other than SL(푣5) and DL(푣3) are omitted for con- +ciseness. Basically, TL, SL and DL offer the functionality of re- +trieving edges by timestamp and linked vertex respectively. For +example, for removing all neighbor edges of a vertex 푣 with de- +gree less than 푘 in TCD operation, we can locate SL(푣) and DL(푣) +to retrieve these edges. Moreover, the linked list of TL can offer +efficient temporal operations. For example, for truncating G to +G[푡푠,푡푒] in TCD operation, we can remove TL(푡) with 푡 < 푡푠 or +푡 > 푡푒 from the linked list of TL conveniently. To get the TTI +of a temporal 푘-core, we only need to check the head and tail +nodes of the linked list of TL in its TEL to get the minimum +and maximum timestamps respectively. The superiority of TEL +is summarized as follows. +• By TCD operation, a TEL will be trimmed to a smaller +TEL, and there is none intermediate TEL produced. Thus, +the memory requirement of (O)TCD algorithm only de- +pends on the size of initial TEL(G[푇푠,푇푒]). +• TEL consumes 푂(|E|) space for storing a temporal graph, +which is optimal because at least 푂(|E|) space is required +for storing a graph (e.g., adjacency lists). Although there +are 6|E|+2|V|+3푛 pointers of TLs, SLs and DLs stored ad- +ditionally, TEL is still compact compared with PHC-Index, +where 푛 is the number of timestamps in the graph. +• TEL supports the basic manipulations listed in Table 1 in +constant time, which are cornerstones of implementing +our algorithms and optimization techniques. +• For dynamic graphs, when a new edge coming, TEL sim- +ply appends a new node representing the current time at +the end of linked list of TL, and then adds this edge as +normal. Thus, TEL can also deal with dynamic graphs. +5.2 +Implement TCD Operation on TEL +Given a TCQ instance, our algorithm starts to work on a copy +of TEL(G[푇푠,푇푒]) in memory, which is obtained by truncating +TEL(G). Then, our algorithm only needs to maintain an instance +of TEL(T 푘 +[푡푠,푇푒]) and another instance of TEL(T 푘 +[푡푠,푡푒]) with [푡푠,푡푒] +⊆ [푇푠,푇푒] in memory. The first instance is used to induce the +first temporal 푘-core T 푘 +[푡푠+1,푇푒] by TCD for each row in Figure 3. +The second instance is used to induce the following temporal +푘-cores T 푘 +[푡푠,푡푒−1] by TCD in each row. Each TCD operation is +decomposed to a series of TEL manipulations, and trims the in- +put TEL without producing any intermediate data. +To assist the implementation of TCD operation, our algorithm +uses a global data structure H푣 that organizes all vertices in the +maintained TEL into a minimum heap ordered by their degree, +so that the vertices with less than 푘 neighbors can be retrieved +directly. Note that, whenever an edge is deleted from the main- +tained TEL, H푣 will also be updated due to the possible decrease +of vertex degrees. The trivial details of updating H푣 is omitted. +Algortithm 4 gives the implementation of TCD operation on +TEL. The algorithm takes as input the TEL of a given graph G, +along with the parameters 푘, 푡푠 and 푡푒 specifying the target tem- +poral 푘-core T 푘 +[푡푠,푡푒]. In truncation phase, TEL(G) is projected +to TEL(G[푡푠,푡푒]) (lines 1-14). Specifically, the linked list of TL +is traversed from the head and tail bidirectionally until meet- +ing 푡푠 and 푡푒 respectively. For each node representing the times- +tamp 푡 traversed, the edges in TL(푡) are removed from TEL, and + +Scalable Time-Range 푘-Core Qery on Temporal Graphs +SL(v1) +SL(v2) +SL(v10) +DL(v1) +DL(v2) +DL(v3) +DL(v10) +… +(v1,v3) +(v1,v3) +TL(1) +(v2,v3) +(v7,v10) +(v5,v7) +(v5,v8) +(v5,v8) +(v7,v8) +(v7,v9) +(v7,v9) +(v2,v3) +(v5,v7) +(v1,v3) +(v7,v8) +(v3,v4) +(v3,v4) +(v3,v5) +(v3,v6) +(v4,v6) +(v4,v7) +(v4,v7) +(v5,v6) +(v5,v7) +(v3,v4) +(v3,v6) +(v4,v5) +(v4,v7) +(v5,v6) +(v7,v10) +(v9,v10) +(v9,v10) +SL(v5) +… +… +Time Lists +Destination +Lists +Source +Lists +TL(2) +TL(3) +TL(4) +TL(5) +TL(6) +TL(7) +TL(8) +Figure 5: The conceptual illustration of a partial TEL of our running example graph. +Table 1: The basic manipulations of TEL. +Name +Functionality +Complexity +next_TL(푇퐿) / prev_TL(푇퐿) +get the next or previous TL in the linked list of TL +푂 (1) +get_SL(푣) / get_DL(푣) +get the SL or DL of a given vertex 푣 from a hash map +푂 (1) +del_TL(푇퐿) +remove the given TL node from the linked list of TL +푂 (1) +del_edge(푒) +delete a given edge 푒 = (푢, 푣, 푡) and update TL(푡), SL(푢) and DL(푣) respectively +푂 (1) +get_TTI() +return the timestamps of head and tail nodes of linked list of TL +푂 (1) +H푣 is updated for each edge removed. In decomposition phase, +TEL(G[푡푠,푡푒]) is further transformed to TEL(T 푘 +[푡푠,푡푒]) (lines 15-24). +Specifically, the algorithm pops the vertex with the least neigh- +bors from H푣 iteratively until the remaining vertices all have at +least 푘 neighbors or the heap is empty. For each popped vertex +푣, it removes the linked edges of 푣 preserved in SL(푣) and DL(푣) +from TEL respectively and updates H푣 accordingly. In particular, +a TL will be removed from the linked list of TL after the last edge +in it has been removed (lines 19 and 23). +To clarify the procedure of Algorithm 4, Figure 6 illustrates an +example of inducing T 2 +[4,5] from T 2 +[3,6]. The edges are going to be +deleted are marked in red color. We can see that, the procedure +is actually a stream of edge deletion, while TEL maintains the +entries to retrieve the remaining edges. +5.3 +Complexity +TCD.Theoretically, the complexity of TCD algorithm is bounded +by �푇푒 +푡=푇푠{(|V[푡,푇푒]|+|E[푡,푇 푒]|) log |V[푡,푇푒]|+푚|E[푡,푇푒]|}, where +푚 is a small constant. For each anchored푡, TCD algorithm gradu- +ally peels T 푘 +[푡,푇푒] like an onion by TCD operation until it contains +none temporal 푘-core. In the process, there are at most |E[푡,푇 푒]| +edges deleted, and deleting each edge takes a small constant time +푂(푚) for TEL updating and at most 푂(log |V[푡,푇푒]|) time for +H푣 maintenance. Similarly, there are at most |V[푡,푇푒]| vertices +deleted, and deleting each vertex takes 푂(log |V[푡,푇푒]|) time for +H푣 maintenance. Therefore, The total time overhead is the sum +of edge and vertex deleting costs. +Note that, the complexity of TCD algorithm can also be rep- +resented by 푂((푇푒 −푇푠)2퐵) according to Algorithm 2, where 퐵 +is the average time overhead of TCD operation. However, 퐵 can- +not be estimated precisely, since each TCD operation may delete +zero to |E[푡,푇 푒]| edges. Therefore, we bound the complexity by +the maximum deleting cost according to Algorithm 4, which is +more reasonable. +OTCD. The complexity of OTCD algorithm is simply bounded +by �(|푉 ∗| + |퐸∗|) log |푉 ∗| +푚|퐸∗|, where 푉 ∗ and 퐸∗ refer to the +sets of vertices and edges that have to be deleted for inducing the +result temporal 푘-cores respectively. Due to the pruning rules, +there are much less temporal 푘-cores induced by OTCD algo- +rithm. Thus, |푉 ∗| and |퐸∗| are orders of magnitude less than the +total number of vertices and edges deleted in TCD algorithm, +most of which are actually used for inducing identical temporal +푘-cores, though they cannot be really estimated. +6 +EXTENSION +To demonstrate the wide applicability of our approach in prac- +tice, we present several typical scenarios that extends the data +model or query model of TCQ, and sketch how to address them +based on our data structure and algorithm in this section. +6.1 +Data Model Extension +Dynamic Graph. Since most real-world graphs are evolving +over time, it is significant to fulfill TCQ on dynamic graphs. Ben- +efitted from its design in “timeline” style, our data structure TEL +can deal with new edges naturally in memory through two new +manipulations add_TL(푡) and add_edge(푢,푣,푡). When a new edge +(푢,푣,푡) arrived, we firstly create an empty TL(푡), and append it +at the end of the linked list of TL since 푡 is obviously greater +than the existing timestamps. Then, we create a new edge node +for (푢,푣,푡) and append it to TL(푡), SL(푢) and DL(푣) respectively. + +Junyong Yang, Ming Zhong, Yuanyuan Zhu, Tieyun Qian, Mengchi Liu, and Jeffery Xu Yu +DL(v1) +DL(v2) +DL(v4) +SL(v1) +SL(v2) +SL(v3) +SL(v4) +DL(v3) +TL(3) +TL(4) +TL(5) +TL(6) +v2 +v1 +v4 +v3 +5 +3 +6 +5 +5 +4 +(v1,v4) +(v1,v2) +(v1,v3) +(v3,v4) +(v2,v3) +(v2,v4) +DL(v1) +DL(v2) +DL(v4) +SL(v1) +SL(v2) +SL(v3) +SL(v4) +DL(v3) +TL(4) +TL(5) +(v1,v2) +(v1,v3) +(v2,v3) +(v2,v4) +DL(v1) +DL(v2) +DL(v4) +SL(v1) +SL(v2) +SL(v3) +SL(v4) +DL(v3) +TL(4) +TL(5) +(v1,v2) +(v1,v3) +(v2,v3) +decomposition +truncation +v2 +v1 +v4 +v3 +5 +5 +5 +4 +v2 +v1 +v3 +5 +5 +4 +Figure 6: An example of TCD operation on TEL. +Algorithm 4: TCD operation in Algorithm 2 +Input: TEL(G), [푡푠,푡푒], 푘 +Output: TEL(T 푘 +[푡푠,푡푒]) +1 푇퐿 ← the head of linked list of TL in TEL(G) +2 while 푇퐿.timestamp ≠ 푡푠 do +3 +for edge 푒 in 푇퐿 do +4 +del_edge(푒) +5 +udpate H푣 +6 +del_TL(푇퐿) +7 +푇퐿 ← next_TL(푇퐿) +8 푇퐿 ← the tail of linked list of TL in TEL(G) +9 while 푇퐿.timestamp ≠ 푡푒 do +10 +for edge 푒 in 푇퐿 do +11 +del_edge(푒) +12 +udpate H푣 +13 +del_TL(푇퐿) +14 +푇퐿 ← prev_TL(푇퐿) +15 while H푣 is not empty and H푣.peek < 푘 do +16 +vertex 푣 ← H푣.pop() +17 +for edge 푒 in SL(푣) do +18 +del_edge(푒) +19 +del_TL(TL(푒.timestamp)) if the TL is empty +20 +update H푣 +21 +for edge 푒 in DL(푣) do +22 +del_edge(푒) +23 +del_TL(TL(푒.timestamp)) if the TL is empty +24 +update H푣 +Both manipulations are finished in constant time. The mainte- +nance of a dynamic TEL is actually consistent with the construc- +tion of a static TEL. Therefore, our (O)TCD algorithm can run +on the dynamic TEL anytime. +In contrast, updating PHC-Index is a non-trivial process. Al- +though there are previous work [20, 29] on coreness updating +for dynamic graphs, the update is only valid for the whole life +time of graph. While, for an arbitrary start time, it is uncertain +whether the coreness of a vertex will be changed by a new edge. +6.2 +Query Model Extension +The existing graph mining tasks regarding 푘-core introduce var- +ious constraints. For temporal graphs, we only focus on the tem- +poral constraints. In the followings, we present two of them +that can be integrated into TCQ model and also be addressed +by our algorithm directly, which demonstrate the generality of +our model and algorithm. +Link Strength Constraint. In the context of temporal graph, +link strength usually refers to the number of parallel edges be- +tween a pair of linked vertices. Obviously, the minimum link +strength in a temporal 푘-core represents some important prop- +erties like validity, since noise interaction may appear over time +and a pair of vertices with low link strength may only have oc- +casional interaction during the time interval. Actually, the previ- +ous work [34] has studied this problem without the time interval +constraint. Therefore, it is reasonable to extend TCQ to retrieve +푘-cores with a lower bound of link strength during a given time +interval. It can be achieved by trivially modifying the TCD Oper- +ation. Specifically, the modified TCD Operation will remove the +edges between two vertices once the number of parallel edges +between them is decreased to be lower than the given lower +bound of link strength, while the original TCD operation will +do this when the number becomes zero. Thus, the modification +brings almost none extra time and space consumption. +Time Span Constraint. In many cases, we prefer to retrieve +temporal 푘-cores with a short time span (between their earliest +and latest timestamps), which is similar to the previous work on +density-bursting subgraphs [5]. Because such a kind of short- +term cohesive subgraphs tend to represent the occurrence of +some special events. TCQ can be conveniently extended for re- +solving the problem by specifying a constraint of time span. Since +the time span of a temporal푘-core is preserved in its TEL, which +is actually the length of its TTI, we can abandon the tempo- +ral 푘-cores returned by TCD operation that cannot satisfy the +time span constraint on the fly. It brings almost no extra time +and space consumption. Moreover, we can also extend TCQ to +find the temporal 푘-core with the shortest or top-푛 shortest time +span. + +Scalable Time-Range 푘-Core Qery on Temporal Graphs +Table 2: Datasets. +Name +|V| +|E| +Span(days) +Youtube +3.2M +9.4M +226 +DBLP +1.8M +29.5M +17532 +Flickr +2.3M +33M +198 +CollegeMsg +1.8K +20K +193 +email-Eu-core-temporal +0.9K +332K +803 +sx-mathoverflow +24.8K +506K +2350 +sx-stackoverflow +2.6M +63.5M +2774 +Table 3: Selected temporal 푘-core queries. +id +G +푡푠 (sec) +푡푒 (sec) +푘 +result # +1 +CollegeMsg +554400 +565200 +2 +61 +2 +CollegeMsg +558000 +568800 +2 +21 +3 +CollegeMsg +561600 +572400 +2 +27 +4 +CollegeMsg +565200 +576000 +2 +26 +5 +CollegeMsg +568800 +579600 +2 +10 +6 +email-Eu-core-temporal +36000 +46800 +3 +2 +7 +email-Eu-core-temporal +39600 +50400 +3 +3 +8 +email-Eu-core-temporal +284400 +295200 +3 +7 +9 +email-Eu-core-temporal +288000 +298800 +3 +25 +10 +email-Eu-core-temporal +291600 +302400 +3 +16 +11 +sx-mathoverflow +864000 +867600 +2 +8 +12 +sx-mathoverflow +1116000 +1119600 +2 +4 +13 +sx-mathoverflow +1389600 +1393200 +2 +5 +14 +sx-mathoverflow +1483200 +1486300 +2 +2 +15 +sx-mathoverflow +1738800 +1742400 +2 +8 +16 +sx-stackoverflow +378000 +381600 +2 +6 +17 +sx-stackoverflow +417600 +421200 +2 +37 +18 +sx-stackoverflow +421200 +424800 +2 +5 +19 +sx-stackoverflow +424800 +428400 +2 +5 +20 +sx-stackoverflow +486000 +489600 +2 +10 +7 +EXPERIMENT +In this section, we conduct experiments to verify both efficiency +and effectiveness of the proposed algorithm on a Windows ma- +chine with Intel Core i7 2.20GHz CPU and 64GB RAM. The al- +gorithms are implemented through C++ Standard Template Li- +brary. Our source codes are shared on GitHub1. +7.1 +Dataset +We choose seven temporal graphs with different sizes and do- +mains for our experiments. The first three graphs are obtained +from KONECT Project [16], and the other four graphs are ob- +tained from the SNAP [17]. The basic statistics of these graphs +are given in Table 2. All timestamps are unified to integers in +seconds. +7.2 +Efficiency +To evaluate the efficiency of our algorithm, we firstly manually +select twenty temporal푘-core queries from tested random queries +with a time span (namely, 푇푒 −푇푠) of 1-3 days, which have been +verified to be valid, namely, there is at least one temporal 푘-core +returned for each query. The setting of time span is moderate, +otherwise other algorithms than OTCD can hardly stop success- +fully. Table 3 gives the details of query parameters, so that other +1https://github.com/ThomasYang-algo/Temporal-k-Core-Query-Project +Table 4: Effect of pruning rules. +id +Triggered Times +Pruned Cell Percentage (%) +PoR +PoU +PoL +PoR +PoU +PoL +Total +1 +54 +72 +2 +0.02 +72 +23.6 +95.62 +6 +2 +4 +1 +0.01 +51.8 +32.1 +83.91 +11 +8 +10 +1 +0.04 +57.1 +24.5 +81.64 +16 +5 +9 +1 +0.04 +56.9 +33.5 +90.44 +1 +2 +3 +4 +5 +0.1 +1 +10 +100 +1000 +0.01 +3600 +Response Time(s) +Query Id + Baseline + TCD + OTCD +(a) CollegeMsg +6 +7 +8 +9 +10 +0.1 +1 +10 +100 +1000 +0.01 +3600 +Response Time(s) +Query Id + Baseline + TCD + OTCD +(b) email-Eu-core-temporal +11 +12 +13 +14 +15 +0.1 +1 +10 +100 +1000 +0.01 +3600 +Response Time(s) +Query Id + Baseline + TCD + OTCD +(c) sx-mathoverflow +16 +17 +18 +19 +20 +0.1 +1 +10 +100 +1000 +0.01 +3600 +Response Time(s) +Query Id + Baseline + TCD + OTCD +(d) sx-stackoverflow +Figure 7: The comparison of response time for selected +queries on SNAP graphs. +researchers can reverify our experimental results or compare +with our approach with the same queries. +Figure 7 compares the response time of Baseline (iPHC-Query), +TCD and OTCD algorithms for each selected query respectively, +which clearly demonstrates the efficiency of ouralgorithm. Firstly, +TCD performs better than baseline for all twenty queries due to +the physical efficiency of TEL, though they both decrementally +or incrementally induce temporal푘-cores. Specifically, TCD spends +around 100 sec for each query. In contrast, baseline spends more +than 1000 sec on CollegeMsg and even cannot finish within an +hour on two other graphs, though it uses a precomputed in- +dex. Furthermore, OTCD runs two or three orders of magnitude +faster than TCD, and only spends about 0.1-1 sec for each query, +which verifies the effectiveness of our pruning method based on +TTI. +To compare the effect of three pruning rules in OTCD algo- +rithm, Table 4 lists their triggered times and the percentage of +subintervals pruned by them for several queries respectively. PoR +and PoU are triggered frequently because their conditions are +more easily to be satisfied. However, PoR actually contributes +pruned subintervals much less than the other two. Because it +only prunes subintervals in the same row, and in contrast, PoU +and PoL can prune an “area” of subintervals. Overall, the three +pruning rules can achieve significant optimization effect together + +Junyong Yang, Ming Zhong, Yuanyuan Zhu, Tieyun Qian, Mengchi Liu, and Jeffery Xu Yu +TCD +OTCD +0.01 +0.1 +1 +10 +100 +1000 +Response Time(s) + TCD- 25%~75% + OTCD- 25%~75% + Range within 1.5IQR + + Median Line + + Mean + Outliers +(a) Youtube +TCD +OTCD +0.1 +1 +10 +100 +1000 +Response Time(s) + TCD- 25%~75% + OTCD- 25%~75% + Range within 1.5IQR + Median Line + Mean + Outliers +(b) Flickr +Figure 8: The statistical distribution of response time for +random queries on KONECT graphs. +Table 5: Memory consumption of (O)TCD algorithm. +Dataset +Process Memory (GB) +CollegeMsg +0.02 +sx-mathoverflow +0.06 +Youtube +1.7 +DBLP +3.1 +Flickr +3.5 +sx-stackoverflow +6.5 +by enabling OTCD algorithm to skip more than 80 percents of +subintervals. +To evaluate the stability of our approach, we conduct statis- +tical analysis of one hundred valid random queries on two new +graphs, namely, Youtube and Flickr. We visualize the distribution +of response time of TCD and OTCD algorithms for these random +queries as boxplots, which are shown by Figure 8. The boxplots +demonstrate that the response time of OTCD varies in a very +limited range, which indicates that the OTCD indeed performs +stable in practice. The outliers represent some queries that have +exceptionally more results, which can be seen as a normal phe- +nomenon in reality. They may reveal that many communities of +the social networks are more active during the period. +Moreover, Table 5 reports the process memory consumption +for different datasets, which depends on the size of TEL mostly. +We can observe that, 1) for the widely-used graphs like Youtube, +DBLP, Flickr and stackoverflow, several gigabytes of memory +are needed for storing TEL, which is acceptable for the ordinary +hardware; and 2) for the very large graphs with billions of edges, +the size of TEL is hundreds of gigabytes approximately, which +would require the distributed memory cluster like Spark. +To verify the scalability of our method with respect to the +query parameters, we test the three algorithms with varing min- +imum degree 푘 and time span (namely, 푇푒 −푇푠) respectively. +Impact of 푘. We select a typical query with span fixed and +푘 ranging from 2 to 6 for different graphs. The response time of +tested algorithms are presented in Figure 9, from which we have +an important observation against common sense. That is, differ- +ent from core decomposition on non-temporal graphs, when the +value of 푘 increases, the response time of TCD and OTCD algo- +rithms decreases gradually. For OTCD, the behind rationale is +clear, namely, its time cost is only bounded by the scale of re- +sults, which decreases sharply with the increase of 푘. To sup- +port the claim, Figure 10 and Figure 11 show the trend of the +amount of result cores and connected components in the result +cores changing with 푘. Intuitively, a greater value of k means +a stricter constraint and thereby filters out some less cohesive +2 +3 +4 +5 +6 +0.1 +1 +10 +100 +1000 +0.01 +3600 +Response Time(s) +k + Baseline + TCD + OTCD +(a) CollegeMsg +2 +3 +4 +5 +6 +0.1 +1 +10 +100 +1000 +0.01 +3600 +Response Time(s) +k + Baseline + TCD + OTCD +(b) sx-mathoverflow +2 +3 +4 +5 +6 +0.1 +1 +10 +100 +1000 +0.01 +3600 +Response Time(s) +k + Baseline + TCD + OTCD +(c) sx-stackoverflow +Figure 9: Trend of response time under a range of 푘. +2 +3 +4 +5 +6 +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +0 +Quantity of Core +k +(a) CollegeMsg +2 +3 +4 +5 +6 +10 +1 +10 +2 +10 +3 +10 +4 +Quantity of Core +k +(b) sx-mathoverflow +2 +3 +4 +5 +6 +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +Quantity of Core +k +(c) sx-stackoverflow +Figure 10: Trend of amount of distinct temporal 푘-cores +under a range of 푘. +2 +3 +4 +5 +6 +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +Connected Component +k +(a) CollegeMsg +2 +3 +4 +5 +6 +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +0 +10 +6 +Connected Component +k +(b) sx-mathoverflow +2 +3 +4 +5 +6 +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +Connected Component +k +(c) sx-stackoverflow +Figure 11: Trend of amount of connected components in +temporal 푘-cores under a range of 푘. +cores. We can see the trend of runtime decrease for OTCD in +Figure 9 is almost the same as the trend of core amount decrease +in Figure 10, which also confirms the scalability of OTCD algo- +rithm. For TCD, the behind rationale is complicated, since it enu- +merates all subintervals and each single decomposition is more +costly with a greater value of 푘. It is just like peeling an onion +layer by layer, which has less layers with a greater value of 푘, so +that the maintenance between layers become less. +Impact of span. Similarly to the test of 푘, we also evalu- +ate the scalability of different algorithms when the query time +span increases. The results are presented in Figure 12. Although +the number of subintervals increases quadratically, the response +time of OTCD still increases moderately following the scale of +query results. In contrast, TCD runs dramatically slower when +the query time span becomes longer. +The above results demonstrate that the efficiency of OTCD +is not sensitive to the change of query parameters, so that it is +scalable in terms of query time interval. +Lastly, for a large graph with a long time span like Youtube, +we test OTCD algorithm by querying temporal 10-cores over +the whole time span. The result is, to find all 19,146 temporal +10-cores within 226 days, the OTCD algorithm spent about 55 +minutes, which is acceptable for such a “full graph scan” task. + +Scalable Time-Range 푘-Core Qery on Temporal Graphs +24 +36 +48 +60 +72 +0.1 +1 +10 +100 +1000 +0.01 +3600 +Response Time(s) +Span(h) + Baseline + TCD + OTCD +(a) CollegeMsg +24 +36 +48 +60 +72 +0.1 +1 +10 +100 +1000 +0.01 +3600 +Response Time(s) +Span(h) + Baseline + TCD + OTCD +(b) sx-mathoverflow +24 +36 +48 +60 +72 +0.1 +1 +10 +100 +1000 +0.01 +3600 +Response Time(s) +Span(h) + Baseline + TCD + OTCD +(c) sx-stackoverflow +Figure 12: Trend of response time under a range of span. +7.3 +Effectiveness +The effectiveness of TCQ is two-fold. Firstly, by given a flexible +time interval, we can find many temporal 푘-cores of different +subintervals, each of which represents a community emerged in +a specific period. Consider the above test on Youtube. Although +it is not feasible to exhibit all 19,146 cores, Figure 13 shows their +distribution by time span. The number of cores generally de- +creases with the increase of time span, which makes sense be- +cause there are always a lot of small communities emerged dur- +ing short periods and then they will interact with each other and +be merged to larger communities within a longer time span. +Secondly, we can continue to filter and analyse the result cores +to gain insights. For example, we record the date in GMT time +for nine of the result cores with a time span less than one day in +Youtube, and try to figure out if they emerged for some special +reasons. Table 6 lists the date and size of the nine cores. We can +see that there is a large core emerged on Dec 10, 2006, which +means more than 40,000 accounts had nearly one million inter- +actions with each other in just a day. That is definitely caused +by a special event. While, most of the rest cores emerged during +summer vacation, which may mean people have more interac- +tions on Youtube in the period. +0 +50 +100 +150 +200 +226 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Number of core +Time span(days) +Figure 13: Distribution of +all +temporal +10-cores +in +Youtube by time span. +Table 6: The date and size +of nine temporal 10-cores +emerged within one day in +Youtube. +Date +|V| +|E| +Dec 10 2006 +46499 +885128 +Feb 08 2007 +1268 +12054 +Mar 25 2007 +21 +139 +Jun 15 2007 +98 +713 +Jun 18 2007 +20 +100 +Jun 20 2007 +124 +1012 +Jun 30 2007 +21 +110 +Jul 02 2007 +21 +110 +Jul 06 2007 +12 +66 +7.4 +Case Study +For case study, we employ OTCD algorithm to query tempo- +ral 10-cores on DBLP. The query interval is set as 2010 to 2018, +which spans over 8 years. By statistics, there exist 43 temporal +10-cores during that period, with 39 of them containing the au- +thor Jian Pei, for whom we further build an ego network from +three selected cores in defferent years. Figure 14 shows the ego +network. The authors in the three cores emerged in 2010, 2012 +and 2014 are shaded by red, yellow and blue respectively. By ob- +serving the evolution of ego network over years, we can infer +the change of author’s research interests or affiliations. +22 vertices of a 10-core +arising in 2010 +14 vertices of a 10-core +arising in 2012 +15 vertices of a 10-core +arising in 2014 +Figure 14: Case Study in DBLP coauthorship network. +A friendship community with 32 members arising in 2007 +114 newly added members on the first day after +124 newly added members on the second day after +Figure 15: Case Study in Youtube friendship network. +To further demonstrate the potential of TCQ, we also employ +TCQ to find temporal 푘-cores that expand quickly over time. +This topic has been addressed in [5]. Since OTCD returns all +distinct cores efficiently, we can conveniently achieve the goal +by identifying the cores contained by other larger cores within +a few of days from the results. Figure 15 shows such a bursting +community on Youtube friendship network. The 32 central ver- +tices colored in red comprise an initial temporal 10-core within +two days. This core is contained by another core about four +times larger, while the TTI of the larger core only expands by +one day. The new vertices in the larger core are colored in or- +ange. Then, the new vertices colored in yellow join them to com- +prise a twice larger new core in the next day. Clearly, these three +temporal 10-cores together represent a community that grows +remarkably fast. In the real world, with more concrete informa- +tion of graphs, such usages of TCQ will facilitate applications +like recommendation, disease control, etc. +7.5 +Discussion on the value of 푘 +TCQ achieves relaxing the constraint on query time interval when +composing푘-core queries on temporal graphs. However, the value +of 푘 is still needed as an input parameter. We give a simple and +rational criteria here for selecting the proper푘 value on different +graphs, though many potential factors have different impacts on +the selection. The criteria is based on two intuitive facts. Firstly, +the number of distinct temporal 푘-cores over a given time in- +terval will decrease with the increase of 푘. Secondly, the size + +uCtelg +Surya Nepal +JianYin +EnhongiChen +Li Xiong +Bin Jiang +ShuhuiWang +Jian Pei +Qingming Huang +Jiawei Han +Siyuan Liu +Ying Zhang +Xindong Wu +Jie Tang +Kai Xu +Chang Liu +Xiang Wang +Rong Jin +Yang Wang +Jinjun Chen +Jeffrey Xu YuJiangchuan Liu +Philip S. Yu +Feng Zhao +Ke Wang +Xuemin Lin +Jian Chen +Hua Wang +Kunbiu +Wenjie Zhang +KeYi +XueLi +Jin Huang +QiangYang +Wei Wang +Hang Li +Yu Yeng +lunduoJunyong Yang, Ming Zhong, Yuanyuan Zhu, Tieyun Qian, Mengchi Liu, and Jeffery Xu Yu +3 +5 +2 +4 +6 +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +0 +CollegeMsg +Quantity of Core +k + Quantity of Core + Average Size +25 +50 +75 +100 +125 +150 +Average Core Size +Figure 16: A statistical chart for selecting the value of 푘. +of returned temporal 푘-cores will shrink with the increase of +푘. Normally, we expect the result cores to be concise and non- +overlapping, especially when detecting the suspicious commu- +nities that are inherently small and isolated, thereby preferring +a greater value of 푘. However, the number of result cores also +matters, which requires the value of 푘 not being too great, oth- +erwise there could be too few results. Therefore, the selection of +푘 should take both size and number of result cores into account, +just like the trade-off between precision and recall. +For example, with 푘 ranging from 2 until 6, Figure 16 shows +the falling curves of both number and average size of result cores +over a specific time interval on CollegeMsg. We can observe that, +setting 푘 = 5 should be a good choice, since the core size has +declined to a relatively small level while the number of results +is still fairly sufficient. +8 +RELATED WORK +Recently, a variety of 푘-core query problems have been stud- +ied on temporal graphs, which involve different temporal objec- +tives or constraints in addition to cohesiveness. The most rele- +vant work to ours is historical 푘-core query [36], which gives a +fixed time interval as query condition. In contrast, our tempo- +ral 푘-core query flexibly find cores of all subintervals. Moreover, +Galimberti et al [12] proposed the span-core query, which also +gives a time interval as query condition. However, the span-core +requires all edges to appear in every moment within the query +interval, which is too strict in practice. Actually, historical푘-core +relaxes span-core, and temporal 푘-core further relaxes historical +푘-core. +Besides, there are the following related work. Wu et al [34] +proposed (푘,ℎ)-core and studied its decomposition algorithm, +which gives an additional constraint on the number of parallel +edges between each pair of linked vertices in the 푘-core, namely, +they should have at least ℎ parallel edges. Li et al [19] proposed +the persistent community search problem and gives a compli- +cated instance called (휃,휏)-persistent 푘-core, which is a 푘-core +persists over a time interval whose span is decided by the pa- +rameters. Similarly, Li et al [21] proposed the continual cohe- +sive subgraph search problem. Chu et al [5] studied the prob- +lem of finding the subgraphs whose density accumulates at the +fastest speed, namely, the subgraphs with bursting density. Qin +et al [27, 28] proposed the periodic community problem to re- +veal frequently happening patterns of social interactions, such +as periodic 푘-core. Wen et al [1] relaxed the constraints of (푘,ℎ)- +core and proposed quasi-(푘,ℎ)-core for better support of main- +tenance. Lastly, Ma et al [25] studied the problem of finding +dense subgraph on weighted temporal graph. These works all +focus on some specific patterns of cohesive substructure on tem- +poral graphs, and propose sophisticated models and methods. +Compared with them, our work addresses a fundamental query- +ing problem, which finds the most general 푘-cores on temporal +graphs with respect to two basic conditions, namely, 푘 and time +interval. As discussed in Section 6.2, we can extend TCQ to find +the more specific 푘-cores by importing the constraints defined +by them, because most of the definitions are special cases of tem- +poral 푘-core, but not vice versa. +Lastly, many research work on cohesive subgraph query for +non-temporal graphs also inspire our work. We categorize them +by the types of graphs as follows: undirected graph [3, 9, 13, 23, +35, 37], directed graph [4, 24, 30], labeled graph [6, 18, 31], attrib- +uted graph [7, 14, 15, 26], spatial graph [8, 10, 39], heterageneous +information network [11]. Besides, many work specific to bipar- +tite graph [22, 32, 33, 38] also contain valuable insights. +9 +CONCLUSION AND FUTURE WORK +For querying communities like푘-cores on temporal graphs, spec- +ifying a time interval in which the communities emerge is the +most fundamental query condition. To the best knowledge we +have, we are the first to study a temporal 푘-core query that al- +lows the users to give a flexible interval and returns all distinct 푘- +cores emerging in any subintervals. Dealing with such a query in +brute force is obviously costly due to the possibly large number +of subintervals. Thus, we propose a novel decremental 푘-core +inducing algorithm and the auxiliary optimization and imple- +mentation methods. Our algorithm only enumerates the neces- +sary subintervals that can induce a final result and reduces re- +dundant computation between subintervals significantly. More- +over, the algorithm is physically decomposed to a series of ef- +ficient data structure manipulations. As a result, although our +algorithm does not use any precomputed index, it still outper- +forms an incremental version of the latest index-based approach +by a remarkable margin. In conclusion, our algorithm is scalable +with respect to the span of given time interval. +In the future, we will study how to leverage our algorithm +as a framework to integrate various temporal 푘-core analytics. +There are a number of related work have considered different +temporal constraints of 푘-cores, most of which can be combined +with the time interval condition to offer more powerful function- +ality. However, their query processing algorithms are essentially +diverse. Therefore, we need to bridge the gap based on a general +and scalable algorithm like ours. +REFERENCES +[1] Wen Bai, Yadi Chen, and Di Wu. 2020. Efficient temporal core maintenance +of massive graphs. Information Sciences 513 (2020), 324–340. +[2] Vladimir Batagelj and Matjaz Zaversnik. 2003. An O (m) algorithm for cores +decomposition of networks. arXiv preprint cs/0310049 (2003). +[3] Francesco Bonchi, Arijit Khan, and Lorenzo Severini. 2019. +Distance- +generalized core decomposition. 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The VLDB +Journal 26, 5 (2017), 709–727. + diff --git a/-dE2T4oBgHgl3EQfQgYx/content/tmp_files/load_file.txt b/-dE2T4oBgHgl3EQfQgYx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3159a4b9681eed6194c0aaee788e9cb9fba78647 --- /dev/null +++ b/-dE2T4oBgHgl3EQfQgYx/content/tmp_files/load_file.txt @@ -0,0 +1,1603 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf,len=1602 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='03770v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='DB] 10 Jan 2023 Scalable Time-Range 푘-Core Qery on Temporal Graphs Junyong Yang Wuhan University Wuhan, China thomasyang@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='cn Ming Zhong∗ Wuhan University Wuhan, China clock@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='cn Yuanyuan Zhu Wuhan University Wuhan, China yyzhu@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='cn Tieyun Qian Wuhan University Wuhan, China qty@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='cn Mengchi Liu South China Normal University Guangzhou, China liumengchi@scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='cn Jeffery Xu Yu The Chinese University of Hong Kong Hong Kong, China yu@se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='hk ABSTRACT Querying cohesive subgraphs on temporal graphs with various time constraints has attractedintensive research interests recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In this paper, we study a novel Temporal 푘-Core Query (TCQ) problem: given a time interval, find all distinct 푘-cores that exist within any subintervals from a temporal graph, which general- izes the previous historical 푘-core query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' This problem is chal- lenging because the number of subintervals increases quadrati- cally to the span of time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For that, we propose a novel Temporal Core Decomposition (TCD) algorithm that decremen- tally induces temporal 푘-cores from the previously induced ones and thus reduces “intra-core” redundant computationsignificantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Then, we introduce an intuitive concept named Tightest Time Interval (TTI) for temporal 푘-core, and design an optimization technique with theoretical guarantee that leverages TTI as a key to predict which subintervals will induce duplicated푘-cores and prunes the subintervals completely in advance, thereby eliminat- ing “inter-core” redundant computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The complexity of op- timized TCD (OTCD) algorithm no longer depends on the span of query time interval but only the scale of final results, which means OTCD algorithm is scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Moreover, we propose a com- pact in-memory data structure named Temporal Edge List (TEL) to implement OTCD algorithm efficiently in physical level with bounded memory requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' TEL organizes temporal edges in a “timeline” and can be updated instantly when new edges ar- rive, and thus our approach can also deal with dynamic temporal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We compare OTCD algorithm with the incremental his- torical 푘-core query on several real-world temporal graphs, and observe that OTCD algorithm outperforms it by three orders of magnitude, even though OTCD algorithm needs none precom- puted index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 1 INTRODUCTION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 Motivation Discovering communities or cohesive subgraphs from temporal graphs has great values in many application scenarios, thereby attracting intensive research interests [1, 5, 12, 19, 25, 27, 34, ∗The corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' This work is licensed under the Creative Commons BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='0 International License.' metadata={'source': 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owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Publication rights licensed to the VLDB Endowment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Proceedings of the VLDB Endowment, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 14, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 1 ISSN 2150-8097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' doi:XX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='XX/XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='XX v10 v1 v2 v3 v4 v5 v6 v7 v8 v9 1 1 6 6 6 6 5 5 2 2 2 2 7 7 2 6 5 5 4 3 5 5 5 5 3 2 2 8 4 1 2-core of time interval [1,8] 2-core of time interval [5,6] 2-core of time interval [2,4] 2-core of time interval [2,6] Figure 1: A running example of temporal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 36] in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Here, a temporal graph refers to an undi- rected multigraph in which each edge has a timestamp to indi- cate when it occurred, as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For example, consider a graph consisting of bank accounts as vertices and fund transfer transactions between accounts as edges with natu- ral timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For applications such as anti-money-laundering, we would like to search communities like 푘-cores that contain a known suspicious account and emerge within a specific time interval like the World Cup, and investigate the associated ac- counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' To address the community query/search problem for a fixed time interval, the concept of historical 푘-core [36] is proposed recently, which is the 푘-core induced from the subgraph of a temporal graph in which all edges occurred out of the time in- terval have been excluded and the parallel edges between each pair of vertices have been merged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Also, the PHC-Query method is proposed to deal with historical 푘-core query/search by using a precomputed index efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' However, we usually do not know the exact time interval of targeted historical 푘-core in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Actually, if we can know the exact time interval, a traditional core decom- position on the projected graph over the given time interval is efficient enough to address the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, it is more reason- able to assume that we can only offer a flexible time interval and need to induce cores from all its subintervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For example, for detecting money laundering by soccer gambling during the World Cup, the 푘-cores emerged over a few of hours around one of the matches are more valuable than a large 푘-core emerging over the whole month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Therefore, we aim to generalize historical 푘-core query by al- lowing the result 푘-cores to be induced by any subinterval of a given time interval, like “flexible versus fixed”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The historical 푘- core query can be seen as a special case of our problem that only evaluates the whole interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Consider the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Junyong Yang, Ming Zhong, Yuanyuan Zhu, Tieyun Qian, Mengchi Liu, and Jeffery Xu Yu Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' As illustrated in Figure 1, given a time interval [1,8], historical 푘-core query only returns the largest core marked by the grey dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In contrast, our temporal 푘-core query re- turns four cores marked by dashed lines with different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' These cores can reveal various insights unseen by the largest one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For ex- ample, some cores like red and blue that emerge in bursty periods may be caused by special events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Also, some persistent or periodic cores may be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Further, we can analyze the interaction be- tween cores and how they evolve over time, such as the small cores like red and blue are merged to the large cores like yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Lastly, some underlying details may be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' During the merge, the ver- tex 푣5 may play a vital role because it appears in all the cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Overall, our general and flexible query model can support many interesting temporal community analytics applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The general and flexible temporal k-core query we study is naturally a generalization of existing query models like histori- cal 푘-core and also potentially a common technique for various temporal graph mining tasks mentioned in the above example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2 Contribution In this paper, we study a novel temporal 푘-core query problem: given a time interval, find all distinct 푘-cores that exist within any subintervals from a temporal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Although the existing PHC-Query returns the historical 푘-core of a fixed time inter- val efficiently, it cannot be trivially applied to deal with the new problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Because inducing 푘-cores for each subinterval individ- ually from scratch is not scalable, since the number of subinter- vals increases quadratically with the span of time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' More- over, PHC-Query suffers from two other intrinsic shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Firstly, it relies on a PHC-Index that precomputes the coreness of all vertices over all time intervals, thereby incurring heavy of- fline time and space overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Secondly, due to its sophisticated construction, it is unclear if PHC-Index can be updated dynami- cally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' It is against the dynamic nature of temporal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In order to overcome the above challenges, we present a novel temporal core decomposition algorithm and auxiliary optimiza- tion and implementation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Our contributions can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We formalize a general time-range cohesive subgraph query problem on ubiquitous temporal graphs, namely, tempo- ral푘-core query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Many previous typical푘-core query mod- els on temporal graphs can be equivalently represented by temporal 푘-core query with particular constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' To address temporal 푘-core query, we propose a simple and yet efficient algorithm framework based on a novel temporal core decomposition operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' By using tempo- ral core decomposition, our algorithm always decremen- tally induces a temporal k-core from the previous induced temporal k-core except the initial one, thereby reducing redundant computation significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Moreover, we propose an intuitive concept named tight- est time interval for temporal k-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' According to the properties of tightest time intervals, we design three prun- ing rules with theoretical guarantee to directly skip subin- tervals that will not induce distinct temporal 푘-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' As a result, the optimized algorithm is scalable in terms of the span of query time interval, since only the necessary subintervals are enumerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For physical implementation of our algorithm, we pro- pose a both space and time efficient data structure named temporal edge list to represent a temporal graph in mem- ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' It can be manipulated to perform temporal core de- composition and tightest time interval based pruning rapidly with bounded memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' More importantly, temporal edge list can be incrementally updated with evolving temporal graphs, so that our approach can support dynamic graph applications naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Lastly, we evaluate the efficiency and effectiveness of our algorithm on real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The experimental re- sults demonstrate that our algorithm outperforms the im- proved PHC-Query by three orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Section 2 for- mally introduces the data model and query model, and also gives a baseline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Sections 3-5 present our algorithm, opti- mization and implementation techniques respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Section 6 briefly discusses some meaningful extension of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Section 7 presents the experiments and analyzes the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Sec- tion 8 investigates the related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Section 9 concludes our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 2 PRELIMINARY In this section, we propose a generalized 푘-core query problem on temporal graphs, which facilitates various temporal commu- nity query/search demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The previous historical푘-core query [36] can be seen as a special case of the proposed problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Specifi- cally, we introduce the data model and query model of the pro- posedproblem in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2 respectively, and then present a nontrivial baseline that addresses the proposed problem based on the existing PHC-Query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 Data Model A temporal graph is normally an undirected graph G = (V, E) with parallel temporal edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Each temporal edge (푢,푣,푡) ∈ E is associated with a timestamp 푡 that indicates when the interac- tion happened between the vertices 푢,푣 ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For example, the temporal edges could be transfer transactions between bank ac- counts in a finance graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Without a loss of generality, we use continuous integers that start from 1 to denote timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Fig- ure 1 illustrates a temporal graph as our running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' There are two useful concepts derived from the temporal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Given a time interval [푡푠,푡푒], we define the projected graph of G over [푡푠,푡푒] as G[푡푠,푡푒] = (V[푡푠,푡푒], E[푡푠,푡푒]), where V[푡푠,푡푒] = V and E[푡푠,푡푒] = {(푢,푣,푡)|(푢,푣,푡) ∈ E, 푡 ∈ [푡푠,푡푒]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Moreover, we define the detemporalized graph of G[푡푠,푡푒] as a simple graph 퐺[푡푠,푡푒] = (푉[푡푠,푡푒], 퐸[푡푠,푡푒]), where 푉[푡푠,푡푒]=V[푡푠,푡푒] and 퐸[푡푠,푡푒] = {(푢,푣)|(푢,푣,푡) ∈ E[푡푠,푡푒] }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2 Query Model For revealing communities in graphs, the 푘-core query is widely adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Given an undirected graph 퐺 and an integer 푘, 푘-core is the maximal induced subgraph of 퐺 in which all vertices have degrees at least 푘, which is denoted by C푘 (퐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The coreness of a vertex 푣 in a graph 퐺 is the largest value of 푘 such that 푣 ∈ C푘 (퐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For temporal graphs, the Historical 푘-Core Query (HCQ) [36] is proposed recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' It aims to find a 푘-core that appears during a specific time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Formally, a historical 푘-core H푘 [푡푠,푡푒] (G) is a 푘-core in the detemporalized projected graph 퐺[푡푠,푡푒] of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, HCQ can be defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Scalable Time-Range 푘-Core Qery on Temporal Graphs Definition 1 (Historical 푘-Core Qery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For a temporal graph G, given an integer 푘 and a time interval [푡푠,푡푒], return H푘 [푡푠,푡푒] (G) = C푘 (퐺[푡푠,푡푒]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In this paper, we propose a novel query model called Tempo- ral 푘-Core Query (TCQ) that generalizes HCQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The main differ- ence is that the query time interval [푇푠,푇푒] of TCQ is a range but not fixed query condition like [푡푠,푡푒] of HCQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In TCQ,푇푠 and 푇푒 are the minimum start time and maximum end time of query time interval respectively, and thereby the 푘-cores induced by each subinterval [푡푠,푡푒] ⊆ [푇푠,푇푒] are all potential results of TCQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Moreover, TCQ directly returns the maximal induced sub- graphs of G in which all vertices have degrees (note that, the number of neighbor vertices but not neighbor edges) at least 푘 as results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We call these subgraphs as temporal 푘-cores and de- note by T 푘 [푡푠,푡푒] (G) a temporal 푘-core that appears over [푡푠,푡푒] on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Obviously, a historical 푘-core H푘 [푡푠,푡푒] (G) is the detempo- ralized temporal 푘-core T 푘 [푡푠,푡푒] (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Therefore, TCQ can be seen as a group of HCQ and HCQ can be seen as a special case of TCQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The formal definition of TCQ is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Definition 2 (Temporal푘-Core Qery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For a temporalgraph G, given an integer 푘 and a time interval [푇푠,푇푒], return all dis- tinct T 푘 [푡푠,푡푒] (G) with [푡푠,푡푒] ⊆ [푇푠,푇푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Note that, TCQ only returns the distinct temporal 푘-cores that are not identical to each other, since multiple subintervals of [푇푠,푇푒] may induce an identical subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For brevity, T 푘 [푡푠,푡푒] (G) is abbreviated as T 푘 [푡푠,푡푒] if the context is self-evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='3 Baseline Algorithm A straightforward solution to TCQ is to enumerate each subin- terval [푡푠,푡푒] ⊆ [푇푠,푇푒] and induce T 푘 [푡푠,푡푒] respectively, which takes 푂(|푇푒 −푇푠|2|E|) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' However, the span of query time in- terval (namely,푇푒−푇푠) can be extremely large in practice, which results in enormous time consumption for inducing all temporal 푘-cores from scratch independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Therefore, we start from a non-trivial baseline based on the existing PHC-Query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 A Short Review of PHC-Qery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' PHC-Query relies on a heavy- weight index called PHC-Index that essentially precomputes the coreness of all vertices in the projected graphs over all possible time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The index is logically a table that stores a set of timestamp pairs for each vertex 푣 ∈ V (column) and each rea- sonable coreness 푘 (row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Given a value of 푘, the coreness of a vertex 푣 is exactly 푘 in the projected graph over [푡푠,푡푒] for each timestamp pair 푡푠 and 푡푒 in the cell (푘, 푣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In particular, due to the monotonicity of coreness of a vertex with respect to 푡푒 when 푡푠 is fixed, PHC-Index can reduce its space cost significantly by only storing the necessary but not all possible timestamp pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Specifically, for a vertex 푣, a coreness 푘 and a start time 푡푠, only a discrete set of core time need to be recorded, since the coreness of the vertex over [푡푠,푡푒] will not change with the increase of 푡푒 until 푡푒 is a core time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Consequently, given an HCQ instance, PHC-Query leverages PHC-Index to directly determine whether a vertex has the coreness no less than the required 푘, by compar- ing the query time interval with the retrieved timestamp pairs, and then induces historical 푘-cores with qualified vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2 Incremental PHC-Qery Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The main idea of our baseline algorithm is to induce temporal 푘-cores incrementally, Algorithm 1: Baseline iPHC-Query algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Input: G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 푇푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 푇푒 Output: all distinct T 푘 [푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푡푒] (G) with [푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푡푒] ⊆ [푇푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푇푒] 1 for 푡푠 ← 푇푠 to 푇푒 do 2 V ← ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' E ← ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' H푣 ← ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' H푒 ← ∅ 3 for 푘 and 푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' retrieve the core time of each vertex in G from PHC-Index and push them into H푣 4 push the temporal edges with timestamps in [푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푇푒] in G into H푒 5 for 푡푒 ← 푡푠 to 푇푒 do 6 pop a vertex from H푣 and add it to V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' until the min core time of H푣 exceeds 푡푒 7 pop an edge from H푒 and add it to E if both vertices linked by this edge are in V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' until the min timestamp of H푒 exceeds 푡푒 8 push all edges that have been popped from H푒 and are not added to E back to H푒 9 collect T 푘 [푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푡푒] = (V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' E) if it is neither empty nor identical to other existing results thereby reducing redundant computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' With a temporal 푘- core T 푘 [푡푠,푡푒], we induce T 푘 [푡푠,푡푒+1] simply by appending new ver- tices to T 푘 [푡푠,푡푒], whose coreness has become no less than푘 due to the expand of time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Those vertices can be directly iden- tified by using core time retrieved from PHC-Index since 푡푠 is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The correctness of baseline algorithm is guaranteed while the correctness of PHC-Query holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The pseudo code of incremental PHC-Query (iPHC-Query) al- gorithm is presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' It enumerates all subinter- vals of a given [푇푠,푇푒] in a particular order for fulfilling efficient incremental temporal 푘-core induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Specifically, it anchors the value of 푡푠 (line 1), and increases the value of 푡푒 from 푡푠 to 푇푒 (line 5), so that T 푘 [푡푠,푡푒+1] can always be incrementally gen- erated from an existing T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For each 푡푠 anchored and the input 푘, the algorithm firstly retrieves the core time of all ver- tices from PHC-Index, and pushes the vertices into a minimum heap H푣 ordered by their core time (line 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Moreover, all tem- poral edges with timestamps in [푡푠,푇푒] are pushed into another minimum heap H푒 ordered by their timestamp (line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Then, the algorithm maintains a vertex set V and an edge set E, which rep- resent the vertices and edges of T 푘 [푡푠,푡푒] respectively, whenever 푡푒 is increased by the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' It pops remaining vertices with core time no greater than 푡푒 from H푣 and adds them to V (line 6), since the corenesss of these vertices are no less than 푘 according to PHC-Index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Also, it pops remaining edges with timestamp no greater than 푡푒 from H푒 and adds them to E if both vertices linked by the edges are in V (line 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Then, it puts back the popped edges that are not in E into H푒 (line 8), because they could still be contained by other temporal 푘-cores induced later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Lastly, a temporal푘-core comprised of V and E that are not empty is collected if it has not been induced before (line 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The complexity of baseline mainly depends on the mainte- nance of both V and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For the maintenance of V, each ver- tex in T 푘 [푡푠,푇푒]is added to V from H푣 at most once in the inner loop (lines 5-9), which takes logarithmic time for a heap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' There- fore, the total cost is bounded by �푇푒 푡=푇푠 |V[푡,푇푒]| log |V[푡,푇푒]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The case is more complicated for the maintenance of E, since Junyong Yang, Ming Zhong, Yuanyuan Zhu, Tieyun Qian, Mengchi Liu, and Jeffery Xu Yu each edge with a timestamp within [푡,푇푒] is likely to be trans- ferred between H푒 and E (lines 7-8), until both its endpoints are contained by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In the worst case, the total cost is bounded by �푇푒 푡=푇푠 |푇푒 − 푡||E[푡,푇 푒]| log |E[푡,푇 푒]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' While, the real cost in prac- tice can be much lower since the |푇푒 − 푡| part should be a more reasonable value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Although the baseline algorithm can achieve incremental in- duction of temporal k-core for each start time, PHC-Index incurs a huge amount of extra space and time overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Moreover, its incremental induction only offers a kind of “intra-core” optimiza- tion that reduces the redundant computation in each temporal 푘-core induction, and lacks of a kind of “inter-core” optimization that can directly avoids inducing some temporal 푘-cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In the following sections, we first propose a novel algorithm that can outperform baseline algorithm without any precomputation and index, and then optimize it significantly to further improve the efficiency by at least three orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 3 ALGORITHM In this section, we propose a novel efficient algorithm to address TCQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Our algorithm leverages a fundamental operation called temporal core decomposition to induce T 푘 [푡푠,푡푒] from T 푘 [푡푠,푡푒+1] decre- mentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' More importantly, our algorithm does not require any precomputation and index space, and can still outperform the baseline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Next, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 introduces the temporal core decomposition operation, and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2 presents our al- gorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 Temporal Core Decomposition (TCD) Firstly, we introduce Temporal Core Decomposition (TCD) as a basic operation on temporal graphs, which is derived from the traditional core decomposition [2] on ordinary graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' TCD refers to a two-step operation of inducing a temporal 푘-core T 푘 [푡푠,푡푒] of a given time interval [푡푠,푡푒] from a given temporal graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The first step is truncation: remove temporal edges with timestamps not in [푡푠,푡푒] from G, namely, induce the projected graph G[푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The second step is decomposition: iteratively peel vertices with degree (the number of neighbor vertices but not neighbor edges) less than 푘 and the edges linked to them to- gether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The correctness of TCD is as intuitive as core decom- position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' An excellent property of TCD operation is that, it can induce a temporal푘-core T 푘 [푡푠,푡푒] from another temporal푘-core T 푘 [푡푠′,푡푒′] with [푡푠,푡푒] ⊂ [푡푠′,푡푒′], so that we can develop a decremental al- gorithm based on TCD operation to achieve efficient processing of TCQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' To prove the correctness of this property, let us consider the following Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Given time intervals [푡푠,푡푒] and [푡푠′,푡푒′] such that [푡푠,푡푒] ⊂ [푡푠′,푡푒′], we have T 푘 [푡푠,푡푒] is a subgraph of T 푘 [푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For each vertex in T 푘 [푡푠,푡푒], its coreness in G[푡푠′,푡푒′] is certainly no less than in G[푡푠,푡푒] (namely, ⩾ 푘), because G[푡푠,푡푒] is a subgraph of G[푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, all vertices in T 푘 [푡푠,푡푒] will be contained by T 푘 [푡푠′,푡푒′] that is a temporal 푘-core of G[푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' □ Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Given a time interval [푡푠, 푡푒] and a temporal 푘- core T 푘 [푡푠′,푡푒′] with [푡푠,푡푒] ⊂ [푡푠′,푡푒′], the subgraph induced by using TCD operation from T 푘 [푡푠′,푡푒′] for [푡푠,푡푒] is T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' v3 v4 v5 v6 v7 v8 6 6 6 6 5 5 2 2 2 6 5 5 5 5 5 5 3 2 4 v3 v4 v5 v6 6 6 6 6 5 5 6 5 5 5 5 5 5 v7 v8 v3 v4 v5 v6 6 6 6 6 5 5 5 5 5 5 truncation decomposition Figure 2: Temporal core decomposition from T 2 [2,6] to T 2 [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Firstly, we prove for any temporal graph G′ satisfy- ing that T 푘 [푡푠,푡푒] is a subgraph of G′ and G′ is a subgraph of G, we can induce T 푘 [푡푠,푡푒] from G′ by using TCD operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For each vertex in T 푘 [푡푠,푡푒], its coreness is not less than 푘 in G′ over [푡푠,푡푒], because this temporal 푘-core is a subgraph of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Mean- while, for each vertex in G′ but not in T 푘 [푡푠,푡푒], its coreness in G′ is not greater than in G, because G′ is a subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, its coreness in G′ over [푡푠,푡푒] is less than 푘, because it is not in the temporal 푘-core T 푘 [푡푠,푡푒] of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' As a result, T 푘 [푡푠,푡푒] is also a temporal 푘-core of G′, and thereby can be induced by using TCD operation from G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Then, consider two temporal 푘-cores T 푘 [푡푠,푡푒] and T 푘 [푡푠′,푡푒′] with [푡푠,푡푒] ⊆ [푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Due to Lemma 1, we have T 푘 [푡푠,푡푒] is a sub- graph of T 푘 [푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Let G′ [푡푠,푡푒] be the temporal graph induced by the first step of TCD from T 푘 [푡푠′,푡푒′], which is certainly a subgraph of T 푘 [푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Since G′ [푡푠,푡푒] only removes the temporal edges not in [푡푠,푡푒], which means these edges are not contained by T 푘 [푡푠,푡푒], it is obviously T 푘 [푡푠,푡푒] is a subgraph of G′ [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, the correct- ness of this theorem holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' □ For example, Figure 2 illustrates the procedure of TCD from T 2 [2,6] to T 2 [5,6] on our running example graph in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The edges with timestamps not in [5, 6] (marked by dashed lines) are firstly removed from T 2 [2,6] by truncation, which results in the decrease of degrees of vertices 푣5, 푣7 and 푣8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Then, the ver- tices with degree less than 2 (marked by dark circles), namely, 푣7 and 푣8 are further peeled by decomposition, together with their edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The remaining temporal graph is T 2 [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2 TCD Algorithm We propose a TCD algorithm to address TCQ by using temporal core decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In general, given a TCQ instance, the TCD algorithm enumerates each subinterval of [푇푠,푇푒] in a particu- lar order, so that the temporal 푘-cores of each subinterval are in- duced decrementally from previously induced temporal 푘-cores except the initial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Specifically, we enumerate a subinterval [푡푠,푡푒] of [푇푠,푇푒] as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Initially, let 푡푠 = 푇푠 and 푡푒 = 푇푒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' It means we induce the largest temporal 푘-core T 푘 [푇푠,푇푒] at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Then, we will anchor the start time 푡푠 = 푇푠 and decrease the end time 푡푒 from 푇푒 until 푡푠 gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' As a result, we can always leverage TCD to induce the temporal푘-core of current subinterval [푡푠,푡푒] from the previously induced temporal 푘-core of [푡푠, 푡푒 + 1] but not from G[푡푠,푡푒] or even G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Whenever the value of 푡푒 is de- creased to 푡푠, the value of 푡푠 will be increased to 푡푠 + 1 until 푡푠 = 푇푒, and the value of 푡푒 will be reset to 푇푒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Then, we in- duce T 푘 [푡푠+1,푡푒] from T 푘 [푡푠,푡푒], and start over the decremental TCD Scalable Time-Range 푘-Core Qery on Temporal Graphs Algorithm 2: TCD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Input: G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' [푇푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푇푒] Output: all distinct T 푘 [푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푡푒] with [푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푡푒] ⊆ [푇푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푇푒] 1 for 푡푠 ← 푇푠 to 푇푒 do // anchor a new start time 2 푡푒 ← 푇푒 // reset the end time 3 if 푡푠 = 푇푠 then 4 T 푘 [푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푡푒] ← TCD(G[푇푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푇푒],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' [푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푡푒]) 5 else 6 T 푘 [푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푡푒] ← TCD(T 푘 [푡푠−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푡푒],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' [푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푡푒]) 7 collect T 푘 [푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푡푒] if it is distinct 8 for 푡푒 ← 푇푒 − 1 to 푡푠 do // iteratively decremental induction 9 T 푘 [푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푡푒] ← TCD(T 푘 [푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푡푒+1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' [푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푡푒]) 10 collect T 푘 [푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푡푒] if it is distinct procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The pseudo code of TCD algorithm is given in Algo- rithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Note that, the details of TCD(G, 푘, [푡푠,푡푒]) function is left to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2, in which we design a specific data structure to implement TCD operation efficiently in physical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Figure 3 gives a demonstration of TCD algorithm for finding temporal 2-cores of time interval [1,8] on our running example graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The temporal 푘-cores are induced line by line and from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Each arrow between temporal 푘-cores represents a TCD operation from tail to head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We can see that, compared with inducing each temporal 푘-core independently, the TCD al- gorithm reduces the computational overhead significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For most induced temporal 푘-cores, a number of vertices and edges have already been excluded while inducing the previous tempo- ral 푘-cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Moreover, with the increase of 푡푠 and the decrease of 푡푒 when 푡푠 is fixed, the size of T 푘 [푡푠,푡푒] will be reduced monotoni- cally until no temporal푘-core exists over [푡푠,푡푒], so that the time and space costs of TCD operation will also be reduced gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Lastly, we compare TCD algorithm with Baseline algorithm abstractly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' When 푡푠 is fixed, Baseline algorithm conducts an in- cremental procedure, in which each vertex is popped once and each edge may be popped and pushed back many times, and in contrast, TCD algorithm conducts a decremental procedure, in which each vertex is peeled once and each edge is also removed once due to Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Therefore, TCD algorithm that is well im- plemented in physical level (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2) can be even more efficient than Baseline algorithm, though it does not need any precomputed index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 4 OPTIMIZATION In this section, we dive deeply into the procedure of TCD al- gorithm and optimize it dramatically by introducing an intu- itive concept called tightest time interval for temporal 푘-cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In a nutshell, we directly prune subintervals without inducing their temporal 푘-cores if we can predict that the temporal 푘- cores are identical to other induced temporal 푘-cores, and tight- est time interval is the key to fulfill prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In this way, the optimized TCD algorithm only performs TCD operations that are necessary for returning all distinct answers to a given TCQ instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Conceptually,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' the new pruning operation of optimized algorithm eliminates the “inter-core” redundant computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v3 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='Figure 3: A demonstration of TCD algorithm for finding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='temporal 2-cores of time interval [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' the original TCD operation eliminates the “intra-core” redun- dant computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, the computational complexity of opti- mized algorithm no longer depends on the span of query time in- terval [푇푠,푇푒] like the baseline algorithm and the original TCD algorithm but only depends on the scale of final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Next, we introduce the concept and properties of tightest time interval in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1, present three pruning rules based on tight- est time interval for TCD algorithm in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2, and briefly conclude and discuss the optimized TCD algorithm in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 Tightest Time Interval (TTI) We have such an observation, a temporal 푘-core of [푡푠,푡푒] may only contain edges with timestamps in a subinterval [푡푠′,푡푒′] ⊂ [푡푠,푡푒], since the edges in [푡푠, 푡푠′) and (푡푒′,푡푒] have been re- moved by core decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For example, consider a tempo- ral 푘-core T 2 [4,8] illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We can see that it does not contain edges with timestamps 4, 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' As a result, if we continue to induce T 2 [4,7] from T 2 [4,8] and to induce T 2 [4,6] from T 2 [4,7], the returned temporal 푘-cores remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The sameness of temporal 푘-cores induced by different subinter- vals inspires us to further optimize TCD algorithm by pruning subintervals directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' As illustrated in Figure 3, the subintervals such as [4,7], [4,6], [5,8], [5,7] and [5,6] all induce the identical temporal 푘-cores to [4,8], so that they can be potentially pruned in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For that, we propose the concept of Tightest Time Interval (TTI) for temporal 푘-cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Given a temporal 푘-core of [푡푠,푡푒], its TTI refers to the minimal time interval [푡푠′,푡푒′] that can in- duce an identical temporal 푘-core to T 푘 [푡푠,푡푒], namely, there is no subinterval of [푡푠′,푡푒′] that can induce an identical temporal 푘- core to T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We formalize the definition of TTI as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Definition 3 (Tightest Time Interval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Given a temporal 푘-core T 푘 [푡푠,푡푒], its tightest time interval T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI is [푡푠′,푡푒′], if and only if 1) T 푘 [푡푠′,푡푒′] is an identical temporal 푘-core to T 푘 [푡푠,푡푒];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 2) there does not exist [푡푠′′,푡푒′′] ⊂ [푡푠′,푡푒′], such that T 푘 [푡푠′′,푡푒′′] is an identical temporal 푘-core to T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Junyong Yang, Ming Zhong, Yuanyuan Zhu, Tieyun Qian, Mengchi Liu, and Jeffery Xu Yu It is easy to prove the TTI of a temporal 푘-core of [푡푠,푡푒] is surely a subinterval of [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' To evaluate the TTI of a given T 푘 [푡푠,푡푒], we have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Given a temporal 푘-core T 푘 [푡푠,푡푒], T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI = [푡푚푖푛,푡푚푎푥 ], where 푡푚푖푛 and 푡푚푎푥 are the minimum and maxi- mum timestamps in T 푘 [푡푠,푡푒] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' On one hand, T 푘 [푡푚푖푛,푡푚푎푥 ] is identical to T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Be- cause we can induce T 푘 [푡푚푖푛,푡푚푎푥 ] by TCD operation from T 푘 [푡푠,푡푒] due to [푡푚푖푛,푡푚푎푥 ] ⊆ [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Meanwhile, during the operation, none edge is actually removed since there is no edge with times- tamp outsides [푡푚푖푛,푡푚푎푥] in T 푘 [푡푠,푡푒], and thus the temporal 푘- core T 푘 [푡푠,푡푒] will remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' On the other hand, any time interval [푡푠′,푡푒′] ⊂ [푡푚푖푛,푡푚푎푥 ] cannot induce a temporal푘 core that is identical to T 푘 [푡푠,푡푒], since the edges with timestamp either 푡푚푖푛 or 푡푚푎푥 in T 푘 [푡푠,푡푒] are excluded at least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' □ With Theorem 2, we can evaluate the TTI of a given tempo- ral 푘-core instantly (by 푂(1) time, see Section 5), which guar- antees the following optimization based on TTI will not incur extra overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Moreover, there are the following important properties of TTI that support our pruning strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Property 1 (Uniqeness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Given a temporal 푘-core T 푘 [푡푠,푡푒], there exists no other time interval than T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI evaluated by Theorem 2 that is also a TTI of T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Let T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI be [푡푠′,푡푒′], and [푡푠′′,푡푒′′] ≠ [푡푠′,푡푒′] be any other time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' There are only two possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Firstly, [푡푠′,푡푒′] ⊄ [푡푠′′,푡푒′′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' However, the edges with timestamp 푡푠′ and 푡푒′ are contained by T 푘 [푡푠,푡푒] according to Theorem 2, and thereby [푡푠′′,푡푒′′] that does not cover [푡푠′,푡푒′] cannot induce T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, the first possibility does not satisfy the first con- dition in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Secondly, [푡푠′,푡푒′] ⊂ [푡푠′′,푡푒′′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' However, since [푡푠′,푡푒′] can induce T 푘 [푡푠,푡푒], [푡푠′′,푡푒′′] is certainly not the tightest even if it can also induce T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, the second possi- bility does not satisfy the second condition in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Con- sequently, [푡푠′′,푡푒′′] ≠ [푡푠′,푡푒′] is not a TTI of T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' □ Property 2 (Eqivalence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Given two temporal푘-cores T 푘 [푡푠,푡푒] and T 푘 [푡푠′,푡푒′], they are identical temporal graphs if and only if T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI = T 푘 [푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' If T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI = T 푘 [푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI, T 푘 [푡푠,푡푒] and T 푘 [푡푠′,푡푒′] are both identical to the temporal 푘-core of the TTI according to Definition 3, and thus are identical to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Conversely, if T 푘 [푡푠,푡푒] and T 푘 [푡푠′,푡푒′] are identical, they must have a same unique TTI according to Theorem 2 and Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' □ Property 3 (Inclusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Given two temporal 푘-cores T 푘 [푡푠,푡푒] and T 푘 [푡푠′,푡푒′], we have T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI ⊆ T 푘 [푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI, if [푡푠,푡푒] ⊆ [푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Since [푡푠,푡푒] ⊆ [푡푠′,푡푒′], we have T 푘 [푡푠,푡푒] is a sub- graph of T 푘 [푡푠′,푡푒′] according to Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, the minimum timestamp in T 푘 [푡푠,푡푒] is certainly no earlier than the the min- imum timestamp in T 푘 [푡푠′,푡푒′], and the maximum timestamp in T 푘 [푡푠,푡푒] is certainly no later than the the maximum timestamp in T 푘 [푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Then, according to Theorem 2, we have T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI ⊆ T 푘 [푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' □ Figure 4a abstracts Figure 3 as a schedule table of subinter- val enumeration, and TCD algorithm will traverse the cells row by row and from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For example, the cell in row 1 and column 6 represents a subinterval [1, 6], in which [2, 6] is the TTI of T 2 [1,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In particular, the grey cells indicate that the temporal 푘-cores of the corresponding subintervals do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Figure 4a clearly reveals that TCD algorithm suffers from induc- ing a number of identical temporal 푘-cores (with the same TTIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For example, the TTI [5, 6] repeats six times, which means six cells will induce identical temporal 푘-cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2 Pruning Rules The main idea of optimizing TCD algorithm is to predict the in- duction of identical temporal푘-cores by leveraging TTI, thereby skipping the corresponding subintervals during the enumera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Specifically, whenever a temporal 푘-core of [푡푠,푡푒] is in- duced, we evaluate its TTI [푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' If 푡푠′ > 푡푠 or/and 푡푒′ < 푡푒, it is triggered that a number of subintervals on the schedule can be pruned in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' According to different relations between [푡푠,푡푒] and [푡푠′,푡푒′], our pruning technique can be categorized into three rules which are not mutually exclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In other words, the three rules may be triggered at the same time, and prune dif- ferent subintervals respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Next, we present these pruning rules in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2 and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 Rule 1: Pruning-on-the-Right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Consider the schedule illus- trated in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For each row, TCD algorithm traverses the cells (namely, subintervals) from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' If the TTI [푡푠′,푡푒′] in the current cell [푡푠,푡푒] meets such a condition, namely, 푡푒′ < 푡푒, a pruning operation will be triggered, and the following cells in this row from [푡푠,푡푒 − 1] until [푡푠,푡푒′] will be skipped be- cause these subintervals will induce identical temporal 푘-cores to T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Since the pruned cells are on the right of trigger cell, we call this rule Pruning-On-the-Right (PoR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The pseudo code of PoR is given in lines 2-4 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The correctness of PoR is guaranteed by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Given a temporal푘-core T 푘 [푡푠,푡푒] whose TTI is [푡푠′,푡푒′], for any time interval [푡푠,푡푒′′] with 푡푒′′ ∈ [푡푒′,푡푒], T 푘 [푡푠,푡푒′′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI = [푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' On one hand, since [푡푠,푡푒′′] ⊆ [푡푠,푡푒], T 푘 [푡푠,푡푒′′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI ⊆ T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI = [푡푠′,푡푒′] according to Inclusion (Property 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' On the other hand, we can prove [푡푠′,푡푒′] ⊆ T 푘 [푡푠,푡푒′′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' If we induce T 푘 [푡푠′,푡푒′] from T 푘 [푡푠,푡푒] by TCD operation, it is easy to know T 푘 [푡푠,푡푒] will remain unchanged, because it only con- tains the edges with timestamps in [푡푠′,푡푒′] according to Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, we have T 푘 [푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI = [푡푠′,푡푒′] according to Equiv- alence (Property 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Also, since [푡푠′,푡푒′] ⊆ [푡푠,푡푒′′], [푡푠′,푡푒′] = T 푘 [푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI ⊆ T 푘 [푡푠,푡푒′′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI according to Inclusion (Property 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' □ With Lemma 2, we can predict that the TTIs in the cells [푡푠,푡푒− 1], · · · , [푡푠,푡푒′] are the same as the trigger cell [푡푠,푡푒], when the Scalable Time-Range 푘-Core Qery on Temporal Graphs ts te 1 2 3 4 5 6 1 2 3 4 5 6 8 7 7 8 [6,6] [5,6] [5,6] [3,6] [2,8] [1,8] [6,6] [5,6] [5,6] [3,6] [2,7] [1,7] [6,6] [5,6] [5,6] [3,6] [2,6] [2,6] [5,5] [5,5] [3,5] [2,5] [2,5] [2,4] [2,4] [2,3] [2,3] [2,2] [2,2] (a) Without pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' ts te 1 2 3 4 5 6 1 2 3 4 5 6 8 7 7 8 [5,6] [3,6] [2,8] [1,8] [2,7] [1,7] [6,6] [2,6] [5,5] [3,5] [2,5] [2,4] [2,3] [2,2] Cell without core induced Pruning-on-the-Right Pruning-on-the-Underside Pruning-on-the-Left [x,y] Cell with core induced, TTI = [x,y] Pruning triggered by cell [1,6] Pruning triggered by cell [3,8] Pruning triggered by cell [4,8] (b) With pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Figure 4: Examples of subinterval pruning based on tightest time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' PoR rule is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, the temporal푘-cores induced by these subintervals are all identical to the induced T 푘 [푡푠,푡푒] according to Equivalence (Property 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For example, Figure 4b illustrates two instances of PoR (the cells in orange and blue colors with left arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' When T 2 [3,8] has been induced, we evaluate its TTI as [3, 6], and thus PoR is trig- gered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' PoR immediately excludes the following two cells [3, 7] and [3, 6] from the schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' As a proof, we can see the TTIs in these two cells are both [3, 6] in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2 Rule 2: Pruning-on-the-Underside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We now consider 푡푠′ > 푡푠, which causes pruning in the following rows but not the cur- rent row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' So we call this rule Pruning-On-the-Underside (PoU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Specifically, if 푡푠′ > 푡푠, for each row 푟 ∈ [푡푠 + 1,푡푠′], the cells [푟,푡푒], [푟,푡푒 − 1], · · · , [푟,푟] will be skipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The pseudo code of PoU is given in lines 5-8 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The correctness of PoU is guaranteed by the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Given a temporal푘-core T 푘 [푡푠,푡푒] whose TTI is [푡푠′,푡푒′], for any time interval [푡푠′′,푡푒] with푡푠′′ ∈ [푡푠,푡푠′], we have the TTI of T 푘 [푡푠′′,푡푒] is [푡푠′,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The proof of this lemma is similar to Lemma 2 and thus is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Given a temporal푘-core T 푘 [푡푠,푡푒] whose TTI is [푡푠′,푡푒′], for any time interval [푟,푐] with 푟 ∈ [푡푠 + 1,푡푠′] and 푐 ∈ [푡푠,푡푒], we have T 푘 [푟,푐] is identical to T 푘 [푡푠,푐].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For 푟 ∈ [푡푠 + 1,푡푠′], we have T 푘 [푟,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI = [푡푠′,푡푒′] according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, T 푘 [푟,푡푒] is identical to T 푘 [푡푠,푡푒] ac- cording to the Equivalence (Property 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Then, we have T 푘 [푟,푐] is identical to T 푘 [푡푠,푐] when 푐 = 푡푒 − 1 since them are induced by the same TCD operation from identical temporal graphs, and so on for the rest [푟,푐] with the decrease of 푐 until 푐 = 푡푠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' □ Lemma 4 indicates that, PoU safely prunes some cells in the following rows, since these cells contain the same TTIs as their upper cells, which even have not been enumerated yet except the trigger cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For example, Figure 4b illustrates two PoU in- stances (the cells in yellow and blue colors with up arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' On enumerating the cell [1, 6], since the contained TTI is [2, 6], the cells [2, 6], · · · , [2, 2] are pruned by PoU, because the TTIs in these cells are the same as the cells [1, 6], · · · , [1, 2] respectively, though the TTIs of cells [1, 5], · · · , [1, 2] have not been evalu- ated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='3 Rule 3: Pruning-on-the-Lef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Lastly, if both 푡푠′ > 푡푠 and 푡푒′ < 푡푒, for each row 푟 ∈ [푡푠′+1, 푡푒′], the cells [푟,푡푒], [푟,푡푒 −1], · · , [푟,푡푒′ + 1] will also be skipped, besides the cells pruned by PoR and PoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Although these cells are in the rows under the current row 푡푠, the temporal 푘-core of each of them is identical to the temporal 푘-core of a cell (namely, [푟,푡푒′]) on the right in the same row but not its upper cell like PoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' So we call this rule Pruning-On-the-Left (PoL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The pseudo code of PoL is given in lines 9-12 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The correctness of PoL is guaranteed by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Given a temporal푘-core T 푘 [푡푠,푡푒] whose TTI is [푡푠′,푡푒′], for any time interval [푟,푐] with 푟 ∈ [푡푠′ + 1,푡푒′] and 푐 ∈ [푡푒′ + 1,푡푒], we have T 푘 [푟,푐] is identical to T 푘 [푟,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Assume T 푘 [푟,푐].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI = [푟 ′,푐′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' According to Inclusion (Property 3), we have [푟 ′,푐′] ⊆ [푡푠′,푡푒′] since [푟,푐] ⊆ [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus,푐′ ⩽ 푡푒′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Then, according to Lemma 2, we have T 푘 [푟,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI = [푟 ′,푐′] since 푡푒′ ∈ [푐′,푐].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Lastly, according to Equivalence (Property 2), we have T 푘 [푟,푐] is identical to T 푘 [푟,푡푒′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' □ For example, Figure 4b illustrates a PoL instance (the cells in blue color with right arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' On enumerating the cell [4, 8], PoL is triggered since the contained TTI is [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Then, the cells [6, 8] and [6, 7] are pruned by PoL because the TTIs contained in them are the same as the cell [6, 6] on the right of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' PoL is more tricky than PoU because the cells are pruned for contain- ing the same TTIs as other cells that are scheduled to traverse after them by TCD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Note that, the cell [4, 8] triggers all three kinds of pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In fact, a cell may trigger PoL only, PoU only, or all three rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='3 Optimized TCD Algorithm Compared with TCD algorithm, the improvement of Optimized TCD (OTCD) algorithm is simply to conduct a pruning opera- tion whenever a temporal 푘-core has been induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Specifically, we evaluate the TTI of this temporal 푘-core, check each pruning rule to determine if it is triggered, and prune the specific subin- tervals on the schedule in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The pseudo code of pruning operation is given in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Note that, the “prune” in Al- gorithm 3 is a logical concept, and can have different physical implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Junyong Yang, Ming Zhong, Yuanyuan Zhu, Tieyun Qian, Mengchi Liu, and Jeffery Xu Yu Algorithm 3: Pruning operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Input: [푡푠, 푡푒] and T 푘 [푡푠,푡푒] 1 [푡푠′,푡푒′] ← T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='TTI // Theorem 2 2 if 푡푒′ < 푡푒 then // Rule 1: PoR 3 for 푐 ← 푡푒 - 1 to 푡푒′ do 4 prune the subinterval [푡푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푐] 5 if 푡푠′ > 푡푠 then // Rule 2: PoU 6 for 푟 ← 푡푠 + 1 to 푡푠′ do 7 for 푐 ← te to r do 8 prune the subinterval [푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푐] 9 if 푡푠′ > 푡푠 and 푡푒′ < 푡푒 then // Rule 3: PoL 10 for r ← ts’+1 to te’ do 11 for c ← te to te’+1 do 12 prune the subinterval [푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='푐] As illustrated in Figure 4b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' OTCD algorithm completely elim- inates repeated inducing of identical temporal 푘-cores,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' each distinct temporal 푘-core is induced exactly once during the whole procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' It means, the real computational complexity of OTCD algorithm is the summation of complexity for induc- ing each distinct temporal 푘-core but not the temporal 푘-core of each subinterval of [푇푠,푇푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Therefore, we say OTCD algo- rithm is scalable with respect to the query time interval [푇푠,푇푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For many real-world datasets, the span of [푇푠,푇푒] could be very large, while there exist only a limited number of distinct tem- poral 푘-cores over this period, so that OTCD algorithm can still process the query efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 5 IMPLEMENTATION In this section, we address the physical implementation of pro- posed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We first introduce a data structure for temporal graph representation in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1, based on which we explain the details of TCD Operation implementation in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 Temporal Edge List (TEL) We propose a novel data structure called Temporal Edge List (TEL) for representing an arbitrary temporal graph (including temporal 푘-cores that are also temporal graphs), which is both the input and output of TCD operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Conceptually, TEL(G) preserves a temporal graph G = (V, E) by organizing its edges in a 3-dimension space, each dimension of which is a set of bidi- rectional linked lists, as illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The first dimen- sion is time, namely, all edges in E are grouped by their times- tamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Each group is stored as a bidirectional linked list called Time List (TL), and TL(푡) denotes the list of edges with a times- tamp 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Then, TEL(G) uses a bidirectional linked list, in which each node represents a timestamp in G, as a timeline in ascend- ing order to link all TLs, so that some temporal operations can be facilitated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Moreover, the other two dimensions are source vertex and destination vertex respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We use a container to store the Source Lists (SL) or Destination Lists (DL) for each vertex 푣 ∈ V, where SL(푣) or DL(푣) is a bidirectional linked list that links all edges whose source or destination vertex is 푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Ac- tually, an SL or DL is an adjacency list of the graph, by which we can retrieve the neighbor vertices and edges of a given vertex efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Given a temporal graph G, TEL(G) is built in mem- ory by adding its edges iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For each edge (푢,푣,푡) ∈ E, it is only stored once, and TL(푡), SL(푢) and DL(푣) will append its pointer at the tail respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Figure 5 illustrates a partial TEL of our example graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The SLs and DLs other than SL(푣5) and DL(푣3) are omitted for con- ciseness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Basically, TL, SL and DL offer the functionality of re- trieving edges by timestamp and linked vertex respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For example, for removing all neighbor edges of a vertex 푣 with de- gree less than 푘 in TCD operation, we can locate SL(푣) and DL(푣) to retrieve these edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Moreover, the linked list of TL can offer efficient temporal operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For example, for truncating G to G[푡푠,푡푒] in TCD operation, we can remove TL(푡) with 푡 < 푡푠 or 푡 > 푡푒 from the linked list of TL conveniently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' To get the TTI of a temporal 푘-core, we only need to check the head and tail nodes of the linked list of TL in its TEL to get the minimum and maximum timestamps respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The superiority of TEL is summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' By TCD operation, a TEL will be trimmed to a smaller TEL, and there is none intermediate TEL produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, the memory requirement of (O)TCD algorithm only de- pends on the size of initial TEL(G[푇푠,푇푒]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' TEL consumes 푂(|E|) space for storing a temporal graph, which is optimal because at least 푂(|E|) space is required for storing a graph (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=', adjacency lists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Although there are 6|E|+2|V|+3푛 pointers of TLs, SLs and DLs stored ad- ditionally, TEL is still compact compared with PHC-Index, where 푛 is the number of timestamps in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' TEL supports the basic manipulations listed in Table 1 in constant time, which are cornerstones of implementing our algorithms and optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For dynamic graphs, when a new edge coming, TEL sim- ply appends a new node representing the current time at the end of linked list of TL, and then adds this edge as normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, TEL can also deal with dynamic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2 Implement TCD Operation on TEL Given a TCQ instance, our algorithm starts to work on a copy of TEL(G[푇푠,푇푒]) in memory, which is obtained by truncating TEL(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Then, our algorithm only needs to maintain an instance of TEL(T 푘 [푡푠,푇푒]) and another instance of TEL(T 푘 [푡푠,푡푒]) with [푡푠,푡푒] ⊆ [푇푠,푇푒] in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The first instance is used to induce the first temporal 푘-core T 푘 [푡푠+1,푇푒] by TCD for each row in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The second instance is used to induce the following temporal 푘-cores T 푘 [푡푠,푡푒−1] by TCD in each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Each TCD operation is decomposed to a series of TEL manipulations, and trims the in- put TEL without producing any intermediate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' To assist the implementation of TCD operation, our algorithm uses a global data structure H푣 that organizes all vertices in the maintained TEL into a minimum heap ordered by their degree, so that the vertices with less than 푘 neighbors can be retrieved directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Note that, whenever an edge is deleted from the main- tained TEL, H푣 will also be updated due to the possible decrease of vertex degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The trivial details of updating H푣 is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Algortithm 4 gives the implementation of TCD operation on TEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The algorithm takes as input the TEL of a given graph G, along with the parameters 푘, 푡푠 and 푡푒 specifying the target tem- poral 푘-core T 푘 [푡푠,푡푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In truncation phase, TEL(G) is projected to TEL(G[푡푠,푡푒]) (lines 1-14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Specifically, the linked list of TL is traversed from the head and tail bidirectionally until meet- ing 푡푠 and 푡푒 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For each node representing the times- tamp 푡 traversed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' the edges in TL(푡) are removed from TEL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' and Scalable Time-Range 푘-Core Qery on Temporal Graphs SL(v1) SL(v2) SL(v10) DL(v1) DL(v2) DL(v3) DL(v10) … (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v3) (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v3) TL(1) (v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v3) (v7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v10) (v5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v7) (v5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v8) (v5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v8) (v7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v8) (v7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v9) (v7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v9) (v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v3) (v5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v7) (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v3) (v7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v8) (v3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v4) (v3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v4) (v3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v5) (v3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v6) (v4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v6) (v4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v7) (v4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v7) (v5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v6) (v5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v7) (v3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v4) (v3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v6) (v4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v5) (v4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v7) (v5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v6) (v7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v10) (v9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v10) (v9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v10) SL(v5) … … Time Lists Destination Lists Source Lists TL(2) TL(3) TL(4) TL(5) TL(6) TL(7) TL(8) Figure 5: The conceptual illustration of a partial TEL of our running example graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Table 1: The basic manipulations of TEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Name Functionality Complexity next_TL(푇퐿) / prev_TL(푇퐿) get the next or previous TL in the linked list of TL 푂 (1) get_SL(푣) / get_DL(푣) get the SL or DL of a given vertex 푣 from a hash map 푂 (1) del_TL(푇퐿) remove the given TL node from the linked list of TL 푂 (1) del_edge(푒) delete a given edge 푒 = (푢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 푣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 푡) and update TL(푡),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' SL(푢) and DL(푣) respectively 푂 (1) get_TTI() return the timestamps of head and tail nodes of linked list of TL 푂 (1) H푣 is updated for each edge removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In decomposition phase, TEL(G[푡푠,푡푒]) is further transformed to TEL(T 푘 [푡푠,푡푒]) (lines 15-24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Specifically, the algorithm pops the vertex with the least neigh- bors from H푣 iteratively until the remaining vertices all have at least 푘 neighbors or the heap is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For each popped vertex 푣, it removes the linked edges of 푣 preserved in SL(푣) and DL(푣) from TEL respectively and updates H푣 accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In particular, a TL will be removed from the linked list of TL after the last edge in it has been removed (lines 19 and 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' To clarify the procedure of Algorithm 4, Figure 6 illustrates an example of inducing T 2 [4,5] from T 2 [3,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The edges are going to be deleted are marked in red color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We can see that, the procedure is actually a stream of edge deletion, while TEL maintains the entries to retrieve the remaining edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='3 Complexity TCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='Theoretically, the complexity of TCD algorithm is bounded by �푇푒 푡=푇푠{(|V[푡,푇푒]|+|E[푡,푇 푒]|) log |V[푡,푇푒]|+푚|E[푡,푇푒]|}, where 푚 is a small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For each anchored푡, TCD algorithm gradu- ally peels T 푘 [푡,푇푒] like an onion by TCD operation until it contains none temporal 푘-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In the process, there are at most |E[푡,푇 푒]| edges deleted, and deleting each edge takes a small constant time 푂(푚) for TEL updating and at most 푂(log |V[푡,푇푒]|) time for H푣 maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Similarly, there are at most |V[푡,푇푒]| vertices deleted, and deleting each vertex takes 푂(log |V[푡,푇푒]|) time for H푣 maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Therefore, The total time overhead is the sum of edge and vertex deleting costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Note that, the complexity of TCD algorithm can also be rep- resented by 푂((푇푒 −푇푠)2퐵) according to Algorithm 2, where 퐵 is the average time overhead of TCD operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' However, 퐵 can- not be estimated precisely, since each TCD operation may delete zero to |E[푡,푇 푒]| edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Therefore, we bound the complexity by the maximum deleting cost according to Algorithm 4, which is more reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' OTCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The complexity of OTCD algorithm is simply bounded by �(|푉 ∗| + |퐸∗|) log |푉 ∗| +푚|퐸∗|, where 푉 ∗ and 퐸∗ refer to the sets of vertices and edges that have to be deleted for inducing the result temporal 푘-cores respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Due to the pruning rules, there are much less temporal 푘-cores induced by OTCD algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, |푉 ∗| and |퐸∗| are orders of magnitude less than the total number of vertices and edges deleted in TCD algorithm, most of which are actually used for inducing identical temporal 푘-cores, though they cannot be really estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 6 EXTENSION To demonstrate the wide applicability of our approach in prac- tice, we present several typical scenarios that extends the data model or query model of TCQ, and sketch how to address them based on our data structure and algorithm in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 Data Model Extension Dynamic Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Since most real-world graphs are evolving over time, it is significant to fulfill TCQ on dynamic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Ben- efitted from its design in “timeline” style, our data structure TEL can deal with new edges naturally in memory through two new manipulations add_TL(푡) and add_edge(푢,푣,푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' When a new edge (푢,푣,푡) arrived, we firstly create an empty TL(푡), and append it at the end of the linked list of TL since 푡 is obviously greater than the existing timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Then, we create a new edge node for (푢,푣,푡) and append it to TL(푡), SL(푢) and DL(푣) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Junyong Yang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Ming Zhong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Yuanyuan Zhu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Tieyun Qian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Mengchi Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' and Jeffery Xu Yu DL(v1) DL(v2) DL(v4) SL(v1) SL(v2) SL(v3) SL(v4) DL(v3) TL(3) TL(4) TL(5) TL(6) v2 v1 v4 v3 5 3 6 5 5 4 (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v4) (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v2) (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v3) (v3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v4) (v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v3) (v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v4) DL(v1) DL(v2) DL(v4) SL(v1) SL(v2) SL(v3) SL(v4) DL(v3) TL(4) TL(5) (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v2) (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v3) (v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v3) (v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v4) DL(v1) DL(v2) DL(v4) SL(v1) SL(v2) SL(v3) SL(v4) DL(v3) TL(4) TL(5) (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v2) (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v3) (v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='v3) decomposition truncation v2 v1 v4 v3 5 5 5 4 v2 v1 v3 5 5 4 Figure 6: An example of TCD operation on TEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Algorithm 4: TCD operation in Algorithm 2 Input: TEL(G), [푡푠,푡푒], 푘 Output: TEL(T 푘 [푡푠,푡푒]) 1 푇퐿 ← the head of linked list of TL in TEL(G) 2 while 푇퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='timestamp ≠ 푡푠 do 3 for edge 푒 in 푇퐿 do 4 del_edge(푒) 5 udpate H푣 6 del_TL(푇퐿) 7 푇퐿 ← next_TL(푇퐿) 8 푇퐿 ← the tail of linked list of TL in TEL(G) 9 while 푇퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='timestamp ≠ 푡푒 do 10 for edge 푒 in 푇퐿 do 11 del_edge(푒) 12 udpate H푣 13 del_TL(푇퐿) 14 푇퐿 ← prev_TL(푇퐿) 15 while H푣 is not empty and H푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='peek < 푘 do 16 vertex 푣 ← H푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='pop() 17 for edge 푒 in SL(푣) do 18 del_edge(푒) 19 del_TL(TL(푒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='timestamp)) if the TL is empty 20 update H푣 21 for edge 푒 in DL(푣) do 22 del_edge(푒) 23 del_TL(TL(푒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='timestamp)) if the TL is empty 24 update H푣 Both manipulations are finished in constant time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The mainte- nance of a dynamic TEL is actually consistent with the construc- tion of a static TEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Therefore, our (O)TCD algorithm can run on the dynamic TEL anytime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In contrast, updating PHC-Index is a non-trivial process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Al- though there are previous work [20, 29] on coreness updating for dynamic graphs, the update is only valid for the whole life time of graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' While, for an arbitrary start time, it is uncertain whether the coreness of a vertex will be changed by a new edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2 Query Model Extension The existing graph mining tasks regarding 푘-core introduce var- ious constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For temporal graphs, we only focus on the tem- poral constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In the followings, we present two of them that can be integrated into TCQ model and also be addressed by our algorithm directly, which demonstrate the generality of our model and algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Link Strength Constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In the context of temporal graph, link strength usually refers to the number of parallel edges be- tween a pair of linked vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Obviously, the minimum link strength in a temporal 푘-core represents some important prop- erties like validity, since noise interaction may appear over time and a pair of vertices with low link strength may only have oc- casional interaction during the time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Actually, the previ- ous work [34] has studied this problem without the time interval constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Therefore, it is reasonable to extend TCQ to retrieve 푘-cores with a lower bound of link strength during a given time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' It can be achieved by trivially modifying the TCD Oper- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Specifically, the modified TCD Operation will remove the edges between two vertices once the number of parallel edges between them is decreased to be lower than the given lower bound of link strength, while the original TCD operation will do this when the number becomes zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, the modification brings almost none extra time and space consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Time Span Constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In many cases, we prefer to retrieve temporal 푘-cores with a short time span (between their earliest and latest timestamps), which is similar to the previous work on density-bursting subgraphs [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Because such a kind of short- term cohesive subgraphs tend to represent the occurrence of some special events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' TCQ can be conveniently extended for re- solving the problem by specifying a constraint of time span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Since the time span of a temporal푘-core is preserved in its TEL, which is actually the length of its TTI, we can abandon the tempo- ral 푘-cores returned by TCD operation that cannot satisfy the time span constraint on the fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' It brings almost no extra time and space consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Moreover, we can also extend TCQ to find the temporal 푘-core with the shortest or top-푛 shortest time span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Scalable Time-Range 푘-Core Qery on 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='EXPERIMENT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='In this section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' we conduct experiments to verify both efficiency and effectiveness of the proposed algorithm on a Windows ma- chine with Intel Core i7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='20GHz CPU and 64GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The al- gorithms are implemented through C++ Standard Template Li- brary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Our source codes are shared on GitHub1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 Dataset We choose seven temporal graphs with different sizes and do- mains for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The first three graphs are obtained from KONECT Project [16], and the other four graphs are ob- tained from the SNAP [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The basic statistics of these graphs are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' All timestamps are unified to integers in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2 Efficiency To evaluate the efficiency of our algorithm, we firstly manually select twenty temporal푘-core queries from tested random queries with a time span (namely, 푇푒 −푇푠) of 1-3 days, which have been verified to be valid, namely, there is at least one temporal 푘-core returned for each query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The setting of time span is moderate, otherwise other algorithms than OTCD can hardly stop success- fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Table 3 gives the details of query parameters, so that other 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='com/ThomasYang-algo/Temporal-k-Core-Query-Project Table 4: Effect of pruning rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' id Triggered Times Pruned Cell Percentage (%) PoR PoU PoL PoR PoU PoL Total 1 54 72 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='02 72 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='6 95.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='5 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='64 16 5 9 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='04 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='9 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='5 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='44 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 1 10 100 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='01 3600 Response Time(s) Query Id Baseline TCD OTCD (a) CollegeMsg 6 7 8 9 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 1 10 100 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='01 3600 Response Time(s) Query Id Baseline TCD OTCD (b) email-Eu-core-temporal 11 12 13 14 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 1 10 100 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='01 3600 Response Time(s) Query Id Baseline TCD OTCD (c) sx-mathoverflow 16 17 18 19 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 1 10 100 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='01 3600 Response Time(s) Query Id Baseline TCD OTCD (d) sx-stackoverflow Figure 7: The comparison of response time for selected queries on SNAP graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' researchers can reverify our experimental results or compare with our approach with the same queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Figure 7 compares the response time of Baseline (iPHC-Query), TCD and OTCD algorithms for each selected query respectively, which clearly demonstrates the efficiency of ouralgorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Firstly, TCD performs better than baseline for all twenty queries due to the physical efficiency of TEL, though they both decrementally or incrementally induce temporal푘-cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Specifically, TCD spends around 100 sec for each query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In contrast, baseline spends more than 1000 sec on CollegeMsg and even cannot finish within an hour on two other graphs, though it uses a precomputed in- dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Furthermore, OTCD runs two or three orders of magnitude faster than TCD, and only spends about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1-1 sec for each query, which verifies the effectiveness of our pruning method based on TTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' To compare the effect of three pruning rules in OTCD algo- rithm, Table 4 lists their triggered times and the percentage of subintervals pruned by them for several queries respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' PoR and PoU are triggered frequently because their conditions are more easily to be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' However, PoR actually contributes pruned subintervals much less than the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Because it only prunes subintervals in the same row, and in contrast, PoU and PoL can prune an “area” of subintervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Overall, the three pruning rules can achieve significant optimization effect together Junyong Yang, Ming Zhong, Yuanyuan Zhu, Tieyun Qian, Mengchi Liu, and Jeffery Xu Yu TCD OTCD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 1 10 100 1000 Response Time(s) TCD- 25%~75% OTCD- 25%~75% Range within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='5IQR Median Line Mean Outliers (a) Youtube TCD OTCD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 1 10 100 1000 Response Time(s) TCD- 25%~75% OTCD- 25%~75% Range within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='5IQR Median Line Mean Outliers (b) Flickr Figure 8: The statistical distribution of response time for random queries on KONECT graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Table 5: Memory consumption of (O)TCD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Dataset Process Memory (GB) CollegeMsg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='02 sx-mathoverflow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='06 Youtube 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='7 DBLP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 Flickr 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='5 sx-stackoverflow 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='5 by enabling OTCD algorithm to skip more than 80 percents of subintervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' To evaluate the stability of our approach, we conduct statis- tical analysis of one hundred valid random queries on two new graphs, namely, Youtube and Flickr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We visualize the distribution of response time of TCD and OTCD algorithms for these random queries as boxplots, which are shown by Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The boxplots demonstrate that the response time of OTCD varies in a very limited range, which indicates that the OTCD indeed performs stable in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The outliers represent some queries that have exceptionally more results, which can be seen as a normal phe- nomenon in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' They may reveal that many communities of the social networks are more active during the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Moreover, Table 5 reports the process memory consumption for different datasets, which depends on the size of TEL mostly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We can observe that, 1) for the widely-used graphs like Youtube, DBLP, Flickr and stackoverflow, several gigabytes of memory are needed for storing TEL, which is acceptable for the ordinary hardware;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' and 2) for the very large graphs with billions of edges, the size of TEL is hundreds of gigabytes approximately, which would require the distributed memory cluster like Spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' To verify the scalability of our method with respect to the query parameters, we test the three algorithms with varing min- imum degree 푘 and time span (namely, 푇푒 −푇푠) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Impact of 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We select a typical query with span fixed and 푘 ranging from 2 to 6 for different graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The response time of tested algorithms are presented in Figure 9, from which we have an important observation against common sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' That is, differ- ent from core decomposition on non-temporal graphs, when the value of 푘 increases, the response time of TCD and OTCD algo- rithms decreases gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For OTCD, the behind rationale is clear, namely, its time cost is only bounded by the scale of re- sults, which decreases sharply with the increase of 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' To sup- port the claim, Figure 10 and Figure 11 show the trend of the amount of result cores and connected components in the result cores changing with 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Intuitively, a greater value of k means a stricter constraint and thereby filters out some less cohesive 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 1 10 100 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='01 3600 Response Time(s) k Baseline TCD OTCD (a) CollegeMsg 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 1 10 100 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='01 3600 Response Time(s) k Baseline TCD OTCD (b) sx-mathoverflow 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 1 10 100 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='01 3600 Response Time(s) k Baseline TCD OTCD (c) sx-stackoverflow Figure 9: Trend of response time under a range of 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 2 3 4 5 6 10 1 10 2 10 3 10 4 10 5 10 6 10 0 Quantity of Core k (a) CollegeMsg 2 3 4 5 6 10 1 10 2 10 3 10 4 Quantity of Core k (b) sx-mathoverflow 2 3 4 5 6 10 0 10 1 10 2 10 3 10 4 10 5 10 6 Quantity of Core k (c) sx-stackoverflow Figure 10: Trend of amount of distinct temporal 푘-cores under a range of 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 2 3 4 5 6 10 0 10 1 10 2 10 3 10 4 10 5 10 6 Connected Component k (a) CollegeMsg 2 3 4 5 6 10 1 10 2 10 3 10 4 10 5 10 0 10 6 Connected Component k (b) sx-mathoverflow 2 3 4 5 6 10 0 10 1 10 2 10 3 10 4 10 5 10 6 Connected Component k (c) sx-stackoverflow Figure 11: Trend of amount of connected components in temporal 푘-cores under a range of 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We can see the trend of runtime decrease for OTCD in Figure 9 is almost the same as the trend of core amount decrease in Figure 10, which also confirms the scalability of OTCD algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For TCD, the behind rationale is complicated, since it enu- merates all subintervals and each single decomposition is more costly with a greater value of 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' It is just like peeling an onion layer by layer, which has less layers with a greater value of 푘, so that the maintenance between layers become less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Impact of span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Similarly to the test of 푘, we also evalu- ate the scalability of different algorithms when the query time span increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The results are presented in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Although the number of subintervals increases quadratically, the response time of OTCD still increases moderately following the scale of query results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In contrast, TCD runs dramatically slower when the query time span becomes longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The above results demonstrate that the efficiency of OTCD is not sensitive to the change of query parameters, so that it is scalable in terms of query time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Lastly, for a large graph with a long time span like Youtube, we test OTCD algorithm by querying temporal 10-cores over the whole time span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The result is, to find all 19,146 temporal 10-cores within 226 days, the OTCD algorithm spent about 55 minutes, which is acceptable for such a “full graph scan” task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Scalable Time-Range 푘-Core Qery on Temporal Graphs 24 36 48 60 72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 1 10 100 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='01 3600 Response Time(s) Span(h) Baseline TCD OTCD (a) CollegeMsg 24 36 48 60 72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 1 10 100 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='01 3600 Response Time(s) Span(h) Baseline TCD OTCD (b) sx-mathoverflow 24 36 48 60 72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='1 1 10 100 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='01 3600 Response Time(s) Span(h) Baseline TCD OTCD (c) sx-stackoverflow Figure 12: Trend of response time under a range of span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='3 Effectiveness The effectiveness of TCQ is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Firstly, by given a flexible time interval, we can find many temporal 푘-cores of different subintervals, each of which represents a community emerged in a specific period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Consider the above test on Youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Although it is not feasible to exhibit all 19,146 cores, Figure 13 shows their distribution by time span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The number of cores generally de- creases with the increase of time span, which makes sense be- cause there are always a lot of small communities emerged dur- ing short periods and then they will interact with each other and be merged to larger communities within a longer time span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Secondly, we can continue to filter and analyse the result cores to gain insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For example, we record the date in GMT time for nine of the result cores with a time span less than one day in Youtube, and try to figure out if they emerged for some special reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Table 6 lists the date and size of the nine cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We can see that there is a large core emerged on Dec 10, 2006, which means more than 40,000 accounts had nearly one million inter- actions with each other in just a day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' That is definitely caused by a special event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' While, most of the rest cores emerged during summer vacation, which may mean people have more interac- tions on Youtube in the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 0 50 100 150 200 226 0 20 40 60 80 100 120 140 160 180 Number of core Time span(days) Figure 13: Distribution of all temporal 10-cores in Youtube by time span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Table 6: The date and size of nine temporal 10-cores emerged within one day in Youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Date |V| |E| Dec 10 2006 46499 885128 Feb 08 2007 1268 12054 Mar 25 2007 21 139 Jun 15 2007 98 713 Jun 18 2007 20 100 Jun 20 2007 124 1012 Jun 30 2007 21 110 Jul 02 2007 21 110 Jul 06 2007 12 66 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='4 Case Study For case study, we employ OTCD algorithm to query tempo- ral 10-cores on DBLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The query interval is set as 2010 to 2018, which spans over 8 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' By statistics, there exist 43 temporal 10-cores during that period, with 39 of them containing the au- thor Jian Pei, for whom we further build an ego network from three selected cores in defferent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Figure 14 shows the ego network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The authors in the three cores emerged in 2010, 2012 and 2014 are shaded by red, yellow and blue respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' By ob- serving the evolution of ego network over years, we can infer the change of author’s research interests or affiliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 22 vertices of a 10-core arising in 2010 14 vertices of a 10-core arising in 2012 15 vertices of a 10-core arising in 2014 Figure 14: Case Study in DBLP coauthorship network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' A friendship community with 32 members arising in 2007 114 newly added members on the first day after 124 newly added members on the second day after Figure 15: Case Study in Youtube friendship network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' To further demonstrate the potential of TCQ, we also employ TCQ to find temporal 푘-cores that expand quickly over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' This topic has been addressed in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Since OTCD returns all distinct cores efficiently, we can conveniently achieve the goal by identifying the cores contained by other larger cores within a few of days from the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Figure 15 shows such a bursting community on Youtube friendship network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The 32 central ver- tices colored in red comprise an initial temporal 10-core within two days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' This core is contained by another core about four times larger, while the TTI of the larger core only expands by one day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The new vertices in the larger core are colored in or- ange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Then, the new vertices colored in yellow join them to com- prise a twice larger new core in the next day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Clearly, these three temporal 10-cores together represent a community that grows remarkably fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In the real world, with more concrete informa- tion of graphs, such usages of TCQ will facilitate applications like recommendation, disease control, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='5 Discussion on the value of 푘 TCQ achieves relaxing the constraint on query time interval when composing푘-core queries on temporal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' However, the value of 푘 is still needed as an input parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We give a simple and rational criteria here for selecting the proper푘 value on different graphs, though many potential factors have different impacts on the selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The criteria is based on two intuitive facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Firstly, the number of distinct temporal 푘-cores over a given time in- terval will decrease with the increase of 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Secondly, the size uCtelg Surya Nepal JianYin EnhongiChen Li Xiong Bin Jiang ShuhuiWang Jian Pei Qingming Huang Jiawei Han Siyuan Liu Ying Zhang Xindong Wu Jie Tang Kai Xu Chang Liu Xiang Wang Rong Jin Yang Wang Jinjun Chen Jeffrey Xu YuJiangchuan Liu Philip S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Yu Feng Zhao Ke Wang Xuemin Lin Jian Chen Hua Wang Kunbiu Wenjie Zhang KeYi XueLi Jin Huang QiangYang Wei Wang Hang Li Yu Yeng lunduoJunyong Yang, Ming Zhong, Yuanyuan Zhu, Tieyun Qian, Mengchi Liu, and Jeffery Xu Yu 3 5 2 4 6 10 1 10 2 10 3 10 4 10 5 10 6 10 0 CollegeMsg Quantity of Core k Quantity of Core Average Size 25 50 75 100 125 150 Average Core Size Figure 16: A statistical chart for selecting the value of 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' of returned temporal 푘-cores will shrink with the increase of 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Normally, we expect the result cores to be concise and non- overlapping, especially when detecting the suspicious commu- nities that are inherently small and isolated, thereby preferring a greater value of 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' However, the number of result cores also matters, which requires the value of 푘 not being too great, oth- erwise there could be too few results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Therefore, the selection of 푘 should take both size and number of result cores into account, just like the trade-off between precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' For example, with 푘 ranging from 2 until 6, Figure 16 shows the falling curves of both number and average size of result cores over a specific time interval on CollegeMsg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We can observe that, setting 푘 = 5 should be a good choice, since the core size has declined to a relatively small level while the number of results is still fairly sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 8 RELATED WORK Recently, a variety of 푘-core query problems have been stud- ied on temporal graphs, which involve different temporal objec- tives or constraints in addition to cohesiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' The most rele- vant work to ours is historical 푘-core query [36], which gives a fixed time interval as query condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In contrast, our tempo- ral 푘-core query flexibly find cores of all subintervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Moreover, Galimberti et al [12] proposed the span-core query, which also gives a time interval as query condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' However, the span-core requires all edges to appear in every moment within the query interval, which is too strict in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Actually, historical푘-core relaxes span-core, and temporal 푘-core further relaxes historical 푘-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Besides, there are the following related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Wu et al [34] proposed (푘,ℎ)-core and studied its decomposition algorithm, which gives an additional constraint on the number of parallel edges between each pair of linked vertices in the 푘-core, namely, they should have at least ℎ parallel edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Li et al [19] proposed the persistent community search problem and gives a compli- cated instance called (휃,휏)-persistent 푘-core, which is a 푘-core persists over a time interval whose span is decided by the pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Similarly, Li et al [21] proposed the continual cohe- sive subgraph search problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Chu et al [5] studied the prob- lem of finding the subgraphs whose density accumulates at the fastest speed, namely, the subgraphs with bursting density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Qin et al [27, 28] proposed the periodic community problem to re- veal frequently happening patterns of social interactions, such as periodic 푘-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Wen et al [1] relaxed the constraints of (푘,ℎ)- core and proposed quasi-(푘,ℎ)-core for better support of main- tenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Lastly, Ma et al [25] studied the problem of finding dense subgraph on weighted temporal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' These works all focus on some specific patterns of cohesive substructure on tem- poral graphs, and propose sophisticated models and methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Compared with them, our work addresses a fundamental query- ing problem, which finds the most general 푘-cores on temporal graphs with respect to two basic conditions, namely, 푘 and time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' As discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content='2, we can extend TCQ to find the more specific 푘-cores by importing the constraints defined by them, because most of the definitions are special cases of tem- poral 푘-core, but not vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Lastly, many research work on cohesive subgraph query for non-temporal graphs also inspire our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' We categorize them by the types of graphs as follows: undirected graph [3, 9, 13, 23, 35, 37], directed graph [4, 24, 30], labeled graph [6, 18, 31], attrib- uted graph [7, 14, 15, 26], spatial graph [8, 10, 39], heterageneous information network [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Besides, many work specific to bipar- tite graph [22, 32, 33, 38] also contain valuable insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' 9 CONCLUSION AND FUTURE WORK For querying communities like푘-cores on temporal graphs, spec- ifying a time interval in which the communities emerge is the most fundamental query condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' To the best knowledge we have, we are the first to study a temporal 푘-core query that al- lows the users to give a flexible interval and returns all distinct 푘- cores emerging in any subintervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Dealing with such a query in brute force is obviously costly due to the possibly large number of subintervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Thus, we propose a novel decremental 푘-core inducing algorithm and the auxiliary optimization and imple- mentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Our algorithm only enumerates the neces- sary subintervals that can induce a final result and reduces re- dundant computation between subintervals significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' More- over, the algorithm is physically decomposed to a series of ef- ficient data structure manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' As a result, although our algorithm does not use any precomputed index, it still outper- forms an incremental version of the latest index-based approach by a remarkable margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In conclusion, our algorithm is scalable with respect to the span of given time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' In the future, we will study how to leverage our algorithm as a framework to integrate various temporal 푘-core analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' There are a number of related work have considered different temporal constraints of 푘-cores, most of which can be combined with the time interval condition to offer more powerful function- ality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' However, their query processing algorithms are essentially diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' Therefore, we need to bridge the gap based on a general and scalable algorithm like ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfQgYx/content/2301.03770v1.pdf'} +page_content=' REFERENCES [1] Wen Bai, Yadi Chen, and Di Wu.' metadata={'source': 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+K9E5T4oBgHgl3EQfYQ9F/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text diff --git a/19AyT4oBgHgl3EQfbvd_/content/tmp_files/2301.00269v1.pdf.txt b/19AyT4oBgHgl3EQfbvd_/content/tmp_files/2301.00269v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..070838b33b20ea3468f3501633defec6d7fce707 --- /dev/null +++ b/19AyT4oBgHgl3EQfbvd_/content/tmp_files/2301.00269v1.pdf.txt @@ -0,0 +1,1475 @@ +WiFi Physical Layer Stays Awake and Responds +When it Should Not +Ali Abedi +Stanford University +USA +abedi@stanford.edu +Haofan Lu +UCLA +USA +haofan@cs.ucla.edu +Alex Chen +University of Waterloo +Canada +zihanchen.ca@gmail.com +Charlie Liu +University of Waterloo +Canada +charlie.liu@uwaterloo.ca +Omid Abari +UCLA +USA +omid@cs.ucla.edu +ABSTRACT +WiFi communication should be possible only between devices in- +side the same network. However, we find that all existing WiFi +devices send back acknowledgments (ACK) to even fake packets +received from unauthorized WiFi devices outside of their network. +Moreover, we find that an unauthorized device can manipulate the +power-saving mechanism of WiFi radios and keep them continu- +ously awake by sending specific fake beacon frames to them. Our +evaluation of over 5,000 devices from 186 vendors confirms that +these are widespread issues. We believe these loopholes cannot be +prevented, and hence they create privacy and security concerns. +Finally, to show the importance of these issues and their conse- +quences, we implement and demonstrate two attacks where an +adversary performs battery drain and WiFi sensing attacks just +using a tiny WiFi module which costs less than ten dollars. +1 +INTRODUCITON +Today’s WiFi networks use advanced authentication and encryption +mechanisms (such as WPA3) to protect our privacy and security +by stopping unauthorized devices from accessing our devices and +data. Despite all these mechanisms, WiFi networks remain vulner- +able to attacks mainly due to their physical layer behaviors and +requirements defined by WiFi standards. In this paper, we find two +loopholes in the IEEE 802.11 standard for the first time and show +how they can put our privacy and security at risk. +a) WiFi radios respond when they should not. In a WiFi +network, when a device sends a packet to another device, the re- +ceiving device sends an acknowledgment back to the transmitter. +In particular, upon receiving a frame, the receiver calculates the +cyclic redundancy check (CRC) of the packet in the physical layer +to detect possible errors. If it passes CRC, then the receiver sends +an Acknowledgment (ACK) to the transmitter to notify the correct +reception of the frame. Surprisingly, we have found that all existing +WiFi devices send back ACKs to even fake packets received from +unauthorized WiFi devices outside of their network. Why should a +WiFi device respond to a fake packet from an unauthorized device?! +b) WiFi radios stay awake when they should not. WiFi chipsets +are mostly in sleep mode to save power. However, to make sure +that they do not miss their incoming packets, they notify their WiFi +access point before entering sleep mode so that the access point +buffers any incoming packets for them. Then, WiFi devices wake up +periodically to receive beacon frames sent by the associated access +point. In regular operation, only the access point sends beacon +frames to notify the devices that have buffered packets. When a +device is notified, it stays awake to receive them. However, these +beacon frames are not encrypted. Hence, we find that an unautho- +rized user can forge those beacon frames to keep a specific device +awake for receiving the (non-existent) buffered frames. +We examine these behaviors and loopholes in detail over dif- +ferent WiFi chipsets from different vendors. Our examination of +over 5,000 WiFi devices from 186 vendors shows that these are +widespread issues. We then study the root cause of these issues +and show that, unfortunately, they cannot be fixed by a simple +solution such as updating WiFi chipsets firmware. Finally, we im- +plement and demonstrate two attacks based on these loopholes. +In the first attack, we show that by forcing WiFi devices to stay +awake and continuously transmit, an adversary can continuously +analyze the signal and extract personal information such as the +breathing rate of the WiFi users. In the second attack, we show that +by forcing WiFi devices to stay awake and continuously transmit, +the adversary can quickly drain the battery, and hence disable WiFi +devices such as home and office security sensors. These attacks +can be performed from outside buildings despite the WiFi network +and devices being completely secured. All the attacker needs is a +$10 microcontroller with integrated WiFi (such as ESP32) and a +battery bank. The attacker device can easily be carried in a pocket +or hidden somewhere near the target building. +The main contributions of this work are: +• We find that WiFi devices respond to fake 802.11 frames with +ACK, even when they are from unauthorized devices. We +also find that WiFi radios can be kept awake by sending them +fake beacon frames indicating they have packets waiting for +them. +• We study these loopholes and their root causes in detail, and +have tested more than 5,000 WiFi access points and client +devices from more than 186 vendors. +• We implement two attacks based on these loopholes using +just a 10-dollar off-the-shelf WiFi module and validate them +in real-world settings. +2 +RELATED WORK +The loopholes we present in this paper are explored using packet +injection, in which an attacker sends fake WiFi packets to devices in +a secured WiFi network. Packet injection has been used in the past +arXiv:2301.00269v1 [cs.NI] 31 Dec 2022 + +to perform various types of attacks against WiFi networks such as +denial of service attacks for a particular client device or total dis- +ruption of the network [14, 15, 17, 41]. These attacks use different +approaches such as beacon stuffing to send false information to +WiFi devices [21, 46], or Traffic Indication Map (TIM) forgery to +prevent clients from receiving data [18, 42]. However, all of these +attacks focus on spoofing 802.11 MAC-layer management frames +to interrupt the normal operation of WiFi networks. To provide a +countermeasure for some of these attacks, the 802.11w standard [7] +introduces a protected management frame that prevents attack- +ers from spoofing 802.11 management frames. Instead of spoofing +802.11 MAC frames, we exploit properties of the 802.11 physical +layer to force a device to stay awake and respond when it should +not. These loopholes open the door to multiple research avenues +including new security and privacy threats. +WiFi sensing attack: Over the past decade, there has been a +significant amount of research on WiFi sensing where WiFi signals +are used to detect human activities [13, 32, 34–36, 38, 43–45, 48]. +However, these systems target applications with social benefits +and cannot be easily used by an attacker to create privacy and +security threats. This is because either these techniques require +cooperation from the target WiFi device or the attacker needs to be +very close to the target to use these systems. A recent study shows +that by capturing WiFi signals coming out of a private building, it +is possible for an adversary to track user movements inside that +building [49]. However, this attack has a bootstrapping stage which +requires the attacker to walk around the target building for a long +time to find the location of the WiFi devices. Furthermore, since +this work relies on only the normal intermittent WiFi activities, it +cannot capture continuous data such as breathing rate. +Battery draining attack: Battery draining attacks date back to +1999 [40] and there have been many studies on such attacks and +potential defense mechanisms since then [20]. Battery discharge +models and energy vulnerability due to operating systems have +been investigated [30, 47]. A more recent study plays multimedia +files implicitly to increase power consumption during web browsing +[27, 28]. In terms of defending, a monitoring agent that searches for +abnormal current draw is discussed in [19]. In contrast, our attack +exploits the loopholes in the 802.11 physical layer protocol and the +power-hungry WiFi transmission to quickly drain a target device’s +battery. We will discuss in Section 3.2 that stopping our proposed +attack is nearly impossible on today’s WiFi devices. +This paper is an extension of our previous workshop publica- +tion [9]. The workshop paper shows preliminary results for our +finding that WiFi devices respond with ACKs to packets received +from outside of their network, and provides a brief discussion on +potential privacy and security concerns of this behavior without +studying them. We have also explored how the WiFi power saving +mechanism can be exploited to keep a target device awake in a +localization attack [12]. In this paper, we provide an in-depth study +of these previously discovered loopholes. We also design and per- +form two privacy and security attacks, based on these loopholes. +Finally, we implement these attacks on off-the-shelve WiFi devices +and present detailed performance evaluations. +Figure 1: WiFi devices send an ACK for any frame they re- +ceive without checking if the frame is valid. +Figure 2: Frames exchanged between attacker and victim +3 +WIFI RESPONDS WHEN IT SHOULD NOT +Most networks use security protocols to prevent unauthorized de- +vices from communicating with their devices. Therefore, one may +assume that a WiFi device only acknowledges frames received from +the associated access point or other devices in the same network. +However, we have found that all today’s WiFi devices acknowledge +even the frames they receive from an unauthorized device from +outside of their network. In particular, as long as the destination +address matches their MAC address, their physical layer acknowl- +edges it, even if the frame has no valid payload. In this section, we +examine this behavior in more detail, and explain why this problem +happens and why it is not preventable. +To better understand this behavior, we run an experiment where +we use two WiFi devices to act as a victim and an attacker. The +attacker sends fake WiFi packets to the victim. We monitor the real +traffic between the attacker and the victim’s device. +Setup: For the victim, we use a tablet, and for the attacker, we +use a USB WiFi dongle that has a Realtek RTL8812AU 802.11ac +chipset. This is a $12 commodity WiFi device. The attacker uses +this device to send fake frames to the victim’s device. To do so, +we develop a python program that uses the Scapy library [37] to +create fake frames. Scapy is a python-based framework that can +generate arbitrary frames with custom data in the header fields. +Note, that the only valid information in the frame is the destination +MAC address (i.e., the victim’s MAC address). The transmitter MAC +address is set to a fake MAC address (i.e., aa:bb:bb:bb:bb:bb), and +the frame has no payload (i.e., null frame) and is not encrypted. +Result: Figure 2 shows the real traffic between the attacker and the +victim device captured using Wireshark packet sniffer [22]. As can +be seen, when the attacker sends a fake frame to the victim, the vic- +tim sends back an ACK to the fake MAC address (aa:bb:bb:bb:bb:bb). +This experiment confirms that WiFi devices acknowledge frames +without checking their validity. Finally, to see if this behavior exists +on other WiFi devices, we have repeated this test with a variety of +devices (such as laptops, smart thermostats, tablets, smartphones, +and access points) with different WiFi chipsets from different ven- +dors, as shown in Table 1. Note, target devices are connected to a +private network and the attacker does not have their secret key. +After performing the same experiment as before, we found that all +2 + +Private WiFi Network +Acknowledgement +Fake 802.11 +Data Frame +Access Point +Target +AttackerSource +Destination +Info +aa:bb:bb:bb:bb:bb +f2:6e:0b: +Null function(No data), +aa:bb:bb:bb:bb:bb...Acknowledqement,Flaqs=.Device +WiFi module +Standard +MSI GE62 laptop +Intel AC 3160 +11ac +Ecobee3 thermostat +Atheros +11n +Surface Pro 2017 +Marvel 88W8897 +11ac +Samsung Galaxy S8 +Murata KM5D18098 +11ac +Google Wifi AP +Qualcomm IPQ 4019 +11ac +Table 1: List of tested chipsets/devices +of these devices also respond to fake packets received from a device +outside of their network. +3.1 +How widespread is this loophole? +In the previous section, we examined a few different WiFi devices +and showed that they are all responding to fake frames from unau- +thorized devices. Here, we examine thousands of devices to see how +widespread this behavior is. In the following, we explain the setup +and results of this experiment. +Setup: To examine thousands of devices, we mounted a WiFi dongle +on the roof of a vehicle and drove around the city to test all nearby +devices. For the WiFi dongle, we use the same Realtek RTL8812AU +USB WiFi dongle, and connect it to a Microsoft Surface, running +Ubuntu 18.04. We develop a multi-threaded program using the +Scapy library [37] to discover nearby devices, send fake 802.11 +frames to the discovered devices, and verify that target devices re- +spond to our fake frames. Specifically, our implementation contains +three threads. The first thread discovers nearby devices by sniffing +WiFi traffic and adding the MAC address of unseen devices to a +target list. The second thread sends fake 802.11 frames to the list of +target devices. Finally, the third thread checks to verify that target +devices respond with an ACK. +Results: We perform this experiment for one hour while driving +around the city. In total, we discovered 5,328 WiFi nodes from +186 vendors. The list includes 1,523 different WiFi client devices +from 147 vendors and 3,805 access points from 94 vendors. Table 2 +shows the top 20 vendors for WiFi devices and WiFi access points +in terms of the number of devices discovered in our experiment. +The list includes devices from major smartphone manufacturers +(such as Apple, Google, and Samsung) and major IoT vendors (such +as Nest, Google, Amazon, and Ecobee). We found that all 5,328 WiFi +Access Points and devices responded to our fake 802.11 frames with +an acknowledgment, and hence we infer that most probably all +of today’s WiFi devices and access points respond to fake frames +when they should not. +3.2 +Can this loophole be fixed? +So far, we have demonstrated that all existing WiFi devices respond +to fake packets received from unauthorized WiFi devices outside of +their network. Now, the next question is why this behavior exists, +and if it can be prevented in future WiFi chipsets. +In a WiFi device, when the physical layer receives a frame, it +checks the correctness of the frame using error-checking mech- +anisms (such as CRC) and transmits an ACK if the frame has no +error. However, checking the validity of the content of a frame is +WiFi Client Device +WiFi Access Point +Vendor +# devices +Vendor +# devices +Apple +143 +Hitron +723 +Google +102 +Sagemcom +601 +Intel +66 +Technicolor +410 +Hitron +65 +eero +195 +HP +63 +Extreme N. +188 +Samsung +56 +Cisco +156 +Espressif +47 +HP +104 +Hon Hai +46 +TP-LINK +101 +Amazon +41 +Google +80 +Sagemcom +38 +D-Link +75 +Liteon +33 +NETGEAR +69 +AzureWave +30 +ASUSTek +51 +Sonos +30 +Aruba +46 +Nest Labs +27 +SmartRG, +44 +Murata +24 +Ubiquiti N. +35 +Belkin +20 +Zebra +35 +TP-LINK +20 +Pegatron +28 +Cisco +16 +Belkin +25 +ecobee +13 +Mitsumi +25 +Microsoft +13 +Apple +19 +Others +630 +Others +789 +Total +1523 +Total +3805 +Table 2: List of WiFi devices and APs that respond to our +fake 802.11 frames. +performed by the MAC and higher layers. Unfortunately, this sepa- +ration of responsibilities and the fact that the physical layer does +not coordinate with higher layers about sending ACKs seem to be +the root cause of the behavior. In particular, we have observed that +when some access points receive fake frames, they start sending +deauthentication frames to the attacker, requesting it to leave the +network. These access points detect the attacker as a “malfunc- +tioning” device and that is why they send deauthentication frames. +Surprisingly, although the access points have detected that they are +receiving fake frames from a “malfunctioning” device, we found +that they still acknowledge the fake frames. +An example traffic that demonstrates this behavior is shown in +Figure 3. As can be seen, although the access point has already sent +three deauthentication frames to the attacker, it still acknowledges +the attacker’s fake frame. We then manually blocked the attacker’s +fake MAC address on the access point. Surprisingly, we observed +that the AP still acknowledges the fake frames. These observations +verify that sending ACK frames happens automatically in the physi- +cal layer without any communication with higher layers. Therefore, +the software running on the access points does not prevent the +physical layer from sending ACKs to fake frames. +The next question is why the software running on WiFi devices +does not prevent this behavior by verifying if the frame is legitimate +before sending an ACK. Unfortunately, this is not possible due to the +WiFi standard timing requirements. Specifically, in the IEEE 802.11 +standard, upon receiving a frame, an ACK must be transmitted +3 + +Figure 3: The attacked access point detects that something +strange is happening, however it still ACKs fake frames +by the end of the Short Interframe Space (SIFS)1 interval which is +10 𝜇s and 16 𝜇s for the 2.4 GHz and 5 GHz bands, respectively. If the +transmitter does not receive an ACK by the end of SIFS, it assumes +that the frame has been lost and retransmits the frame. Therefore, +the WiFi device nefeds to verify the validity of the received frame +in less than 10 𝜇𝑠. This verification must be done by decoding +the frame using the secret shared key. Unfortunately, decoding a +frame in such a short period is not possible. In particular, past work +has shown that the time required to decode a frame is between +200 to 700 𝜇𝑠 when the WPA2 security protocol is used [31, 33, +39]. This processing time is orders of magnitude longer than SIFS. +Hence, existing devices cannot verify the validity of the frame +before sending the ACK, and they acknowledge a frame as long +as it passes the error detection check. One potential approach to +solve this loophole is to implement the security decoder in WiFi +hardware instead of software to significantly speed up its delay. +Although this may solve the problem in future WiFi chipsets, it will +not fix the problem in billions of WiFi chipsets which are already +deployed. +4 +WIFI STAYS AWAKE WHEN IT SHOULD +NOT +We have also found a loophole that allows an unauthorized device +to keep a WiFi device awake all the time. One may think that a +WiFi device can be kept awake by just sending fake back-to-back +packets to it and forcing it to transmit acknowledgment. However, +this approach does not work. Most WiFi radios go to sleep mode +to save energy during inactive states such as screen lock, during +which the attacker is not able to keep them awake by sending back- +to-back packets. Figure 4a show the results of an experiment where +the attacker is continuously transmitting fake packets to a WiFi +device. In this figure, we plot the amplitude of CSI over time for +the ACK packets received from the WiFi device. As can be seen, +the responses are sparse and discontinued even when the attacker +sends back-to-back packets to the WiFi device. This is because the +WiFi device goes to sleep mode frequently. However, we have found +a loophole in the power saving mechanism of WiFi devices which +can be used by an unauthorized device to keep any WiFi device +awake all the time. +1The SIFS is used in the 802.11 standard to give the receiver time to go through different +procedures before it is ready to send the ACK. These procedures include Physical-layer +and MAC-layer header processing, creating the waveform for the ACK, and switching +the RF circuit from receiving to transmitting mode. +(a) Without fake beacon frames +(b) With fake beacon frames +Figure 4: The CSI amplitude of ACKs responded by the tar- +get device when an attacker sends back-to-back fake packets +to it in two scenarios. (a) In this scenario, the attacker is not +using fake beacon frames. Therefore, the target device goes +to sleep mode frequently and does not respond to fake pack- +ets. (b) In this scenario, the attacker infrequently sends fake +beacon frames to keep the target device awake all the time. +4.1 +How does WiFi power saving mechanism +work? +Wireless tranceivers are very power-hungry. Therefore, WiFi radios +spend most of the time in the sleep mode to save power. When a +WiFi radio is in sleep mode, it cannot send or receive WiFi packets. +To avoid missing any incoming packets, when a WiFi device wants +to enter the sleep mode it notifies the WiFi access point so that +the access point buffers any incoming packets for this device. WiFi +devices, however, wake up periodically to receive beacon frames +to find out if packets are waiting for them. In particular, WiFi +access points broadcast beacon frames periodically which includes a +Traffic Indication Map (TIM) field that indicates which devices have +buffered packets on the access point. For example, if the association +ID of a WiFi device is 𝑥, then the (𝑥 + 1)𝑡ℎ bit of TIM is assigned to +that device. Finally, when a device is notified that has some buffered +packets on the access point, it stays awake and replies with a Null- +function packet with a power management bit set to "0". In this way, +the WiFi device informs the access point it is awake and ready to +receive packets. +4.2 +How can one manipulate power saving? +We have found that an unauthorized device can use the power- +saving mechanism of WiFi devices to force them to stay awake. +In particular, an attacker can pretend to be the access point and +broadcasts fake beacon frames indicating that the WiFi device has +buffered traffic, forcing them to stay awake. However, this requires +the attacker to know the MAC address and the SSID of the network’s +access point, as well as the association ID and MAC address of the +targeted device so that it can set the correct bit in TIM. The access +point MAC address and SSID can be easily discovered by sniffing +4 + +Source +Destination +Info +f2:6e:0b: +aa:bb:bb:bb:bb:bb +Deauthentication, +SN=3275 +f2:6e:0b: +aa:bb:bb:bb:bb:bb +Deauthentication, +SN=3275 +f2:6e:0b: +aa:bb:bb:bb:bb:bb +Deauthentication, +SN=3275 +aa:bb:bb:bb:bb:bb +f2:6e:0b: +Null function (No data), +aa:bb:bb:bb:bb:bb +Acknowledgement, +Flags=.. +f2:6e:0b: +aa:bb:bb:bb:bb:bb +Deauthentication, +SN=3281 +f2:6e:0b: +aa:bb:bb:bb:bb:bb +Deauthentication, +SN=328125 +CSI Amplitude +20 +15 +10 +5 +0 +0 +5 +10 +15 +20 +25 +30 +Time (s)25 +CSI Amplitude +20 +15 +10 +5 +0 +0 +5 +10 +15 +20 +25 +30 +Time (s)Figure 5: WiFi devices stay awake on hearing a forged bea- +con frame with TIM flags set up. +the WiFi traffic using software such as Wireshark since the MAC +address is never encrypted and all nodes send packets to the access +point. Note that MAC randomization does not cause any problem +for this process because the attacker finds the randomized MAC +address that is currently being used. Next, the attacker pretends to +be the access point and broadcasts fake beacon frames with TIM set +to "0xFF", indicating all client devices have buffered traffic. Then, it +enters the sniffing mode to sniff for the Null-function packets. The +null-function packets contain the ID and MAC addresses of all WiFi +devices. To avoid keeping all WiFi devices awake, we find that one +can send a fake beacon frame as a unicast packet, instead of the +usual broadcast beacons. This way only the target device receives +the packet and we do not interfere with the operation of other +devices. Interestingly, our experiments show that target devices do +not care if they receive beacons as broadcast or unicast frames. +To better understand this behavior, we run an experiment where +we use two WiFi devices to act as a victim and an attacker, re- +spectively. The attacker sends fake WiFi packets to the victim. We +monitor the real traffic between the attacker and the victim’s device. +Setup: Similar to the experiment described in Section 3, we use an +RTL8812AU USB dongle to inject fake packets to a smartphone held +by a person who is watching YouTube on the phone. The distance +between the smartphone and the user is about 60 cm. The attacking +device and the victim are in two separate rooms. The attacker also +uses an ESP32 WiFi module to record the Channel State Information +(CSI) of received ACKs. +Result: We find that although sending fake beacon frames keeps +the target device awake, sending them very frequently will cause +WiFi devices to recognize the suspicious attacker’s behavior and +disconnect from it. Therefore, to keep the WiFi device awake, in- +stead of just sending beacon frames back-to-back, the attacker can +continuously transmit normal fake packets to a WiFi device and +periodically sends fake beacon frames to keep it awake. Figure 4b +shows the result of an experiment where the attacker is continu- +ously transmitting fake packets to a WiFi device and periodically +sends fake beacon frames. As it can be seen, the target device is +continuously awake and responding to fake packets with ACKs. +5 +PRIVACY IMPLICATION: WIFI SENSING +ATTACK +Recently, there has been a significant amount of work on WiFi +sensing technologies that use WiFi signals to detect events such as +motion, gesture, and breathing rate. In this section, we show how +an adversary can combine WiFi sensing techniques with the above +loopholes to monitor people’s breathing rate whenever she/he +wants from outside buildings despite the WiFi network and de- +vices being completely secured. In particular, an adversary can +force our WiFi devices to stay awake and continuously transmit +WiFi signals. Then she/he can continuously analyze our signals +and extract information such as our breathing rate and presents. +Note, since most of the time, we are close to a WiFi device (such as +a smartwatch, laptop, or tablet), our body will change the ampli- +tude and phase of the signals which can be easily extracted by the +adversary. +5.1 +Attack Design, Scenarios and Setup +5.1.1 +Attack Design. The attacker sends fake packets to a WiFi +device in the target property and pushes it to transmit ACK packets. +In particular, since an adult’s normal breathing rate is around 12 -20 +times per minute (i.e., 0.2- 0.33Hz), receiving several ACK packets +per second is sufficient for the attacker to estimate the breathing +rate, without impacting the performance of the target WiFi network. +The attacker then takes the Fourier transform of the CSI information +of ACK packets to estimate the breathing rate of the person who +is nearby the WiFi device. However, due to the random delays +of the WiFi random access protocol and the operating system’s +scheduling protocol, the collected data samples are not uniformly +spaced in time. Hence, the attacker cannot simply use standard +FFT to estimate the breathing rate. Instead, they need to use a non- +uniform Fourier transform, and a voting algorithm to extract the +breathing rate. The Non-Uniform Fast Fourier Transform (NUFFT) +algorithm 1 used is shown below. +Algorithm 1: Non-uniform FFT +Data: Time indices 𝑡, data samples 𝑥 of length 𝑛 +Result: Magnitude of each frequency component +𝑑 ← min𝑖 (𝑡𝑖 − 𝑡𝑖−1) +𝑖 = 1, 2, ...,𝑛.; +for 𝑖 ← 1 to 𝑛 − 1 do +𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 ← 𝑡 [𝑖] − 𝑡 [𝑖 − 1]; +if 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 > 𝑑 then +𝑐𝑜𝑢𝑛𝑡 ← ⌊𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙/𝑑⌋; +Interpolation(𝑡, 𝑥, 𝑡 [𝑖], 𝑡 [𝑖 − 1], 𝑐𝑜𝑢𝑛𝑡); +end +end +return FFT(𝑡, 𝑥) +The algorithm first finds the minimum time gap between any two +adjacent data points 𝑑, then linearly interpolates any interval that +is larger than the gap with ⌊𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙/𝑑⌋$ samples. Finally, it uses +a regular FFT algorithm to find the magnitude of each frequency +component. A low-pass filter is applied before feeding data to the +FFT analysis to reduce noise (not shown in the algorithm). +Figure 6(a) and 6(b) show the amplitude of CSI before and after +interpolation, respectively, when the attacker sends 10 packets per +second to a WiFi device that is close to the victim. Each figure shows +both the original data (in blue) and the filtered data (in orange). +Figure 6(c) shows the frequency spectrum of the same signals when +a standard FFT or our non-uniform FFT is applied. A prominent +peak at 0.3Hz is shown in the non-uniform FFT spectrum, indicating +a breathing rate of 18 bpm. +5 + +Private WiFi Network +Stay Awake +Fake 802.11 +Beacon +Access Point +Target +Attacker(a) Raw and filtered data before +interpolation +(b) Raw and filtered data after +interpolation +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Frequency (Hz) +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Power +non_uniform_fft +standard_fft +(c) Standard FFT and a non-uniform FFT of +Data +Figure 6: Steps to extract breathing rate from the CSI. +WiFi CSI gives us the amplitude of 52 subcarriers per packet. +We observed that these subcarriers are not equally sensitive to the +motion of the chest. Besides, a subcarrier’s sensitivity may vary +depending on the surrounding environment. For a more reliable +attack, the attacker should identify the most sensitive subcarriers +over a sampling window. Previously proposed voting mechanisms +for coarse-grained motion detection applications [8, 16, 29, 49] +cannot be directly applied in this situation, as chest motion during +respiration is at a much smaller scale. Instead, we developed a soft +voting mechanism, where each subcarrier gives a weighted vote +to a breathing rate value. The breathing rate that gets the most +votes is reported. Specifically, We first find the power of the highest +peak (𝑃𝑝𝑒𝑎𝑘), and then calculate the average power of the rest bins +(𝑃𝑎𝑣𝑒). The exponent of the Peak-to-Average Ratio (PAR): 𝑒 +𝑓𝑝𝑒𝑎𝑘 +𝑓𝑎𝑣𝑒 is +used as the weight of the corresponding subcarrier. In this way, we +guarantee the subcarriers with higher SNR have significantly more +votes than the rest of the subcarriers. +5.1.2 +Attack Scenarios. We evaluate the WiFi sensing attack in +different scenarios, both indoor and outdoor. In the indoor scenario, +the attacker and the target are placed in the same building but on +different floors. The height of one floor in the building is around +2.8 m. This scenario is similar to when the attacker and the target +person are in different units of an apartment or townhouse. In the +outdoor scenario, the attacker is outside the target’s house. For the +outdoor experiments, We place the attacker in another building +which is around 20 m away from the target building. In all of the +experiments, the target WiFi devices are placed 0.5 to 1.4 m away +from the person’s body. The person is either watching a movie, +typing on a laptop, or surfing the web using his cell phone. During +the experiments, other people are walking and living normally in +the house. Finally, we run the attack and compare the estimated +breathing rate with the ground truth. To obtain the ground truth, +we record the target person’s breathing sound by attaching a mi- +crophone near his/her mouth [23]. We then calculate the FFT on +the sound signal to measure the breathing frequency. Note that the +attack does not need this information and this is just to obtain the +ground truth in our experiments. +5.1.3 +Attacker Setup. Hardware Setup: The attacker uses a Linksys +AE6000 WiFi card and an ESP32 WiFi module [25] as the attacking +device. Both devices are connected to a ThinkPad laptop via USB. +The Linksys AE6000 is used to send fake packets and the ESP32 +WiFi module is used to receive acknowledgments (ACK) and extract +CSI. Although we use two different devices for sending and receiv- +ing, one can simply use an ESP32 WiFi module for both purposes. +The use of two separate modules gave us more flexibility in run- +ning many experiments. As for the target device, we use a One Plus +8T smartphone without any software or hardware modifications. +We have also tested our attack on an unmodified Lenovo laptop, a +Microsoft Surface Pro 4 laptop, and a USB WiFi card as the target +device and we obtained similar results. It is worth mentioning that +any WiFi device can be a target without any software or hardware +modification. +Software Setup: We have implemented the CSI collecting script +on the ESP32 WiFi module, and the breathing rate estimation algo- +rithm on the laptop. The collected CSI data is fed to the algorithm +which produces the breathing rate estimation values in real-time. +To process this data in real time, a sliding window (buffer) is used. +The size of the window is 30 s and the stride step is 1 s. 30 seconds +is a large enough window for estimating a stable breathing rate +value. Note that an adult breathes around 6 times during such a +window. The window is a queue of data points, and it updates every +second by including 1 second of new data points to its head and +removing 1 second of old data points from its tail. The breathing +rate estimation runs the analysis algorithm on the data points inside +the window whenever it is updated. The window slides once per +second. Hence, our software reports an estimation of breathing rate +every second. Note that there is a 30-second delay at the beginning +since the window needs to be filled first. +5.2 +Results +We evaluate the effectiveness of the attack in different scenarios +such as when the attacker and the target are in the same building +or different buildings. +5.2.1 +Accuracy in Detecting Breathing Rate. Same Building Sce- +nario: First, we evaluate the accuracy of the attack by estimating +6 + +40 +Raw Data +35 +Filtered Data +30 +1Amplitude +25 +20 +CSI +15 +10 +5 +0 +0 +10 +20 +30 +Time(s)40 +Raw Data +35 +Filtered Data +30 +CSI Amplitude +25 +20 +15 +10 +5 +0 +0 +10 +20 +30 +Time(s)Figure 7: The average accuracy of the at- +tack in estimating the target person’s +breathing rate when he attacker and +target device are in the same building. +Figure 8: The CDF of the error in es- +timating the target person’s breathing +rate when he attacker and target de- +vice are in the same building (different +floor). +Figure 9: The CDF of the error in es- +timating the target person’s breathing +rate when he attacker and target device +are in different buildings (20m away) +the breathing rate in an indoor scenario where the target device +and attacker are in the same building. We evaluate the accuracy +when the target’s breathing rate is 12, 15, 20, and 30 breaths per +minute. Note, that the normal breathing rate for an adult is 12-20 +breaths per minute while resting, and higher when exercising. In +this experiment, the user is watching a video. To make sure the +target person’s breathing rate is close to our desired numbers, we +place a timer in front of the person, where they can adjust their +breathing rate accordingly. This is just to better control the breath- +ing rate during the experiment and is not a requirement nor an +assumption in this attack. We run each experiment for two minutes. +During this time, we collect the estimated breathing rate from both +ground truth and the attack for different locations of the target +device. Figure 7 shows the average accuracy in estimating breath- +ing rate across all experiments. The accuracy is calculated as the +ratio of the estimated breathing rate reported by the attack over the +ground truth breathing rate. The figure shows that the accuracy of +estimating the breathing rate is over 99% in all scenarios. Finally, +Figure 8 plots the Cumulative Distribution Function (CDF) of the +error in detecting breathing rate for over 2400 measurements. The +figure shows that 78% of the estimated results have no error. The +figure also shows that 99% of measurements have less than one +breath per minute error which is negligible. +Different Building Scenario: So far, we have evaluated our at- +tack where the target and the attacker are in different rooms or +floors of the same building. Here we push this further and examine +whether our attack works if the attacker and the target person are +in a different building. We place the target device in a building on +a university campus on a weekday with people around. A person +is sitting around 0.5 m away from the device. We then place the +attacker in another building which is around 20 m away from the +target building. Similar to the previous experiment, we run the +attack and compare the estimated breathing rate with the ground +truth. Figure 9 shows the CDF of error for 180 measurements in +this experiment. Our results show that the attacker successfully +estimates the breathing rate. Note, that the reason that the attack +works even in such a challenging scenario with other people being +around is two-fold. First, using an FFT helps to filter out the effect +Figure 10: The efficacy of estimating the breathing rate when +there is no target near the WiFi device. +of most non-periodic movements and focuses on periodic move- +ments and patterns. Second, wireless channels are more sensitive +to changes as we get closer to the transmitter [11, 24], and since +in these scenarios, the target person is very close to the target de- +vice, their breathing motion has a higher impact on the CSI signal +compared to the other mobility in the environment. +5.2.2 +Human Presence Detection. We next evaluate the efficacy of +detecting whether there is a target person near the WiFi device or +not. In this experiment, the target phone is placed on a desk and the +person stays around the device for 30 seconds, then walks away +from the device, and then comes back near the device. Note, in our +algorithm, when there is no majority vote during the voting phase, +we return −1 to indicate no breathing detected. Figure 10 shows +the results of this experiment. As illustrated in the figure, we can +correctly detect the breathing rate when a person is near the device. +In other words, the algorithm can detect if there is no one near the +target device and refrain from reporting a random value. +5.2.3 +Effect of Distance and Orientation. Next, we evaluate the +effectiveness of the attack for different orientations of the device +with respect to the person. We also evaluate its performance for +different distances between the target device and the target person. +7 + +99.85% +99.44% +99.71% +99.48% +100 +Accuracy (%) +80 +60 +40 +20 +0 +12 +15 +20 +30 +Orientation1.0 +0.8 +CDF +0.6 +0.4 +0.2 +0.8.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Error (RR/min)1.0 +0.8 +DF +0.6 +0.4 +0.2 +0.0 +0 +1 +2 +3 +4 +5 +Error (RR/min)20 +Respiration Rate (bpm) +15 +10 +Target person +Target person +5 +leaves +comes back +0 +-5 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Time (s)(a) various orientations +(b) different distances. +Figure 11: effectiveness of the attack for different orienta- +tion and distance of the targeted WiFi device respect to the +person. +Orientation: We evaluate the effect of orientation of the target +person with respect to the target device (laptop). We run the same +attack as before for different orientations (i.e. sitting in front, back, +left, and right side of a laptop). The user is 0.5m away from the target +device in all cases. Figure 11a shows the result of this experiment. +Each bar shows the average accuracy for 90 measurements. Our +result shows that regardless of the orientation of the person with +respect to the device, the attack is effective and detects the breathing +rate of the person accurately. In particular, even when the person +was behind the target device, the attack still detects the breathing +rate with 99% accuracy. +Distance: Here, we are interested to find out what the maximum +distance between the target device and the person can be while +the attacker still detects the person’s breathing rate. To do so, we +place the attacker device and the target device 5 meters apart in +two different rooms with a wall in between. We then run different +experiments in which the target person stays at different distances +from the target device. In each experiment, we measure the breath- +ing rate for two minutes and calculate the average breathing rate +over this time. Finally, we compare the estimated breathing rate to +the ground truth and calculate the accuracy as mentioned before. +Figure 11b shows the results of this experiment. The accuracy +is over 99% when the distance between the target device and the +target person is less than 60 cm. Note, in reality, people have their +laptops or cellphone very close to themselves most of the time, and +60 cm is representative of these situations. The accuracy drops as +we increase the distance. However, even when the device is at 1.4 m +from the person’s body, the attack can still estimate the breathing +rate with 80% accuracy. Note, this is the accuracy in finding the +absolute breathing rate and the change in the breathing rate can be +detected with much higher accuracy. Finally, the figure shows that +the accuracy suddenly drops to zero for a distance beyond 1.4 m. +This is due to the fact that at that distance the power of the peak +at the output of the FFT goes below the noise floor, and hence, the +peak is not detectable. +5.2.4 +Effect of Multiple People. Last, we evaluate if the attack can +be used to detect the breathing rate of multiple people simultane- +ously. We test our attack in three different scenarios. In the first +scenario, two people are near the laptop while one is working on +the laptop and the other is just sitting next to him, as shown in +Figure 12a. The attacker targets the laptop and tries to estimate +their breathing rate. Note, that the attacker has no prior informa- +tion about how many people are next to the laptop. In the second +scenario, we repeat the same experiment as the first scenario except +that the second person is sitting behind the laptop, as shown in +Figure 12b. In the third scenario, there are two people in the same +space but each person is next to a different device. The attacker +targets the laptops and tries to estimate their breathing rates. In +these experiments, the target device is 0.5-0.7 m away from the +person. +Figure 12c shows the results for this evaluation. The blue bars +show the result for the first person who is working on the laptop, +and the red bars show the results for the second person. Our results +show that the attack effectively detects the breathing rate of both +people regardless of their orientation. However, the accuracy in +detecting the breathing rate for the second person is a bit lower than +the first person for the first and second scenarios. This is because +the second person’s distance to the target device is slightly more +and hence the accuracy has decreased. +6 +SECURITY IMPLICATION: BATTERY +DRAIN ATTACK +In this section, we show how an adversary can drain the battery +of our WiFi devices by using the above loopholes and forcing our +WiFi devices to stay awake and continuously transmit WiFi signals. +6.1 +Attack Design and Setup +6.1.1 +Attack Design. The attacker forces the target device to stay +awake and continuously transmit WiFi packets by sending it back- +to-back fake frames and some periodic fake beacons. However, to +maximize the amount of time the target device spends transmitting, +we study a few different types of fake query packets that the attacker +can send. Note, that the power consumption of transmission is +typically higher than that of reception.2 Hence, to maximize the +battery drain, we want to send a short query packet and receive a +long response. +Table 3 lists some query packets and their corresponding re- +sponses. The best choice for a query packet is Block ACK requests +since the target will respond with a Block ACK that is larger than +other query responses. Another important factor to consider for +maximizing the battery drain is the bitrate. When the bitrate of the +query packet increases, the bitrate of the response will also increase +as specified in the IEEE 802.11 standard. Hence, at first glance, it +2For example, ESP8266 [26] and ESP32 [25] WiFi modules draw 50 and 100 mA when +receiving while they draw 170 and 240 mA when transmitting. These low-power WiFi +modules are very popular for IoT devices [10]. +8 + +99.91% +99.73% +99.72% +99.91% +100 +Accuracy (%) +80 +60 +40 +20 +0 +Front +Back +Left +Right +Orientation100 +Accuracy (%) +80 +60 +40 +20 +0 +0 +20 +40 +60 +80 +100 +120 +140 +160 +Distance (cm)(a) Scenario 1 +(b) Scenario 2 +(c) Breathing Rate Estimation of two persons +Figure 12: Accuracy under three different scenarios: Scenario 1: two people sit side-by-side in front of the target device; Scenario +2: one person sits in front of the target device, the other one sits behind the target device; Scenario 3: two people sit in front +of two target devices, respectively. Attacker attacks one by one. +Query +Query size +Response +Response size +Null +28 bytes +ACK +14 bytes +RTS +20 bytes +CTS +14 bytes +BAR +24 bytes +BA +32 bytes +Table 3: Different types of fake queries and their responses. +Note, Null is a data packet without any payload. BAR and BA +stand for Block ACK Request, and Block ACK, respectivly. +may seem that to maximize the battery drain, the attacker must +use the fastest bitrate possible to transmit query packets, forcing +the target device to transmit as many responses as possible. How- +ever, it turns out that this is not the case. The power consumption +depends mostly on the amount of time the target device spends +transmitting packets. Hence, when a higher rate is used for the +query and response packets, the total time the target spends on +transmission does not increase. In fact, the total time spent trans- +mitting decreases mainly due to overheads such as channel sensing +and backoffs. For example, if we increase the bitrate by 6 times (i.e., +from 1 Mbps to 6 Mbps), the number of packets will increase by +only 3.3 times. As a result, to maximize the transmission time of the +target device, the attacker should use the lowest rate (i.e., 1 Mbps) +for the query packet. +6.1.2 +Attack Setup. +Attacking device: Any WiFi card capable of packet injection can +be used as the attacking device. We use a USB WiFi card connected +to a laptop running Ubuntu 20.04. The WiFi card has an RTL8812AU +chipset [5] that supports IEEE 802.11 a/b/g/n/ac standards. We have +installed the aircrack-ng/rtl8812au driver [1] for this card which +enables robust packet injection. We utilize the Scapy [37] library to +inject fake WiFi packets to the target device. Scapy allows defin- +ing customized packets and multiple options for packet injection. +Since we need to inject many packets in this attack, we use the +sendpfast function to inject packets at high rates. sendpfast relies +on tcpreplay [6] for high performance packet injection. +Target device: Any WiFi-based IoT device can be used as a target. +We choose Amazon Ring Spotlight Cam Battery HD Security Cam- +era [2] for our battery drain experiments. The camera is powered +by a custom 6040 mAh lithium-ion battery. The battery life of this +camera is estimated to be between 6 and 12 months under normal +usage [3, 4]. We leave the camera settings to their defaults which +means most power-consuming options are turned off. This assures +that our measurements will be an upper bound on the battery life +and hence the attack might drain the battery much faster in the real +world. Authors in [41] pointed out the possibility of a battery drain- +ing attack by forging beacon frames. However, they did not provide +any evaluations to test this idea. Moreover, we show how sending +fake packets in addition to fake beacon frames can significantly +increase the power consumption on the victim device. +6.2 +Results +We evaluate the effectiveness of the battery drain attack in terms +of range and using different payload configuration. +6.2.1 +Finding the optimal configuration: As discussed in 6.1.1, send- +ing block ACK requests at the lowest bitrate (i.e., 1 Mbps) should +maximize the power consumption of the target device. To verify +this, we have conducted a series of experiments with different types +of query packets and transmission bitrates. In each experiment, we +continuously transmit query packets to the Ring security camera. +In all experiments, we start with a fully charged battery and the +attacker injects query packets as fast as possible. +Figure 13 (a) shows the maximum number of packets the attacker +could transmit to the target device, and the number of responses +it receives per second. Figure 13 (b) shows the amount of energy +drawn from the battery during one hour of the attack. As expected, +sending Block ACK Requests (BAR) drains more energy from the +battery since the target device spends more time on transmission +than receiving. Moreover, the results verify that although increas- +ing the data rate from 1Mbps to 6Mbps (BAR/1 versus BAR/6) +increases the number of responses, it decreases the energy drained. +As mentioned before, this is because the total time spent transmit- +ting decreases mainly due to overheads such as channel sensing +9 + +D100% +99.48% +100% 99.07% +100 +86.67% +82.05% +Accuracy (%) +80 +60 +40 +20 +0 +Scenario 1 +Scenario 2 +Scenario 3Battery Type +Voltage (V) +Full Capacity (Wh) +100% Drain (min) +25% Drain (min) +CR2032 coin +3.0 +0.68 +14 +3.5 +AAA +1.5 +1.87 +39 +10 +AA +1.5 +4.20 +90 +22 +Table 4: The time it takes for the attack to drain different types of batteries + 0 + 500 + 1000 + 1500 + 2000 + 2500 + 3000 + 3500 +Null/1 +Data/1 +BAR/1 +BAR/6 +Number of Packets +Configurations +Attacker's packets +Target's responses +(a) + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 +Null/1 +Data/1 +BAR/1 +BAR/6 +Watt Hour +Configurations +(b) +Figure 13: The figure shows (a) Average number of packets +sent to and received from the target device. (b) Energy con- +sumption in Watt Hour measured under different configu- +rations (i.e. packet type / bitrate (Mbps) +and backoffs. This result confirms that sending block ACK requests +(BAR) with the lowest datarate is the best option to drain the battery +of the target device. +6.2.2 +Battery drain with optimal configurations. We use the best +setting which is a block ACK request (BAR) query transmitted at +1 Mbps to fully drain the battery of the Ring security camera. We +are able to drain a fully charged battery in 36 hours. Considering +the fact that the typical battery life of this camera is 6 to 12 months, +our attack reduces the battery life by 120 to 240 times! It is worth +mentioning that since a typical user charges the battery every 6-12 +months, on average the batteries are at 40-60%, and therefore it +would take much less for our attack to kill the battery. Moreover, the +RING security camera is using a very large battery, most security +sensors are using smaller batteries. Table 4 shows the amount of +time it takes to drain different batteries. For example, it takes less +than 40 mins to kill a fully charged AAA battery which is a common +battery in many sensors. +6.2.3 +Range of WiFi battery draining attack. A key factor in the +effectiveness of the battery draining attack is how far the attacker +can be from the victim’s device and still be able to carry on the +attack. If the attack can be done from far away, it becomes more +threatening. To evaluate the range of this attack, we design an +experiment in which the attacker transmits packets to the target +from different distances and we measure what percentage of the +attacker’s packets are responded to by the target device. We use +a realistic testbed. The Ring security camera is installed in front +of a house, and the attacker is placed in a car, parked at different +locations on the street. We test the attack at 10 different locations +up to 150 meters away from the target device. Figure 14 shows +these locations and our setup. Each yellow circle represents each +of the locations tested at. The numbers inside the circles show the +percentage of the attacker’s packets responded to by the camera. +Each number is an average of over 60 one-second measurements. +The closest distance is about 5 meters when we park the car in front +of the target house. In this location 97% of the attacker’s packets are +responded to. We conducted other experiments within 10 meters +of the target (not shown here) and we obtained similar results. Our +results show that even within a distance of 100 meters, almost all +attacker’s packets are responded to by the victim’s device. In some +locations such as the rightmost circle (at 150 meters away), we +could still achieve a reply rate as high as 73%, confirming our attack +works even at that distance. The reason for achieving such a long +range is that the attacker transmits at a 1 Mbps bitrate which uses +extremely robust modulation and coding rate (i.e. BPSK modulation +and a 1/11 coding rate). +7 +ETHICAL CONSIDERATIONS +We discussed our project and experiments with our institutions’ +IRB office and they determined that no IRB review nor IRB approval +is required. Moreover, the house and WiFi devices used in most +experiments are owned and controlled by the authors. Finally, in +order to expedite mitigating the attacks presented in this paper, +we have started engagements with WiFi access point and chipset +manufacturers. +8 +CONCLUSION +In this work, we identify two loopholes in the WiFi protocol and +demonstrate their possible privacy and security threats. 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Zero-Effort Cross-Domain Gesture Recognition with Wi-Fi +(MobiSys). 313–325. +[49] Yanzi Zhu, Zhujun Xiao, Yuxin Chen, Zhijing Li, Max Liu, Ben Y Zhao, and Haitao +Zheng. 2020. Et Tu Alexa? When Commodity WiFi Devices Turn into Adversarial +Motion Sensors. In Network and Distributed Systems Security (NDSS) Symposium +2020. +12 + diff --git a/19AyT4oBgHgl3EQfbvd_/content/tmp_files/load_file.txt b/19AyT4oBgHgl3EQfbvd_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9abdb78c29c818080e92db75cad789337192356e --- /dev/null +++ b/19AyT4oBgHgl3EQfbvd_/content/tmp_files/load_file.txt @@ -0,0 +1,870 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf,len=869 +page_content='WiFi Physical Layer Stays Awake and Responds When it Should Not Ali Abedi Stanford University USA abedi@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='edu Haofan Lu UCLA USA haofan@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='edu Alex Chen University of Waterloo Canada zihanchen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='ca@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='com Charlie Liu University of Waterloo Canada charlie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='liu@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='ca Omid Abari UCLA USA omid@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='edu ABSTRACT WiFi communication should be possible only between devices in- side the same network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' However, we find that all existing WiFi devices send back acknowledgments (ACK) to even fake packets received from unauthorized WiFi devices outside of their network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Moreover, we find that an unauthorized device can manipulate the power-saving mechanism of WiFi radios and keep them continu- ously awake by sending specific fake beacon frames to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Our evaluation of over 5,000 devices from 186 vendors confirms that these are widespread issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We believe these loopholes cannot be prevented, and hence they create privacy and security concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Finally, to show the importance of these issues and their conse- quences, we implement and demonstrate two attacks where an adversary performs battery drain and WiFi sensing attacks just using a tiny WiFi module which costs less than ten dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 1 INTRODUCITON Today’s WiFi networks use advanced authentication and encryption mechanisms (such as WPA3) to protect our privacy and security by stopping unauthorized devices from accessing our devices and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Despite all these mechanisms, WiFi networks remain vulner- able to attacks mainly due to their physical layer behaviors and requirements defined by WiFi standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In this paper, we find two loopholes in the IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 standard for the first time and show how they can put our privacy and security at risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' a) WiFi radios respond when they should not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In a WiFi network, when a device sends a packet to another device, the re- ceiving device sends an acknowledgment back to the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In particular, upon receiving a frame, the receiver calculates the cyclic redundancy check (CRC) of the packet in the physical layer to detect possible errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' If it passes CRC, then the receiver sends an Acknowledgment (ACK) to the transmitter to notify the correct reception of the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Surprisingly, we have found that all existing WiFi devices send back ACKs to even fake packets received from unauthorized WiFi devices outside of their network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Why should a WiFi device respond to a fake packet from an unauthorized device?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' b) WiFi radios stay awake when they should not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' WiFi chipsets are mostly in sleep mode to save power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' However, to make sure that they do not miss their incoming packets, they notify their WiFi access point before entering sleep mode so that the access point buffers any incoming packets for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Then, WiFi devices wake up periodically to receive beacon frames sent by the associated access point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In regular operation, only the access point sends beacon frames to notify the devices that have buffered packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' When a device is notified, it stays awake to receive them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' However, these beacon frames are not encrypted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Hence, we find that an unautho- rized user can forge those beacon frames to keep a specific device awake for receiving the (non-existent) buffered frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We examine these behaviors and loopholes in detail over dif- ferent WiFi chipsets from different vendors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Our examination of over 5,000 WiFi devices from 186 vendors shows that these are widespread issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We then study the root cause of these issues and show that, unfortunately, they cannot be fixed by a simple solution such as updating WiFi chipsets firmware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Finally, we im- plement and demonstrate two attacks based on these loopholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In the first attack, we show that by forcing WiFi devices to stay awake and continuously transmit, an adversary can continuously analyze the signal and extract personal information such as the breathing rate of the WiFi users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In the second attack, we show that by forcing WiFi devices to stay awake and continuously transmit, the adversary can quickly drain the battery, and hence disable WiFi devices such as home and office security sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' These attacks can be performed from outside buildings despite the WiFi network and devices being completely secured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' All the attacker needs is a $10 microcontroller with integrated WiFi (such as ESP32) and a battery bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The attacker device can easily be carried in a pocket or hidden somewhere near the target building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The main contributions of this work are: We find that WiFi devices respond to fake 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 frames with ACK, even when they are from unauthorized devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We also find that WiFi radios can be kept awake by sending them fake beacon frames indicating they have packets waiting for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We study these loopholes and their root causes in detail, and have tested more than 5,000 WiFi access points and client devices from more than 186 vendors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We implement two attacks based on these loopholes using just a 10-dollar off-the-shelf WiFi module and validate them in real-world settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 2 RELATED WORK The loopholes we present in this paper are explored using packet injection, in which an attacker sends fake WiFi packets to devices in a secured WiFi network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Packet injection has been used in the past arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='00269v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='NI] 31 Dec 2022 to perform various types of attacks against WiFi networks such as denial of service attacks for a particular client device or total dis- ruption of the network [14, 15, 17, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' These attacks use different approaches such as beacon stuffing to send false information to WiFi devices [21, 46], or Traffic Indication Map (TIM) forgery to prevent clients from receiving data [18, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' However, all of these attacks focus on spoofing 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 MAC-layer management frames to interrupt the normal operation of WiFi networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' To provide a countermeasure for some of these attacks, the 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11w standard [7] introduces a protected management frame that prevents attack- ers from spoofing 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 management frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Instead of spoofing 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 MAC frames, we exploit properties of the 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 physical layer to force a device to stay awake and respond when it should not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' These loopholes open the door to multiple research avenues including new security and privacy threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' WiFi sensing attack: Over the past decade, there has been a significant amount of research on WiFi sensing where WiFi signals are used to detect human activities [13, 32, 34–36, 38, 43–45, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' However, these systems target applications with social benefits and cannot be easily used by an attacker to create privacy and security threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' This is because either these techniques require cooperation from the target WiFi device or the attacker needs to be very close to the target to use these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' A recent study shows that by capturing WiFi signals coming out of a private building, it is possible for an adversary to track user movements inside that building [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' However, this attack has a bootstrapping stage which requires the attacker to walk around the target building for a long time to find the location of the WiFi devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Furthermore, since this work relies on only the normal intermittent WiFi activities, it cannot capture continuous data such as breathing rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Battery draining attack: Battery draining attacks date back to 1999 [40] and there have been many studies on such attacks and potential defense mechanisms since then [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Battery discharge models and energy vulnerability due to operating systems have been investigated [30, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' A more recent study plays multimedia files implicitly to increase power consumption during web browsing [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In terms of defending, a monitoring agent that searches for abnormal current draw is discussed in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In contrast, our attack exploits the loopholes in the 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 physical layer protocol and the power-hungry WiFi transmission to quickly drain a target device’s battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We will discuss in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2 that stopping our proposed attack is nearly impossible on today’s WiFi devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' This paper is an extension of our previous workshop publica- tion [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The workshop paper shows preliminary results for our finding that WiFi devices respond with ACKs to packets received from outside of their network, and provides a brief discussion on potential privacy and security concerns of this behavior without studying them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We have also explored how the WiFi power saving mechanism can be exploited to keep a target device awake in a localization attack [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In this paper, we provide an in-depth study of these previously discovered loopholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We also design and per- form two privacy and security attacks, based on these loopholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Finally, we implement these attacks on off-the-shelve WiFi devices and present detailed performance evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 1: WiFi devices send an ACK for any frame they re- ceive without checking if the frame is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 2: Frames exchanged between attacker and victim 3 WIFI RESPONDS WHEN IT SHOULD NOT Most networks use security protocols to prevent unauthorized de- vices from communicating with their devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Therefore, one may assume that a WiFi device only acknowledges frames received from the associated access point or other devices in the same network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' However, we have found that all today’s WiFi devices acknowledge even the frames they receive from an unauthorized device from outside of their network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In particular, as long as the destination address matches their MAC address, their physical layer acknowl- edges it, even if the frame has no valid payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In this section, we examine this behavior in more detail, and explain why this problem happens and why it is not preventable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' To better understand this behavior, we run an experiment where we use two WiFi devices to act as a victim and an attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The attacker sends fake WiFi packets to the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We monitor the real traffic between the attacker and the victim’s device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Setup: For the victim, we use a tablet, and for the attacker, we use a USB WiFi dongle that has a Realtek RTL8812AU 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11ac chipset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' This is a $12 commodity WiFi device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The attacker uses this device to send fake frames to the victim’s device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' To do so, we develop a python program that uses the Scapy library [37] to create fake frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Scapy is a python-based framework that can generate arbitrary frames with custom data in the header fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Note, that the only valid information in the frame is the destination MAC address (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=', the victim’s MAC address).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The transmitter MAC address is set to a fake MAC address (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=', aa:bb:bb:bb:bb:bb), and the frame has no payload (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=', null frame) and is not encrypted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Result: Figure 2 shows the real traffic between the attacker and the victim device captured using Wireshark packet sniffer [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' As can be seen, when the attacker sends a fake frame to the victim, the vic- tim sends back an ACK to the fake MAC address (aa:bb:bb:bb:bb:bb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' This experiment confirms that WiFi devices acknowledge frames without checking their validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Finally, to see if this behavior exists on other WiFi devices, we have repeated this test with a variety of devices (such as laptops, smart thermostats, tablets, smartphones, and access points) with different WiFi chipsets from different ven- dors, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Note, target devices are connected to a private network and the attacker does not have their secret key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' After performing the same experiment as before, we found that all 2 Private WiFi Network Acknowledgement Fake 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 Data Frame Access Point Target AttackerSource Destination Info aa:bb:bb:bb:bb:bb f2:6e:0b: Null function(No data), aa:bb:bb:bb:bb:bb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='Acknowledqement,Flaqs=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='Device WiFi module Standard MSI GE62 laptop Intel AC 3160 11ac Ecobee3 thermostat Atheros 11n Surface Pro 2017 Marvel 88W8897 11ac Samsung Galaxy S8 Murata KM5D18098 11ac Google Wifi AP Qualcomm IPQ 4019 11ac Table 1: List of tested chipsets/devices of these devices also respond to fake packets received from a device outside of their network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='1 How widespread is this loophole?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In the previous section, we examined a few different WiFi devices and showed that they are all responding to fake frames from unau- thorized devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Here, we examine thousands of devices to see how widespread this behavior is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In the following, we explain the setup and results of this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Setup: To examine thousands of devices, we mounted a WiFi dongle on the roof of a vehicle and drove around the city to test all nearby devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' For the WiFi dongle, we use the same Realtek RTL8812AU USB WiFi dongle, and connect it to a Microsoft Surface, running Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We develop a multi-threaded program using the Scapy library [37] to discover nearby devices, send fake 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 frames to the discovered devices, and verify that target devices re- spond to our fake frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Specifically, our implementation contains three threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The first thread discovers nearby devices by sniffing WiFi traffic and adding the MAC address of unseen devices to a target list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The second thread sends fake 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 frames to the list of target devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Finally, the third thread checks to verify that target devices respond with an ACK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Results: We perform this experiment for one hour while driving around the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In total, we discovered 5,328 WiFi nodes from 186 vendors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The list includes 1,523 different WiFi client devices from 147 vendors and 3,805 access points from 94 vendors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Table 2 shows the top 20 vendors for WiFi devices and WiFi access points in terms of the number of devices discovered in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The list includes devices from major smartphone manufacturers (such as Apple, Google, and Samsung) and major IoT vendors (such as Nest, Google, Amazon, and Ecobee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We found that all 5,328 WiFi Access Points and devices responded to our fake 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 frames with an acknowledgment, and hence we infer that most probably all of today’s WiFi devices and access points respond to fake frames when they should not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2 Can this loophole be fixed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' So far, we have demonstrated that all existing WiFi devices respond to fake packets received from unauthorized WiFi devices outside of their network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Now, the next question is why this behavior exists, and if it can be prevented in future WiFi chipsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In a WiFi device, when the physical layer receives a frame, it checks the correctness of the frame using error-checking mech- anisms (such as CRC) and transmits an ACK if the frame has no error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' However, checking the validity of the content of a frame is WiFi Client Device WiFi Access Point Vendor # devices Vendor # devices Apple 143 Hitron 723 Google 102 Sagemcom 601 Intel 66 Technicolor 410 Hitron 65 eero 195 HP 63 Extreme N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 188 Samsung 56 Cisco 156 Espressif 47 HP 104 Hon Hai 46 TP-LINK 101 Amazon 41 Google 80 Sagemcom 38 D-Link 75 Liteon 33 NETGEAR 69 AzureWave 30 ASUSTek 51 Sonos 30 Aruba 46 Nest Labs 27 SmartRG, 44 Murata 24 Ubiquiti N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 35 Belkin 20 Zebra 35 TP-LINK 20 Pegatron 28 Cisco 16 Belkin 25 ecobee 13 Mitsumi 25 Microsoft 13 Apple 19 Others 630 Others 789 Total 1523 Total 3805 Table 2: List of WiFi devices and APs that respond to our fake 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' performed by the MAC and higher layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Unfortunately, this sepa- ration of responsibilities and the fact that the physical layer does not coordinate with higher layers about sending ACKs seem to be the root cause of the behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In particular, we have observed that when some access points receive fake frames, they start sending deauthentication frames to the attacker, requesting it to leave the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' These access points detect the attacker as a “malfunc- tioning” device and that is why they send deauthentication frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Surprisingly, although the access points have detected that they are receiving fake frames from a “malfunctioning” device, we found that they still acknowledge the fake frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' An example traffic that demonstrates this behavior is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' As can be seen, although the access point has already sent three deauthentication frames to the attacker, it still acknowledges the attacker’s fake frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We then manually blocked the attacker’s fake MAC address on the access point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Surprisingly, we observed that the AP still acknowledges the fake frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' These observations verify that sending ACK frames happens automatically in the physi- cal layer without any communication with higher layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Therefore, the software running on the access points does not prevent the physical layer from sending ACKs to fake frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The next question is why the software running on WiFi devices does not prevent this behavior by verifying if the frame is legitimate before sending an ACK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Unfortunately, this is not possible due to the WiFi standard timing requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Specifically, in the IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 standard, upon receiving a frame, an ACK must be transmitted 3 Figure 3: The attacked access point detects that something strange is happening, however it still ACKs fake frames by the end of the Short Interframe Space (SIFS)1 interval which is 10 𝜇s and 16 𝜇s for the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='4 GHz and 5 GHz bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' If the transmitter does not receive an ACK by the end of SIFS, it assumes that the frame has been lost and retransmits the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Therefore, the WiFi device nefeds to verify the validity of the received frame in less than 10 𝜇𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' This verification must be done by decoding the frame using the secret shared key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Unfortunately, decoding a frame in such a short period is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In particular, past work has shown that the time required to decode a frame is between 200 to 700 𝜇𝑠 when the WPA2 security protocol is used [31, 33, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' This processing time is orders of magnitude longer than SIFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Hence, existing devices cannot verify the validity of the frame before sending the ACK, and they acknowledge a frame as long as it passes the error detection check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' One potential approach to solve this loophole is to implement the security decoder in WiFi hardware instead of software to significantly speed up its delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Although this may solve the problem in future WiFi chipsets, it will not fix the problem in billions of WiFi chipsets which are already deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 4 WIFI STAYS AWAKE WHEN IT SHOULD NOT We have also found a loophole that allows an unauthorized device to keep a WiFi device awake all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' One may think that a WiFi device can be kept awake by just sending fake back-to-back packets to it and forcing it to transmit acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' However, this approach does not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Most WiFi radios go to sleep mode to save energy during inactive states such as screen lock, during which the attacker is not able to keep them awake by sending back- to-back packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 4a show the results of an experiment where the attacker is continuously transmitting fake packets to a WiFi device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In this figure, we plot the amplitude of CSI over time for the ACK packets received from the WiFi device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' As can be seen, the responses are sparse and discontinued even when the attacker sends back-to-back packets to the WiFi device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' This is because the WiFi device goes to sleep mode frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' However, we have found a loophole in the power saving mechanism of WiFi devices which can be used by an unauthorized device to keep any WiFi device awake all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 1The SIFS is used in the 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 standard to give the receiver time to go through different procedures before it is ready to send the ACK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' These procedures include Physical-layer and MAC-layer header processing, creating the waveform for the ACK, and switching the RF circuit from receiving to transmitting mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' (a) Without fake beacon frames (b) With fake beacon frames Figure 4: The CSI amplitude of ACKs responded by the tar- get device when an attacker sends back-to-back fake packets to it in two scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' (a) In this scenario, the attacker is not using fake beacon frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Therefore, the target device goes to sleep mode frequently and does not respond to fake pack- ets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' (b) In this scenario, the attacker infrequently sends fake beacon frames to keep the target device awake all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='1 How does WiFi power saving mechanism work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Wireless tranceivers are very power-hungry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Therefore, WiFi radios spend most of the time in the sleep mode to save power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' When a WiFi radio is in sleep mode, it cannot send or receive WiFi packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' To avoid missing any incoming packets, when a WiFi device wants to enter the sleep mode it notifies the WiFi access point so that the access point buffers any incoming packets for this device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' WiFi devices, however, wake up periodically to receive beacon frames to find out if packets are waiting for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In particular, WiFi access points broadcast beacon frames periodically which includes a Traffic Indication Map (TIM) field that indicates which devices have buffered packets on the access point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' For example, if the association ID of a WiFi device is 𝑥, then the (𝑥 + 1)𝑡ℎ bit of TIM is assigned to that device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Finally, when a device is notified that has some buffered packets on the access point, it stays awake and replies with a Null- function packet with a power management bit set to "0".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In this way, the WiFi device informs the access point it is awake and ready to receive packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2 How can one manipulate power saving?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We have found that an unauthorized device can use the power- saving mechanism of WiFi devices to force them to stay awake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In particular, an attacker can pretend to be the access point and broadcasts fake beacon frames indicating that the WiFi device has buffered traffic, forcing them to stay awake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' However, this requires the attacker to know the MAC address and the SSID of the network’s access point, as well as the association ID and MAC address of the targeted device so that it can set the correct bit in TIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The access point MAC address and SSID can be easily discovered by sniffing 4 Source Destination Info f2:6e:0b: aa:bb:bb:bb:bb:bb Deauthentication, SN=3275 f2:6e:0b: aa:bb:bb:bb:bb:bb Deauthentication, SN=3275 f2:6e:0b: aa:bb:bb:bb:bb:bb Deauthentication, SN=3275 aa:bb:bb:bb:bb:bb f2:6e:0b: Null function (No data), aa:bb:bb:bb:bb:bb Acknowledgement, Flags=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='. f2:6e:0b: aa:bb:bb:bb:bb:bb Deauthentication, SN=3281 f2:6e:0b: aa:bb:bb:bb:bb:bb Deauthentication, SN=328125 CSI Amplitude 20 15 10 5 0 0 5 10 15 20 25 30 Time (s)25 CSI Amplitude 20 15 10 5 0 0 5 10 15 20 25 30 Time (s)Figure 5: WiFi devices stay awake on hearing a forged bea- con frame with TIM flags set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' the WiFi traffic using software such as Wireshark since the MAC address is never encrypted and all nodes send packets to the access point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Note that MAC randomization does not cause any problem for this process because the attacker finds the randomized MAC address that is currently being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Next, the attacker pretends to be the access point and broadcasts fake beacon frames with TIM set to "0xFF", indicating all client devices have buffered traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Then, it enters the sniffing mode to sniff for the Null-function packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The null-function packets contain the ID and MAC addresses of all WiFi devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' To avoid keeping all WiFi devices awake, we find that one can send a fake beacon frame as a unicast packet, instead of the usual broadcast beacons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' This way only the target device receives the packet and we do not interfere with the operation of other devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Interestingly, our experiments show that target devices do not care if they receive beacons as broadcast or unicast frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' To better understand this behavior, we run an experiment where we use two WiFi devices to act as a victim and an attacker, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The attacker sends fake WiFi packets to the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We monitor the real traffic between the attacker and the victim’s device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Setup: Similar to the experiment described in Section 3, we use an RTL8812AU USB dongle to inject fake packets to a smartphone held by a person who is watching YouTube on the phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The distance between the smartphone and the user is about 60 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The attacking device and the victim are in two separate rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The attacker also uses an ESP32 WiFi module to record the Channel State Information (CSI) of received ACKs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Result: We find that although sending fake beacon frames keeps the target device awake, sending them very frequently will cause WiFi devices to recognize the suspicious attacker’s behavior and disconnect from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Therefore, to keep the WiFi device awake, in- stead of just sending beacon frames back-to-back, the attacker can continuously transmit normal fake packets to a WiFi device and periodically sends fake beacon frames to keep it awake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 4b shows the result of an experiment where the attacker is continu- ously transmitting fake packets to a WiFi device and periodically sends fake beacon frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' As it can be seen, the target device is continuously awake and responding to fake packets with ACKs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 5 PRIVACY IMPLICATION: WIFI SENSING ATTACK Recently, there has been a significant amount of work on WiFi sensing technologies that use WiFi signals to detect events such as motion, gesture, and breathing rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In this section, we show how an adversary can combine WiFi sensing techniques with the above loopholes to monitor people’s breathing rate whenever she/he wants from outside buildings despite the WiFi network and de- vices being completely secured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In particular, an adversary can force our WiFi devices to stay awake and continuously transmit WiFi signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Then she/he can continuously analyze our signals and extract information such as our breathing rate and presents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Note, since most of the time, we are close to a WiFi device (such as a smartwatch, laptop, or tablet), our body will change the ampli- tude and phase of the signals which can be easily extracted by the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='1 Attack Design, Scenarios and Setup 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='1 Attack Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The attacker sends fake packets to a WiFi device in the target property and pushes it to transmit ACK packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In particular, since an adult’s normal breathing rate is around 12 -20 times per minute (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='33Hz), receiving several ACK packets per second is sufficient for the attacker to estimate the breathing rate, without impacting the performance of the target WiFi network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The attacker then takes the Fourier transform of the CSI information of ACK packets to estimate the breathing rate of the person who is nearby the WiFi device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' However, due to the random delays of the WiFi random access protocol and the operating system’s scheduling protocol, the collected data samples are not uniformly spaced in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Hence, the attacker cannot simply use standard FFT to estimate the breathing rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Instead, they need to use a non- uniform Fourier transform, and a voting algorithm to extract the breathing rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The Non-Uniform Fast Fourier Transform (NUFFT) algorithm 1 used is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Algorithm 1: Non-uniform FFT Data: Time indices 𝑡, data samples 𝑥 of length 𝑛 Result: Magnitude of each frequency component 𝑑 ← min𝑖 (𝑡𝑖 − 𝑡𝑖−1) 𝑖 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=',𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' for 𝑖 ← 1 to 𝑛 − 1 do 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 ← 𝑡 [𝑖] − 𝑡 [𝑖 − 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' if 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 > 𝑑 then 𝑐𝑜𝑢𝑛𝑡 ← ⌊𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙/𝑑⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Interpolation(𝑡, 𝑥, 𝑡 [𝑖], 𝑡 [𝑖 − 1], 𝑐𝑜𝑢𝑛𝑡);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' end end return FFT(𝑡, 𝑥) The algorithm first finds the minimum time gap between any two adjacent data points 𝑑, then linearly interpolates any interval that is larger than the gap with ⌊𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙/𝑑⌋$ samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Finally, it uses a regular FFT algorithm to find the magnitude of each frequency component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' A low-pass filter is applied before feeding data to the FFT analysis to reduce noise (not shown in the algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 6(a) and 6(b) show the amplitude of CSI before and after interpolation, respectively, when the attacker sends 10 packets per second to a WiFi device that is close to the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Each figure shows both the original data (in blue) and the filtered data (in orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 6(c) shows the frequency spectrum of the same signals when a standard FFT or our non-uniform FFT is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' A prominent peak at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='3Hz is shown in the non-uniform FFT spectrum, indicating a breathing rate of 18 bpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 5 Private WiFi Network Stay Awake Fake 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 Beacon Access Point Target Attacker(a) Raw and filtered data before interpolation (b) Raw and filtered data after interpolation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='0 Frequency (Hz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='00 Power non_uniform_fft standard_fft (c) Standard FFT and a non-uniform FFT of Data Figure 6: Steps to extract breathing rate from the CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' WiFi CSI gives us the amplitude of 52 subcarriers per packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We observed that these subcarriers are not equally sensitive to the motion of the chest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Besides, a subcarrier’s sensitivity may vary depending on the surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' For a more reliable attack, the attacker should identify the most sensitive subcarriers over a sampling window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Previously proposed voting mechanisms for coarse-grained motion detection applications [8, 16, 29, 49] cannot be directly applied in this situation, as chest motion during respiration is at a much smaller scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Instead, we developed a soft voting mechanism, where each subcarrier gives a weighted vote to a breathing rate value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The breathing rate that gets the most votes is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Specifically, We first find the power of the highest peak (𝑃𝑝𝑒𝑎𝑘), and then calculate the average power of the rest bins (𝑃𝑎𝑣𝑒).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The exponent of the Peak-to-Average Ratio (PAR): 𝑒 𝑓𝑝𝑒𝑎𝑘 𝑓𝑎𝑣𝑒 is used as the weight of the corresponding subcarrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In this way, we guarantee the subcarriers with higher SNR have significantly more votes than the rest of the subcarriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2 Attack Scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We evaluate the WiFi sensing attack in different scenarios, both indoor and outdoor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In the indoor scenario, the attacker and the target are placed in the same building but on different floors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The height of one floor in the building is around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='8 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' This scenario is similar to when the attacker and the target person are in different units of an apartment or townhouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In the outdoor scenario, the attacker is outside the target’s house.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' For the outdoor experiments, We place the attacker in another building which is around 20 m away from the target building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In all of the experiments, the target WiFi devices are placed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='5 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='4 m away from the person’s body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The person is either watching a movie, typing on a laptop, or surfing the web using his cell phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' During the experiments, other people are walking and living normally in the house.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Finally, we run the attack and compare the estimated breathing rate with the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' To obtain the ground truth, we record the target person’s breathing sound by attaching a mi- crophone near his/her mouth [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We then calculate the FFT on the sound signal to measure the breathing frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Note that the attack does not need this information and this is just to obtain the ground truth in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='3 Attacker Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Hardware Setup: The attacker uses a Linksys AE6000 WiFi card and an ESP32 WiFi module [25] as the attacking device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Both devices are connected to a ThinkPad laptop via USB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The Linksys AE6000 is used to send fake packets and the ESP32 WiFi module is used to receive acknowledgments (ACK) and extract CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Although we use two different devices for sending and receiv- ing, one can simply use an ESP32 WiFi module for both purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The use of two separate modules gave us more flexibility in run- ning many experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' As for the target device, we use a One Plus 8T smartphone without any software or hardware modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We have also tested our attack on an unmodified Lenovo laptop, a Microsoft Surface Pro 4 laptop, and a USB WiFi card as the target device and we obtained similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' It is worth mentioning that any WiFi device can be a target without any software or hardware modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Software Setup: We have implemented the CSI collecting script on the ESP32 WiFi module, and the breathing rate estimation algo- rithm on the laptop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The collected CSI data is fed to the algorithm which produces the breathing rate estimation values in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' To process this data in real time, a sliding window (buffer) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The size of the window is 30 s and the stride step is 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 30 seconds is a large enough window for estimating a stable breathing rate value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Note that an adult breathes around 6 times during such a window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The window is a queue of data points, and it updates every second by including 1 second of new data points to its head and removing 1 second of old data points from its tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The breathing rate estimation runs the analysis algorithm on the data points inside the window whenever it is updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The window slides once per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Hence, our software reports an estimation of breathing rate every second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Note that there is a 30-second delay at the beginning since the window needs to be filled first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2 Results We evaluate the effectiveness of the attack in different scenarios such as when the attacker and the target are in the same building or different buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='1 Accuracy in Detecting Breathing Rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Same Building Sce- nario: First, we evaluate the accuracy of the attack by estimating 6 40 Raw Data 35 Filtered Data 30 1Amplitude 25 20 CSI 15 10 5 0 0 10 20 30 Time(s)40 Raw Data 35 Filtered Data 30 CSI Amplitude 25 20 15 10 5 0 0 10 20 30 Time(s)Figure 7: The average accuracy of the at- tack in estimating the target person’s breathing rate when he attacker and target device are in the same building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 8: The CDF of the error in es- timating the target person’s breathing rate when he attacker and target de- vice are in the same building (different floor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 9: The CDF of the error in es- timating the target person’s breathing rate when he attacker and target device are in different buildings (20m away) the breathing rate in an indoor scenario where the target device and attacker are in the same building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We evaluate the accuracy when the target’s breathing rate is 12, 15, 20, and 30 breaths per minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Note, that the normal breathing rate for an adult is 12-20 breaths per minute while resting, and higher when exercising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In this experiment, the user is watching a video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' To make sure the target person’s breathing rate is close to our desired numbers, we place a timer in front of the person, where they can adjust their breathing rate accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' This is just to better control the breath- ing rate during the experiment and is not a requirement nor an assumption in this attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We run each experiment for two minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' During this time, we collect the estimated breathing rate from both ground truth and the attack for different locations of the target device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 7 shows the average accuracy in estimating breath- ing rate across all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The accuracy is calculated as the ratio of the estimated breathing rate reported by the attack over the ground truth breathing rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The figure shows that the accuracy of estimating the breathing rate is over 99% in all scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Finally, Figure 8 plots the Cumulative Distribution Function (CDF) of the error in detecting breathing rate for over 2400 measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The figure shows that 78% of the estimated results have no error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The figure also shows that 99% of measurements have less than one breath per minute error which is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Different Building Scenario: So far, we have evaluated our at- tack where the target and the attacker are in different rooms or floors of the same building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Here we push this further and examine whether our attack works if the attacker and the target person are in a different building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We place the target device in a building on a university campus on a weekday with people around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' A person is sitting around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='5 m away from the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We then place the attacker in another building which is around 20 m away from the target building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Similar to the previous experiment, we run the attack and compare the estimated breathing rate with the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 9 shows the CDF of error for 180 measurements in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Our results show that the attacker successfully estimates the breathing rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Note, that the reason that the attack works even in such a challenging scenario with other people being around is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' First, using an FFT helps to filter out the effect Figure 10: The efficacy of estimating the breathing rate when there is no target near the WiFi device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' of most non-periodic movements and focuses on periodic move- ments and patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Second, wireless channels are more sensitive to changes as we get closer to the transmitter [11, 24], and since in these scenarios, the target person is very close to the target de- vice, their breathing motion has a higher impact on the CSI signal compared to the other mobility in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2 Human Presence Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We next evaluate the efficacy of detecting whether there is a target person near the WiFi device or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In this experiment, the target phone is placed on a desk and the person stays around the device for 30 seconds, then walks away from the device, and then comes back near the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Note, in our algorithm, when there is no majority vote during the voting phase, we return −1 to indicate no breathing detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 10 shows the results of this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' As illustrated in the figure, we can correctly detect the breathing rate when a person is near the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In other words, the algorithm can detect if there is no one near the target device and refrain from reporting a random value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='3 Effect of Distance and Orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Next, we evaluate the effectiveness of the attack for different orientations of the device with respect to the person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We also evaluate its performance for different distances between the target device and the target person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 7 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='85% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='44% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='71% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='48% 100 Accuracy (%) 80 60 40 20 0 12 15 20 30 Orientation1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='8 CDF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='5 Error (RR/min)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='8 DF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='0 0 1 2 3 4 5 Error (RR/min)20 Respiration Rate (bpm) 15 10 Target person Target person 5 leaves comes back 0 5 0 10 20 30 40 50 60 70 80 Time (s)(a) various orientations (b) different distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 11: effectiveness of the attack for different orienta- tion and distance of the targeted WiFi device respect to the person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Orientation: We evaluate the effect of orientation of the target person with respect to the target device (laptop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We run the same attack as before for different orientations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' sitting in front, back, left, and right side of a laptop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The user is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='5m away from the target device in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 11a shows the result of this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Each bar shows the average accuracy for 90 measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Our result shows that regardless of the orientation of the person with respect to the device, the attack is effective and detects the breathing rate of the person accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In particular, even when the person was behind the target device, the attack still detects the breathing rate with 99% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Distance: Here, we are interested to find out what the maximum distance between the target device and the person can be while the attacker still detects the person’s breathing rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' To do so, we place the attacker device and the target device 5 meters apart in two different rooms with a wall in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We then run different experiments in which the target person stays at different distances from the target device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In each experiment, we measure the breath- ing rate for two minutes and calculate the average breathing rate over this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Finally, we compare the estimated breathing rate to the ground truth and calculate the accuracy as mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 11b shows the results of this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The accuracy is over 99% when the distance between the target device and the target person is less than 60 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Note, in reality, people have their laptops or cellphone very close to themselves most of the time, and 60 cm is representative of these situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The accuracy drops as we increase the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' However, even when the device is at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='4 m from the person’s body, the attack can still estimate the breathing rate with 80% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Note, this is the accuracy in finding the absolute breathing rate and the change in the breathing rate can be detected with much higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Finally, the figure shows that the accuracy suddenly drops to zero for a distance beyond 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='4 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' This is due to the fact that at that distance the power of the peak at the output of the FFT goes below the noise floor, and hence, the peak is not detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='4 Effect of Multiple People.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Last, we evaluate if the attack can be used to detect the breathing rate of multiple people simultane- ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We test our attack in three different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In the first scenario, two people are near the laptop while one is working on the laptop and the other is just sitting next to him, as shown in Figure 12a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The attacker targets the laptop and tries to estimate their breathing rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Note, that the attacker has no prior informa- tion about how many people are next to the laptop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In the second scenario, we repeat the same experiment as the first scenario except that the second person is sitting behind the laptop, as shown in Figure 12b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In the third scenario, there are two people in the same space but each person is next to a different device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The attacker targets the laptops and tries to estimate their breathing rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In these experiments, the target device is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='7 m away from the person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 12c shows the results for this evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The blue bars show the result for the first person who is working on the laptop, and the red bars show the results for the second person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Our results show that the attack effectively detects the breathing rate of both people regardless of their orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' However, the accuracy in detecting the breathing rate for the second person is a bit lower than the first person for the first and second scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' This is because the second person’s distance to the target device is slightly more and hence the accuracy has decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 6 SECURITY IMPLICATION: BATTERY DRAIN ATTACK In this section, we show how an adversary can drain the battery of our WiFi devices by using the above loopholes and forcing our WiFi devices to stay awake and continuously transmit WiFi signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='1 Attack Design and Setup 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='1 Attack Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The attacker forces the target device to stay awake and continuously transmit WiFi packets by sending it back- to-back fake frames and some periodic fake beacons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' However, to maximize the amount of time the target device spends transmitting, we study a few different types of fake query packets that the attacker can send.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Note, that the power consumption of transmission is typically higher than that of reception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2 Hence, to maximize the battery drain, we want to send a short query packet and receive a long response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Table 3 lists some query packets and their corresponding re- sponses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The best choice for a query packet is Block ACK requests since the target will respond with a Block ACK that is larger than other query responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Another important factor to consider for maximizing the battery drain is the bitrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' When the bitrate of the query packet increases, the bitrate of the response will also increase as specified in the IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Hence, at first glance, it 2For example, ESP8266 [26] and ESP32 [25] WiFi modules draw 50 and 100 mA when receiving while they draw 170 and 240 mA when transmitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' These low-power WiFi modules are very popular for IoT devices [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 8 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='91% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='73% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='72% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='91% 100 Accuracy (%) 80 60 40 20 0 Front Back Left Right Orientation100 Accuracy (%) 80 60 40 20 0 0 20 40 60 80 100 120 140 160 Distance (cm)(a) Scenario 1 (b) Scenario 2 (c) Breathing Rate Estimation of two persons Figure 12: Accuracy under three different scenarios: Scenario 1: two people sit side-by-side in front of the target device;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Scenario 2: one person sits in front of the target device, the other one sits behind the target device;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Scenario 3: two people sit in front of two target devices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Attacker attacks one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Query Query size Response Response size Null 28 bytes ACK 14 bytes RTS 20 bytes CTS 14 bytes BAR 24 bytes BA 32 bytes Table 3: Different types of fake queries and their responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Note, Null is a data packet without any payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' BAR and BA stand for Block ACK Request, and Block ACK, respectivly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' may seem that to maximize the battery drain, the attacker must use the fastest bitrate possible to transmit query packets, forcing the target device to transmit as many responses as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' How- ever, it turns out that this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The power consumption depends mostly on the amount of time the target device spends transmitting packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Hence, when a higher rate is used for the query and response packets, the total time the target spends on transmission does not increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In fact, the total time spent trans- mitting decreases mainly due to overheads such as channel sensing and backoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' For example, if we increase the bitrate by 6 times (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=', from 1 Mbps to 6 Mbps), the number of packets will increase by only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='3 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' As a result, to maximize the transmission time of the target device, the attacker should use the lowest rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=', 1 Mbps) for the query packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2 Attack Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Attacking device: Any WiFi card capable of packet injection can be used as the attacking device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We use a USB WiFi card connected to a laptop running Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The WiFi card has an RTL8812AU chipset [5] that supports IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 a/b/g/n/ac standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We have installed the aircrack-ng/rtl8812au driver [1] for this card which enables robust packet injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We utilize the Scapy [37] library to inject fake WiFi packets to the target device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Scapy allows defin- ing customized packets and multiple options for packet injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Since we need to inject many packets in this attack, we use the sendpfast function to inject packets at high rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' sendpfast relies on tcpreplay [6] for high performance packet injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Target device: Any WiFi-based IoT device can be used as a target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We choose Amazon Ring Spotlight Cam Battery HD Security Cam- era [2] for our battery drain experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The camera is powered by a custom 6040 mAh lithium-ion battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The battery life of this camera is estimated to be between 6 and 12 months under normal usage [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We leave the camera settings to their defaults which means most power-consuming options are turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' This assures that our measurements will be an upper bound on the battery life and hence the attack might drain the battery much faster in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Authors in [41] pointed out the possibility of a battery drain- ing attack by forging beacon frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' However, they did not provide any evaluations to test this idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Moreover, we show how sending fake packets in addition to fake beacon frames can significantly increase the power consumption on the victim device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2 Results We evaluate the effectiveness of the battery drain attack in terms of range and using different payload configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='1 Finding the optimal configuration: As discussed in 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='1, send- ing block ACK requests at the lowest bitrate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=', 1 Mbps) should maximize the power consumption of the target device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' To verify this, we have conducted a series of experiments with different types of query packets and transmission bitrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In each experiment, we continuously transmit query packets to the Ring security camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In all experiments, we start with a fully charged battery and the attacker injects query packets as fast as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 13 (a) shows the maximum number of packets the attacker could transmit to the target device, and the number of responses it receives per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 13 (b) shows the amount of energy drawn from the battery during one hour of the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' As expected, sending Block ACK Requests (BAR) drains more energy from the battery since the target device spends more time on transmission than receiving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Moreover, the results verify that although increas- ing the data rate from 1Mbps to 6Mbps (BAR/1 versus BAR/6) increases the number of responses, it decreases the energy drained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' As mentioned before, this is because the total time spent transmit- ting decreases mainly due to overheads such as channel sensing 9 D100% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='48% 100% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='07% 100 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='67% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='05% Accuracy (%) 80 60 40 20 0 Scenario 1 Scenario 2 Scenario 3Battery Type Voltage (V) Full Capacity (Wh) 100% Drain (min) 25% Drain (min) CR2032 coin 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='68 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='5 AAA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='87 39 10 AA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content="20 90 22 Table 4: The time it takes for the attack to drain different types of batteries 0 500 1000 1500 2000 2500 3000 3500 Null/1 Data/1 BAR/1 BAR/6 Number of Packets Configurations Attacker's packets Target's responses (a) 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='5 3 Null/1 Data/1 BAR/1 BAR/6 Watt Hour Configurations (b) Figure 13: The figure shows (a) Average number of packets sent to and received from the target device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' (b) Energy con- sumption in Watt Hour measured under different configu- rations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' packet type / bitrate (Mbps) and backoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' This result confirms that sending block ACK requests (BAR) with the lowest datarate is the best option to drain the battery of the target device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2 Battery drain with optimal configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We use the best setting which is a block ACK request (BAR) query transmitted at 1 Mbps to fully drain the battery of the Ring security camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We are able to drain a fully charged battery in 36 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Considering the fact that the typical battery life of this camera is 6 to 12 months, our attack reduces the battery life by 120 to 240 times!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' It is worth mentioning that since a typical user charges the battery every 6-12 months, on average the batteries are at 40-60%, and therefore it would take much less for our attack to kill the battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Moreover, the RING security camera is using a very large battery, most security sensors are using smaller batteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Table 4 shows the amount of time it takes to drain different batteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' For example, it takes less than 40 mins to kill a fully charged AAA battery which is a common battery in many sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='3 Range of WiFi battery draining attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' A key factor in the effectiveness of the battery draining attack is how far the attacker can be from the victim’s device and still be able to carry on the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' If the attack can be done from far away, it becomes more threatening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' To evaluate the range of this attack, we design an experiment in which the attacker transmits packets to the target from different distances and we measure what percentage of the attacker’s packets are responded to by the target device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We use a realistic testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The Ring security camera is installed in front of a house, and the attacker is placed in a car, parked at different locations on the street.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We test the attack at 10 different locations up to 150 meters away from the target device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Figure 14 shows these locations and our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Each yellow circle represents each of the locations tested at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The numbers inside the circles show the percentage of the attacker’s packets responded to by the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Each number is an average of over 60 one-second measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The closest distance is about 5 meters when we park the car in front of the target house.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In this location 97% of the attacker’s packets are responded to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' We conducted other experiments within 10 meters of the target (not shown here) and we obtained similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Our results show that even within a distance of 100 meters, almost all attacker’s packets are responded to by the victim’s device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In some locations such as the rightmost circle (at 150 meters away), we could still achieve a reply rate as high as 73%, confirming our attack works even at that distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' The reason for achieving such a long range is that the attacker transmits at a 1 Mbps bitrate which uses extremely robust modulation and coding rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' BPSK modulation and a 1/11 coding rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 7 ETHICAL CONSIDERATIONS We discussed our project and experiments with our institutions’ IRB office and they determined that no IRB review nor IRB approval is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Moreover, the house and WiFi devices used in most experiments are owned and controlled by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Finally, in order to expedite mitigating the attacks presented in this paper, we have started engagements with WiFi access point and chipset manufacturers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 8 CONCLUSION In this work, we identify two loopholes in the WiFi protocol and demonstrate their possible privacy and security threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In partic- ular, we reveal that today’s WiFi radio responds to packets from unauthorized devices outside of the network and it can be easily manipulated to keep awake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' These loopholes can be exploited by malicious attackers to jeopardize our daily use of WiFi devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' As examples, we demonstrate how an attacker can take advantage of these loopholes to extract private information such as breathing rate and quickly exhaust the battery of a typical IoT device, leaving the victim’s device in a disabled state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' REFERENCES [1] [n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' aircrack-ng/rtl8812au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='com/aircrack-ng/rtl8812au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 10 0 m 50 m 100 m 150 m 50 m 100 m 150 m 73 97 90 54 14 84 83 90 70 64 Target Attacker Figure 14: Percentage of attacker’s query packets responded by the target device for different attacker’s locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' [2] [n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Ring Spotlight Cam Battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' https://ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='com/products/spotlight-cam- battery".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' [3] [n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Ring Spotlight Cam Battery Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='org/security- cameras/ring/review/spotlight-cam-battery/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' [4] [n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Ring Spotlight Cam Battery Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='pcmag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='com/reviews/ ring-spotlight-cam-battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' [5] [n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' RTL8812AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='realtek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='com/en/products/communications- network-ics/item/rtl8812au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' [6] [n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Tcpreplay - Pcap editing and replaying utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' https://tcpreplay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='appneta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' com/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' [7] 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' IEEE Standard for Information technology - Telecommunications and information exchange between systems - Local and metropolitan area networks Specific requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications Amendment 4: Protected Management Frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' IEEE Std 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11w-2009 (Amendment to IEEE Std 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11-2007 as amended by IEEE Std 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11k-2008, IEEE Std 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11r-2008, and IEEE Std 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11y-2008) (2009), 1–111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' [8] Heba Abdelnasser, Khaled Harras, and Moustafa Youssef.' metadata={'source': 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In Proceedings of the 19th ACM Workshop on Hot Topics in Networks (HotNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 132–138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' [10] Ali Abedi, Omid Abari, and Tim Brecht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Wi-le: Can wifi replace bluetooth?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='. In Proceedings of the 18th ACM Workshop on Hot Topics in Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 117–124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' [11] Ali Abedi, Farzan Dehbashi, Mohammad Hossein Mazaheri, Omid Abari, and Tim Brecht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Witag: Seamless wifi backscatter communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In Proceedings of the Annual conference of the ACM Special Interest Group on Data Communi- cation on the applications, technologies, architectures, and protocols for computer communication (SIGCOMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 240–252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' [12] Ali Abedi and Deepak Vasisht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Non-Cooperative Wi-Fi Localization and Its Privacy Implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' In Proceedings of the 28th Annual International Conference on Mobile Computing And Networking (MobiCom.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 837–846.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' [14] Mayank Agarwal, Dileep Pasumarthi, Santosh Biswas, and Sukumar Nandi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Machine learning approach for detection of flooding DoS attacks in 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content='11 net- works and attacker localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' International Journal of Machine Learning and Cybernetics 7 (2016), 1035–1051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' [15] Bandar Alotaibi and Khaled Elleithy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Rogue Access Point Detection: Tax- onomy, Challenges, and Future Directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' Wireless Personal Communications 90 (10 2016), 5021– 5028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} +page_content=' https://doi.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfbvd_/content/2301.00269v1.pdf'} diff --git a/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf b/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e3e7110b6e913b9de0a5c26f1ba0df5bce99fe45 --- /dev/null +++ b/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fa02405eabf2c0e80855f98f51faaafc3dbcb7250126557055554db9de9772d8 +size 6562901 diff --git a/2NE1T4oBgHgl3EQf5QXz/content/tmp_files/2301.03511v1.pdf.txt b/2NE1T4oBgHgl3EQf5QXz/content/tmp_files/2301.03511v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c58fe00187e77b1d929700b788939b06b563d1a8 --- /dev/null +++ b/2NE1T4oBgHgl3EQf5QXz/content/tmp_files/2301.03511v1.pdf.txt @@ -0,0 +1,1151 @@ +The Value of Internal Memory for Population Growth in Varying +Environments +Leo Law, BingKan Xue +Department of Physics, University of Florida, Gainesville, FL 32611, USA +Abstract +In varying environments it is beneficial for organisms to utilize available cues to infer the +conditions they may encounter and express potentially favorable traits. However, external cues +can be unreliable or too costly to use. We consider an alternative strategy where organisms +exploit internal sources of information. Even without sensing environmental cues, their internal +states may become correlated with the environment as a result of selection, which then form a +memory that helps predict future conditions. To demonstrate the adaptive value of such internal +memory in varying environments, we revisit the classic example of seed dormancy in annual +plants. Previous studies have considered the germination fraction of seeds and its dependence +on environmental cues. In contrast, we consider a model of germination fraction that depends +on the seed age, which is an internal state that can serve as a memory. We show that, if the +environmental variation has temporal structure, then age-dependent germination fractions will +allow the population to have an increased long-term growth rate. +The more organisms can +remember through their internal states, the higher growth rate a population can potentially +achieve. Our results suggest experimental ways to infer internal memory and its benefit for +adaptation in varying environments. +1 +arXiv:2301.03511v1 [q-bio.PE] 9 Jan 2023 + +1 +Introduction +Organisms can adapt to a varying environment by diversifying their traits among individuals of +the same population. A common form of such diversity is dormancy, where some individuals enter +a dormant state while others remain active [1, 2, 3]. Those that are active will contribute to the +growth of the population under good environmental conditions, but will be vulnerable to periods of +harsh conditions. On the other hand, the dormant individuals are often tolerant to environmental +stress and thus help preserve the population during harsh periods. For example, in a bacterial +population, while most cells grow and divide normally, some cells randomly switch to a reversible +dormant state called persister cells, which makes them tolerant to antibiotics when normal cells +would perish [4, 5, 6]. Other examples include seed dormancy in plants, dauer larva in nematodes, +diapause in insects, etc. +[1, 3, 7, 8]. +These are thought to be a strategy known as diversified +bet-hedging [9, 10], in which organisms express different traits with some probability to create +diversity in the population, so as to increase the long-term growth rate of the population under +environmental variations [11, 12, 13]. +In the simplest form, bet-hedging organisms have fixed probabilities of expressing different traits +[11]. But more generally, organisms can sense cues from the environment that will influence these +probabilities [14, 15]. Such cues may be indicative of future environmental conditions, so that the +organisms may bias the probabilities towards traits that are favorable in the likely environment. It +has been shown that the information contained in the cue about the environment will contribute +to an increase in the population growth rate [14, 16, 17]. However, sensing and responding to +environmental cues may come at a cost, as it requires the expression of specific sensors and signaling +mechanisms [18]. Besides, there may not be enough time for the organisms to respond to the cues +through phenotypic plasticity, as the environment may have changed by the time the trait is +developed [19, 20]. Therefore, it is not always beneficial to rely on environmental cues. +Besides external signals, the behavior of organisms can be influenced by their internal states, such +as physiological or metabolic states [21]. One example is the reserve level – a starved animal may +choose to forage more aggressively despite higher predation risk [22, 21]. Another example is the +age of the organism – it is known that the age of seeds can affect germination in annual plants [23]. +These internal states are not sensors that directly measure the external environment. However, +they may become correlated with the environment as a result of selection, because certain states +are associated with higher fitness in past environmental conditions and thus become more common +in the population. Therefore, the distribution of such internal states among the population can +potentially provide information about the environment, which may be utilized by the organisms. +We will study an example of this situation and show that internal states of the organisms can +indeed serve as internal cues to help them adapt to varying environmental conditions. Such internal +states effectively provide a memory about the past outcomes of selection, which helps predict the +future environment. Moreover, we show that a larger memory capacity enables higher gains in the +2 + +population growth rate. Our results suggest that internal states that were not developed for sensing +the environment could nevertheless be co-opted as internal cues for adaptation, which would save +the cost of sensors and may thus be a more efficient strategy. +To study adaptation in varying environments, we will use seed dormancy as our main example. +Seeds of annual plants will either germinate or stay dormant in a given year. While dormancy +sacrifices the short-term fitness of the seeds, it preserves the population from a catastrophically +bad year with very low yield, and thus results in higher long-term benefit. This has been studied +as a classic model of bet-hedging [11, 14], supported by the fact that dormant seeds eventually ger- +minate under similar environmental conditions [23], and that the germination fraction is negatively +correlated with local environmental variability [24]. It is known that germination is influenced by +environmental cues, such as temperature, humidity, and the number density of surrounding seeds +[15, 25]. Moreover, there is evidence that the probability a seed will germinate also changes with the +age [26, 27, 23]. However, the adaptive value of such age dependence in germination has not been +fully studied [28, 3]. It was shown in [28] that the evolutionarily stable probability of germination +does not depend on seed age if there is no density dependence. Yet, their model did not include +temporal correlation in the environmental variation, which is crucial for memory to be useful in +predicting future environments [29, 30, 31]. We will show that, when there is temporal structure +in the environmental variation, age-dependent germination probabilities can increase the long-term +growth rate of the seed population. +2 +Background +2.1 +Cohen’s model of seed dormancy +Let us first briefly review the idea of bet-hedging and how information emerges as a central quantity +in determining the long-term growth rate of the population. We will follow the classic model of +seed dormancy in annual plants by Cohen [11, 14], as illustrated in Fig. 1A. Each year can be +“good” (denoted as environment ε = 1) or “bad” (ε = 0) for the plant. Seeds that germinate +(“phenotype” φ = 1) in a good year will be able to grow and produce a large number (Y1) of new +seeds. However, in a bad year, germinated plants will have a low yield (Y0). We will set Y0 = 0 and +denote Y1 = Y for simplicity, meaning that germinating in a bad year will result in no offspring. All +germinated plants perish at the end of the year, regardless of their yield. Seeds that stay dormant +(φ = 0) will remain viable the next year with probability V . Thus, the fitness of a seed in a given +environment can be summarized by the matrix fεφ = +� V 0 +V Y +� +. In addition, we assume that the +number of consecutive good years follows a geometric distribution, whereas that of bad years has a +narrow distribution (see Fig. 1B and Appendix A.2). This is meant to describe the scenario where +good growth conditions are disrupted by random occurrence of disasters that affect growth for a +characteristic number of years. +3 + + + +dormant +A +germinate +year 2 +viability +dormant +germinate +year 1 + low yield +viability +bad +... +... +C +B +seeds +⋯ +s1 +s2 +s0 +good +high yield +seeds +Figure 1: (A) Schematic illustration of Cohen’s model of seed dormancy in annual plants. Each year may +be good or bad for plant growth. A seed can either germinate to produce a yield Yε that depends on the +environmental condition ε, or stay dormant with a probability V of still being viable next year. The number +of seeds at the end of year t is Nt. The parameter values used in our calculations are Y0 = 0, Y1 = 4, +V = 0.9. (B) The distribution of duration of consecutive good years and bad years. We choose the duration +of good years to follow a geometric distribution with a mean of 5, and the duration of bad years to have a +Gaussian distribution with a mean and standard deviation of 5 ± 2 cut off at 0 and 10. (C) A state diagram +that represents the seed age. Each state sα represents a seed of age α. Blue arrows represent dormancy that +increases the age by 1; orange arrows represent germination that may produce new seeds of age 0. Weights +on the arrows represent the probability of germination or dormancy. +In the simplest case where seeds receive no environmental cues, the fraction of seeds that germinate +each year is assumed to be a constant, denoted by q. In a good year, the total number of seeds will +grow by a factor (1 − q)V + qY , whereas in a bad year, the number of seeds will reduce to only a +fraction (1 − q)V of the previous year. The long-term growth rate of the population will be given +by (see derivation in Appendix A.1) +Λ = p log +� +(1 − q)V + qY +� ++ (1 − p) log +� +(1 − q)V +� +, +(1) +where p is the frequency of good years and (1 − p) is that for bad years. The germination fraction +that maximizes the long-term growth rate is +q∗ = p Y − V +Y − V +(2) +for p > V/Y and 0 otherwise. In the limit of high yield (Y ≫ V ), this leads to the classic result +q∗ ≈ p, which means the optimal germination fraction should match the frequency of good years +[11]. The model can be extended to seeds that receive some external cue (ξ) about the environment +[14]. +In this case, the optimal germination fraction will depend on the cue. +As a result, the +population can grow faster than without the cue (see Appendix A.1). +4 + +0.25 +Good years +Bad years +0.20 +distribution +0.15 +0.10 : +0.05 +0.00 +12345678910 +12345678910 +duration +duration + +growth rate +perfect information +external cue +no cue +perfect memory +internal state +no memory +A +B +Figure 2: The long-term growth rate Λ of populations with different sources of information. +(A) The +value of external cues: Λmax is the maximum possible growth rate attainable if the population has perfect +information about the future environment. +Λbet is the highest growth rate achievable by a bet-hedging +population without receiving cues, which is suppressed by the entropy of the environment H(ε). Λcue is the +growth rate when the population utilizes a cue ξ that has a mutual information I(ε; ξ) with the environment. +(B) The value of internal memory: Organisms can utilize their internal states as memory, such that their +behavior depends on which state they are in. Λbet from bet-hedging also represents the case with no memory, +which corresponds to having only one internal state (L = 1). More states (L > 1) provides larger memory +capacity and allows a higher growth rate Λint for the population. Λmem is the highest growth rate achievable +by organisms with a perfect memory (L → ∞) of their lineage history. +These well-known results are summarized schematically in Fig. 2A. At the top level is the maximum +possible growth rate Λmax, which is attainable only if individuals have perfect information about +future environmental conditions and respond accordingly, i.e., germinate if it will be a good year +and go dormant if it will be bad. On the other hand, if there is no environmental cue, the best +strategy is bet-hedging with fixed probabilities, which achieves a growth rate Λbet. This is less than +Λmax by an amount H(ε), which is the Shannon entropy from information theory that quantifies +the uncertainty of the varying environment (See Appendix A.1). However, if a cue ξ is used to +help predict the environment, the population can increase the growth rate from Λbet to Λcue, up +by an amount I(ε; ξ) that is equal to the mutual information between the cue and the environment +(Appendix A.1). Note that Λcue is still not as high as Λmax unless the cue is fully accurate. The +relations between these growth rates illustrated here (similar to plots in [16, 32]) show that, in +order for the population to better adapt to varying environments, it must utilize available sources +of information about the environment. +2.2 +Internal source of information +Instead of sensing external cues, below we consider another possibility for organisms to use their +internal states as a source of information. We will use the age of seeds as an example. The state +diagram representing seed ages are illustrated in Fig. 1C, where a state sα represents a seed of age +α. A blue arrow represents a seed going into dormancy for one year, so that the age is increased +by 1. +An orange arrow represents a seed that germinates and potentially produces new seeds, +5 + +which will have age 0. +The weights on the arrows represent the probability of germination or +dormancy. For a simple bet-hedging strategy without any cues, the probability of germination will +be a constant, which equals q∗ from Eq. (2), independent of the seed age. We will study the case +where the germination fraction can depend on the seed age, and show that the population can +acquire information from this internal state to achieve a higher growth rate. +3 +Results +3.1 +Seed age as an internal cue +We first study whether the seed age as an internal state contains useful information about the +environment. Let αt−1 be the seed age at the beginning of year t, and εt be the coming environment +that year. If αt−1 has no information about the environment, then it will be statistically independent +of εt, i.e., P(εt|αt−1) = P(εt). Therefore, whether seed age is informative about the environment +can be inferred from the conditional probability P(εt|αt−1). +To calculate that, we simulate a +sufficiently long sequence of environments, denoted by εt for each year t. We also simulate a single +lineage of plants that uses the constant germination fraction q∗. Each year the seed can either +germinate or stay dormant, and the probability of choosing the phenotype φt is further weighted +by the fitness f(εt, φt) to account for selection (see procedure in Appendix A.3). The seed age along +the lineage is recorded as αt. From the sequences of εt and αt, we estimate the joint probability +distribution P(εt, αt−1), from which the conditional probability P(εt|αt−1) is calculated. As shown +in Fig. 3, the probability of the environment εt does depend on the seed age αt−1. This means that +knowing the seed age allows a more accurate prediction of the coming environment. Therefore, it +is possible for the population to “co-opt” the seed age as an “internal cue” for the environment. In +analogy to the case of external cues, we expect that such information can be used to increase the +long-term population growth rate. +We therefore consider a strategy where the germination fraction depends on the seed age, denoted +by qα and represented by weights on the arrows in Fig. 1C. To calculate the long-term growth rate, +let N be a vector that represents the age-structured population, with components Nα being the +number of seeds of age α. The dynamics of N is described by a matrix M(ε; q) that depends on +the environment ε and the germination fractions q (with components qα), +M(ε; q) = +� +� +� +� +� +� +� +q0 Yε +q0 Yε +· · · +(1−q1)V +0 +· · · +0 +(1−q2)V +... +... +... +... +� +� +� +� +� +� +� +(3) +Each year, the population vector is multiplied by the matrix that corresponds to the current +6 + +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +seed age, +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +probability, P( t| +t +1) +bet-hedging +age-dependent +Figure 3: Probability of the coming environment εt conditioned on the seed age αt−1 at the beginning +of year t, as calculated by simulating a lineage of seeds. Dashed line is the marginal probability of the +environment, which would indicate that the seed age is uncorrelated with the environment. Blue bars are +when the population uses a bet-hedging strategy with a constant germination fraction. Orange bars are +when the germination fraction depends on the seed age to maximize population growth rate. In both cases +the seed age is correlated with the environment and thus useful as an internal cue. +environment εt, +N t = M(εt; q) · N t−1 , +(4) +Here M(εt; q) is a random matrix because εt is a random variable. The temporal sequence of εt +is randomly drawn according to the distributions of good and bad years. The long-term growth +rate Λ of the population is then given by the Lyapunov exponent of the product of these random +matrices [33], which is calculated numerically (see methods in Appendix A.2). +We vary the age-dependent germination fractions qα to maximize Λ. As expected, this growth rate +using seed age as an internal cue (Λint) is greater than that of bet-hedging without cues (Λbet), as +illustrated in Fig. 2B (see also Fig. 6 below). The optimal germination fraction as a function of seed +age is shown in Fig. 4. An intuitive explanation for the age dependence is that, in this example, the +bad environment typically lasts a number of years, so it is advantageous for a seed to stay dormant +for a similar period of time to wait it out. Those that germinate in the wrong phase of the bad +year cycle will be eliminated by selection, and the remaining individuals tend to be synchronized +with the environment. In contrast, if there is no temporal structure in the environment, such as +when the environment is randomly and independently chosen each year, then the seed age will no +longer be correlated with the environment. In that case, the best strategy is to have a constant +germination fraction (equal to q∗ in the bet-hedging case, see Fig. 4), as argued in [28]. +Note that the information about the environment is contained in the distribution of seed ages +within the population, which results from selection in previous years. Compared to the case of an +external cue that is shared by all individuals, the seed age varies among individuals (which prevents +7 + +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +seed age, +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +germination fraction, q +temporally structured +uncorrelated environment +Figure 4: Dependence of the germination fraction q on the seed age α that maximizes the population growth +rate. Blue bars are when the environment is temporally structured, as described by the duration of good and +bad years in Fig. 1B. Orange bars are when the environment is drawn independently each year, for which the +germination fraction need not depend on seed age and is equal to the bet-hedging solution in Eq. 2 (dashed). +an analytic expression for Λ). It acts as an individual’s memory of its own lineage history, which +helps it infer the likely environment in the future. Importantly, the increase in population growth +rate does not come at any cost associated with sensing external cues. Thus, such an internal source +of information proves to be beneficial for the population. +3.2 +Internal states as memory +We have shown that internal states of organisms may help them “remember” the past outcomes of +selection to be able to predict the future environment, leading to an increased population growth +rate. Intuitively, the more the organisms can remember, the better they may predict and adapt to +the environment. To test this in our model, we can vary the memory size by changing the number +of possible internal states. The state diagram in Fig. 1C has potentially an infinite number of +states. They can be truncated at a finite number L, such that seeds exceeding age (L − 1) will +remain in the state sL−1 until they germinate or perish (Fig. 5A). This allows us to study how the +population growth rate depends on the number of states L. +We first note that having only one internal state (L = 1, Fig. 5B) is effectively having no memory, +because the system will always be in that same state regardless of the past events. In this case, +the germination fraction is always equal to q0 associated with the only state s0. Having a constant +germination fraction means that this case corresponds to the simple bet-hedging strategy. The +maximum long-term growth rate will just be Λbet achieved at q0 = q∗ found in Eq. (2). +For two internal states (L = 2, Fig. 5C), the model reduces to “phenotypic switching”, in which +8 + + + +sL−1 +⋯ +A +B +C +s0 +s1 +s0 +s0 +s1 +Figure 5: State diagrams for age-dependent germination. (A) The germination fraction q depends on the +seed age α up to α = L−1, beyond which it remains the same. Varying the length L effectively varies +the memory capacity of the organisms. (B) With only one state (L = 1), the organism effectively has no +memory, and the germination fraction is a constant, corresponding to simple bet-hedging. (C) The two-state +case corresponds to a Markov process where the organisms switch back and forth between two phenotypes, +with transition probabilities P(φ1|φ0) = q1 and P(φ0|φ1) = 1−q0. +the organisms randomly switch between two phenotypes (germination or dormancy) with fixed +transition probabilities. Specifically, the probability for a dormant seed to germinate next year is +q1, and the probability for a new seed (that came from a germinated plant) to go dormant is 1−q0. +This is a Markov process, for which the transition between phenotypes does not depend on how +long a phenotype has lasted. It implies that the germination fraction only depends on whether the +seed is fresh (age 0) or has been dormant (age > 0), but not on how long it has been dormant. As a +result of being Markovian, the duration of the dormant phenotype will be geometrically distributed. +A larger L will allow the germination fraction to depend more sensitively on the seed age (L > 2, +Fig. 5A). The number of states L roughly represents how many dormant years a seed can remember. +For each number L, we search for the maximum long-term growth rate Λ over the parameters +{q0, · · · , qL−1} (see methods in Appendix A.2). As shown in Fig. 6, Λ increases monotonically as +more states are incorporated. Therefore, more memory allows faster population growth and hence +better adaptation to environmental variation. Note that Λ quickly approaches a limit Λmem when +L becomes greater than the typical duration of the bad environment (equal to 5 in this example, +see Fig. 1B). Intuitively, there is no need to remember longer dormancy because there is no benefit +in staying dormant for longer than the duration of bad years. The relation between the growth +rate and memory is illustrated schematically in Fig. 2B. +If we think of seed age as an internal cue for the environment, we can calculate the mutual infor- +mation I(εt; αt−1) between the environment εt and the seed age αt−1, using the joint probability +P(εt, αt−1) calculated the same way as in Sec. 2.2. Fig. 6 shows that the mutual information also +increases with the number of states L, as more memory is available. When plotted against each +other, the long-term growth rate Λ increases with the mutual information I (Fig. 6 inset), just like +9 + +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +number of states, L +0.04 +0.10 +0.16 +0.22 +0.28 +long-term growth rate, +-0.04 +0.02 +0.08 +0.14 +0.20 +mutual information, I +-0.04 +0.08 +0.20 +info, I +0.04 +0.16 +0.28 +growth, +growth rate +mutual info +Figure 6: Long-term growth rate Λ of populations that have different memory capacity as measured by the +number of internal states L. For each L, the age-dependent germination fractions qα are chosen to maximize +Λ. Also plotted is the mutual information I between the previous seed age αt−1 and the environment εt. +Both Λ and I increase monotonically with the memory capacity L, approaching their respective limits as +L ≫ 5 (mean duration of bad years). (Inset) Long-term growth rate Λ increases monotonically with the +mutual information I. Gray diagonal line represents Cohen’s model with external cues, in which Λ = Λbet+I. +for an external cue. Note that in Cohen’s model with external cues [14], Λ is simply proportional +to I (see Eq. (A14) in Appendix A.1). In comparison, for the same amount of information I, the +population achieves a higher growth rate Λ using seed age as an internal cue (Fig. 6 inset). +So far we have considered a very specific structure for the state diagrams (Fig. 5A, “age-diagram”). +It might be possible that, given the number of internal states, there are other diagrams that can +lead to a high long-term growth rate. Such diagrams could represent other types of internal states +instead of the age. For example, the reserve level of an organism can be represented by a linear +diagram, such that the organism moves up one or more states if it succeeds in foraging or moves +down one state if it fails [21]. +To find which structure of internal states provides the highest +long-term growth rate for the population, we searched all possible diagrams of a given number of +states (up to L = 6, beyond which it is computationally difficult), optimizing the weights qα for +each diagram (see Appendix A.4). It turns out that the age-diagram in Fig. 5A is optimal for the +temporal structure of the environment that we assumed (Fig. 1B). In general, the state diagram is +a mathematical representation of memory, known as the “ϵ-machine” of a stochastic process [34]; +a formal treatment and application to population growth in varying environments is given by [30]. +10 + +1 2 3 4 5 6 7 8 9 10 +duration +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +distribution +Germination +A +1 2 3 4 5 6 7 8 9 10 +duration +Dormancy +B +L = 2 +L = 5 +L = 10 +Figure 7: The distribution of the duration of consecutive germinations or dormant years along a lineage of +seeds. Different colors correspond to age-dependent germination fractions qα for different memory capacities +L. (A) For each L, the duration of germinations matches a geometric distribution with a mean of 1/q0 +(dashed line for L = 2 and solid line for L = 10), meaning that there is no memory of previous germinations. +(B) The duration of dormancy has a distribution that changes shape depending on the memory capacity L. +L = 2 (phenotypic switching) results in a geometric distribution with a mean of 1/(1−q1) (dashed line). +Larger L’s result in deviation from a geometric distribution, which is indicative of having internal memory. +4 +Discussion +4.1 +Characterization of internal memory +Memory arising from age-dependent germination fractions can be characterized by the distribution +of the duration of dormancy. That is, given a large number of fresh seeds, what is the distribution +of the time that each seed stays dormant before germinating. To calculate this distribution, we +simulate one lineage of seeds over a long time in the absence of selection (see Appendix A.3), and +record the sequence of phenotypes, i.e., whether a seed germinated or not each year. Fig. 7 shows +the distribution of the number of consecutive years that successive seeds germinate or that a seed +stays dormant. The number of consecutive germination years is geometrically distributed with a +mean of 1/q0 (Fig. 7A), because every new seed has the same probability q0 of germinating. In +other words, a new seed has no memory of the age of the plant that it came from. Thus, the +absence of phenotypic memory is signified by the geometric distribution. +On the other hand, the distribution of the consecutive dormant years (i.e., the duration of dor- +mancy) depends on the number of internal states L. For L = 2, as discussed in Sec. 3.2, there is +no memory of how long a seed has been dormant. Indeed, the distribution of dormancy durations +is geometric with a mean of 1/(1−q1) (Fig. 7B). But as L increases, the distribution becomes +more bell-shaped and closer to the distribution of consecutive bad years (Fig. 1B). (In the limit +where the fitness matrix fεφ is diagonal, the optimal strategy will be such that the duration of each +phenotype exactly matches the distribution of the corresponding environment; see Appendix A.5). +The deviation of the distribution from being geometric indicates that the seed has memory of how +long it has been dormant, which is necessary for the germination fraction to depend on the seed +11 + +age. Thus, the shape of the dormancy distribution can be used as an experimental signature of +internal memory. +The best demonstration of memory in phenotypic changes is found in experiments on the bacteria +Bacillus subtilis [35]. During its growth, B. subtilis can switch between two phenotypes, either as +a free-moving cell by making flagela or as part of an aggregate by producing extracellular matrix +[36, 37]. It is thought that the aggregate cells have an advantage for colonization and can better +cope with a harsh environment by sharing resources, whereas the motile cells are better at dispersing +and searching for nutrients. The durations of these two cell types along continuous cell lineages are +measured in a constant environmental condition [35]. It was found that the time a lineage stays in +the motile cell type follows an exponential distribution with a mean of ∼ 81 generations, while the +aggregate cell type is maintained for a narrowly distributed duration with a mean and standard +deviation of 7.6 ± 2.1 generations (see Fig. 2(d,f) of [35]). This implies that the motile cell type +is memoryless while the aggregate cell type has memory. That is, an aggregate cell keeps track +of how long it has been part of an aggregate, whereas a motile cell turns off motility with a fixed +probability at every cell division. These two distributions of phenotype durations look similar to +those found in our model (Fig. 7). Importantly, since the switching of cell types is measured in +a constant environment, it is evident that the phenotypic changes are influenced by some internal +states of the cell, rather than external cues. This method of inferring the existence of internal +memory by measuring the duration of phenotypes can be potentially applied to seeds. It would +require measuring the duration of seed dormancy by planting seeds in separate pots under the same +environmental condition and recording how soon they germinate. +4.2 +Evidence for age-dependent dormancy +Our model assumes that the probability of a seed entering or exiting dormancy depends on the +age. +If the bad environment typically persists for a number of years, then the model predicts +that the probability of exiting dormancy should be small initially and increase over a timescale +that matches the duration of bad years (Fig. 4). Data from past experiments have shown that +for different species the germination fraction can either increase or decrease between the first and +second years [23], while data going beyond the second year are scarce. To test the above prediction +also requires knowing the statistics of bad years. Alternatively, age-dependent germination can be +tested by measuring the distribution of dormancy durations, as discussed in Sec. 4.1 (Fig. 7B). For +that purpose, one has to measure the final age of seeds right before they germinate. Studies on +seed age structure have been done in the past [26, 27], but with the goal of measuring the current +age of seeds in a population at a given time, even though some seeds will continue to be dormant. +We are not aware of existing studies that measured the distribution of final seed ages. +Dormancy in other organisms can also be studied using our model. One example is insect diapause +[38], which is considered another example of bet-hedging. In many insect species, the larvae can +12 + +enter diapause at a certain developmental stage to avoid unfavorable conditions, instead of pro- +ceeding with normal development to become adults. In a simple model of diapause [39], the larvae +may undergo multiple years of diapause and have a fixed probability of (re)entering diapause each +year (see Fig. 1 of [39]), similar to Cohen’s model of seed dormancy [11]. This would correspond +to our model with L = 1, such that the decision to enter diapause is memoryless. Another model +assumes that the larvae can only undergo one period of diapause and must exit after that [40]. This +pattern is a special case of our model with L = 2, where the state s0 would correspond to a new +larva and s1 to diapause. The larva can either develop to an adult with probability q0 and produce +offspring (arrow from s0 back to itself), or enter diapause with probability 1 − q0 (arrow to s1). +However, once it undergoes diapause, it must exit and develop, so there is only one arrow leaving +s1, which goes to s0 with probability q1 = 1. In this scenario, it was found that diapause is ben- +eficial in varying environments that are temporally correlated [40], in agreement with our results. +More generally, one may study situations where diapause can be repeated for a number of times, +which would correspond to a diagram like Fig. 5A. Our results suggest that which form of diapause +is evolutionarily favored depends on the complexity of temporal structure in the environmental +variation, which could potentially be tested in empirical studies. +5 +Conclusion +We have shown that the internal states of organisms can serve as a memory to help the population +adapt in varying environments. In order for this strategy to be useful, the environment must be +temporally structured, and the internal states must become correlated with the environment. We +have demonstrated that such correlation can arise from selection alone, without direct interaction +with the environment. More generally, some internal states of organisms may be correlated with +the environment as a result of phenotypic plasticity. For example, seeds produced in a good year +may be bigger than those produced in a bad year, so seed size could provide a memory of the +past environment. It is known that seed size can affect germination probability [41], and it will be +interesting to study if such dependence can benefit population growth in varying environments. +Organisms are complex systems with a lot of internal degrees of freedom, some of which might +happen to become correlated with the environment through selection or plasticity. Even though +these internal states might not have developed as sensors for environmental cues, they could be +co-opted as information sources to guide the organism’s behavior. To test whether seed age could +be co-opted to affect germination, one might compare accessions of annual plants in temporally +structured environments and those in unpredictable environments. Our model predicts that the +germination fraction would evolve to depend on the seed age in the former case. +Dormancy has been proposed to cause a “storage effect” that promotes species coexistence in vary- +ing environments [42]. Our model of age-dependent dormancy may be studied in such community +13 + +ecology context. If the presence of other species is viewed as part of the environment for the focal +species, then internal states such as seed age could potentially provide a memory of past interac- +tion with those other species. For example, reserve level of the predator may be an indicator of +past encounters with prey [21]. History-dependent ecological interactions have been experimentally +indicated in microbial communities [43]. It will be interesting to use our framework to study such +ecological dynamics of organisms whose phenotypes depend on their memory. +A +Methods +A.1 +Analytic derivation of Cohen’s model +Consider a population of annual plant seeds, each of which can either germinate (φ = 1) or stay +dormant (φ = 0) each year. The environment can be either good (ε = 1) or bad (ε = 0). If a seed +germinates in a good year, it will reproduce and yield Y1 number of seeds; but a seed germinating +in a bad year will only yield Y0 seeds, with Y1 > Y0 (in the main text we set Y0 to 0 for simplicity). +If a seed stays dormant, then the probability that it will remain viable is V . For Y1 > V > Y0, it is +favorable for a seed to germinate in a good year but stay dormant in a bad year. The number of +seeds at year t is denoted by Nt and obeys the equation: +Nt = Nt−1 +� +(1 − q)V + qYεt +� +, +(A1) +where εt is the environment in that year and q is the fraction of seeds that germinates. The number +of seeds at year T can be calculated recursively as: +NT = N0 +T +� +t=1 +� +(1 − q)V + qYεt +� += N0 +� +(1 − q)V + qY0 +�T0� +(1 − q)V + qY1 +�T1, +(A2) +where Tε is the total number of years that the environment is ε. The long-term growth rate Λ is +defined as the asymptotic rate of logarithmic increase: +Λ ≡ lim +T→∞ +1 +T log NT +N0 += P0 log +� +(1 − q)V + qY0 +� ++ P1 log +� +(1 − q)V + qY1 +� +, +(A3) +where Pε ≡ lim +T→∞ +Tε +T is the frequency of environment ε. The germination fraction q∗ that maximizes +Λ is found by setting the derivative ∂Λ +∂q to zero, which gives (assuming q∗ > 0): +q∗ = +V P1 +V − Y0 +− +V P0 +Y1 − V . +(A4) +And the corresponding maximum growth rate Λbet is: +Λbet = P0 log P0(Y1 − Y0)V +(Y1 − V ) ++ P1 log P1(Y1 − Y0)V +(V − Y0) +. +(A5) +14 + +If the seeds have perfect information about the future environment, then they should all germinate in +good years and stay dormant in bad years. This would result in a total population NT = N0 V T0 Y T1 +1 +instead of Eq. (A2), which gives the maximum possible growth rate: +Λmax = P0 log V + P1 log Y1 . +(A6) +The difference between Λmax and Λbet is then given by: +Λmax − Λbet = −P0 log P0(Y1 − Y0) +(Y1 − V ) +− P1 log P1(Y1 − Y0)V +(V − Y0)Y1 +. +(A7) +In the limit Y0 → 0 and Y1 ≫ V , it simplifies to: +Λmax − Λbet = −P0 log P0 − P1 log P1 ≡ H(ε) , +(A8) +which is the entropy of the environment. +The model above can be generalized to include an external cue ξ that is correlated with the +environment ε. Assume that, given ξ, the seeds will germinate with probability P(φ = 1|ξ) ≡ qξ. +The total number of seeds then obeys the equation: +Nt = Nt−1 +� +(1 − qξt)V + qξt Yεt +� +, +(A9) +where ξt is the cue received in year t. Repeating the same procedure as above, one finds that the +population after T years becomes: +NT = N0 +� +ε,ξ +� +(1 − qξ)V + qξYε +�Tεξ, +(A10) +where Tεξ is the number of years that the environment is ε while the cue is ξ. The long-term growth +rate is then given by: +Λ = +� +ε,ξ +Pεξ log +� +(1 − qξ)V + qξYε +� +, +(A11) +where Pεξ = lim +T→∞ +Tεξ +T +is the joint probability of the environment ε and the cue ξ. The optimal +germination fraction q∗ +ξ that maximizes Eq. (A11) is given by (assuming q∗ +ξ > 0): +q∗ +ξ = V P1|ξ +V − Y0 +− V P0|ξ +Y1 − V , +(A12) +which is the same as Eq. (A4) except that Pε is replaced by the conditional probability Pε|ξ = Pεξ +Pξ . +The maximum growth rate achieved by using the external cue is then given by plugging Eq. (A12) +into Eq. (A11), which gives: +Λcue = +� +ε,ξ +Pεξ log Pε|ξ + P0 log (Y1 − Y0)V +(Y1 − V ) ++ P1 log (Y1 − Y0)V +(V − Y0) . +(A13) +The difference between Λcue and Λbet is then: +Λcue − Λbet = +� +ε,ξ +Pεξ log Pε|ξ +Pε +≡ I(ε; ξ) , +(A14) +which is precisely the mutual information between the environment ε and the cue ξ. +15 + +A.2 +Numerical solution for age-dependent germination +In our model where the germination fraction depends on the seed age, neither the growth rate nor +the optimal germination fraction has an analytic solution. Here we describe how they are calculated +numerically. Since the seeds are heterogeneous in age, the population is described by a vector N +with components Nα that represents the number of seeds of age α. As described in the main text, +the vector N t at year t obeys the equation: +N t = M(εt; q) · N t−1 , +(A15) +where the matrix M depends on the current environment εt and the germination fractions qα ≡ +P(φ=1|α), as given in Eq. (3). Thus, the population vector after a long time T is: +N T = +� T +� +t=1 +M(εt; q) +� +· N 0 , +(A16) +and the long-term growth rate is formally given by the largest Lyapunov exponent of the product +of matrices: +Λ = lim +T→∞ +1 +T log +����� +T +� +t=1 +M(εt; q) +����� , +(A17) +where | · | is the matrix norm, which we choose to define as the largest eigenvalue for non-negative +matrices. Compared to Cohen’s model, here Λ cannot be calculated analytically because the matrix +multiplications are non-commutative. To numerically calculate Λ, we simply use the above equation +with a very large T, as the limit is expected to converge [33]. +We first draw a sequence of T random environments as follows. Define an epoch of time τε as the +number of consecutive years that the environment remains to be ε until it switches. The good and +bad epochs are drawn from the distributions: +P(τ1 =k) = 1 +µ1 +� +1 − 1 +µ1 +�k−1 +, +k = 1, 2, · · · , ∞ +(A18) +P(τ0 =k) = 1 +Z exp +� +− (k − µ0)2 +2σ2 +� +, +k = 1, 2, · · · , 2µ0−1. +(A19) +Here µε is the mean duration for the epochs, σ characterizes the variability of the bad epochs, and +Z is a normalization constant. For the example used in the main text (Fig. 1B), µ1 = µ0 = 5 and +σ = 2. 50000 epochs are drawn for each environment, with a total length T ≈ 500000. +To calculate Λ, we need to calculate the product �T +t=1 M(εt; q). For convenience, we define M (s) ≡ +�s +t=1 M(εt; q). Then M (T) can be calculated recursively by +M (t) = M(εt; q) · M (t−1), +(A20) +We normalize M (t) at every time step by the value of its largest entry, and this normalization +factor nt is stored. The Lyapunov exponent is then given by Λ = 1 +T +� �T +t=1 log nt + log w +� +, where +16 + +w is the largest eigenvalue of the normalized M (T) (which does not matter for Λ when T is large, +but matters for its derivative that we calculate below). +To find the germination fractions q∗ +α that maximizes Λ, we use the optimization routine L-BFGS-B, +which allows us to impose the constraint 0 ≤ q∗ +α ≤ 1. Besides the numerical function that calculates +Λ as described above, we also supply the Jacobian of the function, i.e., the derivative +∂Λ +∂qα . This +requires calculating the derivative of M (T) with respect to qα, which can be done using the recursive +relation +∂M (t) +∂qα += ∂M(εt; q) +∂qα +· M (t−1) + M(εt; q) · ∂M (t−1) +∂qα +, +(A21) +together with that for M (t) in Eq. (A20), from t = 1 all the way to T. We normalize ∂M(t) +∂qα +by the +same factor nt as for M (t) at every time step. The derivative of Λ is then given by +∂Λ +∂qα += 1 +T +1 +|M (T)| +∂|M (T)| +∂qα += 1 +T +1 +w +� +u · ∂M (T) +∂qα +· v +� +, +(A22) +where u and v are the left and right eigenvectors of M (T) corresponding to its largest eigenvalue +w. This derivative is then supplied as the Jacobian to the L-BFGS-B optimization routine to find +the optimal q∗ that maximizes Λ. +A.3 +Simulating a lineage +Simulation of a continuous lineage of seeds is used to estimate the joint probability P(εt, αt−1) of +the environment εt and the seed age αt−1 in Sec. 2.2, which is then used to calculate their mutual +information I(εt; αt−1) in Sec. 3.2. For a given set of germination fractions qα, the simulation is +done as follows. We start from a fresh seed of age 0. The sequence of environments, {ε1, · · · , εT }, +is drawn beforehand as described in Sec. A.2. +In each year, we decide whether the seed germinates or not using the germination probability that +corresponds to its age. To account for selection bias, we weight the probabilities by the fitness +values in the current environment. That is, in year t, the seed along the lineage has probability +qαt−1Yεt +qαt−1Yεt + (1 − qαt−1)V +to germinate and reset the age to 0, and otherwise stays dormant with its age increased from +αt−1 to αt = αt−1 + 1. We repeat this procedure from t = 1 to T, recording the sequence of αt. +Afterwards, the number of times that the pair (εt, αt−1) takes a particular combination of values is +counted, which is then normalized to be the joint probability distribution P(εt, αt−1), from which +the mutual information I(εt; αt−1) is calculated. +Lineage simulation is also used to calculate the distribution of dormancy duration in Sec. 4.1, i.e., +the distribution of how many consecutive years a seed stays dormant in the absence of environmental +17 + +variation. To calculate this distribution, we once again start with a fresh seed of age 0 and use +the probability q0 to decide if the seed germinates. This time the probability is not weighted by +the fitness because we are calculating the dormancy durations in the absence of selection. The +above procedure is repeated for a long period of time T and the sequence of phenotypes at each +time step is recorded as φt. The duration of germination or dormancy is calculated by parsing the +sequence of phenotypes {φt} into consecutive epochs of germination or dormancy. The distribution +of their durations is then calculated by normalizing the histograms of these epochs. Note that these +distributions can also be calculated using Eq. (A30) in Appendix A.5. +A.4 +Exhaustive search of state diagrams +To verify that the age-diagram in Fig. 5A is the optimal topology, we test all possible state diagrams +for up to 6 internal states. For a diagram with L states, we label the states as s0, s1, · · · , sL−1. +Each state has two outgoing arrows, corresponding to either dormancy or germination. Each arrow +can go to any other state or loop back. Therefore, naively, there can be L2L possible diagrams. +However, many of these diagrams are equivalent in the sense that they are simply permutations +of the states. To remove the redundant diagrams, we use a “sieve” method as follows. We first +represent a diagram by a (L × 2) integer matrix, whose entry of the α-th row and φ-th column +represents which state the system will transition to if it is at age α and expresses phenotype φ. +The diagrams are then indexed by a number that results from flattening the matrix and treating +it as a base-L number. +Then, we enumerate all L2L diagrams starting from the index 0. +For +each diagram, we find all its permutations and remove their indices from the list. Furthermore, +we exclude diagrams that have two or more disjoint parts to keep only connected diagrams. We +go over the list of diagrams, skipping the indices that have been removed. In the end, the total +number of non-degenerate diagrams for L = 1, 2, · · · is +n(L) = 1, 6, 52, 892, 21291, 658885, · · · +which is the number of unlabeled, strongly connected, L-state, 2-input automata (Sequence A027835 +from OLEIS). This number grows quickly and we are only able to study diagrams for up to L = 6. +For each of the diagrams with L states, we numerically find the optimal qα and the maximum Λ +as in Sec. A.2. This is computationally intensive and is done on a computer cluster. Then, among +all diagrams of L states, we find the optimal diagram with the largest Λ. For up to L = 6, it turns +out that the age-diagram is the optimal diagram for our model. +A.5 +Analytical results for extreme selection +In the limit of extreme selection, the fitness matrix is diagonal, i.e., fεφ = +� V 0 +0 Y +� +. This means, +hypothetically, that a seed can survive only if it germinates in a good year or stays dormant in a +18 + +bad year. In this case, the long-term growth rate Λ and the optimal germination fractions q∗ +α have +analytical solutions. Indeed, the population becomes homogeneous because, once it encounters +a good year, only the seeds that germinate will survive, and subsequently the population will +consist of only fresh seeds. From then on, the seed age will be synchronized with the number of +consecutive bad years, and will be reset to 0 whenever there is a good year. Let βt−1 be the number +of consecutive bad years right before year t (which is 0 if the previous year is good). It will be equal +to the seed age αt−1 of the population at the beginning of year t. Therefore, the seed population +changes over time according to: +Nt = Nt−1(1 − qβt−1)V +or +Nt−1 qβt−1Y , +(A23) +depending on whether the environment εt = 0 or 1. Over a period of time T, the number of seeds +will be: +NT = N0 +� +β +� +(1 − qβ)V +�T0β� +qβY +�T1β , +(A24) +where Tεβ is the number of years that the environment is ε while the previous number of consecutive +bad years is β. This equation has the same form as Eq. (A10), with the external cue ξ replaced by +β. The long-term growth rate has the expression +Λ ≡ lim +T→∞ +1 +T log NT +N0 += +� +β +P0β log[(1 − qβ)V ] + +� +β +P1β log[qβY ] +(A25) +where Pεβ = lim +T→∞ +Tεβ +T +is the joint probability of the environment εt and the number of bad years +βt−1. Setting the derivative +∂Λ +∂qα = 0, the optimal germination fractions q∗ +α are found to be +q∗ +α = +P1α +P0α + P1α +≡ P1|α ≡ P(εt =1|βt−1 =α) . +(A26) +Here P(εt =1|βt−1 =α) represents the conditional probability that the coming year is good, given +that there has been α consecutive bad years. It is related to the duration distribution of bad years, +P(τ0) from Eq. (A19), through +P(εt =1|βt−1 =α) = P(τ0 =α) +P(τ0 ≥α). +(A27) +An important consequence of this result is that, for the germination fractions q∗ +α, the dormancy +duration of the seeds (as in Fig. 7B) will have the same distribution as the duration of bad years +(Fig. 1B). This is because, by definition, qα ≡ P(φt = 1|αt−1 = α). Let δ0 denote the duration of +dormancy, then similar to Eq. (A27), we have +P(φt =1|αt−1 =α) = P(δ0 =α) +P(δ0 ≥α) . +(A28) +Equating the left-hand sides of Eqs. (A27) and (A28) leads to, as stated above, +P(δ0 =α) = P(τ0 =α) . +(A29) +19 + +Incidentally, for a general qα, it can be shown that +P(δ0 =α) = qα +α−1 +� +k=1 +(1 − qk) . +(A30) +References +[1] Baskin CC, Baskin JM. +Seeds: Ecology, Biogeography, and Evolution of Dormancy and +Germination. 2nd ed. Elsevier Science; 2014. +[2] Lennon JT, Jones SE. Microbial seed banks: The ecological and evolutionary implications of +dormancy. Nat Rev Microbiol. 2011;9(2):119-30. +[3] Lennon J, den Hollander F, Wilke-Berenguer M, Blath J. Principles of seed banks and the +emergence of complexity from dormancy. Nat Commun. 2021;12(4807). +[4] Balaban NQ, Merrin J, Chait R, Kowalik L, Leibler S. Bacterial persistence as a phenotypic +switch. Science. 2004;12(5690):305. +[5] Harms A, Maisonneuve E, Gerdes K. Mechanisms of bacterial persistence during stress and +antibiotic exposure. Science. 2016;354(6318):aaf4268. +[6] Manuse S, Shan Y, Canas-Duarte SJ, Bakshi S, Sun WS, Mori H, et al. 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Phys Rev X. 2015;5(4):041014. +22 + diff --git a/2NE1T4oBgHgl3EQf5QXz/content/tmp_files/load_file.txt b/2NE1T4oBgHgl3EQf5QXz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b08d817376ce6db689a0cc7608e96ed55c32c4b8 --- /dev/null +++ b/2NE1T4oBgHgl3EQf5QXz/content/tmp_files/load_file.txt @@ -0,0 +1,757 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf,len=756 +page_content='The Value of Internal Memory for Population Growth in Varying Environments Leo Law, BingKan Xue Department of Physics, University of Florida, Gainesville, FL 32611, USA Abstract In varying environments it is beneficial for organisms to utilize available cues to infer the conditions they may encounter and express potentially favorable traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' However, external cues can be unreliable or too costly to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We consider an alternative strategy where organisms exploit internal sources of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Even without sensing environmental cues, their internal states may become correlated with the environment as a result of selection, which then form a memory that helps predict future conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' To demonstrate the adaptive value of such internal memory in varying environments, we revisit the classic example of seed dormancy in annual plants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Previous studies have considered the germination fraction of seeds and its dependence on environmental cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In contrast, we consider a model of germination fraction that depends on the seed age, which is an internal state that can serve as a memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We show that, if the environmental variation has temporal structure, then age-dependent germination fractions will allow the population to have an increased long-term growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The more organisms can remember through their internal states, the higher growth rate a population can potentially achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Our results suggest experimental ways to infer internal memory and its benefit for adaptation in varying environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='03511v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='PE] 9 Jan 2023 1 Introduction Organisms can adapt to a varying environment by diversifying their traits among individuals of the same population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' A common form of such diversity is dormancy, where some individuals enter a dormant state while others remain active [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Those that are active will contribute to the growth of the population under good environmental conditions, but will be vulnerable to periods of harsh conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' On the other hand, the dormant individuals are often tolerant to environmental stress and thus help preserve the population during harsh periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For example, in a bacterial population, while most cells grow and divide normally, some cells randomly switch to a reversible dormant state called persister cells, which makes them tolerant to antibiotics when normal cells would perish [4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Other examples include seed dormancy in plants, dauer larva in nematodes, diapause in insects, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' [1, 3, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' These are thought to be a strategy known as diversified bet-hedging [9, 10], in which organisms express different traits with some probability to create diversity in the population, so as to increase the long-term growth rate of the population under environmental variations [11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In the simplest form, bet-hedging organisms have fixed probabilities of expressing different traits [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' But more generally, organisms can sense cues from the environment that will influence these probabilities [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Such cues may be indicative of future environmental conditions, so that the organisms may bias the probabilities towards traits that are favorable in the likely environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' It has been shown that the information contained in the cue about the environment will contribute to an increase in the population growth rate [14, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' However, sensing and responding to environmental cues may come at a cost, as it requires the expression of specific sensors and signaling mechanisms [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Besides, there may not be enough time for the organisms to respond to the cues through phenotypic plasticity, as the environment may have changed by the time the trait is developed [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Therefore, it is not always beneficial to rely on environmental cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Besides external signals, the behavior of organisms can be influenced by their internal states, such as physiological or metabolic states [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' One example is the reserve level – a starved animal may choose to forage more aggressively despite higher predation risk [22, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Another example is the age of the organism – it is known that the age of seeds can affect germination in annual plants [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' These internal states are not sensors that directly measure the external environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' However, they may become correlated with the environment as a result of selection, because certain states are associated with higher fitness in past environmental conditions and thus become more common in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Therefore, the distribution of such internal states among the population can potentially provide information about the environment, which may be utilized by the organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We will study an example of this situation and show that internal states of the organisms can indeed serve as internal cues to help them adapt to varying environmental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Such internal states effectively provide a memory about the past outcomes of selection, which helps predict the future environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Moreover, we show that a larger memory capacity enables higher gains in the 2 population growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Our results suggest that internal states that were not developed for sensing the environment could nevertheless be co-opted as internal cues for adaptation, which would save the cost of sensors and may thus be a more efficient strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' To study adaptation in varying environments, we will use seed dormancy as our main example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Seeds of annual plants will either germinate or stay dormant in a given year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' While dormancy sacrifices the short-term fitness of the seeds, it preserves the population from a catastrophically bad year with very low yield, and thus results in higher long-term benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This has been studied as a classic model of bet-hedging [11, 14], supported by the fact that dormant seeds eventually ger- minate under similar environmental conditions [23], and that the germination fraction is negatively correlated with local environmental variability [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' It is known that germination is influenced by environmental cues, such as temperature, humidity, and the number density of surrounding seeds [15, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Moreover, there is evidence that the probability a seed will germinate also changes with the age [26, 27, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' However, the adaptive value of such age dependence in germination has not been fully studied [28, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' It was shown in [28] that the evolutionarily stable probability of germination does not depend on seed age if there is no density dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Yet, their model did not include temporal correlation in the environmental variation, which is crucial for memory to be useful in predicting future environments [29, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We will show that, when there is temporal structure in the environmental variation, age-dependent germination probabilities can increase the long-term growth rate of the seed population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='1 Cohen’s model of seed dormancy Let us first briefly review the idea of bet-hedging and how information emerges as a central quantity in determining the long-term growth rate of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We will follow the classic model of seed dormancy in annual plants by Cohen [11, 14], as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Each year can be “good” (denoted as environment ε = 1) or “bad” (ε = 0) for the plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Seeds that germinate (“phenotype” φ = 1) in a good year will be able to grow and produce a large number (Y1) of new seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' However, in a bad year, germinated plants will have a low yield (Y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We will set Y0 = 0 and denote Y1 = Y for simplicity, meaning that germinating in a bad year will result in no offspring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' All germinated plants perish at the end of the year, regardless of their yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Seeds that stay dormant (φ = 0) will remain viable the next year with probability V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Thus, the fitness of a seed in a given environment can be summarized by the matrix fεφ = � V 0 V Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In addition, we assume that the number of consecutive good years follows a geometric distribution, whereas that of bad years has a narrow distribution (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 1B and Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This is meant to describe the scenario where good growth conditions are disrupted by random occurrence of disasters that affect growth for a characteristic number of years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 3 dormant A germinate year 2 viability dormant germinate year 1 low yield viability bad .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' C B seeds ⋯ s1 s2 s0 good high yield seeds Figure 1: (A) Schematic illustration of Cohen’s model of seed dormancy in annual plants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Each year may be good or bad for plant growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' A seed can either germinate to produce a yield Yε that depends on the environmental condition ε, or stay dormant with a probability V of still being viable next year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The number of seeds at the end of year t is Nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The parameter values used in our calculations are Y0 = 0, Y1 = 4, V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (B) The distribution of duration of consecutive good years and bad years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We choose the duration of good years to follow a geometric distribution with a mean of 5, and the duration of bad years to have a Gaussian distribution with a mean and standard deviation of 5 ± 2 cut off at 0 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (C) A state diagram that represents the seed age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Each state sα represents a seed of age α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Blue arrows represent dormancy that increases the age by 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' orange arrows represent germination that may produce new seeds of age 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Weights on the arrows represent the probability of germination or dormancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In the simplest case where seeds receive no environmental cues, the fraction of seeds that germinate each year is assumed to be a constant, denoted by q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In a good year, the total number of seeds will grow by a factor (1 − q)V + qY , whereas in a bad year, the number of seeds will reduce to only a fraction (1 − q)V of the previous year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The long-term growth rate of the population will be given by (see derivation in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='1) Λ = p log � (1 − q)V + qY � + (1 − p) log � (1 − q)V � , (1) where p is the frequency of good years and (1 − p) is that for bad years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The germination fraction that maximizes the long-term growth rate is q∗ = p Y − V Y − V (2) for p > V/Y and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In the limit of high yield (Y ≫ V ), this leads to the classic result q∗ ≈ p, which means the optimal germination fraction should match the frequency of good years [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The model can be extended to seeds that receive some external cue (ξ) about the environment [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In this case, the optimal germination fraction will depend on the cue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' As a result, the population can grow faster than without the cue (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='25 Good years Bad years 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='20 distribution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='10 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='00 12345678910 12345678910 duration duration growth rate perfect information external cue no cue perfect memory internal state no memory A B Figure 2: The long-term growth rate Λ of populations with different sources of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A) The value of external cues: Λmax is the maximum possible growth rate attainable if the population has perfect information about the future environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Λbet is the highest growth rate achievable by a bet-hedging population without receiving cues, which is suppressed by the entropy of the environment H(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Λcue is the growth rate when the population utilizes a cue ξ that has a mutual information I(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' ξ) with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (B) The value of internal memory: Organisms can utilize their internal states as memory, such that their behavior depends on which state they are in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Λbet from bet-hedging also represents the case with no memory, which corresponds to having only one internal state (L = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' More states (L > 1) provides larger memory capacity and allows a higher growth rate Λint for the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Λmem is the highest growth rate achievable by organisms with a perfect memory (L → ∞) of their lineage history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' These well-known results are summarized schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' At the top level is the maximum possible growth rate Λmax, which is attainable only if individuals have perfect information about future environmental conditions and respond accordingly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=', germinate if it will be a good year and go dormant if it will be bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' On the other hand, if there is no environmental cue, the best strategy is bet-hedging with fixed probabilities, which achieves a growth rate Λbet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This is less than Λmax by an amount H(ε), which is the Shannon entropy from information theory that quantifies the uncertainty of the varying environment (See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' However, if a cue ξ is used to help predict the environment, the population can increase the growth rate from Λbet to Λcue, up by an amount I(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' ξ) that is equal to the mutual information between the cue and the environment (Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Note that Λcue is still not as high as Λmax unless the cue is fully accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The relations between these growth rates illustrated here (similar to plots in [16, 32]) show that, in order for the population to better adapt to varying environments, it must utilize available sources of information about the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='2 Internal source of information Instead of sensing external cues, below we consider another possibility for organisms to use their internal states as a source of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We will use the age of seeds as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The state diagram representing seed ages are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 1C, where a state sα represents a seed of age α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' A blue arrow represents a seed going into dormancy for one year, so that the age is increased by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' An orange arrow represents a seed that germinates and potentially produces new seeds, 5 which will have age 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The weights on the arrows represent the probability of germination or dormancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For a simple bet-hedging strategy without any cues, the probability of germination will be a constant, which equals q∗ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (2), independent of the seed age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We will study the case where the germination fraction can depend on the seed age, and show that the population can acquire information from this internal state to achieve a higher growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 3 Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='1 Seed age as an internal cue We first study whether the seed age as an internal state contains useful information about the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Let αt−1 be the seed age at the beginning of year t, and εt be the coming environment that year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' If αt−1 has no information about the environment, then it will be statistically independent of εt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=', P(εt|αt−1) = P(εt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Therefore, whether seed age is informative about the environment can be inferred from the conditional probability P(εt|αt−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' To calculate that, we simulate a sufficiently long sequence of environments, denoted by εt for each year t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We also simulate a single lineage of plants that uses the constant germination fraction q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Each year the seed can either germinate or stay dormant, and the probability of choosing the phenotype φt is further weighted by the fitness f(εt, φt) to account for selection (see procedure in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The seed age along the lineage is recorded as αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' From the sequences of εt and αt, we estimate the joint probability distribution P(εt, αt−1), from which the conditional probability P(εt|αt−1) is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 3, the probability of the environment εt does depend on the seed age αt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This means that knowing the seed age allows a more accurate prediction of the coming environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Therefore, it is possible for the population to “co-opt” the seed age as an “internal cue” for the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In analogy to the case of external cues, we expect that such information can be used to increase the long-term population growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We therefore consider a strategy where the germination fraction depends on the seed age, denoted by qα and represented by weights on the arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 1C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' To calculate the long-term growth rate, let N be a vector that represents the age-structured population, with components Nα being the number of seeds of age α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The dynamics of N is described by a matrix M(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' q) that depends on the environment ε and the germination fractions q (with components qα), M(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' q) = � � � � � � � q0 Yε q0 Yε · · (1−q1)V 0 · · 0 (1−q2)V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' � � � � � � � (3) Each year, the population vector is multiplied by the matrix that corresponds to the current 6 0 1 2 3 4 5 6 7 8 9 seed age, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='0 probability, P( t| t 1) bet-hedging age-dependent Figure 3: Probability of the coming environment εt conditioned on the seed age αt−1 at the beginning of year t, as calculated by simulating a lineage of seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Dashed line is the marginal probability of the environment, which would indicate that the seed age is uncorrelated with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Blue bars are when the population uses a bet-hedging strategy with a constant germination fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Orange bars are when the germination fraction depends on the seed age to maximize population growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In both cases the seed age is correlated with the environment and thus useful as an internal cue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' environment εt, N t = M(εt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' q) · N t−1 , (4) Here M(εt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' q) is a random matrix because εt is a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The temporal sequence of εt is randomly drawn according to the distributions of good and bad years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The long-term growth rate Λ of the population is then given by the Lyapunov exponent of the product of these random matrices [33], which is calculated numerically (see methods in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We vary the age-dependent germination fractions qα to maximize Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' As expected, this growth rate using seed age as an internal cue (Λint) is greater than that of bet-hedging without cues (Λbet), as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 2B (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 6 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The optimal germination fraction as a function of seed age is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' An intuitive explanation for the age dependence is that, in this example, the bad environment typically lasts a number of years, so it is advantageous for a seed to stay dormant for a similar period of time to wait it out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Those that germinate in the wrong phase of the bad year cycle will be eliminated by selection, and the remaining individuals tend to be synchronized with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In contrast, if there is no temporal structure in the environment, such as when the environment is randomly and independently chosen each year, then the seed age will no longer be correlated with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In that case, the best strategy is to have a constant germination fraction (equal to q∗ in the bet-hedging case, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 4), as argued in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Note that the information about the environment is contained in the distribution of seed ages within the population, which results from selection in previous years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Compared to the case of an external cue that is shared by all individuals, the seed age varies among individuals (which prevents 7 0 1 2 3 4 5 6 7 8 9 seed age, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='0 germination fraction, q temporally structured uncorrelated environment Figure 4: Dependence of the germination fraction q on the seed age α that maximizes the population growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Blue bars are when the environment is temporally structured, as described by the duration of good and bad years in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Orange bars are when the environment is drawn independently each year, for which the germination fraction need not depend on seed age and is equal to the bet-hedging solution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 2 (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' an analytic expression for Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' It acts as an individual’s memory of its own lineage history, which helps it infer the likely environment in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Importantly, the increase in population growth rate does not come at any cost associated with sensing external cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Thus, such an internal source of information proves to be beneficial for the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='2 Internal states as memory We have shown that internal states of organisms may help them “remember” the past outcomes of selection to be able to predict the future environment, leading to an increased population growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Intuitively, the more the organisms can remember, the better they may predict and adapt to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' To test this in our model, we can vary the memory size by changing the number of possible internal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The state diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 1C has potentially an infinite number of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' They can be truncated at a finite number L, such that seeds exceeding age (L − 1) will remain in the state sL−1 until they germinate or perish (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 5A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This allows us to study how the population growth rate depends on the number of states L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We first note that having only one internal state (L = 1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 5B) is effectively having no memory, because the system will always be in that same state regardless of the past events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In this case, the germination fraction is always equal to q0 associated with the only state s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Having a constant germination fraction means that this case corresponds to the simple bet-hedging strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The maximum long-term growth rate will just be Λbet achieved at q0 = q∗ found in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For two internal states (L = 2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 5C), the model reduces to “phenotypic switching”, in which 8 sL−1 ⋯ A B C s0 s1 s0 s0 s1 Figure 5: State diagrams for age-dependent germination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A) The germination fraction q depends on the seed age α up to α = L−1, beyond which it remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Varying the length L effectively varies the memory capacity of the organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (B) With only one state (L = 1), the organism effectively has no memory, and the germination fraction is a constant, corresponding to simple bet-hedging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (C) The two-state case corresponds to a Markov process where the organisms switch back and forth between two phenotypes, with transition probabilities P(φ1|φ0) = q1 and P(φ0|φ1) = 1−q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' the organisms randomly switch between two phenotypes (germination or dormancy) with fixed transition probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Specifically, the probability for a dormant seed to germinate next year is q1, and the probability for a new seed (that came from a germinated plant) to go dormant is 1−q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This is a Markov process, for which the transition between phenotypes does not depend on how long a phenotype has lasted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' It implies that the germination fraction only depends on whether the seed is fresh (age 0) or has been dormant (age > 0), but not on how long it has been dormant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' As a result of being Markovian, the duration of the dormant phenotype will be geometrically distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' A larger L will allow the germination fraction to depend more sensitively on the seed age (L > 2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 5A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The number of states L roughly represents how many dormant years a seed can remember.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For each number L, we search for the maximum long-term growth rate Λ over the parameters {q0, · · · , qL−1} (see methods in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 6, Λ increases monotonically as more states are incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Therefore, more memory allows faster population growth and hence better adaptation to environmental variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Note that Λ quickly approaches a limit Λmem when L becomes greater than the typical duration of the bad environment (equal to 5 in this example, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Intuitively, there is no need to remember longer dormancy because there is no benefit in staying dormant for longer than the duration of bad years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The relation between the growth rate and memory is illustrated schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' If we think of seed age as an internal cue for the environment, we can calculate the mutual infor- mation I(εt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' αt−1) between the environment εt and the seed age αt−1, using the joint probability P(εt, αt−1) calculated the same way as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 6 shows that the mutual information also increases with the number of states L, as more memory is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' When plotted against each other, the long-term growth rate Λ increases with the mutual information I (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 6 inset), just like 9 1 2 3 4 5 6 7 8 9 10 number of states, L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='28 long-term growth rate, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='20 mutual information, I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='20 info, I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='28 growth, growth rate mutual info Figure 6: Long-term growth rate Λ of populations that have different memory capacity as measured by the number of internal states L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For each L, the age-dependent germination fractions qα are chosen to maximize Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Also plotted is the mutual information I between the previous seed age αt−1 and the environment εt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Both Λ and I increase monotonically with the memory capacity L, approaching their respective limits as L ≫ 5 (mean duration of bad years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (Inset) Long-term growth rate Λ increases monotonically with the mutual information I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Gray diagonal line represents Cohen’s model with external cues, in which Λ = Λbet+I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' for an external cue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Note that in Cohen’s model with external cues [14], Λ is simply proportional to I (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A14) in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In comparison, for the same amount of information I, the population achieves a higher growth rate Λ using seed age as an internal cue (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 6 inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' So far we have considered a very specific structure for the state diagrams (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 5A, “age-diagram”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' It might be possible that, given the number of internal states, there are other diagrams that can lead to a high long-term growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Such diagrams could represent other types of internal states instead of the age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For example, the reserve level of an organism can be represented by a linear diagram, such that the organism moves up one or more states if it succeeds in foraging or moves down one state if it fails [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' To find which structure of internal states provides the highest long-term growth rate for the population, we searched all possible diagrams of a given number of states (up to L = 6, beyond which it is computationally difficult), optimizing the weights qα for each diagram (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' It turns out that the age-diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 5A is optimal for the temporal structure of the environment that we assumed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In general, the state diagram is a mathematical representation of memory, known as the “ϵ-machine” of a stochastic process [34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' a formal treatment and application to population growth in varying environments is given by [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 10 1 2 3 4 5 6 7 8 9 10 duration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='30 distribution Germination A 1 2 3 4 5 6 7 8 9 10 duration Dormancy B L = 2 L = 5 L = 10 Figure 7: The distribution of the duration of consecutive germinations or dormant years along a lineage of seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Different colors correspond to age-dependent germination fractions qα for different memory capacities L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A) For each L, the duration of germinations matches a geometric distribution with a mean of 1/q0 (dashed line for L = 2 and solid line for L = 10), meaning that there is no memory of previous germinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (B) The duration of dormancy has a distribution that changes shape depending on the memory capacity L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' L = 2 (phenotypic switching) results in a geometric distribution with a mean of 1/(1−q1) (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Larger L’s result in deviation from a geometric distribution, which is indicative of having internal memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 4 Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='1 Characterization of internal memory Memory arising from age-dependent germination fractions can be characterized by the distribution of the duration of dormancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' That is, given a large number of fresh seeds, what is the distribution of the time that each seed stays dormant before germinating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' To calculate this distribution, we simulate one lineage of seeds over a long time in the absence of selection (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='3), and record the sequence of phenotypes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=', whether a seed germinated or not each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 7 shows the distribution of the number of consecutive years that successive seeds germinate or that a seed stays dormant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The number of consecutive germination years is geometrically distributed with a mean of 1/q0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 7A), because every new seed has the same probability q0 of germinating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In other words, a new seed has no memory of the age of the plant that it came from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Thus, the absence of phenotypic memory is signified by the geometric distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' On the other hand, the distribution of the consecutive dormant years (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=', the duration of dor- mancy) depends on the number of internal states L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For L = 2, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='2, there is no memory of how long a seed has been dormant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Indeed, the distribution of dormancy durations is geometric with a mean of 1/(1−q1) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 7B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' But as L increases, the distribution becomes more bell-shaped and closer to the distribution of consecutive bad years (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (In the limit where the fitness matrix fεφ is diagonal, the optimal strategy will be such that the duration of each phenotype exactly matches the distribution of the corresponding environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The deviation of the distribution from being geometric indicates that the seed has memory of how long it has been dormant, which is necessary for the germination fraction to depend on the seed 11 age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Thus, the shape of the dormancy distribution can be used as an experimental signature of internal memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The best demonstration of memory in phenotypic changes is found in experiments on the bacteria Bacillus subtilis [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' During its growth, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' subtilis can switch between two phenotypes, either as a free-moving cell by making flagela or as part of an aggregate by producing extracellular matrix [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' It is thought that the aggregate cells have an advantage for colonization and can better cope with a harsh environment by sharing resources, whereas the motile cells are better at dispersing and searching for nutrients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The durations of these two cell types along continuous cell lineages are measured in a constant environmental condition [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' It was found that the time a lineage stays in the motile cell type follows an exponential distribution with a mean of ∼ 81 generations, while the aggregate cell type is maintained for a narrowly distributed duration with a mean and standard deviation of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='1 generations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 2(d,f) of [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This implies that the motile cell type is memoryless while the aggregate cell type has memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' That is, an aggregate cell keeps track of how long it has been part of an aggregate, whereas a motile cell turns off motility with a fixed probability at every cell division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' These two distributions of phenotype durations look similar to those found in our model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Importantly, since the switching of cell types is measured in a constant environment, it is evident that the phenotypic changes are influenced by some internal states of the cell, rather than external cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This method of inferring the existence of internal memory by measuring the duration of phenotypes can be potentially applied to seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' It would require measuring the duration of seed dormancy by planting seeds in separate pots under the same environmental condition and recording how soon they germinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='2 Evidence for age-dependent dormancy Our model assumes that the probability of a seed entering or exiting dormancy depends on the age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' If the bad environment typically persists for a number of years, then the model predicts that the probability of exiting dormancy should be small initially and increase over a timescale that matches the duration of bad years (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Data from past experiments have shown that for different species the germination fraction can either increase or decrease between the first and second years [23], while data going beyond the second year are scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' To test the above prediction also requires knowing the statistics of bad years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Alternatively, age-dependent germination can be tested by measuring the distribution of dormancy durations, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 7B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For that purpose, one has to measure the final age of seeds right before they germinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Studies on seed age structure have been done in the past [26, 27], but with the goal of measuring the current age of seeds in a population at a given time, even though some seeds will continue to be dormant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We are not aware of existing studies that measured the distribution of final seed ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Dormancy in other organisms can also be studied using our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' One example is insect diapause [38], which is considered another example of bet-hedging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In many insect species, the larvae can 12 enter diapause at a certain developmental stage to avoid unfavorable conditions, instead of pro- ceeding with normal development to become adults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In a simple model of diapause [39], the larvae may undergo multiple years of diapause and have a fixed probability of (re)entering diapause each year (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 1 of [39]), similar to Cohen’s model of seed dormancy [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This would correspond to our model with L = 1, such that the decision to enter diapause is memoryless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Another model assumes that the larvae can only undergo one period of diapause and must exit after that [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This pattern is a special case of our model with L = 2, where the state s0 would correspond to a new larva and s1 to diapause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The larva can either develop to an adult with probability q0 and produce offspring (arrow from s0 back to itself), or enter diapause with probability 1 − q0 (arrow to s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' However, once it undergoes diapause, it must exit and develop, so there is only one arrow leaving s1, which goes to s0 with probability q1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In this scenario, it was found that diapause is ben- eficial in varying environments that are temporally correlated [40], in agreement with our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' More generally, one may study situations where diapause can be repeated for a number of times, which would correspond to a diagram like Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 5A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Our results suggest that which form of diapause is evolutionarily favored depends on the complexity of temporal structure in the environmental variation, which could potentially be tested in empirical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 5 Conclusion We have shown that the internal states of organisms can serve as a memory to help the population adapt in varying environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In order for this strategy to be useful, the environment must be temporally structured, and the internal states must become correlated with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We have demonstrated that such correlation can arise from selection alone, without direct interaction with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' More generally, some internal states of organisms may be correlated with the environment as a result of phenotypic plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For example, seeds produced in a good year may be bigger than those produced in a bad year, so seed size could provide a memory of the past environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' It is known that seed size can affect germination probability [41], and it will be interesting to study if such dependence can benefit population growth in varying environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Organisms are complex systems with a lot of internal degrees of freedom, some of which might happen to become correlated with the environment through selection or plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Even though these internal states might not have developed as sensors for environmental cues, they could be co-opted as information sources to guide the organism’s behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' To test whether seed age could be co-opted to affect germination, one might compare accessions of annual plants in temporally structured environments and those in unpredictable environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Our model predicts that the germination fraction would evolve to depend on the seed age in the former case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Dormancy has been proposed to cause a “storage effect” that promotes species coexistence in vary- ing environments [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Our model of age-dependent dormancy may be studied in such community 13 ecology context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' If the presence of other species is viewed as part of the environment for the focal species, then internal states such as seed age could potentially provide a memory of past interac- tion with those other species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For example, reserve level of the predator may be an indicator of past encounters with prey [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' History-dependent ecological interactions have been experimentally indicated in microbial communities [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' It will be interesting to use our framework to study such ecological dynamics of organisms whose phenotypes depend on their memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' A Methods A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='1 Analytic derivation of Cohen’s model Consider a population of annual plant seeds, each of which can either germinate (φ = 1) or stay dormant (φ = 0) each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The environment can be either good (ε = 1) or bad (ε = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' If a seed germinates in a good year, it will reproduce and yield Y1 number of seeds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' but a seed germinating in a bad year will only yield Y0 seeds, with Y1 > Y0 (in the main text we set Y0 to 0 for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' If a seed stays dormant, then the probability that it will remain viable is V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For Y1 > V > Y0, it is favorable for a seed to germinate in a good year but stay dormant in a bad year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The number of seeds at year t is denoted by Nt and obeys the equation: Nt = Nt−1 � (1 − q)V + qYεt � , (A1) where εt is the environment in that year and q is the fraction of seeds that germinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The number of seeds at year T can be calculated recursively as: NT = N0 T � t=1 � (1 − q)V + qYεt � = N0 � (1 − q)V + qY0 �T0� (1 − q)V + qY1 �T1, (A2) where Tε is the total number of years that the environment is ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The long-term growth rate Λ is defined as the asymptotic rate of logarithmic increase: Λ ≡ lim T→∞ 1 T log NT N0 = P0 log � (1 − q)V + qY0 � + P1 log � (1 − q)V + qY1 � , (A3) where Pε ≡ lim T→∞ Tε T is the frequency of environment ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The germination fraction q∗ that maximizes Λ is found by setting the derivative ∂Λ ∂q to zero, which gives (assuming q∗ > 0): q∗ = V P1 V − Y0 − V P0 Y1 − V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A4) And the corresponding maximum growth rate Λbet is: Λbet = P0 log P0(Y1 − Y0)V (Y1 − V ) + P1 log P1(Y1 − Y0)V (V − Y0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A5) 14 If the seeds have perfect information about the future environment, then they should all germinate in good years and stay dormant in bad years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This would result in a total population NT = N0 V T0 Y T1 1 instead of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A2), which gives the maximum possible growth rate: Λmax = P0 log V + P1 log Y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A6) The difference between Λmax and Λbet is then given by: Λmax − Λbet = −P0 log P0(Y1 − Y0) (Y1 − V ) − P1 log P1(Y1 − Y0)V (V − Y0)Y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A7) In the limit Y0 → 0 and Y1 ≫ V , it simplifies to: Λmax − Λbet = −P0 log P0 − P1 log P1 ≡ H(ε) , (A8) which is the entropy of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The model above can be generalized to include an external cue ξ that is correlated with the environment ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Assume that, given ξ, the seeds will germinate with probability P(φ = 1|ξ) ≡ qξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The total number of seeds then obeys the equation: Nt = Nt−1 � (1 − qξt)V + qξt Yεt � , (A9) where ξt is the cue received in year t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Repeating the same procedure as above, one finds that the population after T years becomes: NT = N0 � ε,ξ � (1 − qξ)V + qξYε �Tεξ, (A10) where Tεξ is the number of years that the environment is ε while the cue is ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The long-term growth rate is then given by: Λ = � ε,ξ Pεξ log � (1 − qξ)V + qξYε � , (A11) where Pεξ = lim T→∞ Tεξ T is the joint probability of the environment ε and the cue ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The optimal germination fraction q∗ ξ that maximizes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A11) is given by (assuming q∗ ξ > 0): q∗ ξ = V P1|ξ V − Y0 − V P0|ξ Y1 − V , (A12) which is the same as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A4) except that Pε is replaced by the conditional probability Pε|ξ = Pεξ Pξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The maximum growth rate achieved by using the external cue is then given by plugging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A12) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A11), which gives: Λcue = � ε,ξ Pεξ log Pε|ξ + P0 log (Y1 − Y0)V (Y1 − V ) + P1 log (Y1 − Y0)V (V − Y0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A13) The difference between Λcue and Λbet is then: Λcue − Λbet = � ε,ξ Pεξ log Pε|ξ Pε ≡ I(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' ξ) , (A14) which is precisely the mutual information between the environment ε and the cue ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 15 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='2 Numerical solution for age-dependent germination In our model where the germination fraction depends on the seed age, neither the growth rate nor the optimal germination fraction has an analytic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Here we describe how they are calculated numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Since the seeds are heterogeneous in age, the population is described by a vector N with components Nα that represents the number of seeds of age α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' As described in the main text, the vector N t at year t obeys the equation: N t = M(εt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' q) · N t−1 , (A15) where the matrix M depends on the current environment εt and the germination fractions qα ≡ P(φ=1|α), as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Thus, the population vector after a long time T is: N T = � T � t=1 M(εt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' q) � N 0 , (A16) and the long-term growth rate is formally given by the largest Lyapunov exponent of the product of matrices: Λ = lim T→∞ 1 T log ����� T � t=1 M(εt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' q) ����� , (A17) where | · | is the matrix norm, which we choose to define as the largest eigenvalue for non-negative matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Compared to Cohen’s model, here Λ cannot be calculated analytically because the matrix multiplications are non-commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' To numerically calculate Λ, we simply use the above equation with a very large T, as the limit is expected to converge [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We first draw a sequence of T random environments as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Define an epoch of time τε as the number of consecutive years that the environment remains to be ε until it switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The good and bad epochs are drawn from the distributions: P(τ1 =k) = 1 µ1 � 1 − 1 µ1 �k−1 , k = 1, 2, · · · , ∞ (A18) P(τ0 =k) = 1 Z exp � − (k − µ0)2 2σ2 � , k = 1, 2, · · · , 2µ0−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A19) Here µε is the mean duration for the epochs, σ characterizes the variability of the bad epochs, and Z is a normalization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For the example used in the main text (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 1B), µ1 = µ0 = 5 and σ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 50000 epochs are drawn for each environment, with a total length T ≈ 500000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' To calculate Λ, we need to calculate the product �T t=1 M(εt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For convenience, we define M (s) ≡ �s t=1 M(εt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Then M (T) can be calculated recursively by M (t) = M(εt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' q) · M (t−1), (A20) We normalize M (t) at every time step by the value of its largest entry, and this normalization factor nt is stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The Lyapunov exponent is then given by Λ = 1 T � �T t=1 log nt + log w � , where 16 w is the largest eigenvalue of the normalized M (T) (which does not matter for Λ when T is large, but matters for its derivative that we calculate below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' To find the germination fractions q∗ α that maximizes Λ, we use the optimization routine L-BFGS-B, which allows us to impose the constraint 0 ≤ q∗ α ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Besides the numerical function that calculates Λ as described above, we also supply the Jacobian of the function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=', the derivative ∂Λ ∂qα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This requires calculating the derivative of M (T) with respect to qα, which can be done using the recursive relation ∂M (t) ∂qα = ∂M(εt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' q) ∂qα M (t−1) + M(εt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' q) · ∂M (t−1) ∂qα , (A21) together with that for M (t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A20), from t = 1 all the way to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We normalize ∂M(t) ∂qα by the same factor nt as for M (t) at every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The derivative of Λ is then given by ∂Λ ∂qα = 1 T 1 |M (T)| ∂|M (T)| ∂qα = 1 T 1 w � u · ∂M (T) ∂qα v � , (A22) where u and v are the left and right eigenvectors of M (T) corresponding to its largest eigenvalue w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This derivative is then supplied as the Jacobian to the L-BFGS-B optimization routine to find the optimal q∗ that maximizes Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='3 Simulating a lineage Simulation of a continuous lineage of seeds is used to estimate the joint probability P(εt, αt−1) of the environment εt and the seed age αt−1 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='2, which is then used to calculate their mutual information I(εt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' αt−1) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For a given set of germination fractions qα, the simulation is done as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We start from a fresh seed of age 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The sequence of environments, {ε1, · · · , εT }, is drawn beforehand as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In each year, we decide whether the seed germinates or not using the germination probability that corresponds to its age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' To account for selection bias, we weight the probabilities by the fitness values in the current environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' That is, in year t, the seed along the lineage has probability qαt−1Yεt qαt−1Yεt + (1 − qαt−1)V to germinate and reset the age to 0, and otherwise stays dormant with its age increased from αt−1 to αt = αt−1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We repeat this procedure from t = 1 to T, recording the sequence of αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Afterwards, the number of times that the pair (εt, αt−1) takes a particular combination of values is counted, which is then normalized to be the joint probability distribution P(εt, αt−1), from which the mutual information I(εt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' αt−1) is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Lineage simulation is also used to calculate the distribution of dormancy duration in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=', the distribution of how many consecutive years a seed stays dormant in the absence of environmental 17 variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' To calculate this distribution, we once again start with a fresh seed of age 0 and use the probability q0 to decide if the seed germinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This time the probability is not weighted by the fitness because we are calculating the dormancy durations in the absence of selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The above procedure is repeated for a long period of time T and the sequence of phenotypes at each time step is recorded as φt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The duration of germination or dormancy is calculated by parsing the sequence of phenotypes {φt} into consecutive epochs of germination or dormancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The distribution of their durations is then calculated by normalizing the histograms of these epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Note that these distributions can also be calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A30) in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='4 Exhaustive search of state diagrams To verify that the age-diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 5A is the optimal topology, we test all possible state diagrams for up to 6 internal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For a diagram with L states, we label the states as s0, s1, · · · , sL−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Each state has two outgoing arrows, corresponding to either dormancy or germination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Each arrow can go to any other state or loop back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Therefore, naively, there can be L2L possible diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' However, many of these diagrams are equivalent in the sense that they are simply permutations of the states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' To remove the redundant diagrams, we use a “sieve” method as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We first represent a diagram by a (L × 2) integer matrix, whose entry of the α-th row and φ-th column represents which state the system will transition to if it is at age α and expresses phenotype φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The diagrams are then indexed by a number that results from flattening the matrix and treating it as a base-L number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Then, we enumerate all L2L diagrams starting from the index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For each diagram, we find all its permutations and remove their indices from the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Furthermore, we exclude diagrams that have two or more disjoint parts to keep only connected diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' We go over the list of diagrams, skipping the indices that have been removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In the end, the total number of non-degenerate diagrams for L = 1, 2, · · · is n(L) = 1, 6, 52, 892, 21291, 658885, · · · which is the number of unlabeled, strongly connected, L-state, 2-input automata (Sequence A027835 from OLEIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This number grows quickly and we are only able to study diagrams for up to L = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For each of the diagrams with L states, we numerically find the optimal qα and the maximum Λ as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This is computationally intensive and is done on a computer cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Then, among all diagrams of L states, we find the optimal diagram with the largest Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' For up to L = 6, it turns out that the age-diagram is the optimal diagram for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='5 Analytical results for extreme selection In the limit of extreme selection, the fitness matrix is diagonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=', fεφ = � V 0 0 Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This means, hypothetically, that a seed can survive only if it germinates in a good year or stays dormant in a 18 bad year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' In this case, the long-term growth rate Λ and the optimal germination fractions q∗ α have analytical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Indeed, the population becomes homogeneous because, once it encounters a good year, only the seeds that germinate will survive, and subsequently the population will consist of only fresh seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' From then on, the seed age will be synchronized with the number of consecutive bad years, and will be reset to 0 whenever there is a good year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Let βt−1 be the number of consecutive bad years right before year t (which is 0 if the previous year is good).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' It will be equal to the seed age αt−1 of the population at the beginning of year t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Therefore, the seed population changes over time according to: Nt = Nt−1(1 − qβt−1)V or Nt−1 qβt−1Y , (A23) depending on whether the environment εt = 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Over a period of time T, the number of seeds will be: NT = N0 � β � (1 − qβ)V �T0β� qβY �T1β , (A24) where Tεβ is the number of years that the environment is ε while the previous number of consecutive bad years is β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This equation has the same form as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A10), with the external cue ξ replaced by β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' The long-term growth rate has the expression Λ ≡ lim T→∞ 1 T log NT N0 = � β P0β log[(1 − qβ)V ] + � β P1β log[qβY ] (A25) where Pεβ = lim T→∞ Tεβ T is the joint probability of the environment εt and the number of bad years βt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Setting the derivative ∂Λ ∂qα = 0, the optimal germination fractions q∗ α are found to be q∗ α = P1α P0α + P1α ≡ P1|α ≡ P(εt =1|βt−1 =α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A26) Here P(εt =1|βt−1 =α) represents the conditional probability that the coming year is good, given that there has been α consecutive bad years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' It is related to the duration distribution of bad years, P(τ0) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A19), through P(εt =1|βt−1 =α) = P(τ0 =α) P(τ0 ≥α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A27) An important consequence of this result is that, for the germination fractions q∗ α, the dormancy duration of the seeds (as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 7B) will have the same distribution as the duration of bad years (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' 1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' This is because, by definition, qα ≡ P(φt = 1|αt−1 = α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Let δ0 denote the duration of dormancy, then similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A27), we have P(φt =1|αt−1 =α) = P(δ0 =α) P(δ0 ≥α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A28) Equating the left-hand sides of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A27) and (A28) leads to, as stated above, P(δ0 =α) = P(τ0 =α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A29) 19 Incidentally, for a general qα, it can be shown that P(δ0 =α) = qα α−1 � k=1 (1 − qk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' (A30) References [1] Baskin CC, Baskin JM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQf5QXz/content/2301.03511v1.pdf'} +page_content=' Seeds: Ecology, Biogeography, and Evolution of Dormancy and Germination.' metadata={'source': 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Wiersma1,2,6, Francesco Riboli*2,7 + 1 Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135 Torino, Italy; 2 European Laboratory for +Nonlinear Spectroscopy, Via Nello Carrara 1, 50019 Sesto Fiorentino (FI), Italy; 3 Physics Dept. LMU Munchen, +Schellingstraße 4/III D-80799 Munchen, Germany; 4Electrical and Computer Engineering (ECE) Dept., University +of Connecticut, Storrs, CT, USA; 5 Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università +di Siena, via Roma 56, 53100 Siena; 6 Dipartimento di Fisica, Università di Firenze, Via Sansone 1, 50019, Sesto +Fiorentino, Italia; 7CNR-INO, Via N. Carrara 1 Sesto Fiorentino, 50019, Italy. +*nocentini@lens.unifi.it; riboli@lens.unifi.it + +Disordered photonic structures are promising materials for the realization of physical +unclonable functions (PUF) – physical objects that can overcome the limitations of +conventional digital security methods1–6 and that enable cryptographic protocols +immune against attacks by future quantum computers 7,8. One PUF limitation, so far, has +been that their physical configuration is either fixed or can only be permanently +modified9, and hence allowing only one token per device. We show that it is possible to +overcome this limitation by creating a reconfigurable structure made by light- +transformable polymers, in which the physical structure of the unclonable function itself +can be reversibly reconfigured. We term this novel concept Hyper PUF or HPUF in that +it allows a large number of physical unclonable functions to co-exist simultaneously +within one and the same device. The physical transformation of the structure is done all- +optically in a reversible and spatially controlled fashion. Our novel technology provides +a massive enhancement in security generating more complex keys containing a larger +amount of information. At the same time, it allows for new applications, for example +serving multiple clients on a single encryption device and the practical implementation +of quantum secure authentication of data10. + +Complex photonic systems11–17 are characterized by a multitude of spatial degrees of freedom that in +presence of coherent light illumination produce in the far field a complex intensity pattern (called speckle +pattern) as the result of the interference of a large number of independent transmission channels18. In +particular, the optical speckle pattern that is generated by disordered materials is extremely sensitive to +minute changes in the physical structure of the material19,20, to the level that it is nearly impossible to clone +such disordered structures and obtain the same optical response without resorting to cloning techniques at +the molecular level. Such structural characteristics make them ideal candidates for cryptographic primitives +such as physical unclonable functions for authentication and communication purposes 2,3. Among the other +types of PUFs, electrical Strong PUFs have been examined intensively by the PUF community 4,21,22, but +most of them have been attacked successfully via various digital and physical techniques over the years23,24. +Due to their promise of higher three-dimensional complexities and entropy levels, this has put optical PUFs +back in the focus of recent PUF research. + + +2 + +Optical physical unclonable functions have been introduced by Pappu6 with the name of Physical One-Way +Functions. In this first instantiation, the PUF interrogation and the resulting challenge-response pair (CRP) +protocol4,13–15 relied on different angles of incidence of the laser and allowed to extract cryptographic keys +with 230 independent bits (over a total bit string length of 2400 bits). While the optical setup based on +moveable mechanical components limits the reproducibility of measurements, in later works the employ of +modulators as challenge generators in the spatial25,26 or spectral27,28 domain provided a significant +improvement. However, those PUFs rely on a static hardware whose properties cannot be reconfigured in +case of detected attack. To overcome this limitation, Kursawe et al. showed that permanent modifications +can be created by melting the polymer aggregates with a net entropy decrease in every new reconfiguration +29. Horstmeyer and coauthors showed that it is possible to reconfigure an optical PUF by exploiting +electrical driven polymer dispersed liquid crystals 25,23and John et al. managed to do this electrically by +using halide perovskite memristors30. In all these cases, the internal states of the PUF cannot be recovered +after reconfiguration, and their entropy remains constant25. To increase the information entropy of the PUF, +it is necessary to provide a reversible transformation among the possible microscopic configurations. A +preliminary result in this direction was obtained by Gan and coauthors, who reported that the temperature- +controlled phase transition of Vanadium oxides nanocrystals can be used to create a reversible switching +among two states (crystalline and amorphous) 31. +In this context, we introduce a new concept and technology platform that provides interchangeable multi- +level operation by reversibly transforming the scattering properties of a complex photonic medium based +on photosensitive polymeric film. The operation principles of this cryptographic primitive – that we term +Hyper PUF (HPUF) – is illustrated in Fig. 1a. A “standard” PUF (left panel of Fig. 1) is characterized by +an authentication process via a single challenge CiProbe, while the HPUF (right panel of Fig. 1) is interrogated +by a challenge Cik = (CiProbe, LkTrans) consisting of two sub-challenges. First, a configuration pattern LkTrans +(a spatially modulated parametric matrix) transforms the internal configuration of the PUF between +different levels in an all-optical and reversible manner. The configuration pattern determines the scattering +potential. Each scattering potential is associated to a different level of the HPUF. Secondly, a standard +interrogation challenge CiProbe produces a measurable unique optical interference pattern as the PUF +response Rik(CiProbe, LkTrans). Mathematically, the HPUF can be modelled as a parametric function that maps +its domain to a larger codomain, whose dimension depends not only on the number of CiProbe but also on +the number of transformer challenges LkTrans, i.e. f : (CiProbe, LkTrans) → Rik. The same internal configuration +can be restored by applying the same transformer challenge, allowing back-and-forth switching between +the PUF’s internal levels. This marks a significant difference between HPUFs and existing reconfigurable +PUF designs,9,30,32 in which internal changes are permanent and non-reversible. +The practical usage of physical unclonable functions is governed by a registration and verification protocol +of the challenge-response-pairs33 that for standard and HPUFs differ in the library dimensionality and the +type of challenge sent to the claimant. To discriminate between legitimate and fraudulent authentication +requests, the similarity of two binary keys needs to be evaluated. Among the several metrics (such as +standard error, Pearson correlation coefficient, and mutual information), our analysis exploits the fractional +hamming distance (FHD) – i.e. the percentage of bits that differs between two binary strings. This is a +common choice both in biometrics and in PUF characterization 6,34. The FHD distribution between the +responses to the same challenge (like FHD) quantifies the stability of the system, while that one between +the responses to different random challenges (unlike FHD) is used to evaluate the correlation of the +independent responses. Indeed, following the method introduced by Daugman 27,28, the number N of +independent bits (the entropy) of the generated keys – i.e. the number of independent degrees of freedom – +can be estimated by assuming that the unlike FHD can be modeled with an equivalent binomial distribution +B(N, p) and expressed as a function of the mean value p and standard distribution of the curve σ, 𝑁 = + + +3 + +�∗(���) +�� + 34,35. We refer to intra-device FHDs when comparing responses from the same PUF or inter-device +FHDs when comparing responses from different PUFs. + +Figure 1. Schematic representation of the interrogation process for standard and Hyper PUFs. Working +mechanism of the deterministic behavior for the challenge response pair generation for standard PUFs (left panel) +and HPUFs (right panel). For the standard PUF, the challenge CiProbe probes the only possible internal configuration +of the hardware, producing only one response Ri to a given challenge. In the HPUF, each configuration pattern +reversibly transforms the PUF level into a new one, producing different responses Rik to a given challenge CiProbe. +The HPUF is a 3D disordered photonic medium that is responsive to the transformer challenge while +unperturbed to the probing challenge. It consists of a polymer film where liquid crystal (LC) droplets are +randomly dispersed via an emulsion process resulting in polymer dispersed and polymer stabilized liquid +crystals (PD&SLC)36 as shown in Fig. 2a. The response selectivity between the transformer and the stimulus +challenges is achieved by doping the common liquid crystal 5CB with a blue absorbing dye (dispersed red +1, DR1). Blue incoherent light (LkTrans) transforms the internal state of the PUF by absorbing light and +thereby generating a temperature driven LC phase transition, while red coherent light (CiProbe) probes the +transformed PUFs. The LC droplets are further stabilized with cross linker molecules (Fig. 2a) that create +a fixed polymeric network37 to favor the recovery of the LC alignment in the nematic phase after the phase +transition (Fig. 2b,c). Fig. 2d-f show the polarized optical microscope characterization of the nematic- +isotropic-nematic phase transition of the LC within the illuminated spot of blue light. The presence of the +cross linker molecules guarantees an hysteresis-free process – i.e. a reversible switching between the two +LC phases38. The transformation between different internal configurations is deterministic, stable, and +repeatable, regardless of the history of the system. + +Standard PUF +HyperPUF +Challenge +Levels +C.Probe +Trans +Trans +Trans +Challenge +CProbe +PUF +PUFL +Trans +PUF (L95% for the fundamental mode. +A sketch of the experimental setup is given in Fig. 1(b). The cavity was fabricated by splicing +each pigtail of the tapered fiber to a commercial fiber Bragg grating (FBG) mirror (Omega +Optical). The two FBG mirrors consisted of stacked dielectric mirrors coated on the end faces +of fiber patchcords (SM1250 (9/80), Fibercore) and had a reflectivity of 97% at 𝜆 = 776 nm. +Both mirrors had almost the same reflectivity over all input polarization angles (< 1% variation). +The cavity also contained an in-line polarization controller (IPC, see Fig.1(b)) to manipulate the +birefringence inside the cavity. Moving the paddles of the IPC induced stress and strain in the +fiber, thereby changing the effective cavity length. A typical cavity length was ∼ 2 m, which was +physically measured and estimated from the cavity free-spectral range (FSR). + +DigiLockA linearly polarized Gaussian beam from a laser at 𝜆 = 776 nm (Toptica DL100 pro) was +launched into the fiber cavity. The laser frequency was either scanned or locked to a mode of +interest using a Pound-Drever-Hall locking module (Toptica Digilock110). The cavity output +beam was split into three paths: one for the laser feedback controller to observe the cavity spectra +and to lock to specific modes, one for imaging the spatial profile of the modes with a CCD +camera, and one for analyzing the transverse SOP of each mode using a removable quarter wave +plate (QWP), a rotating linear polarizer, and a CCD camera, see Fig. 1(b). Six intensity profile +images were taken in total for each mode. Four images were taken without the QWP and with the +linear polarizer angle set to 0◦ (I𝐻), 45◦ (I𝐷), 90◦ (I𝑉 ), and 135◦ (I𝐴), and two images were taken +by inserting the QWP set to 90◦ while the polarizer was set to 45◦ (I𝑅) and 135◦ (I𝐿). The SOPs +were determined by analyzing the six profile images using Stokes polarimetry. Furthermore, the +Stokes phase and Stokes index were determined [41], see Section 2 2.3. +2.2. +Simulations +Each mode experiences arbitrary birefringence as it propagates along the fiber. The total field +in the fiber at any point is the sum of the propagating modes with a corresponding phase shift. +The addition of FBG mirrors to the fiber induces an additional birefringence [56, 57], which +can be incorporated in a single birefringence matrix. Note, this model does not include cavity +boundary conditions since we only aim to simulate the spatial profiles of the fiber modes. We can +calculate an arbitrary fiber field, E, due to interference and birefringence by taking a summation +over different fiber modes, such that +E = +𝑛 +∑︁ +𝑀=1 +𝐽𝑀 𝐴𝑀E𝑀𝑒𝑖𝜙𝑀 , +(1) +where n is the number of eigenmodes to be interfered, E𝑀 is the electric field of a fiber eigenmode +M ∈ TE0,𝑚, TM0,𝑚, HEℓ,𝑚 and EHℓ,𝑚, with ℓ ∈ Z+ being the azimuthal mode order, which +defines the helical phase front and the associated phase gradient in the fiber transverse plane. +m ∈ Z+ is the radial mode order, which indicates the m𝑡ℎ solution of the corresponding eigenvalue +equation [5]. A𝑀 is the amplitude, 𝜙𝑀 is the phase between modes, and J𝑀 represents the +arbitrary birefringence Jones matrix of each eigenmode E𝑀, such that +𝐽𝑀 = 𝑒𝑖𝜂𝑀/2 �� +� +𝑐𝑜𝑠2𝜃𝑀 + 𝑒𝑖𝜂𝑀 𝑠𝑖𝑛2𝜃𝑀 +(1 − 𝑒𝑖𝜂𝑀 )𝑐𝑜𝑠𝜃𝑀 𝑠𝑖𝑛𝜃𝑀 +(1 − 𝑒𝑖𝜂𝑀 )𝑐𝑜𝑠𝜃𝑀 𝑠𝑖𝑛𝜃𝑀 +𝑠𝑖𝑛2𝜃𝑀 + 𝑒𝑖𝜂𝑀 𝑐𝑜𝑠2𝜃𝑀 +�� +� +, +(2) +where 𝜂𝑀 is the relative phase retardation induced between the fast axis and the slow axis, and +𝜃𝑀 is the orientation of the fast axis with respect to the horizontal-axis, i.e., perpendicular to +mode propagation. +Let us now consider the system with an ONF supporting HE𝑜 +11, HE𝑒 +11, TE01, TM01, HE𝑜 +21 and +HE𝑒 +21, so that the number of modes that can be interfered is n ≤ 6. The cross-sectional profiles +and SOPs of TE01 and HE𝑒 +21 are shown in Fig. 2(a, b), respectively. The TM01 and HE𝑜 +21 modes +are not shown here but their vector fields are orthogonal to the TE01 and HE𝑒 +21 at every point, +respectively. These modes have donut-shape mode profiles with linearly polarized vector fields +at any point in the mode cross-section. As an example of possible fiber modes using Eq. 1, Fig. +2(c) illustrates in-phase interference of the TE01 and HE𝑒 +21 modes with equal amplitudes. The +resulting mode has a lobe-shape intensity pattern with scalar fields. Fig. 2(d) is an example of +a mode resulting from the interference of the circularly polarized HE11 and an out-of-phase (a +𝜋/2 phase difference) TE01 and TM01 with equal amplitudes. The SOP, which is overlapped on +the intensity profile images, are marked as red and blue ellipse, corresponding to right and left +handed orientation, respectively. This mode is the co-called lemon [55], which contains not only +linear polarization but also elliptical and circular polarization components in one mode. + +Fig. 2. Simulations of (a) TE01, (b) HE𝑒 +21, (c) TE01 + HE𝑒 +21 and (d) lemon. The red +and blue SOPs indicate right-handed and left-handed ellipticities, respectively. The +scale bars show the normalized intensity (from 0 to 1) and the Stokes phase (from 0 to +2𝜋). Stokes singularity points of 𝜎12, 𝜎23, and 𝜎31 are indicated as pink, orange, and +blue dots, respectively. An L-line is indicated in green. + +(a) +(b) +(c) +(d) +Φ12 +D. +DWhen using Eq. 1 to simulate mode profiles, a number of eigenmodes with similar intensity +patterns and SOPs to an experimentally observed cavity mode were selected as the initial +conditions. Next, the variables A𝑀, 𝜙𝑀, 𝜂𝑀, and 𝜃𝑀 were tuned to match the experimentally +observed cavity mode intensities, SOPs, and Stokes phases. Polarization topological defects in +the simulated modes were then identified, using the method described in the following Section 2 +2.3. +2.3. +Analysis +The polarization gradient was calculated in order to identify Stokes singularities in the cross- +section of the mode. The gradient map is known as the Stokes phase, 𝜙𝑖 𝑗, which is given +by [42,45] +𝜙𝑖 𝑗 = 𝐴𝑟𝑔(𝑆𝑖 + 𝑖𝑆 𝑗), +(3) +where 𝑆𝑖 and 𝑆 𝑗 are Stokes parameters with {i, j} ∈ {1, 2, 3} in order, and i ≠ j. The phase +uncertainty points, i.e., Stokes singularities, were identified by obtaining the Stokes indices, 𝜎𝑖 𝑗, +which are defined as [42,45] +𝜎𝑖 𝑗 = 1 +2𝜋 +∮ +𝑐 +𝜙𝑖 𝑗 · 𝑑𝑐, +(4) +where +∮ +𝑐 𝜙𝑖 𝑗 · 𝑑𝑐 = Δ 𝜙𝑖 𝑗 is the counterclockwise azimuthal change of the Stokes phase around the +Stokes singularity. Singularities of 𝜎12 are known as V-points and C-points, in vector and ellipse +fields, respectively [42]. Singularities of 𝜎23 and 𝜎31 are known as Poincaré vortices [43–45]. +L-lines are located where 𝜙23 = {0, 𝜋, 2𝜋}. Table 1 is a summary of the classification of the Stokes +singularity types in terms of the Stokes phases and singularity indices with the corresponding +polarizations in the vector and ellipse fields [43,45,46,58]. +Table 1. List of Stokes singularities in vector fields (v) and ellipse fields (e) by the +singularity index, 𝜎𝑖 𝑗, using the Stokes phase, 𝜙𝑖 𝑗, with {i, j} ∈ {1, 2, 3} in order. +Stokes +Stokes phase +Stokes index/ +Polarization +singularity +Phase values +V-point (v) +𝜙12 +𝜎12 +Null +C-point (e) +𝜙12 +𝜎12 +R/L +Poincaré +𝜙23 +𝜎23 +H/V +vortex (e) +𝜙31 +𝜎31 +D/A +L-line (e) +𝜙23 +0, 𝜋, 2𝜋 +Linear +The Stokes singularity points and L-lines were found from the Stokes phases, then superimposed +and marked on the mode profiles. As examples, from Figs. 2(a, b), the center of the mode profiles +for both TE01 and HE𝑒 +21 contain a V-point, with 𝜎12 = -2 and +2 (pink dot), respectively. These +points were found from their Stokes phases 𝜙12 (lower panels in Figs. 2(a, b)). In contrast, the +lemon mode in Fig. 2(d) has a closed loop representing an L-line (green) and all three types of +Stokes singularities: a C-point with 𝜎12 = -1 (pink dot), Poincaré vortices with 𝜎23 = -1 and +1 +(orange dots), and 𝜎31 = -1 and +1 (blue dots) were found from 𝜙12, 𝜙23, and 𝜙31, respectively. +The lobe-shaped scalar mode in Fig. 2(c) does not have a 2𝜋 gradient in any associated Stoke +phases, since topological defects can only exist in non-scalar fields [41]. + +3. +Results and discussion +3.1. +Cavity with a single-mode optical nanofiber +As an initial experimental test, the spectrum for a HOM cavity containing an ONF of waist +diameter ∼ 450 nm was obtained, see Fig. 3(a). This ONF waist can only support the fundamental +modes. The IPC paddle angles were set so that two distinct, well-separated modes with minimal +spectral overlap were observed. The finesses of Modes 1 and 2 in Fig. 3(a) were 12 and 15, +respectively. The laser was locked to each of these two cavity modes consecutively and the +mode profiles were observed at the output end face of the fiber cavity. The corresponding mode +intensity profiles, SOPs, and Stokes phases are shown in Figs. 3(b)(i, ii). The intensity profiles for +both Modes 1 and 2 were slightly skewed Gaussian shapes. The HE11 eigenmode intensity shape +is Gaussian, so the slight deviation from the expected shape may be attributed to aberrations in +the optical beam path. In terms of polarization distribution, the Stokes phases of Modes 1 and 2 +were uniform; in other words, their SOPs were scalar fields, regardless of the IPC paddle angles +chosen, as expected for the HE11 mode. +Although the pretapered fiber supported the full set of eigenmodes in LP11, LP02, and LP21, +when the ONF with a diameter ∼ 450 nm was inserted between the two sets of mirrors, only one +or two modes with quasi-Gaussian profiles were observed, no matter which IPC paddle angles +were chosen. The HOMs were filtered out due to the tapered fiber waist being SM, analogous to +an intracavity pinhole spatial filter. Mode filtering as a function of the ONF waist diameter was +observed experimentally [17]. However, here, we could additionally observe the mode filtering +effect on the cavity spectrum and SOP of each mode. +In an ideal SM-ONF cavity with no birefringence, there are two degenerate orthogonal modes. +However, due to random birefringence of the fiber and the cavity mirrors, the two modes +become non-degenerate, i.e., separated in frequency, leading to coupling between the modes [59]. +Mode coupling of orthogonal modes can occur in a birefringent medium and this effect can +increase in a cavity configuration [60]. Mode coupling in an ONF cavity due to asymmetrical +mirrors has been discussed previously [56] and experimental evidence of mode coupling due to +intrinsic birefringence in a SM-ONF cavity has already been reported [57]. In our experiments, +non-orthogonal combinations of SOPs were observed, as seen in Figs. 3(b)(i, ii). Mode 1 was +horizontally polarized (red/blue lines in Fig. 3(b)(i)), while Mode 2 was left elliptically polarized +(blue ellipse in Fig. 3(b)(ii)). By adjusting the IPC angles, it was possible to change the phase +relationship and coupling between the HE𝑜 +11 and HE𝑒 +11 modes, and shift between orthogonal and +non-orthogonal combinations of SOPs. +3.2. +Cavity with a higher-order mode optical nanofiber +Next, the spectrum for a HOM cavity containing an ONF of waist diameter ∼ 840 nm was +obtained, see Fig. 4(a). This ONF can support the HE11, TE01, TM01, HE𝑜 +21, and HE𝑒 +21 modes. +The IPC paddle angles were set to obtain the maximum number of well-resolved modes in a +single FSR, see Fig. 4(a). One can clearly see five distinct peaks indicating that the HOM-ONF +does not degrade the modes in the cavity and the finesses of the cavity modes are high enough to +resolve them. The finesses of Modes 1 to 5 were 12, 16, 13, 22, and 13, respectively. The mode +finesse values of the cavity with a HOM-ONF were in the same range as those for the cavity +with a SM-ONF (Fig. 3(a)), implying that the HOM-ONF was adiabatic for the LP11 group of +modes. The laser was locked to each of the cavity modes consecutively and the mode profiles +were observed at the output of the fiber cavity. The corresponding mode intensity profiles, SOPs, +and Stokes phases are shown in Figs. 4(b)(i-iv). In the spectrum shown in Fig. 4(a), there were +five distinctive modes, but locking to Mode 3 was not possible because of its close proximity to +the dominant Mode 4. +Two flat-top intensity profiles were observed in Modes 1 and 4, Figs. 4(b)(i, iii) respectively. + +Fig. 3. (a) A typical spectrum for a HOM cavity with a SM-ONF as the laser is scanned +over 150 MHz. The spectrum over a single FSR is indicated by the red box. (b) Mode +intensity profiles showing the SOPs (top) and corresponding Stokes phases (bottom) +for (i) Mode 1 and (ii) Mode 2. The red and blue SOPs indicate right-handed and +left-handed ellipticities, respectively. The scale bars show the normalized intensity +(from 0 to 1) and the Stokes phase (from 0 to 2𝜋). + +Laser scan frequency (MHz) +(i) +(ii) +Φ12Fig. 4. (a) A typical spectrum for a cavity with a HOM-ONF as the laser is scanned +over 150 MHz. The spectrum over a single FSR is indicated by the red box. (b) Mode +intensity profiles showing the SOP (top) and the corresponding Stokes phases (bottom) +for (i) Mode 1, (ii) Mode 2, (iii) Mode 4, and (iv) Mode 5. The red and blue SOPs +indicate right-handed and left-handed ellipticities, respectively. The scale bars show +the normalized intensity (from 0 to 1) and the Stokes phase (from 0 to 2𝜋). Stokes +singularity points of 𝜎12, 𝜎23, and 𝜎31 are indicated as pink, orange, and blue dots, +respectively. L-lines are indicated in green. (c) Corresponding simulated results. + +Laser scan frequency (MHz) +(i) +(ii) +(i) +(iv +D +D +(i) +(iv) +Φ12 +Φ23 +Φ31 +Φ23 +Φ31 +D +中 +23 +D +3The SOPs of these modes are markedly different to those for the Gaussian-type modes in Figs. +3(b)(i, ii), which have simple scalar SOPs. Modes 1 and 4 were inhomogeneously polarized +ellipse fields, showing regions of left and right circular polarizations divided by an L-line (Figs. +4(b)(i, iii)). The center of these two modes exhibited diagonal and anti-diagonal polarizations, +respectively, i.e., the SOPs at the center of the modes were orthogonal to each other. Going +towards the edges of the modes, the polarization changes from linear to circular, with opposite +handedness either side of the L-lines. Notice also in Fig. 4(a) that Modes 1 and 4 are not well +frequency separated from neighboring modes. This suggests that the mode profiles and SOPs of +these modes were not only affected by birefringence and degenerate modal interference, but also +some non-degenerate modal interference with neighboring cavity modes [60]. Additionally, for +Mode 4, we identified two C-points (𝜎12 = -1), indicated by the pink dots in Fig. 4(b)(iii), where +the value of 𝜙12 changed by 2𝜋 (see Table 1). Interference of HE11 with modes from the LP11 +group can generate C-points in a few-mode fiber [55], see Fig. 2(d). +We performed basic simulations to determine if combinations of HE11 and some mode(s) in the +LP11 family could generate similar mode profiles and SOP structures as those in Figs. 4(b)(i, iii). +The simulated results are shown in Figs. 4(c)(i, iii). The HE11 and TM01 modes were selected +as possible contributors and their amplitudes, phase, and birefringence fitting parameters were +tuned to match the experimental results. Modes 1 and 4, see Figs. 4(b)(i, iii), could have been +formed from different mode combinations rather than our assumed HE11 and TM01; however, +these modes were very likely formed by interference between HE11 and some mode(s) of the +LP11 group, resulting in their inhomogeneous SOPs and flat-top shapes. +We also observed two distorted lobe-shaped modes, Modes 2 and 5, see Figs. 4(b)(ii, iv). +The lobe-shaped pattern also arises from modal interference between modes in the LP11 family +(as an example, see Fig. 2(c)). With reference to Table 1, Mode 2, Fig. 4(b)(ii), showed +all three types of Stokes singularities, indicated by pink dots for C-points (𝜎12 = +1) and +orange/blue dots for Poincaré vortices (𝜎23 = -1 /𝜎31 = +1), as presented in 𝜙12, 𝜙23, and 𝜙31, +respectively. A single mode containing all Stokes singularities has been demonstrated using +free-space interferometers [43,46]; here, we generated them within a single mode using a fiber +cavity system. Mode 5, Fig. 4(b)(iv), also had two C-points (𝜎12 = +1) and a Poincaré vortex +(𝜎23 = +1), as seen in 𝜙12, and 𝜙23, respectively. Fig. 4(a) shows that Modes 2 and 5 are not well +frequency separated from Modes 1 and 4, respectively. Therefore, there is a likely contribution +from the HE11 mode resulting in distortion of the lobe shape. +To simulate Mode 2 in Fig. 4(b)(ii), we combined TE01, HE𝑒 +21, and HE11, and to simulate +Mode 5 in Fig. 4(b)(iv), we used TM01, HE𝑒 +21, and HE11. The amplitude of each mode, phase +shift, and birefringence parameters were adjusted to achieve a close fit. The simulated results +are shown in Figs. 4(c)(ii, iv). These plots are not exact replications of the experimental results +since the parameter space is large and the exact initial conditions are not known; nevertheless, +the match is reasonably close. +Interestingly, many of the cavity modes obtained in different sets of spectra, which were +generated using different IPC angles, exhibited Stokes singularities. Polarization singularities are +known to propagate through a birefringent medium as C-lines and L-surfaces and their evolution +is affected by the homogeneity of the birefringence along the propagation path [47–49]. This +phenomenon is due to the conservation of the topological charge [49,58,61], and the Stokes index +value, 𝜎𝑖 𝑗, remains constant [58]. However, our cavity is an inhomogeneous birefringent medium +as it contains a number of different birefringent elements such as the FBG mirrors and the IPC, as +such, the degree of birefringence varies along the propagation direction. Therefore, the presence +of Stokes singularities in the imaged field at the cavity output does not necessarily guarantee the +existence of such topological defects in the ONF region. Nonetheless, singularity points can +enter, move and exit with a smooth and continuous variation of birefringence [50]. Therefore, +the SOP is expected to evolve along the length of the cavity, with singularity points shifting and + +making numerous entries and exits in the cross-section profile of the modes. However, since the +ONF waist is relatively straight and uniform, the birefringence variation at the waist should be +minimal [62] and topological features appearing at the start of the waist should be preserved +every 2𝜋 along the waist. +Theoretically, the HOM-ONF can support a total of six eigenmodes as mentioned earlier. +Therefore, one might expect that the spectrum should show six distinct modes. However, we +typically observed three to five distinct peaks in a single FSR depending on the IPC paddle angles. +This could be explained by the lack of sufficient finesse to resolve all modes, some of which are +closely overlapped [60]. However, it may be feasible to increase the mode finesses by increasing +the mirror reflectivity and using an ONF with lower transmission loss than the one used (the +estimated loss of Mode 4, the highest finesse in Fig. 4(a), was ∼ 20%). Nonetheless, the finesse +values of our ∼ 2 m long cavity with a HOM-ONF should be sufficient for cQED experiments +with narrow line-width emitters such as cold atoms. +Fig. 5. (a) Mode intensity profiles for quasi-donut-shaped cavity modes from the cavity +containing a HOM-ONF with their SOPs (top) and Stokes phases (bottom) similar to +the fiber eigenmodes of (i) HE𝑒 +21, (ii) HE𝑜 +21, (iii) TE01, and (iv) TM01. The red and +blue SOPs indicate right-handed and left-handed ellipticities, respectively. Scale bars +show intensity (from 0 to 1) and Stokes phase (from 0 to 2𝜋). Stokes singularities of +𝜎12, 𝜎23, and 𝜎31 are indicated as pink, orange, and blue dots, respectively. L-lines are +illustrated as green lines. (b) Corresponding simulated results. +3.3. +In situ higher-order cavity mode tuning +A key feature of this setup is the ability to tune the spectrum and SOP to create the desired mode +in the cavity. We aimed to observe modes with donut-shaped intensity patterns and SOPs similar +to the fiber eigenmodes TE01 (Fig. 2(a)), TM01, HE𝑜 +21, and HE𝑒 +21 (Fig. 2(b)). To achieve this, the +laser was locked to a well-resolved lobe-shaped mode. The paddle angles of the IPC were then +adjusted, and the mode shape was monitored with a CCD camera until a donut mode profile was +observed. Unlocking and scanning the laser revealed a new spectrum with each mode containing + +(i) +(ii) +(iii) +(iv) +(D) +(iv) +D +Da new profile. The IPC was adjusted again to maximize another mode and the laser was locked to +this new mode. The IPC paddle angles were tuned to once more convert the mode profile to a +donut shape. This procedure was repeated for four different modes, see Figs. 5(a)(i-iv), and these +modes look similar to the true fiber eigenmodes of HE𝑒 +11 (Fig. 2(b)), HE𝑜 +11, TE01 (Fig. 2(a)), and +TM01, respectively. There was a slight deformation from a perfect donut shape and their SOPs +were not vector fields, but rather ellipse fields with alternating regions of opposite handiness. +While the donut eigenmodes possessed a V-point at the center as indicated by pink dots in Figs. +2(a, b), the observed quasi-donut modes in Figs. 5(a)(i-iv) had some nominal intensity at the +center. These modes had two C-points of 𝜎12 = -1 or +1 near the center (see pink dots in Figs. +5 (a)(i-iv)), as opposed to a single point of 𝜎12 = -2 or +2 in the true eigenmodes (Figs. 2(a, +b)). Indeed, perturbation of vector field polarization singularities can occur when scalar linearly +polarized beams are interfered [63]. +These donut-shaped cavity modes were also simulated, as shown in Figs. 5(b)(i-iv). To +obtain a good fit for the experimentally observed intensities, SOPs, and Stokes phases in Figs. +5(a)(i-iv), the simulated modes included a slight deformation of the donut shape by adding some +components of the HE11 mode to modes in the LP11 group. Moreover, the simulated results +show that the Stokes phases are very similar to those obtained experimentally. The number of +possible combinations of modal interference with varying birefringence is large and this leads +to discrepancies between the experiment and simulation. However, these findings indicate that +the experimentally observed quasi-donut modes are likely the result of residual interference +between the HE11 mode and modes in the LP11 group. Degeneracy of multiple modes may be +avoided by increasing the cavity mode finesses so that each mode can be well separated. The +system demonstrated here shows that, even in a complex system, the HOMs and their SOPs can +be controlled to create exotic topological states. +4. +Conclusion +We have experimentally demonstrated a Fabry-Pérot fiber cavity with a HOM-ONF and performed +cavity spectroscopy. The cavity mode profiles and transverse polarization topology were also +determined by imaging and analyzing the individual cavity modes at the output. These modes +had inhomogeneous polarization distributions with a number of Stokes singularities. We also +simulated the fiber modes which closely match those observed at the output of the cavity. +Moreover, in situ intracavity manipulation of the modal birefringence and interference to select +a specific mode of interest was demonstrated. This indicates that the evanescent field of an +HON-ONF could be tuned by adjusting the IPC paddle angles. +These findings are a step toward investigating the interactions between SAM and OAM of +a HOM-ONF. Research into the interference of HOMs at the waist of an ONF is an exciting +opportunity to uncover the nature of light-matter interactions in tightly confining geometries +with topological singularities. Additionally, the realization of a (de)multiplexing system using +degenerate HOMs in an ONF-based cavity may be possible by improving the tunability of the +modal birefringence and interference. Such a system is attractive for future quantum information +platforms as efficient and secure storage. +The interference of higher-order cavity modes with fixed ratios in the evanescent field of an +ONF may also be used to trap and manipulate cold atoms. Adjusting the overlap and SOP of +the HOMs should result in movement of the trapping sites relative to each other, enabling some +trap dynamics to be studied [4,15,16]. This cavity could be also used with quantum emitters +to study multimode cQED effects using degenerate HOMs. The HOM cavity studied here had +moderate finesse to enter the cQED experiments for interactions with cold atoms. In free-space +optics, strong coupling of multiple transverse HOMs with atoms has been achieved [38], whereas +this has not been achieved using an ONF-type cavity. Our work is a significant step towards this +realization. + +Moreover, the ability of our cavity to generate all three types of Stokes singularities may be +useful to realize not only a C-point laser but also an all-Stokes singularity laser using a few-mode +fiber. The combinations of fiber modes that we used in the simulations were found via manual +trial-and-error estimates to obtain a visual match with the experimentally observed modes. More +accurate control could be achieved by using machine learning techniques to fully cover the +parameter space of permitted modes in the cavity. This may enable us to determine the correct +combination of modes that lead to the observed cavity outputs and facilitate feedback to optimize +the input to the system to generate desired modes in the cavity. +Funding. +Okinawa Institute of Science and Technology Graduate University. +Acknowledgments. +The authors acknowledge F. Le Kien, L. Ruks, V. G. Truong, and J. M. Ward for +discussions and K. Karlsson for technical assistance. +Disclosures. +The authors declare no conflicts of interest. +Data availability. +Data underlying the results presented in this paper are not publicly available at this +time but may be obtained from the authors upon reasonable request. +References +1. +K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Reports 592, 1–38 (2015). +2. +P. Solano, J. A. Grover, J. E. 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Senthilkumaran, “Perturbation of V-point polarization singular vector +beams,” Opt. & Laser Technol. 158, 108842 (2023). + diff --git a/8dFQT4oBgHgl3EQf4jbF/content/tmp_files/load_file.txt b/8dFQT4oBgHgl3EQf4jbF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1d898d2fdb3adb047b93c1c67d0bb22e44b25377 --- /dev/null +++ b/8dFQT4oBgHgl3EQf4jbF/content/tmp_files/load_file.txt @@ -0,0 +1,937 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf,len=936 +page_content='Manipulation of polarization topology using a Fabry-Pérot fiber cavity with a higher-order mode optical nanofiber MAKI MAEDA,1,* JAMEESH KELOTH,1 AND SÍLE NIC CHORMAIC1 1Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan maki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='maeda@oist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='jp Abstract: Optical nanofiber cavity research has mainly focused on the fundamental mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Here, a Fabry-Pérot fiber cavity with an optical nanofiber supporting the higher-order modes, TE01, TM01, HE𝑜 21, and HE𝑒 21, is demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Using cavity spectroscopy, with mode imaging and analysis, we observe cavity resonances that exhibit complex, inhomogeneous states of polarization with topological features containing Stokes singularities such as C-points, Poincaré vortices, and L-lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' In situ tuning of the intracavity birefringence enables the desired profile and polarization of the cavity mode to be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' These findings open new research possibilities for cold atom manipulation and multimode cavity quantum electrodynamics using the evanescent fields of higher-order mode optical nanofibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Introduction Novel phenomena that can be revealed in non-paraxial light, such as transverse spin and spin-orbit coupling, have led to increasing interest in the tightly confined light observed in nano-optical devices [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Optical nanofibers (ONFs), where the waist is subwavelength in size, are useful in this context because they provide very tight radial confinement of the electric field and facilitate diffraction-free propagation over several centimeters [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Most ONF research focuses on single-mode ONFs (SM-ONFs) that only support the fundamental mode, HE11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' In contrast, higher-order mode ONFs (HOM-ONFs), fabricated from a few-mode optical fiber, can guide HOMs, such as TE01, TM01, HE𝑒 21, and HE𝑜 21 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' In the weakly guided regime, which is generally used to describe light propagation in standard optical fiber, this group of modes can be viewed to form the linearly polarized mode, LP11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' To date, there has been a lot more attention paid to HOM-ONFs in theoretical work [4–10] than experimental work due to the difficulty in precisely controlling the fiber waist size and obtaining selective mode excitation at the waist [3,11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' In principle, there are many interesting phenomena which can be explored with a HOM-ONF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' For example, it has been proposed that the relationship between spin angular momentum (SAM) and orbital angular momentum (OAM) can be studied [5,10,13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Additionally, it was proposed that a HOM-ONF could be used to trap and manipulate cold atoms [4, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Fabrication of an ONF that supports the HOMs was achieved [3,17,18] and subsequently shown to more efficiently manipulate dielectric microbeads in the evanescent field than SM-ONFs [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Other experimental work has shown that when cold atoms also interact with HOMs, detected signals are stronger than when one uses a SM-ONF only [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Introducing a cavity system to the ONF could further increase light-matter interactions due to cavity quantum electrodynamics (cQED) effects [22–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' To date, numerous types of SM- ONF-based cavities have been proposed [25–30] and the interactions of their resonance modes with various quantum emitters have been studied [31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Strong light-atom coupling using SM-ONF-based Fabry-Pérot and ring resonators has already been achieved [34,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Superstrong coupling of cold atoms and multiple longitudinal modes of a long fiber-ring resonator consisting of a SM-ONF section was demonstrated [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Utilizing multiple degenerate higher-order transverse modes in free-space has shown to exhibit strong coupling [37,38], further illustrating the importance of realizing a HOM-ONF-based cavity system at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The advantages are arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='13432v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='optics] 31 Jan 2023 not only for enhanced interactions via cQED effects, but also for a better overall understanding of the behavior of the modes in such a cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Studying the behavior of the HOM-ONF cavity spectrum and the cavity mode profiles gives additional insight into the nature of the HOMs themselves, as well as how they interfere with each other and interact with the external environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The generation of TE01 and TM01 modes in a laser cavity consisting of a microfiber directional coupler-based mode converter was demonstrated previously [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' However, earlier attempts to realize a passive HOM optical microfiber cavity did not yield any resonant peaks in the cavity spectrum apart from the fundamental modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' in other words, the typical donut- or lobe-shaped intensity profiles associated with HOMs were not observed [40], primarily due to challenges when engineering the taper profile to minimize losses at the taper transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The inhomogeneous polarization structure of HOMs needs to be taken into account when studying a fiber cavity system with a HOM-ONF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' In recent years, complex polarization dis- tributions and the generation of polarization singularities have been investigated using various methods, giving rise to the relatively new field of singular optics [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Polarization singularities are a subset of Stokes singularities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=', phase singularity points in Stokes phases [42,43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' In fact, higher-order fiber eigenmodes are vector optical fields with a polarization singularity called a V-point, where the state of polarization (SOP), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=', how the polarization is distributed in the cross-section of a given mode, is undefined [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Other types of Stokes singularities can be formed in elliptical optical fields, such as the polarization singularity of C-points, where the polarization orientation is undefined [41, 42], and Poincaré vortices, where the polarization handedness is undefined [43–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Moreover, points of linear polarization can form continuous lines, which are classified as L-lines [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The generation of all Stokes singularities within a single beam has been demonstrated using a free-space interferometer [43,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Modal interference in a birefringent crystal can facilitate the creation of polarization singularities [47,48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' As a result, the SOP can significantly vary along the propagation length, with C-points and L-lines propagating as C-lines, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=', continuous lines of circular polarization, and L-surfaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=', surfaces of linear polarization, respectively [47–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Moreover, polarization singularities can appear, move or disappear from a given cross-sectional region with a smooth and continuous change of birefringence [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Birefringent media were used to create laser cavity modes containing a polarization singularity [51,52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' These experiments were limited to the generation of low-order V-points due to a lack of control in the amplitude, phase, and SOP, all of which would be required to create other types of polarization singularities [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' A few-mode optical fiber cavity has the potential to generate complex laser modes by its highly variable degree of birefringence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Interference and birefringence are generally inseparable properties in fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The modal interference pattern in a fiber changes continually with a periodicity of 2𝜋 when the relative phase between modes is changed between 0 to 2𝜋 as the eigenmodes propagate along the fiber [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' This effect was used in a few-mode optical fiber to generate ellipse fields containing a C-point [54,55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Due to the increasing complexities of modal interference in few-mode fibers, filtering for the desired set of HOMs, and selectively exciting them to generate and manipulate polarization singularities, are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Realizing a fiber cavity containing an ONF should enable both spatial and frequency filtering for selective excitation of HOMs, as well as enhancement of the resonant mode coupling effect [56,57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' In this paper, we experimentally demonstrate a HOM-ONF-based Fabry-Pérot fiber cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The transverse polarization topology of any given resonant mode is determined by selecting modes from the cavity spectra and analyzing the images of the transmitted mode profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' We also demonstrate in situ intracavity manipulation of the modal birefringence to change the amplitude, frequency position, and the SOP of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' This work is a significant step towards gaining full control of the evanescent field at the HOM-ONF waist and extends the range of applications Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' (a) Sketch of tapered optical fiber with trilinear shape, d𝑤𝑎𝑖𝑠𝑡: waist diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' (b) Schematic of experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' L: lens, HWP: half-wave plate, PBS: polarizing beam splitter, M: mirror, M𝐶: cavity mirror, IPC: in-line polarization controller, BS: beam splitter, QWP: quarter-wave plate, which was inserted to calculate S3, LP: linear polarizer, CCD: camera, MMF: multimode fiber, PD: photodiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' for which such nanodevices could be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Experiments For the HOMs described in Section 1 to propagate throughout the cavity with a HOM-ONF, the nanofiber must be low loss for the entire LP11 set of modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Tapered fibers were drawn from SM1250 (9/80) fiber (Fibercore) using an oxy-hydrogen flame pulling rig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The untapered fiber supports the LP01, LP11, LP21, and LP02 modes at a wavelength, 𝜆 = 776 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The modes supported by the tapered fiber depend on the tapering profile and the waist diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' We used two different tapered fibers with waist diameters of (i) ∼ 450 nm for SM behavior (HE𝑜 11 and HE𝑒 11) and (ii) ∼ 840 nm for the HOM-ONF, which supports HE𝑜 11, HE𝑒 11, TE01, TM01, HE𝑜 21, and HE𝑒 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The shape of the tapered fibers was chosen to be trilinear, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 1(a), with angles of Ω1 = 2 mrad, Ω2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='5 mrad and Ω3 = 1 mrad in order to be adiabatic for the LP11 and LP01 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Fiber transmission following the tapering process was >95% for the fundamental mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' A sketch of the experimental setup is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The cavity was fabricated by splicing each pigtail of the tapered fiber to a commercial fiber Bragg grating (FBG) mirror (Omega Optical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The two FBG mirrors consisted of stacked dielectric mirrors coated on the end faces of fiber patchcords (SM1250 (9/80), Fibercore) and had a reflectivity of 97% at 𝜆 = 776 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Both mirrors had almost the same reflectivity over all input polarization angles (< 1% variation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The cavity also contained an in-line polarization controller (IPC, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='1(b)) to manipulate the birefringence inside the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Moving the paddles of the IPC induced stress and strain in the fiber, thereby changing the effective cavity length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' A typical cavity length was ∼ 2 m, which was physically measured and estimated from the cavity free-spectral range (FSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' DigiLockA linearly polarized Gaussian beam from a laser at 𝜆 = 776 nm (Toptica DL100 pro) was launched into the fiber cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The laser frequency was either scanned or locked to a mode of interest using a Pound-Drever-Hall locking module (Toptica Digilock110).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The cavity output beam was split into three paths: one for the laser feedback controller to observe the cavity spectra and to lock to specific modes, one for imaging the spatial profile of the modes with a CCD camera, and one for analyzing the transverse SOP of each mode using a removable quarter wave plate (QWP), a rotating linear polarizer, and a CCD camera, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Six intensity profile images were taken in total for each mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Four images were taken without the QWP and with the linear polarizer angle set to 0◦ (I𝐻), 45◦ (I𝐷), 90◦ (I𝑉 ), and 135◦ (I𝐴), and two images were taken by inserting the QWP set to 90◦ while the polarizer was set to 45◦ (I𝑅) and 135◦ (I𝐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The SOPs were determined by analyzing the six profile images using Stokes polarimetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Furthermore, the Stokes phase and Stokes index were determined [41], see Section 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Simulations Each mode experiences arbitrary birefringence as it propagates along the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The total field in the fiber at any point is the sum of the propagating modes with a corresponding phase shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The addition of FBG mirrors to the fiber induces an additional birefringence [56, 57], which can be incorporated in a single birefringence matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Note, this model does not include cavity boundary conditions since we only aim to simulate the spatial profiles of the fiber modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' We can calculate an arbitrary fiber field, E, due to interference and birefringence by taking a summation over different fiber modes, such that E = 𝑛 ∑︁ 𝑀=1 𝐽𝑀 𝐴𝑀E𝑀𝑒𝑖𝜙𝑀 , (1) where n is the number of eigenmodes to be interfered, E𝑀 is the electric field of a fiber eigenmode M ∈ TE0,𝑚, TM0,𝑚, HEℓ,𝑚 and EHℓ,𝑚, with ℓ ∈ Z+ being the azimuthal mode order, which defines the helical phase front and the associated phase gradient in the fiber transverse plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' m ∈ Z+ is the radial mode order, which indicates the m𝑡ℎ solution of the corresponding eigenvalue equation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' A𝑀 is the amplitude, 𝜙𝑀 is the phase between modes, and J𝑀 represents the arbitrary birefringence Jones matrix of each eigenmode E𝑀, such that 𝐽𝑀 = 𝑒𝑖𝜂𝑀/2 �� � 𝑐𝑜𝑠2𝜃𝑀 + 𝑒𝑖𝜂𝑀 𝑠𝑖𝑛2𝜃𝑀 (1 − 𝑒𝑖𝜂𝑀 )𝑐𝑜𝑠𝜃𝑀 𝑠𝑖𝑛𝜃𝑀 (1 − 𝑒𝑖𝜂𝑀 )𝑐𝑜𝑠𝜃𝑀 𝑠𝑖𝑛𝜃𝑀 𝑠𝑖𝑛2𝜃𝑀 + 𝑒𝑖𝜂𝑀 𝑐𝑜𝑠2𝜃𝑀 �� � , (2) where 𝜂𝑀 is the relative phase retardation induced between the fast axis and the slow axis, and 𝜃𝑀 is the orientation of the fast axis with respect to the horizontal-axis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=', perpendicular to mode propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Let us now consider the system with an ONF supporting HE𝑜 11, HE𝑒 11, TE01, TM01, HE𝑜 21 and HE𝑒 21, so that the number of modes that can be interfered is n ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The cross-sectional profiles and SOPs of TE01 and HE𝑒 21 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2(a, b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The TM01 and HE𝑜 21 modes are not shown here but their vector fields are orthogonal to the TE01 and HE𝑒 21 at every point, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' These modes have donut-shape mode profiles with linearly polarized vector fields at any point in the mode cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' As an example of possible fiber modes using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2(c) illustrates in-phase interference of the TE01 and HE𝑒 21 modes with equal amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The resulting mode has a lobe-shape intensity pattern with scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2(d) is an example of a mode resulting from the interference of the circularly polarized HE11 and an out-of-phase (a 𝜋/2 phase difference) TE01 and TM01 with equal amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The SOP, which is overlapped on the intensity profile images, are marked as red and blue ellipse, corresponding to right and left handed orientation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' This mode is the co-called lemon [55], which contains not only linear polarization but also elliptical and circular polarization components in one mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Simulations of (a) TE01, (b) HE𝑒 21, (c) TE01 + HE𝑒 21 and (d) lemon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The red and blue SOPs indicate right-handed and left-handed ellipticities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The scale bars show the normalized intensity (from 0 to 1) and the Stokes phase (from 0 to 2𝜋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Stokes singularity points of 𝜎12, 𝜎23, and 𝜎31 are indicated as pink, orange, and blue dots, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' An L-line is indicated in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' (a) (b) (c) (d) Φ12 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' DWhen using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 1 to simulate mode profiles, a number of eigenmodes with similar intensity patterns and SOPs to an experimentally observed cavity mode were selected as the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Next, the variables A𝑀, 𝜙𝑀, 𝜂𝑀, and 𝜃𝑀 were tuned to match the experimentally observed cavity mode intensities, SOPs, and Stokes phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Polarization topological defects in the simulated modes were then identified, using the method described in the following Section 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Analysis The polarization gradient was calculated in order to identify Stokes singularities in the cross- section of the mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The gradient map is known as the Stokes phase, 𝜙𝑖 𝑗, which is given by [42,45] 𝜙𝑖 𝑗 = 𝐴𝑟𝑔(𝑆𝑖 + 𝑖𝑆 𝑗), (3) where 𝑆𝑖 and 𝑆 𝑗 are Stokes parameters with {i, j} ∈ {1, 2, 3} in order, and i ≠ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The phase uncertainty points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=', Stokes singularities, were identified by obtaining the Stokes indices, 𝜎𝑖 𝑗, which are defined as [42,45] 𝜎𝑖 𝑗 = 1 2𝜋 ∮ 𝑐 𝜙𝑖 𝑗 · 𝑑𝑐, (4) where ∮ 𝑐 𝜙𝑖 𝑗 · 𝑑𝑐 = Δ 𝜙𝑖 𝑗 is the counterclockwise azimuthal change of the Stokes phase around the Stokes singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Singularities of 𝜎12 are known as V-points and C-points, in vector and ellipse fields, respectively [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Singularities of 𝜎23 and 𝜎31 are known as Poincaré vortices [43–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' L-lines are located where 𝜙23 = {0, 𝜋, 2𝜋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Table 1 is a summary of the classification of the Stokes singularity types in terms of the Stokes phases and singularity indices with the corresponding polarizations in the vector and ellipse fields [43,45,46,58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' List of Stokes singularities in vector fields (v) and ellipse fields (e) by the singularity index, 𝜎𝑖 𝑗, using the Stokes phase, 𝜙𝑖 𝑗, with {i, j} ∈ {1, 2, 3} in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Stokes Stokes phase Stokes index/ Polarization singularity Phase values V-point (v) 𝜙12 𝜎12 Null C-point (e) 𝜙12 𝜎12 R/L Poincaré 𝜙23 𝜎23 H/V vortex (e) 𝜙31 𝜎31 D/A L-line (e) 𝜙23 0, 𝜋, 2𝜋 Linear The Stokes singularity points and L-lines were found from the Stokes phases, then superimposed and marked on the mode profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' As examples, from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2(a, b), the center of the mode profiles for both TE01 and HE𝑒 21 contain a V-point, with 𝜎12 = -2 and +2 (pink dot), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' These points were found from their Stokes phases 𝜙12 (lower panels in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2(a, b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' In contrast, the lemon mode in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2(d) has a closed loop representing an L-line (green) and all three types of Stokes singularities: a C-point with 𝜎12 = -1 (pink dot), Poincaré vortices with 𝜎23 = -1 and +1 (orange dots), and 𝜎31 = -1 and +1 (blue dots) were found from 𝜙12, 𝜙23, and 𝜙31, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The lobe-shaped scalar mode in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2(c) does not have a 2𝜋 gradient in any associated Stoke phases, since topological defects can only exist in non-scalar fields [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Results and discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Cavity with a single-mode optical nanofiber As an initial experimental test, the spectrum for a HOM cavity containing an ONF of waist diameter ∼ 450 nm was obtained, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' This ONF waist can only support the fundamental modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The IPC paddle angles were set so that two distinct, well-separated modes with minimal spectral overlap were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The finesses of Modes 1 and 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 3(a) were 12 and 15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The laser was locked to each of these two cavity modes consecutively and the mode profiles were observed at the output end face of the fiber cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The corresponding mode intensity profiles, SOPs, and Stokes phases are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 3(b)(i, ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The intensity profiles for both Modes 1 and 2 were slightly skewed Gaussian shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The HE11 eigenmode intensity shape is Gaussian, so the slight deviation from the expected shape may be attributed to aberrations in the optical beam path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' In terms of polarization distribution, the Stokes phases of Modes 1 and 2 were uniform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' in other words, their SOPs were scalar fields, regardless of the IPC paddle angles chosen, as expected for the HE11 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Although the pretapered fiber supported the full set of eigenmodes in LP11, LP02, and LP21, when the ONF with a diameter ∼ 450 nm was inserted between the two sets of mirrors, only one or two modes with quasi-Gaussian profiles were observed, no matter which IPC paddle angles were chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The HOMs were filtered out due to the tapered fiber waist being SM, analogous to an intracavity pinhole spatial filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Mode filtering as a function of the ONF waist diameter was observed experimentally [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' However, here, we could additionally observe the mode filtering effect on the cavity spectrum and SOP of each mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' In an ideal SM-ONF cavity with no birefringence, there are two degenerate orthogonal modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' However, due to random birefringence of the fiber and the cavity mirrors, the two modes become non-degenerate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=', separated in frequency, leading to coupling between the modes [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Mode coupling of orthogonal modes can occur in a birefringent medium and this effect can increase in a cavity configuration [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Mode coupling in an ONF cavity due to asymmetrical mirrors has been discussed previously [56] and experimental evidence of mode coupling due to intrinsic birefringence in a SM-ONF cavity has already been reported [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' In our experiments, non-orthogonal combinations of SOPs were observed, as seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 3(b)(i, ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Mode 1 was horizontally polarized (red/blue lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 3(b)(i)), while Mode 2 was left elliptically polarized (blue ellipse in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 3(b)(ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' By adjusting the IPC angles, it was possible to change the phase relationship and coupling between the HE𝑜 11 and HE𝑒 11 modes, and shift between orthogonal and non-orthogonal combinations of SOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Cavity with a higher-order mode optical nanofiber Next, the spectrum for a HOM cavity containing an ONF of waist diameter ∼ 840 nm was obtained, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' This ONF can support the HE11, TE01, TM01, HE𝑜 21, and HE𝑒 21 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The IPC paddle angles were set to obtain the maximum number of well-resolved modes in a single FSR, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' One can clearly see five distinct peaks indicating that the HOM-ONF does not degrade the modes in the cavity and the finesses of the cavity modes are high enough to resolve them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The finesses of Modes 1 to 5 were 12, 16, 13, 22, and 13, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The mode finesse values of the cavity with a HOM-ONF were in the same range as those for the cavity with a SM-ONF (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 3(a)), implying that the HOM-ONF was adiabatic for the LP11 group of modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The laser was locked to each of the cavity modes consecutively and the mode profiles were observed at the output of the fiber cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The corresponding mode intensity profiles, SOPs, and Stokes phases are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(b)(i-iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' In the spectrum shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(a), there were five distinctive modes, but locking to Mode 3 was not possible because of its close proximity to the dominant Mode 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Two flat-top intensity profiles were observed in Modes 1 and 4, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(b)(i, iii) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' (a) A typical spectrum for a HOM cavity with a SM-ONF as the laser is scanned over 150 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The spectrum over a single FSR is indicated by the red box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' (b) Mode intensity profiles showing the SOPs (top) and corresponding Stokes phases (bottom) for (i) Mode 1 and (ii) Mode 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The red and blue SOPs indicate right-handed and left-handed ellipticities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The scale bars show the normalized intensity (from 0 to 1) and the Stokes phase (from 0 to 2𝜋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Laser scan frequency (MHz) (i) (ii) Φ12Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' (a) A typical spectrum for a cavity with a HOM-ONF as the laser is scanned over 150 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The spectrum over a single FSR is indicated by the red box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' (b) Mode intensity profiles showing the SOP (top) and the corresponding Stokes phases (bottom) for (i) Mode 1, (ii) Mode 2, (iii) Mode 4, and (iv) Mode 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The red and blue SOPs indicate right-handed and left-handed ellipticities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The scale bars show the normalized intensity (from 0 to 1) and the Stokes phase (from 0 to 2𝜋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Stokes singularity points of 𝜎12, 𝜎23, and 𝜎31 are indicated as pink, orange, and blue dots, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' L-lines are indicated in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' (c) Corresponding simulated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Laser scan frequency (MHz) (i) (ii) (i) (iv D D (i) (iv) Φ12 Φ23 Φ31 Φ23 Φ31 D 中 23 D 3The SOPs of these modes are markedly different to those for the Gaussian-type modes in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 3(b)(i, ii), which have simple scalar SOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Modes 1 and 4 were inhomogeneously polarized ellipse fields, showing regions of left and right circular polarizations divided by an L-line (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(b)(i, iii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The center of these two modes exhibited diagonal and anti-diagonal polarizations, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=', the SOPs at the center of the modes were orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Going towards the edges of the modes, the polarization changes from linear to circular, with opposite handedness either side of the L-lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Notice also in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(a) that Modes 1 and 4 are not well frequency separated from neighboring modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' This suggests that the mode profiles and SOPs of these modes were not only affected by birefringence and degenerate modal interference, but also some non-degenerate modal interference with neighboring cavity modes [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Additionally, for Mode 4, we identified two C-points (𝜎12 = -1), indicated by the pink dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(b)(iii), where the value of 𝜙12 changed by 2𝜋 (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Interference of HE11 with modes from the LP11 group can generate C-points in a few-mode fiber [55], see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' We performed basic simulations to determine if combinations of HE11 and some mode(s) in the LP11 family could generate similar mode profiles and SOP structures as those in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(b)(i, iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The simulated results are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(c)(i, iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The HE11 and TM01 modes were selected as possible contributors and their amplitudes, phase, and birefringence fitting parameters were tuned to match the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Modes 1 and 4, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(b)(i, iii), could have been formed from different mode combinations rather than our assumed HE11 and TM01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' however, these modes were very likely formed by interference between HE11 and some mode(s) of the LP11 group, resulting in their inhomogeneous SOPs and flat-top shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' We also observed two distorted lobe-shaped modes, Modes 2 and 5, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(b)(ii, iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The lobe-shaped pattern also arises from modal interference between modes in the LP11 family (as an example, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' With reference to Table 1, Mode 2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(b)(ii), showed all three types of Stokes singularities, indicated by pink dots for C-points (𝜎12 = +1) and orange/blue dots for Poincaré vortices (𝜎23 = -1 /𝜎31 = +1), as presented in 𝜙12, 𝜙23, and 𝜙31, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' A single mode containing all Stokes singularities has been demonstrated using free-space interferometers [43,46];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' here, we generated them within a single mode using a fiber cavity system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Mode 5, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(b)(iv), also had two C-points (𝜎12 = +1) and a Poincaré vortex (𝜎23 = +1), as seen in 𝜙12, and 𝜙23, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(a) shows that Modes 2 and 5 are not well frequency separated from Modes 1 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Therefore, there is a likely contribution from the HE11 mode resulting in distortion of the lobe shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' To simulate Mode 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(b)(ii), we combined TE01, HE𝑒 21, and HE11, and to simulate Mode 5 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(b)(iv), we used TM01, HE𝑒 21, and HE11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The amplitude of each mode, phase shift, and birefringence parameters were adjusted to achieve a close fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The simulated results are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(c)(ii, iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' These plots are not exact replications of the experimental results since the parameter space is large and the exact initial conditions are not known;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' nevertheless, the match is reasonably close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Interestingly, many of the cavity modes obtained in different sets of spectra, which were generated using different IPC angles, exhibited Stokes singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Polarization singularities are known to propagate through a birefringent medium as C-lines and L-surfaces and their evolution is affected by the homogeneity of the birefringence along the propagation path [47–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' This phenomenon is due to the conservation of the topological charge [49,58,61], and the Stokes index value, 𝜎𝑖 𝑗, remains constant [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' However, our cavity is an inhomogeneous birefringent medium as it contains a number of different birefringent elements such as the FBG mirrors and the IPC, as such, the degree of birefringence varies along the propagation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Therefore, the presence of Stokes singularities in the imaged field at the cavity output does not necessarily guarantee the existence of such topological defects in the ONF region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Nonetheless, singularity points can enter, move and exit with a smooth and continuous variation of birefringence [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Therefore, the SOP is expected to evolve along the length of the cavity, with singularity points shifting and making numerous entries and exits in the cross-section profile of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' However, since the ONF waist is relatively straight and uniform, the birefringence variation at the waist should be minimal [62] and topological features appearing at the start of the waist should be preserved every 2𝜋 along the waist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Theoretically, the HOM-ONF can support a total of six eigenmodes as mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Therefore, one might expect that the spectrum should show six distinct modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' However, we typically observed three to five distinct peaks in a single FSR depending on the IPC paddle angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' This could be explained by the lack of sufficient finesse to resolve all modes, some of which are closely overlapped [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' However, it may be feasible to increase the mode finesses by increasing the mirror reflectivity and using an ONF with lower transmission loss than the one used (the estimated loss of Mode 4, the highest finesse in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4(a), was ∼ 20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Nonetheless, the finesse values of our ∼ 2 m long cavity with a HOM-ONF should be sufficient for cQED experiments with narrow line-width emitters such as cold atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' (a) Mode intensity profiles for quasi-donut-shaped cavity modes from the cavity containing a HOM-ONF with their SOPs (top) and Stokes phases (bottom) similar to the fiber eigenmodes of (i) HE𝑒 21, (ii) HE𝑜 21, (iii) TE01, and (iv) TM01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The red and blue SOPs indicate right-handed and left-handed ellipticities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Scale bars show intensity (from 0 to 1) and Stokes phase (from 0 to 2𝜋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Stokes singularities of 𝜎12, 𝜎23, and 𝜎31 are indicated as pink, orange, and blue dots, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' L-lines are illustrated as green lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' (b) Corresponding simulated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' In situ higher-order cavity mode tuning A key feature of this setup is the ability to tune the spectrum and SOP to create the desired mode in the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' We aimed to observe modes with donut-shaped intensity patterns and SOPs similar to the fiber eigenmodes TE01 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2(a)), TM01, HE𝑜 21, and HE𝑒 21 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' To achieve this, the laser was locked to a well-resolved lobe-shaped mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The paddle angles of the IPC were then adjusted, and the mode shape was monitored with a CCD camera until a donut mode profile was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Unlocking and scanning the laser revealed a new spectrum with each mode containing (i) (ii) (iii) (iv) (D) (iv) D Da new profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The IPC was adjusted again to maximize another mode and the laser was locked to this new mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The IPC paddle angles were tuned to once more convert the mode profile to a donut shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' This procedure was repeated for four different modes, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 5(a)(i-iv), and these modes look similar to the true fiber eigenmodes of HE𝑒 11 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2(b)), HE𝑜 11, TE01 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2(a)), and TM01, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' There was a slight deformation from a perfect donut shape and their SOPs were not vector fields, but rather ellipse fields with alternating regions of opposite handiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' While the donut eigenmodes possessed a V-point at the center as indicated by pink dots in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2(a, b), the observed quasi-donut modes in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 5(a)(i-iv) had some nominal intensity at the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' These modes had two C-points of 𝜎12 = -1 or +1 near the center (see pink dots in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 5 (a)(i-iv)), as opposed to a single point of 𝜎12 = -2 or +2 in the true eigenmodes (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 2(a, b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Indeed, perturbation of vector field polarization singularities can occur when scalar linearly polarized beams are interfered [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' These donut-shaped cavity modes were also simulated, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 5(b)(i-iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' To obtain a good fit for the experimentally observed intensities, SOPs, and Stokes phases in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 5(a)(i-iv), the simulated modes included a slight deformation of the donut shape by adding some components of the HE11 mode to modes in the LP11 group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Moreover, the simulated results show that the Stokes phases are very similar to those obtained experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The number of possible combinations of modal interference with varying birefringence is large and this leads to discrepancies between the experiment and simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' However, these findings indicate that the experimentally observed quasi-donut modes are likely the result of residual interference between the HE11 mode and modes in the LP11 group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Degeneracy of multiple modes may be avoided by increasing the cavity mode finesses so that each mode can be well separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The system demonstrated here shows that, even in a complex system, the HOMs and their SOPs can be controlled to create exotic topological states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Conclusion We have experimentally demonstrated a Fabry-Pérot fiber cavity with a HOM-ONF and performed cavity spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The cavity mode profiles and transverse polarization topology were also determined by imaging and analyzing the individual cavity modes at the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' These modes had inhomogeneous polarization distributions with a number of Stokes singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' We also simulated the fiber modes which closely match those observed at the output of the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Moreover, in situ intracavity manipulation of the modal birefringence and interference to select a specific mode of interest was demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' This indicates that the evanescent field of an HON-ONF could be tuned by adjusting the IPC paddle angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' These findings are a step toward investigating the interactions between SAM and OAM of a HOM-ONF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Research into the interference of HOMs at the waist of an ONF is an exciting opportunity to uncover the nature of light-matter interactions in tightly confining geometries with topological singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Additionally, the realization of a (de)multiplexing system using degenerate HOMs in an ONF-based cavity may be possible by improving the tunability of the modal birefringence and interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Such a system is attractive for future quantum information platforms as efficient and secure storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The interference of higher-order cavity modes with fixed ratios in the evanescent field of an ONF may also be used to trap and manipulate cold atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Adjusting the overlap and SOP of the HOMs should result in movement of the trapping sites relative to each other, enabling some trap dynamics to be studied [4,15,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' This cavity could be also used with quantum emitters to study multimode cQED effects using degenerate HOMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The HOM cavity studied here had moderate finesse to enter the cQED experiments for interactions with cold atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' In free-space optics, strong coupling of multiple transverse HOMs with atoms has been achieved [38], whereas this has not been achieved using an ONF-type cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Our work is a significant step towards this realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Moreover, the ability of our cavity to generate all three types of Stokes singularities may be useful to realize not only a C-point laser but also an all-Stokes singularity laser using a few-mode fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The combinations of fiber modes that we used in the simulations were found via manual trial-and-error estimates to obtain a visual match with the experimentally observed modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' More accurate control could be achieved by using machine learning techniques to fully cover the parameter space of permitted modes in the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' This may enable us to determine the correct combination of modes that lead to the observed cavity outputs and facilitate feedback to optimize the input to the system to generate desired modes in the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Okinawa Institute of Science and Technology Graduate University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The authors acknowledge F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Le Kien, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Ruks, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Truong, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Ward for discussions and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Karlsson for technical assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Disclosures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFQT4oBgHgl3EQf4jbF/content/2301.13432v1.pdf'} +page_content=' K.' metadata={'source': 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II. Stage I release of NIRCam imaging and +catalogs in the Abell 2744 region. +Diego Paris +,1 Emiliano Merlin +,1 Adriano Fontana +,1 Andrea Bonchi +,2, 1 Gabriel Brammer +,3, 4 +Matteo Correnti,2, 1 Tommaso Treu +,5 Kristan Boyett +,6, 7 Antonello Calabr`o +,1 Marco Castellano +,1 +Wenlei Chen +,8 Lilan Yang +,9 K. Glazebrook +,10 Patrick Kelly +,8 Anton M. Koekemoer +,11 +Nicha Leethochawalit +,12 Sara Mascia +,1 Charlotte Mason +,3, 4 Takahiro Morishita +,13 +Mario Nonino +,14 Laura Pentericci +,1 Gianluca Polenta +,2 Guido Roberts-Borsani +,5 Paola Santini +,1 +Michele Trenti +,6, 7 Eros Vanzella +,15 Benedetta Vulcani +,16 Rogier A. Windhorst +,17 +Themiya Nanayakkara +,10 and Xin Wang +18, 19, 20 +1INAF Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Rome, Italy +2Space Science Data Center, Italian Space Agency, via del Politecnico, 00133, Roma, Italy +3Cosmic Dawn Center (DAWN), Denmark +4Niels Bohr Institute, University of Copenhagen, Jagtvej 128, DK-2200 Copenhagen N, Denmark +5Department of Physics and Astronomy, University of California, Los Angeles, 430 Portola Plaza, Los Angeles, CA 90095, USA +6School of Physics, University of Melbourne, Parkville 3010, VIC, Australia +7ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia +8Minnesota Institute for Astrophysics, University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455, USA +9Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo, Kashiwa, Japan 277-8583 +10Centre for Astrophysics and Supercomputing, Swinburne University of Technology, PO Box 218, Hawthorn, VIC 3122, Australia +11Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA +12National Astronomical Research Institute of Thailand (NARIT), Mae Rim, Chiang Mai, 50180, Thailand +13IPAC, California Institute of Technology, MC 314-6, 1200 E. California Boulevard, Pasadena, CA 91125, USA +14(INAF - Osservatorio Astronomico di Trieste, Via Tiepolo 11, I-34131 Trieste, Italy) +15INAF – OAS, Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Gobetti 93/3, I-40129 Bologna, Italy +16INAF Osservatorio Astronomico di Padova, vicolo dell’Osservatorio 5, 35122 Padova, Italy +17School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287-1404, USA +18School of Astronomy and Space Science, University of Chinese Academy of Sciences (UCAS), Beijing 100049, China +19National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China +20Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, China +ABSTRACT +We present images and a multi–wavelength photometric catalog based on all of the JWST NIRCam +observations obtained to date in the region of the Abell 2744 galaxy cluster. These data come from +three different programs, namely the GLASS-JWST Early Release Science Program, UNCOVER, and +Director’s Discretionary Time program 2756. The observed area in the NIRCam wide-band filters - +covering the central and extended regions of the cluster, as well as new parallel fields - is 46.5 arcmin2 in +total. All images in eight bands (F090W, F115W, F150W, F200W, F277W, F356W, F410M, F444W) +have been reduced adopting the latest calibration and references files available to date. Data reduction +has been performed using an augmented version of the official JWST pipeline, with improvements +aimed at removing or mitigating defects in the raw images and improve the background subtraction +and photometric accuracy. We obtain a F444W-detected multi–band catalog including all NIRCam +data and available HST data, adopting forced aperture photometry on PSF-matched images. The +catalog is intended to enable early scientific investigations, and is optimized for the study of faint +galaxies; it contains 24389 sources, with a 5σ limiting magnitude in the F444W band ranging from +28.5 AB to 30.5 AB, as a result of the varying exposure times of the surveys that observed the field. We +Corresponding author: Diego Paris +diego.paris@inaf.it +arXiv:2301.02179v1 [astro-ph.GA] 5 Jan 2023 + +ID2 +Paris et al. +publicly release the reduced NIRCam images, associated multi-wavelength catalog, and code adopted +for 1/f noise removal with the aim of aiding users to familiarize themselves with JWST NIRCam data +and identify suitable targets for follow-up observations. +Keywords: galaxies: high-redshift, galaxies: photometry +1. INTRODUCTION +In just a few months of observations, JWST has +demonstrated its revolutionary scientific capabilites. +Early observations have shown that its performance is +equal or better than expected, with image quality and +overall efficiency that match or surpass pre-launch esti- +mates (Rigby et al. 2022). Publicly available datasets +obtained by the Early Release Observations and Early +Release Science programs have already enabled a large +number of publications based on JWST data, ranging +from exoplanets to the distant Universe. +In particular, a number of works exploited the power +of NIRCAM to gather the first sizeable sample of can- +didates at z ≥ 10 (e.g., Castellano et al. 2022a; Don- +nan et al. 2023; Finkelstein et al. 2022; Morishita & Sti- +avelli 2022; Naidu et al. 2022; Yan et al. 2022; Roberts- +Borsani et al. 2022; Robertson et al. 2022; Castellano +et al. 2022b; Bouwens et al. 2022), showing the power of +JWST in exploring the Universe during the re-ionization +epoch. +In this paper we present the full data set obtained +with NIRCam in the region of of the z = 0.308 cluster +Abell 2744 that will significantly expands the available +area for deep extragalactic observations. +The central +region of the cluster, with the assistance of lensing mag- +nification, allows an insight into the distant Universe +at depth and resolution superior of those of NIRCam +in blank fields. +The data set analyzed here are ob- +tained through three public programs: i) GLASS-JWST +ERS (Treu et al. 2022), ii) UNCOVER (Bezanson et al. +2022), and iii) Director’s Discretionary Time Program +2756, aimed at following up a Supernova discovered in +GLASS-JWST NIRISS imaging. We have analyzed and +combined the imaging data of all these programs and +obtained a multi-wavelength catalog of the objects de- +tected in the F444W band. +In order to facilitate exploitation of these data, we +release reduced images and associated catalog on our +website and through the Mikulski Archives for Space +Telescopes (MAST). This release fulfills and exceeds the +requirements of the Stage I data release planned as part +of the GLASS-JWST program. +It is anticipated that +a final (Stage II) release will follow in approximately +a year, combining additional images scheduled in 2023, +and taking advantage of future improvements in data +processing and calibrations. +This paper is organized as follows. In Section 2 we +present the data-set and discuss the image processing +pipeline. In Section 3 the methods applied for the de- +tection of the sources and the photometric techniques +used to compute the fluxes are presented. +Finally in +Section 4 we summarize the results. +Throughout the +paper we adopt AB magnitudes (Oke & Gunn 1983). +2. DATA REDUCTION +2.1. Data Set +The NIRCam data analyzed in this paper are taken +from three programs that targeted the z = 0.308 clus- +ter Abell 2744 (A2744 hereafter) and its surroundings. +The first set of NIRCam images were taken as part of +the GLASS-JWST survey (Treu et al. 2022, hereafter +T22), in parallel to primary NIRISS observations on +June 28–29 2022 and to NIRSpec observations on Nov. +10–11, 2022. We refer to these data sets as GLASS1 and +GLASS2, or collectively as GLASS, both of which con- +sist of imaging in seven broad-band filters from F090W +to F444W (see Treu et al. 2022 for details). We note that +the final pointing is different from the scheduled one pre- +sented by Treu et al. (2022) due to the adoption of an +alternate position angle (PA) during the NIRSpec spec- +troscopic observations. +As the primary spectroscopic +target was the A2744 cluster, these parallel images are +offset to the North-West. By virtue of the long exposure +times, these images are the deepest presented here. +The second set of NIRCam observations considered +here were taken as part of the UNCOVER program +(Bezanson et al. 2022), which targets the center of the +A2744 cluster and the immediate surroundings. These +images are composed of four pointings and result in a +relatively homogeneous depth, as discussed below. They +were taken on November 2-4-7 and 15, and adopt the +same filter set as GLASS-JWST, except for the adop- +tion of the F410M filter instead of F090W. +Finally, NIRCam imaging of the A2744 center was +also obtained as part of DDT program 2756 (PI W. +Chen, DDT hereafter) on October 20 and December +6 2022 (UT). These two data sets are dubbed DDT1 +and DDT2 hereafter. The DDT filter set is the same +as GLASS-JWST with the exception of the F090W fil- +ter, and overall shorter exposure times. One of the two +NIRCam modules overlaps with UNCOVER. + +GLASS-JWST: Abell 2744 NIRCam photometric catalog +3 +Figure 1. Full view of the F444W mosaics. Colored boxes show the position of the three different data sets used here: GLASS +(green), UNCOVER (blue) and DDT (red). The entire image (including the empty space) is approximately 12.7 × 10.9 arc +minutes wide. +In Table 1 we list the exposure times adopted in the +various filters for each of the aforementioned programs, +while the footprints of the fields are illustrated in Fig- +ure 1. +As a result of the overlap between programs and of +their different observation strategies, the resulting ex- +posure map is complex and inhomogenous across bands +and area. An analysis of the depth resulting from this +exposure map is reported below. +2.2. Data reduction +2.2.1. Pre-reduction steps +Image pre-reduction was executed using the official +JWST calibration pipeline, provided by the Space Tele- +scope Science Institute (STScI) as a Python software +suite1. We adopted Version 1.8.2 of the pipeline and Ver- +sions between cjwst 1014.pmap and cjwst 1019.pmap +of the CRDS files (the only changes between these +versions is the astrometric calibration, that is dealt +with as described below). +We executed the first two +stages of the pipeline (i.e. +calwebb detector1 and +1 https://jwst-pipeline.readthedocs.io/en/latest/jwst/ +introduction.html +Table 1. NIRCam Exposure time +Filter +GLASS1 +GLASS2 +DDT1/2 +UNCOVER +F090W +11520 +16492 +- +- +F115W +11520 +16492 +2104 +10822 +F150W +6120 +8246 +2104 +10822 +F200W +5400 +8246 +2104 +6700 +F277W +5400 +8246 +2104 +6700 +F356W +6120 +8246 +2104 +6700 +F410M +- +- +- +6700 +F444W +23400 +32983 +2104 +8246 +Note—Exposure time (in seconds) for each pointing of the +three programs considered here. +calwebb image2), adopting the optimized parameters +for the NIRCam imaging mode, that convert single de- +tector raw images into photometric calibrated images. +Using the first pipeline stage calwebb detector1 we +processed the raw uncalibrated data (uncal.fits) in +order to apply detector-level corrections performed on +a group-by-group basis, as dark subtractions, reference +pixels corrections, non-linearity corrections and jump + +4 +Paris et al. +detection that allows to identify cosmic rays (CR) events +on the single groups. The last step of this pipeline stage +allows us to derive the mean count rate, in units of +counts per second, for each pixel by performing a lin- +ear fit to the data in the input image (the so-called +ramp-fitting) excluding the group masked due to the +identification of a cosmic ray jump. +The output files of the previous steps (rate.fits) +are +processed +through +the +second +pipeline +stage +calwebb image2, +which +consists +of +additional +instrument-level and observing-mode corrections and +calibrations, as the geometric-distortion correction, the +flat-fielding, and the photometric calibrations that con- +verting the data from units of countrate to surface +brightness (i.e. +MJy per steradian) generates a fully +calibrated exposure (cal.fits). +The cal.fits file contains also an RMS layer, which +combines the contribution of all pixel noise sources, and +a DQ mask where the first bit (DO NOT USE) identifies pix- +els that should not be used during the resampling phase. +We then applied a number of custom procedures to +remove instrumental defects that are not dealt with +the STScI pipeline. Some of them have already been +adopted in (Merlin et al. 2022, hereafter M22) and +described there: +we illustrate below only the major +changes to the STScI pipeline in default configuration +and/or to the procedure adopted in M22. +• “Snowballs”, i.e. circular artifacts observed in the +in-flight data caused by a large cosmic ray impacts. +Those hits leave a bright ring-shaped defect in the +image since the affected pixels are just partially +identified and masked. In M22, we developed a +technique to fully mask out these features, which +was not necessary here. Indeed, version 1.8.1 of +the JWST pipeline introduced the option to iden- +tify snowball events, expanding the typical mask- +ing area to include all the pixels affected. +This +new implementation provides the opportunity to +correct these artifacts directly at the ramp fitting +stage, at the cost of a larger noise on the corre- +sponding pixels. We activated this non-default op- +tion, and fine tuned the corresponding parameters +to completely mask all the observed snowballs and, +at the same time, minimize the size of high noise +areas. +• “NL Mask”, on cal images of the NIRCam Mod- +ule B Long Wavelength detector are visible bright +groups of pixels not well corrected during prere- +duction. These pixels are more evident on deeper +pointing and are identified as “well not defined” +pixels2 in the Non Linearity Calibration file 3. +We selected those pixels and masked them as +DO NOT USE to not to be used during stacking +phase. +• 1/f noise, which introduces random vertical and +horizontal stripes into the images (see Schlawin +et al. 2020). We remove this by subtracting the +median value from each line/column, after mask- +ing out all objects and bad pixels. +The masks +were obtained by running SExtractor (Version +2.25.0) (Bertin & Arnouts 1996) and then dilat- +ing the resulting segmentation image, applying a +differential procedure to dilate objects depending +on their ISOAREA: the segmentation of objects +with ISOAREA<5000 pixels was dilated using a +3 × 3 convolution kernel and a dilation of 15 pix- +els, while for the segmentation of objects with +ISOAREA⩾5000 pixels a 9 × 9 convolution ker- +nel and a dilation of 4 × 15 pixels was used. The +procedure was executed separately for each am- +plifier in the SW detectors (i.e. 4 times for each +individual image) with the exception of the denser +areas corresponding to the centers of the clusters +and the brightest field star, where objects are sig- +nificantly larger than the amplifier width (500 pix- +els, corresponding to about 30”) and could not be +masked efficiently. In this case we removed the 1/f +noise over the entire row. As this extension of the +STScI pipeline could be useful for other programs, +we publicly release the code adopted for this step. +• Scattered light: +we identify additive features in +the F115W, F150W and F200W images. +These +low-surface brightness features have already been +revealed by commissioning data (see Rigby et al. +2022) and are due to scattered light entering into +optical path. These anomalies have been dubbed +wisps or claws, depending on their origin and mor- +phology. +Wisps have a nearly constant shape +and a template pattern is available for subtraction +from the images. We removed these features by +extracting their 2D profile from the available tem- +plate (we do not use the entire template image to +avoid subtracting its empty but noisy regions) and +then normalizing the residual template to match +the feature intensity in each image. Claws have +been first identified and singled out in images. +2 https://www.stsci.edu/files/live/sites/www/files/home/jwst/ +documentation/technical-documents/ documents/JWST-STScI- +004714.pdf +3 https://jwst-crds.stsci.edu/browse/jwst nircam linearity 0011.rmap + +GLASS-JWST: Abell 2744 NIRCam photometric catalog +5 +Figure 2. Examples of custom procedures to remove resid- +ual instrumental defects, not dealt with the current STScI +pipeline. Top: 1/f stripes removal on a GLASS F200W single +exposure. Bottom: A portion of the GLASS F150W mosaic +before and after the claws treatment. +Their shape on each image has been reconstructed +by interpolating a 2D mesh with box size 32 pix- +els and then eventually subtracted from the same +image. We find that these procedures efficiently +remove most of these features, as shown in Fig- +ure 2. +Other defects were found in the F090W image, and +to a lesser extent in the F115W one, which are +due to a so-called “wing-tilt event” that happened +during the observations. These defects have been +masked as in M22. +We then re-scaled the single exposures to units of +µJy/pixel, using the conversion factors output by the +pipeline. +2.2.2. Astrometry +The astrometric calibration was performed using +SCAMP (Bertin 2006), with 3rd order distortion correc- +tions (PV coefficients up to j = 10). At variance with the +procedure we adopted in M22, we started from the dis- +tortion coefficient computed by the STScI pipeline and +stored in the cal images, and refine the astrometric so- +lution by running scamp in cal mode, which optimizes +the solution with limited variations from the starting so- +lution. We have found this procedure both accurate and +reliable, as described below. We first obtained a global +astrometric solution for the F444W image, which is usu- +ally the deepest, tied to a ground-based catalog obtained +in the i-band with the Magellan telescope in good see- +ing condition (see T22 for details) of the same region, +which had been previously aligned to GAIA-DR3 stars +(Gaia Collaboration et al. 2016, 2022 in prep.). We then +took the resulting high-resolution catalog in F444W as +reference for the other JWST bands, using compact, iso- +lated sources detected at high signal-to-noise at all wave- +lengths. Each NIRCam detector has been analysed inde- +pendently, in order to simplify the treatment of distor- +tions and minimise the offsets of the sources in different +exposures. Finally, we used SWarp (Bertin et al. 2002) +to combine the single exposures into mosaics projected +onto a common aligned grid of pixels, and SExtractor +to further clean the images by subtracting the residual +sky background. The pixel scale of all the images was +set to 0.031′′ (the approximate native value of the short +wavelength bands), to allow for simple processing with +photometric algorithms. +The final image, computed as a weighted stack of all +the images from the three programs, has a size of 24397× +21040 pixels, corresponding to 12.6 × 10.87 arcmin2. In +this frame, the area covered by the wide-band NIRCam +images (F115W, F150W, F200W, F277W, F356W and +F444W) is of exactly 46.5 arcmin2. The F444W image +is shown in Figure 1. +Given the especially deep and sharp nature of the +JWST images, where most of the faint objects have sizes +below 0.5′′, the requirements on the final astrometric ac- +curacy are extremely tight, to avoid errors in the multi- +band photometry (where a displacement of as little as +0.1′′ can bias color estimates). These requirements must +be met also in the overlapping regions of the various +surveys, which have often been observed with different +detectors. +To verify the final astrometric solution we conducted +a number of validation tests, where we compare the +positions of cross-matched objects in catalogues ex- +tracted from different images. +For each of these cat- +alogues we used SExtractor in single image mode +and adopted the XWIN and YWIN estimators of +the object center, which are more accurate than other +choices. +At the unprecedented image quality of NIR- +Cam, the accurate center of extra–galactic objects with +complex morphology may be difficult to estimate with +high accuracy, especially when observed across a large +wavelength interval. +To minimize errors, we limited +the comparison to objects with well defined positions, +using the ∆X, ∆Y +=ERRAWIN WORLD, ERRB- +WIN WORLD estimators of the error and limiting the +analysis to objects with (∆X2 + ∆Y 2)1/2 ≤ 0.018”. +From these catalogues we estimated both the average +offset of the object centers ∆α and ∆δ, and the median +average deviation madα and madδ, which measure the + +6 +Paris et al. +Figure 3. Validation tests on the astrometric registration. Left: scatter diagram reporting the displacement δRA and δDEC +of sources between the Magellan i–band catalog registered to Gaia DR3 used as global reference for calibration and the final +F444W NIRCam catalog. Middle left: As above, applied to the scatter between the AstroDeep catalog and the final F444W +NIRCam catalog obtained on the central region of the A2744 cluster, as obtained in the context of the Frontier Fields initiative +(Merlin et al. 2016a). Middle right: Offset between the position of sources in the F444W and the F115W images. Right: +Positional offset between the objects detected in the UNCOVER–only images and those in the GLASS and DDT samples, on +two overlapping regions. In all diagrams the average value ∆α and ∆δ and the median average deviation mad∆α and mad∆δ +are reported. +intrinsic scatter in the alignment. In Figure 3 we report +the main outcome of these tests: +• (Left) We first compared the positions of objects +in the original Magellan i-band and the resulting +F444W of the entire mosaic. We find an essentially +zero offset and madα ≃ madδ ≃ 0.02”, which is 2/3 +of a pixel. +• (Middle left) We compared the F444W catalog +with the AstroDeep H160 catalog obtained on the +central region of the A2744 cluster, as obtained in +the context of the Frontier Fields initiative (Mer- +lin et al. 2016a). While the intrinsic scatter is still +good (madα ≃ madδ ≃ 0.02”), we find a system- +atic offset by about 1 pixel in RA and 2.5 pixels in +DEC, which is most likely due to different choices +in the absolute calibration of the ACS/WFC3 data +released within the Frontier Fields. +• (Middle right) We compare here the relative cali- +bration of filters at the two extremes of the spec- +tral range, F444W and F115W, where morphologi- +cal variations and color terms may change the cen- +ter position and affect the astrometric procedure. +We find again very good alignment with negligible +offset and small madα. ≃ madδ ≃ 0.01” +• (Right) Finally, we compare the astrometric solu- +tions on the overlapping areas by summing inde- +pendently the data of the three different programs +and checking the accuracy in the overlapping area. +Again we find very good alignment with negligible +offset and small madα ≃ madδ ≃ 0.01”. +We therefore conclude that the astrometric procedure +is accurate and adequate to the goals of this Stage I +release. In the future we plan to further explore and +validate other options for astrometric registration and +also release images with a smaller pixel scale, to better +exploit the unprecedented image quality of the JWST +data. We note that the GLASS-JWST data have a very +limited dithering pattern (which was driven by spectro- +scopic requirements) and so may benefit only marginally +from moving to smaller pixels. +2.3. Estimating the Final Depth +The final coaddition of the different images is weighted +according to their depth, as estimated by the RMS im- +age produced by the pipeline. We therefore obtain an +optimally averaged image with the resulting RMS im- +age. We a posteriori verified whether the noise estimate +encoded in the RMS effectively reproduced the photo- +metric noise. +To do this, we injected artificial point +sources of known magnitude in empty regions of the im- +age, and measured their fluxes and uncertainties with +a-phot (Merlin et al. 2019), using apertures of radius +0.1′′. To take into account the fact that the mosaics are +the result of a complex pattern of different exposures, +we divided the maps into regions of similar total expo- +sure time, and performed this analysis separately in each +region. +In general, we find that the RMS of the resulting flux +distribution is 1.1× larger than the value we would ex- + +Magellan i vs F444w (1810 obj.) +△α = 0.000", mad^α = 0.02 +A6 = -0.002", madAs = 0.01 +0.09 +0.06 +0.03 +0.00 +2-0.03 +-0.06 +-0.09 +△α (")H160 vs F444W (728 0bi. +△α = 0.033", madAα = 0.02 +A6 = -0.072", madAs = 0.02 +0.18 +0.12 +0.06 +0.00 +1-0.06 +-0.12 +-0.18 +△α (")F115W vs F444W (3649 0bj.) +△α = 0.000", madAα= 0.01 +A6 = -0.002", madas = 0.01 +0.09 +0.06 +0.03 +0.00 +2-0.03 +-0.06 +-0.09 +△α (")F444w overlap regions (1529 obj. +Aα = -0.002",mad^α = 0.01 +△6 = 0.002", madAs = 0.01 +0.09 +0.06 +0.03 +0.00 +2-0.03 +-0.06 +-0.09 +△α (")GLASS-JWST: Abell 2744 NIRCam photometric catalog +7 +Figure 4. Depth of the full mosaic F444W image, as pro- +duced by our pipeline on the basis of the variance image +of each exposure and with the re-normalization described in +the text. Each pixel has been converted into 5σ limiting flux +computed on a circular aperture of 0.2”. +pect from the SExtractor errors, which are computed +from the RMS image. A larger difference (1.4×) is found +for the F444W GLASS image, which is affected by a +residual pattern due to poor flat–fielding with the cur- +rent calibration data. We therefore re-scaled the RMS +maps produced by the pipeline according to these fac- +tors. +The resulting depth of this procedure is shown in Fig- +ure 4. The RMS image is converted into a 5σ limiting +flux computed on a circular aperture with a diameter +of 0.2′′, that is the size adopted to estimate colors of +faint sources. The depth ranges from ≃ 28.6 AB on the +DDT2 footprint (in particular the area not overlapping +with DDT1) to ≃ 30.2 AB in the area where GLASS1 +and GLASS2 overlap, arguably one of the deepest im- +ages obtained so far by JWST. +A more quantitative assessment of the depth in the +various filters is reported in Figure 5, where we show the +distribution of the limiting magnitudes in each image +resulting from the different strategies adopted by the +surveys,computed as described above. A clear pattern is +seen, illustrating the large, mid–depth area obtained by +UNCOVER and the shallower and deeper parts obtained +by DDT and GLASS respectively. +2.4. HST Imaging +We have also used the existing images obtained with +HST in previous programs, namely with the F435W, +F606W, F775W and F814W bands with ACS and the +Figure 5. Distribution of the limiting magnitude for each +band, as shown in the legend. Limiting magnitudes per pixel +have been computed as for Figure 4. +F105W, F125W, F140W and F160W bands with WFC3 +- other HST data are available from MAST but are ei- +ther too shallow and/or limited in area and are not con- +sidered here. Among these data are included also the +images that we obtained with DDT Program HST-GO- +17231 (PI: Treu), which was specifically aimed at obtain- +ing ACS coverage for the majority of the GLASS1 and +GLASS 2 fields. We have used calibrated stacked image +and weights (G. Brammer, private communication) that +we have realigned (after checking that the astrometric +solution is consistent) onto our reference grid to allow a +straightforward computation of colors. +3. PHOTOMETRIC CATALOG +3.1. Detection + +2.5 +F090W +F115W +2.0 +ou +1.5 +(el, +1.0 +N +0.5 +9.9 +F150W +F200W +2.0 +QU +1.5 +el, +1.0 +N +0.5 +2:9 +F277W +F356W +2.0 +xel, norm +1.5 +1.0 +0.5 +2:9 +F41QM +F444W +2.0 +ou +1.5 +xel, +1.0 +N +0.5 +0.0 +28 +29 +30 +31 +28 +29 +30 +31 +magim +maglim26.5 +27 +27.5 +28 +28.5 +29 +29.5 +30 +30.58 +Paris et al. +We follow here the same prescriptions adopted by M22 +and Castellano et al. (2022a,b). We performed source +detections on the F444W band, since it is generally the +deepest or among the deepest image for each data set, +and because high-redshift sources (which are the main +focus of these observations) are typically brighter at +longer wavelengths. This approach has the advantage of +delivering a clear-cut criterion for the object detections, +that can easily be translated into a cut of rest-frame +properties for high redshift sources. +We used SExtractor, adopting a double–pass ob- +ject detection as applied for the HST-CANDELS cam- +paign (see Galametz et al. 2013), to detect the objects, +following the recipes and parameters described in M22. +We note in particular that we adopt a detection thresh- +old corresponding to a signal-to-noise ratio (SNR) of 2. +This is based on simulations, as discussed in M22. The +other SExtractor parameters used are listed in M22. +The final SExtractor catalogue on the entire A2744 +area contains 24389 objects. +Estimating the completeness and purity in a patchy +(in terms of area and exposure) mosaic derived from +the large number of observations adopted here, is in- +trinsically ambiguous. As shown in Figure 5 the depth +of these images spans approximately 2 magnitudes, and +the completeness is therefore inhomogenoues - not to +mention the existence of the cluster that complicates +both the detection and the estimate of the foreground +volume (C22b). For these reasons, we do not attempt +the traditional estimate of the completeness and refer to +Figure 4 and to Figure 5 for an evaluation of the depth. +For a proper analysis of the completeness we refer the +reader to the methodology adopted by C22b were we +estimate the completeness separately on the individual +mosaics of the three data sets, which were processed in- +dependently. We make the three mosaics available upon +request for this purpose. +3.2. Photometry +We have compiled a multi-wavelength photometric +catalog following again the prescriptions of M22, which +in turn is based on previous experience with Hubble +Space Telescope (HST) images in CANDELS (see e.g. +Galametz et al. 2013) and in AstroDeep (Merlin et al. +2016b, 2021). The catalog is based on a detection per- +formed on the F444W image described above, and PSF– +matched aperture photometry of all the sources. +We +include all the NIRCam images presented here and ex- +isting images obtained with HST in previous programs, +namely with the F435W, F606W, F775W and F814W +bands with ACS and the F105W, F125W, F140W and +F160W bands with WFC3. +The images considered here have PSFs that range +from 0.035” to 0.2”. Considering that most of the ob- +jects have small sizes, with half–light–radii less than +0.2”, it is necessary to apply a PSF homogenization to +avoid bias in the derivation of color across the spectral +range. +3.2.1. PSF matching +Since the detection band is the one with the coars- +est resolution, we PSF-matched all the other NIRCam +images to it for color fidelity. We created convolution +kernels using the WebbPSF models publicly provided +by STScI4, combining them with a Wiener filtering al- +gorithm based on the one described in Boucaud et al. +(2016); and we used a customised version of the con- +volution module in t-phot (Merlin et al. 2015, 2016a), +which uses FFTW3 libraries, to smooth the images. This +approach delivers consistent results with those obtained +using the software Galight (Ding et al. 2020). +We +note that this approach is inevitably approximated. The +JWST PSF is time– and position–dependent (Nardiello +et al. 2022), and our dataset is the inhomogeneous com- +bination of data obtained at different times and with +different PA, so that the PSF definitely changes over +the field. For this version of the catalog we used the +Uncover PSF models as average PSFs, and we plan to +improve our PSF estimation in the future versions of the +catalog that will be released in Stage II. +Similarly, concerning the HST images, we note that all +of them have too few stars to obtain a robust estimate +of the PSF directly from the images, so that we adopt +in all cases existing HST PSFs, taken from CANDELS. +This approximation may introduce small biases in the +final catalog. ACS images have been PSF-matched to +F444W, while for the WFC3 F105W, F125W, F140W +and F160W images, which have a PSF larger than the +F444W one, we have done the inverse - smoothed the +F444W image and the WFC3 F105W, F125W, F140W +to the F160W and followed a slightly different procedure +that we describe below. +3.2.2. Flux estimate +The total flux is measured with a-phot on the detec- +tion image F444W by means of a Kron elliptical aper- +ture (Kron 1980). As we have shown in M22, simula- +tions suggest that Kron fluxes measured with a-phot +are somewhat less affected by systematic errors, while +being slightly more noisy. +4 https://jwst-docs.stsci.edu/jwst-near-infrared-camera/ +nircam-predicted-performance/nircam-point-spread-functions + +GLASS-JWST: Abell 2744 NIRCam photometric catalog +9 +Then, we used a-phot to measure the fluxes at the po- +sitions of the detected sources on the PSF-matched im- +ages, masking neighboring objects using the SExtrac- +tor segmentation map. Given the wide range of magni- +tudes and sizes of the target galaxies we have measured +the flux in a range of apertures: the segmentation area +(the images being on the same grid and PSF-matched) +and five circular apertures with diameters that are inte- +ger multiples (2×, 3×, 8×, 16×, ) of the FWHM in the +F444W band, that correspond to 0.28′′, 0.42′′, 1.12′′ and +2.24′′ diameters. For the four WFC3 images (which have +a PSF larger than F444W) we first filtered the F444W +to their FWHM and then measured colors between the +filtered F444W and the WFC3 images. To minimize bi- +ases when these colors are combined with those of the +other bands, we use in this case apertures the same mul- +tiples of the WFC3 PSF adopted for the other bands. +We remark that this procedure is only approximate, and +delivers a first order correction of the systematic effects +due to different PSFs. In a future release we plan to +adopt more sophisticated approaches to optimize pho- +tometry, including but not limited to the improvement +of the PSF estimate and applying T-PHOT on WFC3 +images that have a larger PSF. +Total fluxes are obtained in the other bands by +normalizing the colors in a given aperture to the +F444W total flux, +i.e. +by computing fm,total += +fm,aper/fF 444W,aper × fF 444W,total, as described in M22. +We release the five catalogues described above (one +computed on segmentation and four on the different +apertures) and we leave the user to choose which is the +most suitable for a given science application. In general +small-aperture catalogues are more appropriate for faint +sources as they match their small sizes and minimize de- +belending. Larger apertures may be more appropriate +for brighter sources and especially cluster members. +3.2.3. Validation tests +We have performed a few validation tests to verify pri- +marily the flux calibration, that has been the subject of +many revisions in these first months, and to a lesser ex- +tent of the procedure adopted to derive the photometric +catalog. +The overlap between GLASS1 and GLASS2 southern +quadrants offers us a nice opportunity to test the NIR- +Cam flux calibration. Indeed, the two GLASS observa- +tions have been observed in two epochs (July and Octo- +ber 2022) with a PA difference of nearly 150 degrees. As +a result, the southern quadrant of GLASS1 and GLASS2 +are largely overlapping but have been observed with +modules B and A, respectively. We have therefore ob- +tained stacked images of the two epochs separately, built +Figure 6. Stability of the photometric calibration between +different detectors, as measured by comparing the photom- +etry of high S/N objects (S/N > 25) detected in the two +epochs of observations in the SE quadrant of GLASS (lower +leftmost green square in Figure 1). Objects in this area have +been observed in two epochs (July and October 2022) and +with modules B and A, respectively. For each filter difference +in magnitude ∆M = M1 − M2 for objects between epoch1 +and epoch2 as a function of M1 is reported. Red dashed lines +represent the median offsets, namely we found: ∆M ≈ 0.06 +with mad ≈ 0.05 for F090W, ∆M ≈ 0.05 with mad ≈ 0.04 +for F115W, ∆M ≈ 0.04 with mad ≈ 0.04 for F150W, +∆M ≈ 0.02 with mad ≈ 0.04 for F200W, ∆M ≈ 0.05 with +mad ≈ 0.04 for F277W, and negligible in F356W and F444W +with mad ≈ 0.03 and mad ≈ 0.02 respectively. We have vi- +sually inspected the bright objects with |∆M| > 0.05 and +verified that they mostly originate from saturated stars or +objects with incomplete coverage. +a photometric catalog with the same recipes and checked +the magnitude difference between objects observed with +different detectors. The result of this exercise, that has +been done on all bands, is reported in Figure 6. We note +that in the short bands the two modules are made of 4 +detectors, each with an independent calibration, that we +plot all together in Figure 6. The comparison, that is +limited to objects observed with high S/N > 25, shows +that the average magnitude difference between the two + +GLASS 1 vS. 2 +0.2 +F090W +0.0 +-0.2 +0.2 +F115W +0.0 +-0.2 +0.2 +F150W +0.0 +-0.2 +M +0.2 +F200W +0.0 +0.2 +F277W +0.0 +-0.2 +0.2 +F356W +0.0 +-0.2 +0.2 +F444W +0.0 +-0.2 +20 +21 +22 +23 +24 +25 +26 +27 +M110 +Paris et al. +modules is in general quite small, in all cases below 0.05 +mags (see Figure 6 and its captions for details). This +confirms that the flux calibration between the different +modules is reasonably stable at this stage. +As a further check to validate the photometric +pipeline, we have compared the m606 and m150 mag- +nitudes for the sources in the core of the A2744 cluster +with those measured in the same F606W and in the +nearby F160W bands measured on HST images, that +we published within the AstroDeep project (Merlin et al. +2016b; Castellano et al. 2016). This comparison is shown +in Figure 7. Magnitudes in the NIRCam F150W band +have been shifted by ≃ 0.05 in order to correct for the +small bandpass difference: the term was estimated using +theoretical SEDs from a simulated photometric catalog, +created using Egg (Schreiber et al. 2017). The com- +parison shows that - when the same approach is used to +estimate colours, i.e. isophotal magnitudes are adopted +- the agreement between the two catalogues is excel- +lent. When we use instead relatively smaller aperture +in 8×FWHM for the NIRCam photometry we tend to +underestimate the F150W and - even more - the F606W +flux of the brightest sources, which are considerably +more extended than 8×FWHM. We ascribe this effect to +the existence of colour gradients in bright objects, such +that small-sized apertures tend to sample the central, +redder part of the galaxies. +From this comparison we conclude that - quite reas- +suringly - the overall photometric chain is consistent be- +tween the well established Frontier Fields data and these +new data. At the same time, we remark that the choice +of which aperture is optimal depends on the size and +kind of objects under study. +For faint sources, small +apertures tend to have higher S/N and should be pre- +ferred. +For brightest sources, larger apertures should +be preferred. It is also possible to estimate rough color +gradients by comparing the various apertures that we re- +lease. We also tested that applying the same technique +without PSF matching introduces an offset of the order +of ∼0.2 mags in the final colors, which would clearly af- +fect the derived photometric redshifts and SED fitting +results. +Finally, in an effort to cross-validate our results prior +to release, in the lead up to this paper we compared +our catalogs to those under development by the UN- +COVER team (Weaver et al. 2023, in prep) based on +the same raw datasets. The image processing and pho- +tometric procedures adopted by the two teams have sig- +nificant differences. +The main are: i) image coaddi- +tion (UNCOVER team adopts grizli, while we use a +custom pipeline which uses scamp and swarp; ii) ob- +ject detection (UNCOVER uses an optimally stacked +Figure 7. +Comparison of photometry between this work +and AstroDeep HST catalogs in the core of the A2744 clus- +ter. Upper: Difference between the magnitude in the F160W +WFC3 band in AstroDeep and the F150W NIRCam of this +work for objects in common between the two catalogues. The +F150W magnitude has been corrected for the ≃ 0.05 mag- +nitude shift between the two bands. Filled point represent +the difference between magnitudes computed in isophotal ar- +eas in both catalogues. Empty points represent the magni- +tude difference adopting the F150W magnitude computed in +8×FWHM. Bottom: As above, for the F606W band. The +systematic bias between isophotal and 8×FWHM colours is +due to color gradients in the center of bright sources. +F277W+F356W+F444W image after removing the intr- +acluster light, while we use F444W); iii) techniques and +tools for PSF matching and photometry. For these rea- +sons, we expect some differences between the catalogs, +especially for faint sources at the detection limit. How- +ever, our comparison of working versions of the catalogs +produced by the two teams shows overall a good agree- +ment in the colors and magnitudes of the vast majority +of objects, with no evidence of significant bias beyond +what can be explained by the different choices. We defer +a detailed comparison to future versions of the catalog +(Stage II). +4. SUMMARY +We present in this paper the data obtained by three +NIRCam programs on the A2744 cluster: the GLASS- +JWST Early Release Science Program, UNCOVER, and +Directory Discretionary Time 2756. All the data, taken +with eight different filters (F090W, F115W, F150W, +F200W, F277W, F356W, F410M, F444W), have been +reduced with an updated pipelines that builds upon the +official STScI pipeline but includes a number of improve- +ment to better remove some instrumental signature and +streamline the process. +All frames have been aligned onto a common frame +with 0.031” pixel scale, approximately matching the na- + +1.0 +F160WAstrodeep +-F150Wuncover,corr +0.5 +0.0 +88 +D +0.5 +-1.0 +1.0 +0.5 +0.0 +0.5 +O +O +1.0 +18 +19 +20 +21 +22 +23 +24 +F160WAstrodeepGLASS-JWST: Abell 2744 NIRCam photometric catalog +11 +tive pixel scale of the short wavelength data. The final +images on the whole A2744 region cover an area of 46.5 +arcmin2 with PSF ranging from 0.035” (for the F090W +image) to 0.14” (F444W), and reach astonishingly deep +5σ magnitude limits from 28.5 to 30.5, depending on +location and filter. +We exploit also other HST publicly available programs +which have targeted the area, including also the avail- +able HST ACS and WFC3 data in the F435W, F606W, +F775W and F814W (ACS) and F105W, F125W, F140W +and F160W (WFC3) bands, to expand the coverage of +the visible-to-IR wavelength range. +On these data we derive a photometric catalog by +detecting objects in the F444W image and comput- +ing PSF-matched forced photometry on the remaining +bands. +We made a number of tests to validate the photometric +calibrations, either internal, based on overlapping parts +observed in different epochs with different modules, and +external, based on cross-correlation with the AstroDeep +catalog of the cluster region. They both confirm that +photometric offset are limited to at most 0.05 mags or +less. Slightly larger (0.1 mags) systematic biases, espe- +cially when HST bands are concerned, could be due to +the simplified PSF matching that we adopt in this first +release. +As we do not explicitly remove the intra-cluster light, +photometry of faint sources in the cluster core might +also be affected by poor background subtraction. +We publicly release the entire mosaic of the NIRCam +images. The three individual images of each program, +which are more homogeneous in terms of PSF orienta- +tion and coverage/depth, and potentially more suitable +for accurate photometry and for accurate estimate of +incompleteness, are also available upon request. +We also publicly release the multi-wavelength cata- +logue on the entire A2744 area, which includes 24389 +objects. We release 5 independent catalogues, based on +a different aperture (2×, 3×, 8×, 16× the PSF) and in +the isophotal area. This catalog is optimized for high +redshift galaxies, and in general for faint extragalactic +sources, and aimed at allowing a first look at the data +and the selection of targets for Cycle 2 proposals. In +future releases we plan to include updated calibrations +and procedures for the image processing and to optimize +the photometry with more sophisticated approaches for +PSF matching. +Finally we also release the code developed to remove +the 1/f noise from the NIRCam images, that improves +upon the current implementation in the STScI pipeline +with a more effective masking of sources in the image. +Images, catalogues and software are immediately +available for download from the GLASS-ERS collabora- +tion website5 and from the AstroDeep website6. They +will also be made available at the MAST archive upon +acceptance of the paper. +All the JWST data used in this paper can be found in +MAST: 10.17909/fqaq-p393. +ACKNOWLEDGEMENT +We warmly thank J. Weaver, K. Withaker, I. Labb`e +and R. Bezanson for sharing their data with us prior +to publication, which made it possible to compare the +two processes for data analysis. +This work is based +on observations made with the NASA/ESA/CSA James +Webb Space Telescope, and with the NASA/ESA Hub- +ble Space Telescope. +The data were obtained from +the Mikulski Archive for Space Telescopes at the Space +Telescope Science Institute, which is operated by the +Association of Universities for Research in Astronomy, +Inc., under NASA contract NAS 5-03127 for JWST +and NAS 5–26555 for HST. These observations are as- +sociated with program JWST-ERS-1324, JWST-DDT- +2756, and JWST-GO-2561, and several HST programs. +We acknowledge financial support from NASA through +grant JWST-ERS-1324. +This research is supported +in part by the Australian Research Council Centre of +Excellence for All Sky Astrophysics in 3 Dimensions +(ASTRO 3D), through project number CE170100013. +KG and TN acknowledge support from Australian Re- +search Council Laureate Fellowship FL180100060. MB +acknowledges support from the Slovenian national re- +search agency ARRS through grant N1-0238. +We +acknowledge financial support through grants PRIN- +MIUR 2017WSCC32 and 2020SKSTHZ. We acknowl- +edge support from the INAF Large Grant 2022 “Ex- +tragalactic Surveys with JWST” (PI Pentericci). CM +acknowledges support by the VILLUM FONDEN under +grant 37459. RAW acknowledges support from NASA +JWST Interdisciplinary Scientist grants NAG5-12460, +NNX14AN10G and 80NSSC18K0200 from GSFC. The +Cosmic Dawn Center (DAWN) is funded by the Danish +National Research Foundation under grant DNRF140. +This work has made use of data from the Euro- +pean Space Agency (ESA) mission Gaia (https://www. +cosmos.esa.int/gaia), processed by the Gaia Data Pro- +cessing and Analysis Consortium (DPAC, https://www. +cosmos.esa.int/web/gaia/dpac/consortium). +Funding +for the DPAC has been provided by national institu- +tions, in particular the institutions participating in the +5 https://glass.astro.ucla.edu +6 http://www.astrodeep.eu + +12 +Paris et al. +Gaia Multilateral Agreement. The authors thank Paola +Marrese and Silvia Marinoni (Space Science Data Cen- +ter, Italian Space Agency) for their contribution to the +work. +REFERENCES +Bertin, E. 2006, in Astronomical Society of the Pacific +Conference Series, Vol. 351, Astronomical Data Analysis +Software and Systems XV, ed. C. Gabriel, C. Arviset, +D. Ponz, & E. 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A., et al. 2022, arXiv +e-prints, arXiv:2209.04092. +https://arxiv.org/abs/2209.04092 + diff --git a/ANE0T4oBgHgl3EQfPgBb/content/tmp_files/load_file.txt b/ANE0T4oBgHgl3EQfPgBb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8164f9877bd36b954505d9d61b108f99cab12ef1 --- /dev/null +++ b/ANE0T4oBgHgl3EQfPgBb/content/tmp_files/load_file.txt @@ -0,0 +1,856 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf,len=855 +page_content='Draft version January 6, 2023 Typeset using LATEX twocolumn style in AASTeX63 The GLASS-JWST Early Release Science Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Stage I release of NIRCam imaging and catalogs in the Abell 2744 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' China 19National Astronomical Observatories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Beijing 100101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' China 20Institute for Frontiers in Astronomy and Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Beijing Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Beijing 102206,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' China ABSTRACT We present images and a multi–wavelength photometric catalog based on all of the JWST NIRCam observations obtained to date in the region of the Abell 2744 galaxy cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' These data come from three different programs, namely the GLASS-JWST Early Release Science Program, UNCOVER, and Director’s Discretionary Time program 2756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The observed area in the NIRCam wide-band filters - covering the central and extended regions of the cluster, as well as new parallel fields - is 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 arcmin2 in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' All images in eight bands (F090W, F115W, F150W, F200W, F277W, F356W, F410M, F444W) have been reduced adopting the latest calibration and references files available to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Data reduction has been performed using an augmented version of the official JWST pipeline, with improvements aimed at removing or mitigating defects in the raw images and improve the background subtraction and photometric accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We obtain a F444W-detected multi–band catalog including all NIRCam data and available HST data, adopting forced aperture photometry on PSF-matched images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The catalog is intended to enable early scientific investigations, and is optimized for the study of faint galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' it contains 24389 sources, with a 5σ limiting magnitude in the F444W band ranging from 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 AB to 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 AB, as a result of the varying exposure times of the surveys that observed the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We Corresponding author: Diego Paris diego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='paris@inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='it arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='02179v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='GA] 5 Jan 2023 ID2 Paris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' publicly release the reduced NIRCam images, associated multi-wavelength catalog, and code adopted for 1/f noise removal with the aim of aiding users to familiarize themselves with JWST NIRCam data and identify suitable targets for follow-up observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Keywords: galaxies: high-redshift, galaxies: photometry 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' INTRODUCTION In just a few months of observations, JWST has demonstrated its revolutionary scientific capabilites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Early observations have shown that its performance is equal or better than expected, with image quality and overall efficiency that match or surpass pre-launch esti- mates (Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Publicly available datasets obtained by the Early Release Observations and Early Release Science programs have already enabled a large number of publications based on JWST data, ranging from exoplanets to the distant Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' In particular, a number of works exploited the power of NIRCAM to gather the first sizeable sample of can- didates at z ≥ 10 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=', Castellano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Don- nan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Morishita & Sti- avelli 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Roberts- Borsani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Castellano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Bouwens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022), showing the power of JWST in exploring the Universe during the re-ionization epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' In this paper we present the full data set obtained with NIRCam in the region of of the z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='308 cluster Abell 2744 that will significantly expands the available area for deep extragalactic observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The central region of the cluster, with the assistance of lensing mag- nification, allows an insight into the distant Universe at depth and resolution superior of those of NIRCam in blank fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The data set analyzed here are ob- tained through three public programs: i) GLASS-JWST ERS (Treu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022), ii) UNCOVER (Bezanson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022), and iii) Director’s Discretionary Time Program 2756, aimed at following up a Supernova discovered in GLASS-JWST NIRISS imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We have analyzed and combined the imaging data of all these programs and obtained a multi-wavelength catalog of the objects de- tected in the F444W band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' In order to facilitate exploitation of these data, we release reduced images and associated catalog on our website and through the Mikulski Archives for Space Telescopes (MAST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' This release fulfills and exceeds the requirements of the Stage I data release planned as part of the GLASS-JWST program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' It is anticipated that a final (Stage II) release will follow in approximately a year, combining additional images scheduled in 2023, and taking advantage of future improvements in data processing and calibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' In Section 2 we present the data-set and discuss the image processing pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' In Section 3 the methods applied for the de- tection of the sources and the photometric techniques used to compute the fluxes are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Finally in Section 4 we summarize the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Throughout the paper we adopt AB magnitudes (Oke & Gunn 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' DATA REDUCTION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Data Set The NIRCam data analyzed in this paper are taken from three programs that targeted the z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='308 clus- ter Abell 2744 (A2744 hereafter) and its surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The first set of NIRCam images were taken as part of the GLASS-JWST survey (Treu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022, hereafter T22), in parallel to primary NIRISS observations on June 28–29 2022 and to NIRSpec observations on Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 10–11, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We refer to these data sets as GLASS1 and GLASS2, or collectively as GLASS, both of which con- sist of imaging in seven broad-band filters from F090W to F444W (see Treu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We note that the final pointing is different from the scheduled one pre- sented by Treu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' (2022) due to the adoption of an alternate position angle (PA) during the NIRSpec spec- troscopic observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' As the primary spectroscopic target was the A2744 cluster, these parallel images are offset to the North-West.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' By virtue of the long exposure times, these images are the deepest presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The second set of NIRCam observations considered here were taken as part of the UNCOVER program (Bezanson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022), which targets the center of the A2744 cluster and the immediate surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' These images are composed of four pointings and result in a relatively homogeneous depth, as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' They were taken on November 2-4-7 and 15, and adopt the same filter set as GLASS-JWST, except for the adop- tion of the F410M filter instead of F090W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Finally, NIRCam imaging of the A2744 center was also obtained as part of DDT program 2756 (PI W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Chen, DDT hereafter) on October 20 and December 6 2022 (UT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' These two data sets are dubbed DDT1 and DDT2 hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The DDT filter set is the same as GLASS-JWST with the exception of the F090W fil- ter, and overall shorter exposure times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' One of the two NIRCam modules overlaps with UNCOVER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' GLASS-JWST: Abell 2744 NIRCam photometric catalog 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Full view of the F444W mosaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Colored boxes show the position of the three different data sets used here: GLASS (green), UNCOVER (blue) and DDT (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The entire image (including the empty space) is approximately 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='7 × 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='9 arc minutes wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' In Table 1 we list the exposure times adopted in the various filters for each of the aforementioned programs, while the footprints of the fields are illustrated in Fig- ure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' As a result of the overlap between programs and of their different observation strategies, the resulting ex- posure map is complex and inhomogenous across bands and area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' An analysis of the depth resulting from this exposure map is reported below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Data reduction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Pre-reduction steps Image pre-reduction was executed using the official JWST calibration pipeline, provided by the Space Tele- scope Science Institute (STScI) as a Python software suite1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We adopted Version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2 of the pipeline and Ver- sions between cjwst 1014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='pmap and cjwst 1019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='pmap of the CRDS files (the only changes between these versions is the astrometric calibration, that is dealt with as described below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We executed the first two stages of the pipeline (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' calwebb detector1 and 1 https://jwst-pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='io/en/latest/jwst/ introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='html Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' NIRCam Exposure time Filter GLASS1 GLASS2 DDT1/2 UNCOVER F090W 11520 16492 F115W 11520 16492 2104 10822 F150W 6120 8246 2104 10822 F200W 5400 8246 2104 6700 F277W 5400 8246 2104 6700 F356W 6120 8246 2104 6700 F410M 6700 F444W 23400 32983 2104 8246 Note—Exposure time (in seconds) for each pointing of the three programs considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' calwebb image2), adopting the optimized parameters for the NIRCam imaging mode, that convert single de- tector raw images into photometric calibrated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Using the first pipeline stage calwebb detector1 we processed the raw uncalibrated data (uncal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='fits) in order to apply detector-level corrections performed on a group-by-group basis, as dark subtractions, reference pixels corrections, non-linearity corrections and jump 4 Paris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' detection that allows to identify cosmic rays (CR) events on the single groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The last step of this pipeline stage allows us to derive the mean count rate, in units of counts per second, for each pixel by performing a lin- ear fit to the data in the input image (the so-called ramp-fitting) excluding the group masked due to the identification of a cosmic ray jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The output files of the previous steps (rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='fits) are processed through the second pipeline stage calwebb image2, which consists of additional instrument-level and observing-mode corrections and calibrations, as the geometric-distortion correction, the flat-fielding, and the photometric calibrations that con- verting the data from units of countrate to surface brightness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' MJy per steradian) generates a fully calibrated exposure (cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='fits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='fits file contains also an RMS layer, which combines the contribution of all pixel noise sources, and a DQ mask where the first bit (DO NOT USE) identifies pix- els that should not be used during the resampling phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We then applied a number of custom procedures to remove instrumental defects that are not dealt with the STScI pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Some of them have already been adopted in (Merlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022, hereafter M22) and described there: we illustrate below only the major changes to the STScI pipeline in default configuration and/or to the procedure adopted in M22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' “Snowballs”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' circular artifacts observed in the in-flight data caused by a large cosmic ray impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Those hits leave a bright ring-shaped defect in the image since the affected pixels are just partially identified and masked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' In M22, we developed a technique to fully mask out these features, which was not necessary here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Indeed, version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='1 of the JWST pipeline introduced the option to iden- tify snowball events, expanding the typical mask- ing area to include all the pixels affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' This new implementation provides the opportunity to correct these artifacts directly at the ramp fitting stage, at the cost of a larger noise on the corre- sponding pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We activated this non-default op- tion, and fine tuned the corresponding parameters to completely mask all the observed snowballs and, at the same time, minimize the size of high noise areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' “NL Mask”, on cal images of the NIRCam Mod- ule B Long Wavelength detector are visible bright groups of pixels not well corrected during prere- duction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' These pixels are more evident on deeper pointing and are identified as “well not defined” pixels2 in the Non Linearity Calibration file 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We selected those pixels and masked them as DO NOT USE to not to be used during stacking phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 1/f noise, which introduces random vertical and horizontal stripes into the images (see Schlawin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We remove this by subtracting the median value from each line/column, after mask- ing out all objects and bad pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The masks were obtained by running SExtractor (Version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0) (Bertin & Arnouts 1996) and then dilat- ing the resulting segmentation image, applying a differential procedure to dilate objects depending on their ISOAREA: the segmentation of objects with ISOAREA<5000 pixels was dilated using a 3 × 3 convolution kernel and a dilation of 15 pix- els, while for the segmentation of objects with ISOAREA⩾5000 pixels a 9 × 9 convolution ker- nel and a dilation of 4 × 15 pixels was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The procedure was executed separately for each am- plifier in the SW detectors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 4 times for each individual image) with the exception of the denser areas corresponding to the centers of the clusters and the brightest field star, where objects are sig- nificantly larger than the amplifier width (500 pix- els, corresponding to about 30”) and could not be masked efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' In this case we removed the 1/f noise over the entire row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' As this extension of the STScI pipeline could be useful for other programs, we publicly release the code adopted for this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Scattered light: we identify additive features in the F115W, F150W and F200W images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' These low-surface brightness features have already been revealed by commissioning data (see Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022) and are due to scattered light entering into optical path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' These anomalies have been dubbed wisps or claws, depending on their origin and mor- phology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Wisps have a nearly constant shape and a template pattern is available for subtraction from the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We removed these features by extracting their 2D profile from the available tem- plate (we do not use the entire template image to avoid subtracting its empty but noisy regions) and then normalizing the residual template to match the feature intensity in each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Claws have been first identified and singled out in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='edu/files/live/sites/www/files/home/jwst/ documentation/technical-documents/ documents/JWST-STScI- 004714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='pdf 3 https://jwst-crds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='edu/browse/jwst nircam linearity 0011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='rmap GLASS-JWST: Abell 2744 NIRCam photometric catalog 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Examples of custom procedures to remove resid- ual instrumental defects, not dealt with the current STScI pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Top: 1/f stripes removal on a GLASS F200W single exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Bottom: A portion of the GLASS F150W mosaic before and after the claws treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Their shape on each image has been reconstructed by interpolating a 2D mesh with box size 32 pix- els and then eventually subtracted from the same image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We find that these procedures efficiently remove most of these features, as shown in Fig- ure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Other defects were found in the F090W image, and to a lesser extent in the F115W one, which are due to a so-called “wing-tilt event” that happened during the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' These defects have been masked as in M22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We then re-scaled the single exposures to units of µJy/pixel, using the conversion factors output by the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Astrometry The astrometric calibration was performed using SCAMP (Bertin 2006), with 3rd order distortion correc- tions (PV coefficients up to j = 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' At variance with the procedure we adopted in M22, we started from the dis- tortion coefficient computed by the STScI pipeline and stored in the cal images, and refine the astrometric so- lution by running scamp in cal mode, which optimizes the solution with limited variations from the starting so- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We have found this procedure both accurate and reliable, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We first obtained a global astrometric solution for the F444W image, which is usu- ally the deepest, tied to a ground-based catalog obtained in the i-band with the Magellan telescope in good see- ing condition (see T22 for details) of the same region, which had been previously aligned to GAIA-DR3 stars (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2016, 2022 in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We then took the resulting high-resolution catalog in F444W as reference for the other JWST bands, using compact, iso- lated sources detected at high signal-to-noise at all wave- lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Each NIRCam detector has been analysed inde- pendently, in order to simplify the treatment of distor- tions and minimise the offsets of the sources in different exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Finally, we used SWarp (Bertin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2002) to combine the single exposures into mosaics projected onto a common aligned grid of pixels, and SExtractor to further clean the images by subtracting the residual sky background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The pixel scale of all the images was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='031′′ (the approximate native value of the short wavelength bands), to allow for simple processing with photometric algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The final image, computed as a weighted stack of all the images from the three programs, has a size of 24397× 21040 pixels, corresponding to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='6 × 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='87 arcmin2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' In this frame, the area covered by the wide-band NIRCam images (F115W, F150W, F200W, F277W, F356W and F444W) is of exactly 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 arcmin2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The F444W image is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Given the especially deep and sharp nature of the JWST images, where most of the faint objects have sizes below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5′′, the requirements on the final astrometric ac- curacy are extremely tight, to avoid errors in the multi- band photometry (where a displacement of as little as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='1′′ can bias color estimates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' These requirements must be met also in the overlapping regions of the various surveys, which have often been observed with different detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' To verify the final astrometric solution we conducted a number of validation tests, where we compare the positions of cross-matched objects in catalogues ex- tracted from different images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' For each of these cat- alogues we used SExtractor in single image mode and adopted the XWIN and YWIN estimators of the object center, which are more accurate than other choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' At the unprecedented image quality of NIR- Cam, the accurate center of extra–galactic objects with complex morphology may be difficult to estimate with high accuracy, especially when observed across a large wavelength interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' To minimize errors, we limited the comparison to objects with well defined positions, using the ∆X, ∆Y =ERRAWIN WORLD, ERRB- WIN WORLD estimators of the error and limiting the analysis to objects with (∆X2 + ∆Y 2)1/2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='018”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' From these catalogues we estimated both the average offset of the object centers ∆α and ∆δ, and the median average deviation madα and madδ, which measure the 6 Paris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Validation tests on the astrometric registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Left: scatter diagram reporting the displacement δRA and δDEC of sources between the Magellan i–band catalog registered to Gaia DR3 used as global reference for calibration and the final F444W NIRCam catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Middle left: As above, applied to the scatter between the AstroDeep catalog and the final F444W NIRCam catalog obtained on the central region of the A2744 cluster, as obtained in the context of the Frontier Fields initiative (Merlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Middle right: Offset between the position of sources in the F444W and the F115W images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Right: Positional offset between the objects detected in the UNCOVER–only images and those in the GLASS and DDT samples, on two overlapping regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' In all diagrams the average value ∆α and ∆δ and the median average deviation mad∆α and mad∆δ are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' intrinsic scatter in the alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' In Figure 3 we report the main outcome of these tests: (Left) We first compared the positions of objects in the original Magellan i-band and the resulting F444W of the entire mosaic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We find an essentially zero offset and madα ≃ madδ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='02”, which is 2/3 of a pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' (Middle left) We compared the F444W catalog with the AstroDeep H160 catalog obtained on the central region of the A2744 cluster, as obtained in the context of the Frontier Fields initiative (Mer- lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' While the intrinsic scatter is still good (madα ≃ madδ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='02”), we find a system- atic offset by about 1 pixel in RA and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 pixels in DEC, which is most likely due to different choices in the absolute calibration of the ACS/WFC3 data released within the Frontier Fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' (Middle right) We compare here the relative cali- bration of filters at the two extremes of the spec- tral range, F444W and F115W, where morphologi- cal variations and color terms may change the cen- ter position and affect the astrometric procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We find again very good alignment with negligible offset and small madα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' ≃ madδ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='01” (Right) Finally, we compare the astrometric solu- tions on the overlapping areas by summing inde- pendently the data of the three different programs and checking the accuracy in the overlapping area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Again we find very good alignment with negligible offset and small madα ≃ madδ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='01”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We therefore conclude that the astrometric procedure is accurate and adequate to the goals of this Stage I release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' In the future we plan to further explore and validate other options for astrometric registration and also release images with a smaller pixel scale, to better exploit the unprecedented image quality of the JWST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We note that the GLASS-JWST data have a very limited dithering pattern (which was driven by spectro- scopic requirements) and so may benefit only marginally from moving to smaller pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Estimating the Final Depth The final coaddition of the different images is weighted according to their depth, as estimated by the RMS im- age produced by the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We therefore obtain an optimally averaged image with the resulting RMS im- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We a posteriori verified whether the noise estimate encoded in the RMS effectively reproduced the photo- metric noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' To do this, we injected artificial point sources of known magnitude in empty regions of the im- age, and measured their fluxes and uncertainties with a-phot (Merlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2019), using apertures of radius 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' To take into account the fact that the mosaics are the result of a complex pattern of different exposures, we divided the maps into regions of similar total expo- sure time, and performed this analysis separately in each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' In general, we find that the RMS of the resulting flux distribution is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='1× larger than the value we would ex- Magellan i vs F444w (1810 obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=') △α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='000", mad^α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='02 A6 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='002", madAs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='00 2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='09 △α (")H160 vs F444W (728 0bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' △α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='033", madAα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='02 A6 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='072", madAs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='00 1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='18 △α (")F115W vs F444W (3649 0bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=') △α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='000", madAα= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='01 A6 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='002", madas = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='00 2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='09 △α (")F444w overlap regions (1529 obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Aα = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='002",mad^α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='01 △6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='002", madAs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='00 2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='09 △α (")GLASS-JWST: Abell 2744 NIRCam photometric catalog 7 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Depth of the full mosaic F444W image, as pro- duced by our pipeline on the basis of the variance image of each exposure and with the re-normalization described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Each pixel has been converted into 5σ limiting flux computed on a circular aperture of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' pect from the SExtractor errors, which are computed from the RMS image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' A larger difference (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='4×) is found for the F444W GLASS image, which is affected by a residual pattern due to poor flat–fielding with the cur- rent calibration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We therefore re-scaled the RMS maps produced by the pipeline according to these fac- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The resulting depth of this procedure is shown in Fig- ure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The RMS image is converted into a 5σ limiting flux computed on a circular aperture with a diameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2′′, that is the size adopted to estimate colors of faint sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The depth ranges from ≃ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='6 AB on the DDT2 footprint (in particular the area not overlapping with DDT1) to ≃ 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2 AB in the area where GLASS1 and GLASS2 overlap, arguably one of the deepest im- ages obtained so far by JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' A more quantitative assessment of the depth in the various filters is reported in Figure 5, where we show the distribution of the limiting magnitudes in each image resulting from the different strategies adopted by the surveys,computed as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' A clear pattern is seen, illustrating the large, mid–depth area obtained by UNCOVER and the shallower and deeper parts obtained by DDT and GLASS respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' HST Imaging We have also used the existing images obtained with HST in previous programs, namely with the F435W, F606W, F775W and F814W bands with ACS and the Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Distribution of the limiting magnitude for each band, as shown in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Limiting magnitudes per pixel have been computed as for Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' F105W, F125W, F140W and F160W bands with WFC3 other HST data are available from MAST but are ei- ther too shallow and/or limited in area and are not con- sidered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Among these data are included also the images that we obtained with DDT Program HST-GO- 17231 (PI: Treu), which was specifically aimed at obtain- ing ACS coverage for the majority of the GLASS1 and GLASS 2 fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We have used calibrated stacked image and weights (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Brammer, private communication) that we have realigned (after checking that the astrometric solution is consistent) onto our reference grid to allow a straightforward computation of colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' PHOTOMETRIC CATALOG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Detection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 F090W F115W 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 ou 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 (el, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='9 F150W F200W 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 QU 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 el, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 2:9 F277W F356W 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 xel, norm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 2:9 F41QM F444W 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 ou 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 xel, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 28 29 30 31 28 29 30 31 magim maglim26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 27 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 28 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 29 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 30 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='58 Paris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We follow here the same prescriptions adopted by M22 and Castellano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' (2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We performed source detections on the F444W band, since it is generally the deepest or among the deepest image for each data set, and because high-redshift sources (which are the main focus of these observations) are typically brighter at longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' This approach has the advantage of delivering a clear-cut criterion for the object detections, that can easily be translated into a cut of rest-frame properties for high redshift sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We used SExtractor, adopting a double–pass ob- ject detection as applied for the HST-CANDELS cam- paign (see Galametz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2013), to detect the objects, following the recipes and parameters described in M22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We note in particular that we adopt a detection thresh- old corresponding to a signal-to-noise ratio (SNR) of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' This is based on simulations, as discussed in M22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The other SExtractor parameters used are listed in M22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The final SExtractor catalogue on the entire A2744 area contains 24389 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Estimating the completeness and purity in a patchy (in terms of area and exposure) mosaic derived from the large number of observations adopted here, is in- trinsically ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' As shown in Figure 5 the depth of these images spans approximately 2 magnitudes, and the completeness is therefore inhomogenoues - not to mention the existence of the cluster that complicates both the detection and the estimate of the foreground volume (C22b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' For these reasons, we do not attempt the traditional estimate of the completeness and refer to Figure 4 and to Figure 5 for an evaluation of the depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' For a proper analysis of the completeness we refer the reader to the methodology adopted by C22b were we estimate the completeness separately on the individual mosaics of the three data sets, which were processed in- dependently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We make the three mosaics available upon request for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Photometry We have compiled a multi-wavelength photometric catalog following again the prescriptions of M22, which in turn is based on previous experience with Hubble Space Telescope (HST) images in CANDELS (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Galametz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2013) and in AstroDeep (Merlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2016b, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The catalog is based on a detection per- formed on the F444W image described above, and PSF– matched aperture photometry of all the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We include all the NIRCam images presented here and ex- isting images obtained with HST in previous programs, namely with the F435W, F606W, F775W and F814W bands with ACS and the F105W, F125W, F140W and F160W bands with WFC3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The images considered here have PSFs that range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='035” to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Considering that most of the ob- jects have small sizes, with half–light–radii less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2”, it is necessary to apply a PSF homogenization to avoid bias in the derivation of color across the spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' PSF matching Since the detection band is the one with the coars- est resolution, we PSF-matched all the other NIRCam images to it for color fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We created convolution kernels using the WebbPSF models publicly provided by STScI4, combining them with a Wiener filtering al- gorithm based on the one described in Boucaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' and we used a customised version of the con- volution module in t-phot (Merlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2015, 2016a), which uses FFTW3 libraries, to smooth the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' This approach delivers consistent results with those obtained using the software Galight (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We note that this approach is inevitably approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The JWST PSF is time– and position–dependent (Nardiello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2022), and our dataset is the inhomogeneous com- bination of data obtained at different times and with different PA, so that the PSF definitely changes over the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' For this version of the catalog we used the Uncover PSF models as average PSFs, and we plan to improve our PSF estimation in the future versions of the catalog that will be released in Stage II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Similarly, concerning the HST images, we note that all of them have too few stars to obtain a robust estimate of the PSF directly from the images, so that we adopt in all cases existing HST PSFs, taken from CANDELS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' This approximation may introduce small biases in the final catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' ACS images have been PSF-matched to F444W, while for the WFC3 F105W, F125W, F140W and F160W images, which have a PSF larger than the F444W one, we have done the inverse - smoothed the F444W image and the WFC3 F105W, F125W, F140W to the F160W and followed a slightly different procedure that we describe below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Flux estimate The total flux is measured with a-phot on the detec- tion image F444W by means of a Kron elliptical aper- ture (Kron 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' As we have shown in M22, simula- tions suggest that Kron fluxes measured with a-phot are somewhat less affected by systematic errors, while being slightly more noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 4 https://jwst-docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='edu/jwst-near-infrared-camera/ nircam-predicted-performance/nircam-point-spread-functions GLASS-JWST: Abell 2744 NIRCam photometric catalog 9 Then, we used a-phot to measure the fluxes at the po- sitions of the detected sources on the PSF-matched im- ages, masking neighboring objects using the SExtrac- tor segmentation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Given the wide range of magni- tudes and sizes of the target galaxies we have measured the flux in a range of apertures: the segmentation area (the images being on the same grid and PSF-matched) and five circular apertures with diameters that are inte- ger multiples (2×, 3×, 8×, 16×, ) of the FWHM in the F444W band, that correspond to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='28′′, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='42′′, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='12′′ and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='24′′ diameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' For the four WFC3 images (which have a PSF larger than F444W) we first filtered the F444W to their FWHM and then measured colors between the filtered F444W and the WFC3 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' To minimize bi- ases when these colors are combined with those of the other bands, we use in this case apertures the same mul- tiples of the WFC3 PSF adopted for the other bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We remark that this procedure is only approximate, and delivers a first order correction of the systematic effects due to different PSFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' In a future release we plan to adopt more sophisticated approaches to optimize pho- tometry, including but not limited to the improvement of the PSF estimate and applying T-PHOT on WFC3 images that have a larger PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Total fluxes are obtained in the other bands by normalizing the colors in a given aperture to the F444W total flux, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' by computing fm,total = fm,aper/fF 444W,aper × fF 444W,total, as described in M22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We release the five catalogues described above (one computed on segmentation and four on the different apertures) and we leave the user to choose which is the most suitable for a given science application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' In general small-aperture catalogues are more appropriate for faint sources as they match their small sizes and minimize de- belending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Larger apertures may be more appropriate for brighter sources and especially cluster members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Validation tests We have performed a few validation tests to verify pri- marily the flux calibration, that has been the subject of many revisions in these first months, and to a lesser ex- tent of the procedure adopted to derive the photometric catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The overlap between GLASS1 and GLASS2 southern quadrants offers us a nice opportunity to test the NIR- Cam flux calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Indeed, the two GLASS observa- tions have been observed in two epochs (July and Octo- ber 2022) with a PA difference of nearly 150 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' As a result, the southern quadrant of GLASS1 and GLASS2 are largely overlapping but have been observed with modules B and A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We have therefore ob- tained stacked images of the two epochs separately, built Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Stability of the photometric calibration between different detectors, as measured by comparing the photom- etry of high S/N objects (S/N > 25) detected in the two epochs of observations in the SE quadrant of GLASS (lower leftmost green square in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Objects in this area have been observed in two epochs (July and October 2022) and with modules B and A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' For each filter difference in magnitude ∆M = M1 − M2 for objects between epoch1 and epoch2 as a function of M1 is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Red dashed lines represent the median offsets, namely we found: ∆M ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='06 with mad ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='05 for F090W, ∆M ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='05 with mad ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='04 for F115W, ∆M ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='04 with mad ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='04 for F150W, ∆M ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='02 with mad ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='04 for F200W, ∆M ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='05 with mad ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='04 for F277W, and negligible in F356W and F444W with mad ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='03 and mad ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='02 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We have vi- sually inspected the bright objects with |∆M| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='05 and verified that they mostly originate from saturated stars or objects with incomplete coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' a photometric catalog with the same recipes and checked the magnitude difference between objects observed with different detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The result of this exercise, that has been done on all bands, is reported in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We note that in the short bands the two modules are made of 4 detectors, each with an independent calibration, that we plot all together in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The comparison, that is limited to objects observed with high S/N > 25, shows that the average magnitude difference between the two GLASS 1 vS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2 F090W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2 F115W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2 F150W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2 F200W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2 F277W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2 F356W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2 F444W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2 20 21 22 23 24 25 26 27 M110 Paris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' modules is in general quite small, in all cases below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='05 mags (see Figure 6 and its captions for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' This confirms that the flux calibration between the different modules is reasonably stable at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' As a further check to validate the photometric pipeline, we have compared the m606 and m150 mag- nitudes for the sources in the core of the A2744 cluster with those measured in the same F606W and in the nearby F160W bands measured on HST images, that we published within the AstroDeep project (Merlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Castellano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' This comparison is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Magnitudes in the NIRCam F150W band have been shifted by ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='05 in order to correct for the small bandpass difference: the term was estimated using theoretical SEDs from a simulated photometric catalog, created using Egg (Schreiber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The com- parison shows that - when the same approach is used to estimate colours, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' isophotal magnitudes are adopted the agreement between the two catalogues is excel- lent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' When we use instead relatively smaller aperture in 8×FWHM for the NIRCam photometry we tend to underestimate the F150W and - even more - the F606W flux of the brightest sources, which are considerably more extended than 8×FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We ascribe this effect to the existence of colour gradients in bright objects, such that small-sized apertures tend to sample the central, redder part of the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' From this comparison we conclude that - quite reas- suringly - the overall photometric chain is consistent be- tween the well established Frontier Fields data and these new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' At the same time, we remark that the choice of which aperture is optimal depends on the size and kind of objects under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' For faint sources, small apertures tend to have higher S/N and should be pre- ferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' For brightest sources, larger apertures should be preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' It is also possible to estimate rough color gradients by comparing the various apertures that we re- lease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We also tested that applying the same technique without PSF matching introduces an offset of the order of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='2 mags in the final colors, which would clearly af- fect the derived photometric redshifts and SED fitting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Finally, in an effort to cross-validate our results prior to release, in the lead up to this paper we compared our catalogs to those under development by the UN- COVER team (Weaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 2023, in prep) based on the same raw datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The image processing and pho- tometric procedures adopted by the two teams have sig- nificant differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The main are: i) image coaddi- tion (UNCOVER team adopts grizli, while we use a custom pipeline which uses scamp and swarp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' ii) ob- ject detection (UNCOVER uses an optimally stacked Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Comparison of photometry between this work and AstroDeep HST catalogs in the core of the A2744 clus- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Upper: Difference between the magnitude in the F160W WFC3 band in AstroDeep and the F150W NIRCam of this work for objects in common between the two catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The F150W magnitude has been corrected for the ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='05 mag- nitude shift between the two bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Filled point represent the difference between magnitudes computed in isophotal ar- eas in both catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Empty points represent the magni- tude difference adopting the F150W magnitude computed in 8×FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Bottom: As above, for the F606W band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The systematic bias between isophotal and 8×FWHM colours is due to color gradients in the center of bright sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' F277W+F356W+F444W image after removing the intr- acluster light, while we use F444W);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' iii) techniques and tools for PSF matching and photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' For these rea- sons, we expect some differences between the catalogs, especially for faint sources at the detection limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' How- ever, our comparison of working versions of the catalogs produced by the two teams shows overall a good agree- ment in the colors and magnitudes of the vast majority of objects, with no evidence of significant bias beyond what can be explained by the different choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We defer a detailed comparison to future versions of the catalog (Stage II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' SUMMARY We present in this paper the data obtained by three NIRCam programs on the A2744 cluster: the GLASS- JWST Early Release Science Program, UNCOVER, and Directory Discretionary Time 2756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' All the data, taken with eight different filters (F090W, F115W, F150W, F200W, F277W, F356W, F410M, F444W), have been reduced with an updated pipelines that builds upon the official STScI pipeline but includes a number of improve- ment to better remove some instrumental signature and streamline the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' All frames have been aligned onto a common frame with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='031” pixel scale, approximately matching the na- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 F160WAstrodeep F150Wuncover,corr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 88 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 O O 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='0 18 19 20 21 22 23 24 F160WAstrodeepGLASS-JWST: Abell 2744 NIRCam photometric catalog 11 tive pixel scale of the short wavelength data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The final images on the whole A2744 region cover an area of 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 arcmin2 with PSF ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='035” (for the F090W image) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='14” (F444W), and reach astonishingly deep 5σ magnitude limits from 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5 to 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='5, depending on location and filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We exploit also other HST publicly available programs which have targeted the area, including also the avail- able HST ACS and WFC3 data in the F435W, F606W, F775W and F814W (ACS) and F105W, F125W, F140W and F160W (WFC3) bands, to expand the coverage of the visible-to-IR wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' On these data we derive a photometric catalog by detecting objects in the F444W image and comput- ing PSF-matched forced photometry on the remaining bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We made a number of tests to validate the photometric calibrations, either internal, based on overlapping parts observed in different epochs with different modules, and external, based on cross-correlation with the AstroDeep catalog of the cluster region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' They both confirm that photometric offset are limited to at most 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='05 mags or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Slightly larger (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='1 mags) systematic biases, espe- cially when HST bands are concerned, could be due to the simplified PSF matching that we adopt in this first release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' As we do not explicitly remove the intra-cluster light, photometry of faint sources in the cluster core might also be affected by poor background subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We publicly release the entire mosaic of the NIRCam images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The three individual images of each program, which are more homogeneous in terms of PSF orienta- tion and coverage/depth, and potentially more suitable for accurate photometry and for accurate estimate of incompleteness, are also available upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We also publicly release the multi-wavelength cata- logue on the entire A2744 area, which includes 24389 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We release 5 independent catalogues, based on a different aperture (2×, 3×, 8×, 16× the PSF) and in the isophotal area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' This catalog is optimized for high redshift galaxies, and in general for faint extragalactic sources, and aimed at allowing a first look at the data and the selection of targets for Cycle 2 proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' In future releases we plan to include updated calibrations and procedures for the image processing and to optimize the photometry with more sophisticated approaches for PSF matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Finally we also release the code developed to remove the 1/f noise from the NIRCam images, that improves upon the current implementation in the STScI pipeline with a more effective masking of sources in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Images, catalogues and software are immediately available for download from the GLASS-ERS collabora- tion website5 and from the AstroDeep website6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' They will also be made available at the MAST archive upon acceptance of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' All the JWST data used in this paper can be found in MAST: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='17909/fqaq-p393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' ACKNOWLEDGEMENT We warmly thank J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Weaver, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Withaker, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Labb`e and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Bezanson for sharing their data with us prior to publication, which made it possible to compare the two processes for data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope, and with the NASA/ESA Hub- ble Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=', under NASA contract NAS 5-03127 for JWST and NAS 5–26555 for HST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' These observations are as- sociated with program JWST-ERS-1324, JWST-DDT- 2756, and JWST-GO-2561, and several HST programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We acknowledge financial support from NASA through grant JWST-ERS-1324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' This research is supported in part by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' KG and TN acknowledge support from Australian Re- search Council Laureate Fellowship FL180100060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' MB acknowledges support from the Slovenian national re- search agency ARRS through grant N1-0238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We acknowledge financial support through grants PRIN- MIUR 2017WSCC32 and 2020SKSTHZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' We acknowl- edge support from the INAF Large Grant 2022 “Ex- tragalactic Surveys with JWST” (PI Pentericci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' CM acknowledges support by the VILLUM FONDEN under grant 37459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' RAW acknowledges support from NASA JWST Interdisciplinary Scientist grants NAG5-12460, NNX14AN10G and 80NSSC18K0200 from GSFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under grant DNRF140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' This work has made use of data from the Euro- pean Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='int/gaia), processed by the Gaia Data Pro- cessing and Analysis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='int/web/gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Funding for the DPAC has been provided by national institu- tions, in particular the institutions participating in the 5 https://glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='edu 6 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='astrodeep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content='eu 12 Paris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfPgBb/content/2301.02179v1.pdf'} +page_content=' The authors thank Paola Marrese and Silvia Marinoni (Space Science Data Cen- ter, Italian Space Agency) for their contribution to the work.' metadata={'source': 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b/AdAyT4oBgHgl3EQf3_pa/content/tmp_files/2301.00778v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8549e451d7270731f656e7c6635cdf62e91fce8 --- /dev/null +++ b/AdAyT4oBgHgl3EQf3_pa/content/tmp_files/2301.00778v1.pdf.txt @@ -0,0 +1,2579 @@ +arXiv:2301.00778v1 [math.PR] 2 Jan 2023 +LECTURE NOTES ON TREE-FREE REGULARITY +STRUCTURES +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +Abstract. These lecture notes are intended as reader’s digest of +recent work on a diagram-free approach to the renormalized cen- +tered model in Hairer’s regularity structures. More precisely, it is +about the stochastic estimates of the centered model, based on Malli- +avin calculus and a spectral gap assumption. We focus on a specific +parabolic partial differential equation in quasi-linear form driven by +(white) noise. +We follow a natural renormalization strategy based on preserving +symmetries, and carefully introduce Hairer’s notion of a centered +model, which provides the coefficients in a formal series expansion +of a general solution. We explain how the Malliavin derivative in +conjunction with Hairer’s re-expansion map allows to reformulate +this definition in a way that is stable under removing the small-scale +regularization. +A few exemplary proofs are provided, both of analytic and of alge- +braic character. The working horse of the analytic arguments is an +“annealed” Schauder estimate and related Liouville principle, which +is provided. The algebra of formal power series, in variables that +play the role of coordinates of the solution manifold, and its algebra +morphisms are the key algebraic objects. +Keywords: Singular SPDE, Regularity Structures, BPHZ renor- +malization, Malliavin calculus, quasi-linear PDE. +MSC 2020: 60H17, 60L30, 60H07, 81T16, 35K59. +Contents +1. +A singular quasi-linear SPDE +3 +2. +Annealed Schauder theory +5 +3. +Symmetry-motivated postulates on the form of the counter +terms +8 +4. +Algebrizing the counter term +10 +5. +Algebrizing the solution manifold: The centered model +12 +6. +The main result: A stochastic estimate of the centered +model +17 +7. +Malliavin derivative and Spectral gap (SG) +19 +8. +The structure group and the re-expansion map +26 +1 + +2 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +References +33 +The theory of regularity structures by Hairer provides a systematic way +to treat the small-scale divergences in singular semi-linear stochastic +PDEs. Quintessential models of mathematical physics like the dynam- +ical Φ4 +3 model or the KPZ equation have been treated. Inspired by +Lyon’s theory of rough paths, this theory separates probabilistic and +analytical aspects: +• Centered model. In a first probabilistic step, the coefficients of +a local formal power series representation of a general solution +of the renormalized PDE are constructed and estimated; the co- +efficients are indexed by (decorated) trees, and their stochastic +estimate follows the diagrammatic approach to renormalization +of quantum field theories. +• Modelled distribution. In a second analytical step, inspired by +Gubinelli’s controlled rough path, the solution of a specific ini- +tial value problem is found as a fixed point based on modulating +and truncating the formal power series . This step is purely de- +terministic. +This automated two-pronged approach relies on an understanding of +the algebraic nature of the re-expansion maps that allow to pass from +one base-point to another in the local power series representation, in +form of the “structure group”. The main progress of regularity struc- +tures over the term-by-term treatment in the mathematical physics +literature is that thanks to centering and re-expansion, the second step +yields a rigorous (small data) well-posedness result. As an introductory +text to the theory of regularity structures we recommend [9]. +In [17], motivated by the extension to a quasi-linear setting featuring +a general non-linearity a(u), an alternative realization of Hairer’s reg- +ularity structures was proposed; it replaces trees with a more greedy +index set. This index set of multi-indices naturally comes up when +writing a general solution u as a functional of a, or rather as a func- +tion of the coefficients of a in its power law expansion. In [17] it was +established that any solution of the renormalized PDE can be locally +approximated by a modelled distribution. This a-priori estimate was +obtained under the assumption that the natural stochastic estimates +on the centered model are available. +In [15] this program was continued: Based on scaling and other symme- +tries, a canonical renormalization of the PDE and its centered model +was proposed, and the centered model was stochastically constructed +and estimated. These notes present selected aspects of [15], providing +additional motivation. For a simpler setting where no renormalization +and thus only purely deterministic estimates are needed, we recommend + +LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES +3 +to also have a look at1 [13]. The algebraic aspects of the multi-index +based regularity structures are worked out in [14], where in line with +Hairer’s postulates the underlying Hopf-algebraic nature of the struc- +ture group was uncovered. In fact, the Hopf algebra arises from a Lie +algebra generated by natural actions on the space of non-linearities a +and solutions u. +Other approaches to singular SPDEs include the theory of paracon- +trolled distributions by Gubinelli, Imkeller, and Perkowski, we rec- +ommend [8] for a first reading, and the renormalization group flow +approach introduced by Kupiainen and generalized by Duch; we rec- +ommend [12] and [6] for an introduction. The para-controlled calculus +provides an alternative to the separation into model and modelled dis- +tribution, replacing localization in physical space-time by localization +on the Fourier side; it is (typically) also indexed by trees. The flow +approach blends the stochastic and the deterministic step of regularity +structures, and has an index set closer to multi-indices. While these +alternative approaches might be more efficient in specific situations, +they presumably lack the full flexibility of the two-pronged approach +of regularity structures with its conceptual clarity. +1. A singular quasi-linear SPDE +We are interested in nonlinear elliptic or parabolic equations with a +random and thus typically rough right hand side ξ. Our approach is +guided by moving beyond the well-studied semi-linear case. We con- +sider a mildly quasi-linear case where the coefficients of the leading- +order derivatives depend on the solution u itself. To fix ideas, we focus +on the parabolic case in a single space dimension; since we treat the +parabolic equation in the whole space-time like an anisotropic ellip- +tic equation, we denote by x1 the space-like and by x2 the time-like +variable. Hence we propose to consider +(∂2 − ∂2 +1)u = a(u)∂2 +1u + ξ, +(1) +where we think of the values of a(u) to be such that the equation +is parabolic. +We are interested in laws / ensembles of ξ where the +solutions v to the linear equation +(∂2 − ∂2 +1)v = ξ +(2) +1however, the setting in [13] is different in the sense that it imposes an artificial +space-time periodicity: on the one hand, this allows to separate construction from +estimation, on the other hand, it obfuscates the quintessential scaling + +4 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +are (almost surely) H¨older continuous, where it will turn out to be +convenient to express this in the “annealed” form2 of +sup +x̸=y +1 +|y − x|αE +1 +2|v(y) − v(x)|2 < ∞ +(3) +for some exponent α ∈ (0, 1). +In view of the anisotropic nature of +∂2 − ∂2 +1 and its invariance under the rescaling x1 = sˆx1 and x2 = s2ˆx2, +H¨older continuity in (3) is measured w. r. t. the Carnot-Carath´eodory +distance +“|y − x|” := +4� +(y1 − x1)4 + (y2 − x2)2 ∼ |y1 − x1| + |y2 − x2| +1 +2. +(4) +By Schauder theory for ∂2 − ∂2 +1, on which we shall expand on in Sub- +section 2, this is the case for white noise ξ with α = 1 +2. The rationale +is that white noise has order of regularity −D +2 , where D is the effective +dimension, which in case of (2) is D = 1 + 2 = 3 since in view of (4) +the time-like variable x2 counts twice, and that (∂2 − ∂2 +1)−1 increases +regularity by two, leading to −D +2 + 2 = 1 +2. +In the range of α ∈ (0, 1), the SPDE (1) is what is called “singular”: +We cannot expect that the order of regularity of u and thus a(u) is +better than the one of v, which is α, and hence the order of regularity +of ∂2 +1u is no better than α − 2. Since α + (α − 2) < 0 for α < 1, the +product a(u)∂2 +1u cannot be classically/deterministically defined3. As +discussed at the end of Section 2, a renormalization is needed4. +The same feature occurs for the (semi-linear) multiplicative heat equa- +tion (∂2 −∂2 +1)u = a(u)ξ; in fact, our approach also applies to this semi- +linear case, which already has been treated by (standard) regularity +structures in [10]. A singular product is already present in the case +when the x1-dependence is suppressed, so that the above semi-linear +equation turns into the SDE du +dx2 = a(u)ξ with white noise ξ in the time- +like variable x2. In this case, the analogue of v from (2) is Brownian +motion, which is characterized by E(v(y2)−v(x2))2 = |y2−x2| and thus +annealed H¨older exponent 1 +2 in x2, which in view of (4) corresponds to +the border-line setting α = 1. Ito’s integral and, more recently, Lyons’ +rough paths [16] and Gubinelli’s controlled rough paths [7] have been +devised to tackle the issue in this SDE setting. +2Think of Brownian motion which satisfies E +1 +2 (B(s) − B(t))2 = |s − t| +1 +2 while +not being H¨older continuous of exponent 1 +2 almost surely. Following the jargon an- +nealed/quenched from statistical mechanics models (which itself is borrowed from +metallurgy), we speak of annealed norms when the inner norm is an Lp-norm +w. r. t. probability E and the outer norm is a space-time one. +3It is a classical result that the multiplication extends naturally from Cα × Cβ +into D′ if and only if α + β > 0, see [1, Section 2.6]. +4The range α > 1, while still subtle for α < 2, does not require a renormalization, +see [13]. + +LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES +5 +2. Annealed Schauder theory +This section provides the main (linear) PDE ingredient for our result. +At the same time, it will allow us to discuss (2). +In view of (2), we are interested in the fundamental solution of the +differential operator A := ∂2 − ∂2 +1. It turns out to be convenient to use +the more symmetric5 fundamental solution of the non-negative A∗A += (−∂2 −∂2 +1)(∂2 −∂2 +1) = ∂4 +1 −∂2 +2. Moreover, it will be more transparent +to “disintegrate” the latter fundamental solution, by which we mean +writing it as +´ ∞ +0 dtψt(z), where {ψt}t>0 are the kernels of the semi- +group exp(−tA∗A). Clearly, the Fourier transform is given by +Fψt(q) = exp(−t(q4 +1 + q2 +2)) +(4) += exp(−t|q|4). +(5) +In particular, ψt is a Schwartz function. For a Schwartz distribution f +like realizations of white noise, we thus define ft(y) as the pairing of f +with ψt(y − ·); ft is a smooth function. On the level of these kernels, +the semi-group property translates into +ψs ∗ ψt = ψs+t +and +ˆ +ψt = 1. +(6) +By construction, {ψt}t satisfies the PDE +∂tψt + (∂4 +1 − ∂2 +2)ψt = 0. +(7) +By scale invariance of (7) under x1 = sˆx1, x2 = s2ˆx2, and t = s4ˆt, we +have +ψt(x1, x2) = +1 +( +4√ +t)D=3 ψ1( x1 +4√ +t, +x2 +( +4√ +t)2). +(8) +Lemma 1. Let 0 < α ≤ η < ∞ with η ̸∈ Z, p < ∞, and x ∈ R2 be +given. For a random Schwartz distribution f with +E +1 +p|ft(y)|p ≤ ( +4√ +t)α−2( +4√ +t + |y − x|)η−α +for all t > 0, y ∈ R2, +(9) +there exists a unique random function u of the class +sup +y∈R2 +1 +|y − x|η E +1 +p|u(y)|p < ∞ +(10) +satisfying (distributionally in R2) +(∂2 − ∂2 +1)u = f + (polynomial of degree ≤ η − 2). +(11) +It actually satisfies (11) without the polynomial. Moreover, the l. h. s. of +(10) is bounded by a constant only depending on α and η. +5It is symmetric under reflection not just in space but also in time + +6 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +Now white noise ξ is an example of such a random Schwartz distri- +bution: +Since ξt(y) is a centered Gaussian, we have E +1 +p|ξt(y)|p ≲p +E +1 +2(ξt(y))2. By using the characterizing property of white noise in terms +of its pairing with a test function ζ +E(ξ, ζ)2 = +ˆ +ζ2, +(12) +we have E +1 +2(ξt(y))2 = +� ´ +ψ2 +t (y − ·) +� 1 +2, which by scaling (8) is equal +to ( +4√ +t)− D +2 ( +´ +ψ2 +1) +1 +2 ∼ ( +4√ +t)− D +2 . This specifies the sense in which white +noise ξ has order of regularity −D +2 . +Fixing a “base point” x, Lemma 1 thus constructs the solution of (2) +distinguished by v(x) = 0. Note that the output (10) takes the form of +E +1 +p|v(y)−v(x)|p ≲p |y−x| +1 +2, which extends (3) from p = 2 to general p. +Hence Lemma 1 provides an annealed version of a Schauder estimate, +alongside a Liouville-type uniqueness result. +Proof of Lemma 1. By construction, +´ ∞ +0 dt(−∂2 − ∂2 +1)ψt is the funda- +mental solution of ∂2 − ∂2 +1, so that we take the convolution of it with +f. However, in order to obtain a convergent expression for t ↑ ∞, we +need to pass to a Taylor remainder: +u = +ˆ ∞ +0 +dt(id − Tη +x)(−∂2 − ∂2 +1)ft, +(13) +where Tη +x is the operation of taking the Taylor polynomial of order ≤ η; +as we shall argue the additional Taylor polynomial does not affect the +PDE. +We claim that (13) is well-defined and estimated as +E +1 +p|u(y)|p ≲ |y − x|η. +To this purpose, we first note that +E +1 +p|∂nft(y)|p ≲ ( +4√ +t)α−2−|n|( +4√ +t + |y − x|)η−α, +(14) +where +∂nf := ∂n1 +1 ∂n2 +2 f +and +|n| = n1 + 2n2. +(15) +Indeed, by the semi-group property (6) we may write ∂nft(y) = +´ +dz +∂nψ t +2(y−z) f t +2(z), so that E +1 +p|∂nft(y)|p ≤ +´ +dz|∂nψ t +2(y−z)|E +1 +p|f t +2(z)|p. +Hence by (9), (14) follows from the kernel bound +´ +dz |∂nψ t +2(y − z)| +( +4√ +t + |y − x|)η−α ≲ ( +4√ +t)−|n|( +4√ +t + |y − x|)η−α, which itself is a conse- +quence of the scaling (8) and the fact that ψ 1 +2 is a Schwartz function. +Equipped with (14), we now derive two estimates for the integrand +of (13), namely for +4√ +t ≥ |y − x| (“far field”) and for +4√ +t ≤ |y − x| +(“near field”). We write the Taylor remainder (id − Tη +x)(∂2 + ∂2 +1)ft(y) + +LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES +7 +as a linear combination of6 (y − x)n∂n(∂2 + ∂2 +1)ft(z) with |n| > η and +at some point z intermediate to y and x. +By (14) such a term is +estimated by |y − x||n|( +4√ +t)α−4−|n|( +4√ +t + |y − x|)η−α, which in the far +field is ∼ |y − x||n|( +4√ +t)η−4−|n|. Since the exponent on t is < −1, we +obtain as desired +E +1 +p| +ˆ ∞ +|y−x|4 dt(id − Tη +x)(∂2 + ∂2 +1)ft(y)|p ≲ |y − x|η. +For the near-field term, i. e. for +4√ +t ≤ |y − x|, we proceed as follows: +E +1 +p |(id − Tη +x)(∂2 + ∂2 +1)ft(y)|p +≤ E +1 +p|(∂2 + ∂2 +1)ft(y)|p + +� +|n|≤η +|y − x||n|E +1 +p|∂n(∂2 + ∂2 +1)ft(x)|p +(14) +≲ ( +4√ +t)α−4|y − x|η−α + +� +|n|≤η +|y − x||n|( +4√ +t)η−4−|n|. +Since η is not an integer, the sum restricts to |n| < η, so that all +exponents on t are > −1. Hence we obtain as desired +E +1 +p| +ˆ |y−x|4 +0 +dt(id − Tη +x)(∂2 + ∂2 +1)ft(y)|p ≲ |y − x|η. +It can be easily checked that (13) is indeed a solution of (11), even +without a polynomial. For a detailed proof we refer to [15, Proposi- +tion 4.3]. +We turn to the uniqueness of u in the class (10) satisfying (11). Given +two such solutions u1, u2, we observe that ¯u := u1 − u2 satisfies (10) +and (11) with f = 0. In particular ∂n(∂2 − ∂2 +1)¯u = 0 for |n| > η − 2, +and thus from (7) we obtain ∂t∂n¯ut = 0 provided |n| > η − 4. Thus, +∂n¯ut is independent of t > 0. Moreover, (10) implies that E|∂n¯ut| → 0 +as t → ∞ for |n| > η. Hence we learn from t → 0 that ∂n¯u = 0 +for |n| > η, i.e. ¯u is a polynomial of degree ≤ η. Since η ̸∈ Z this +strengthens to ¯u is a polynomial of degree < η, and by (10) it vanishes +at x to order η which yields the desired ¯u = 0. +□ +We return to the discussion of the singular product a(u)∂2 +1u, in its +simplest form of +v∂2 +1v = ∂2 +1 +1 +2v2 − (∂1v)2. +While in view of Lemma 1 the first r. h. s. term is well-defined as +a random Schwartz distribution, we now argue that the second term +6where xn := xn1 +1 xn2 +2 + +8 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +diverges. Indeed, applying ∂1 to the representation formula (13), so +that the constant Taylor term drops out, we have +∂1v = +ˆ ∞ +0 +dt∂1(−∂2 − ∂2 +1)ξt. +(16) +Hence for space-time white noise +E(∂1v(x))2 +(16) += +ˆ ∞ +0 +dt +ˆ ∞ +0 +ds E +� +∂1(−∂2 − ∂2 +1)ξt(x)∂1(−∂2 − ∂2 +1)ξs(x) +� +(12) += +ˆ ∞ +0 +dt +ˆ ∞ +0 +ds +ˆ +R2 dy ∂1(−∂2 − ∂2 +1)ψt(x − y)∂1(−∂2 − ∂2 +1)ψs(x − y) +(6) += +ˆ ∞ +0 +dt +ˆ ∞ +0 +ds ∂2 +1(∂2 +2 − ∂4 +1)ψs+t(0) +(8) +∼ +ˆ ∞ +0 +dt +ˆ ∞ +0 +ds +4√ +t + s +−D−6. +Note that since 1 +4(−D−6) < −2 for D = 3, the double integral diverges. +This divergence arises from t ↓ 0 and s ↓ 0, that is, from small space- +time scales, and thus is called an ultra-violet (UV) divergence. A quick +fix is to introduce an UV cut-off, which for instance can be implemented +by mollifying ξ. Using the semi-group convolution ξτ specifies the UV +cut-off scale to be of the order of +4√τ. It is easy to check that in this +case +E(∂1v(x))2 ∼ +ˆ ∞ +τ +dt +ˆ ∞ +τ +ds +4√ +t + s +−D−6 ∼ ( +4√τ)−1. +The goal is to modify the equation (1) by “counter terms” such that +• the solution manifold stays under control as the ultra-violet +cut-off τ ↓ 0, +• invariances of the solution manifold are preserved i.e. the solu- +tion manifold keeps as many symmetries as possible. +In view of the above discussion, we expect the coefficients of the counter +terms to diverge as the cut-off tends to zero. +3. Symmetry-motivated postulates on the form of the +counter terms +In view of α ∈ (0, 1), u is a function while we think of all derivatives +∂nu as being only Schwartz distributions. Hence it is natural to start +from the very general Ansatz that the counter term is a polynomial in +{∂nu}n̸=0 with coefficients that are general (local) functions in u: +(∂2 − ∂2 +1)u + +� +β +hβ(u) +� +n̸=0 +( 1 +n!∂nu)β(n) = a(u)∂2 +1u + ξ, +(17) + +LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES +9 +where β runs over all multi-indices7 in n ̸= 0 and n! := (n1!)(n2!). For +simplicity of this heuristic discussion, we drop the regularization on ξ +and don’t index the counter term with τ. +Only counter terms that have an order strictly below the order of the +leading ∂2 − ∂2 +1 are desirable, so that one postulates that the sum in +(17) restricts to those multi-indices for which +|β|p := +� +n̸=0 +|n|β(n) < 2 +where +|n| := n1 + 2n2. +(18) +This leaves only β = 0 and β = e(1,0), where the latter means β(n) = +δ(1,0) +n +, so that (17) collapses to +(∂2 − ∂2 +1)u + h(u) + h′(u)∂1u = a(u)∂2 +1u + ξ. +(19) +One also postulates that h and h′ depend on the noise ξ only through +its law / distribution / ensemble, hence are deterministic. Since we +assume that the law is invariant under space-time translation, i. e. is +stationary, it was natural to postulate that h and h′ do not explicitly +depend on x, hence are homogeneous. +Reflection symmetry. +Let us now assume that the law of ξ is +invariant under +space-time translation y �→ y + x, +space reflection y �→ (−y1, y2). +(20) +We now argue that under this assumption, it is natural to postulate +that the term h′(u)∂1u in (19) is not present, so that we are left with +(∂2 − ∂2 +1)u + h(u) = a(u)∂2 +1u + ξ. +(21) +To this purpose, let x ∈ R2 be arbitrary yet fixed, and consider the +reflection at the line {y1 = x1} given by Ry = (2x1 − y1, y2), which +by pull back acts on functions as ˜u(y) := u(Ry). Since (1) features no +explicit y-dependence, and only involves even powers of ∂1, which like +∂2 commute with R, we have +(u, ξ) satisfies (1) +=⇒ +(u(R·), ξ(R·)) satisfies (1). +(22) +Since we postulated that h and h′ depend on ξ only via its law, and +since in view of the assumption (20), ˜ξ = ξ(R·) has the same law as ξ, +it is natural to postulate that the symmetry (22) extends from (1) to +(19). Spelled out, this means that (19) implies +(∂2 − ∂2 +1)˜u + h(˜u) + h′(˜u)∂1˜u = a(˜u)∂2˜u + ˜ξ. +Evaluating both identities at y = x, and taking the difference, we get +for any solution of (19) that h′(u(x))∂1u(x) = h′(u(x))(−∂1u(x)), and +thus h′(u(x))∂1u(x) = 0, as desired. +7which associate to every index n a β(n) ∈ N0 such that β(n) vanishes for all +but finitely many n’s + +10 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +Covariance under u-shift. We now come to our most crucial pos- +tulate, which restricts how the nonlinearity h depends on the non- +linearity / constitutive law a. Hence we no longer think of a single +nonlinearity a, but consider all non-linearities at once, in the spirit +of rough paths. This point of view reveals another invariance of (1), +namely for any shift v ∈ R +(u, a) satisfies (1) +=⇒ +(u − v, a(· + v)) satisfies (1). +(23) +A priori, h is a function of the u-variable that has a functional de- +pendence on a, as denoted by h = h[a](u). +We postulate that the +symmetry (23) extends from (1) to (21). This is the case provided we +have the following shift-covariance property +h[a](u + v) = h[a(· + v)](u) +for all u ∈ R. +(24) +This property can also be paraphrased as: Whatever algorithm one +uses to construct h from a, it should not depend on the choice of origin +in what is just an affine space R ∋ u. Property (24) implies that the +counter term is determined by a functional c = c[a] on the space of +nonlinearities a: +h[a](v) = c[a(· + v)]. +(25) +Renormalization now amounts to choosing c such that the solution +manifold stays under control as the UV regularization of ξ fades away. +4. Algebrizing the counter term +In this section, we algebrize the relationship between a and the counter +term h given by a functional c as in (25). To this purpose, we introduce +the following coordinates8 on the space of analytic functions a of the +variable u: +zk[a] := 1 +k! +dka +duk (0) +for k ≥ 0. +(26) +These are made such that by Taylor’s +a(u) = +� +k≥0 +ukzk[a] +for a ∈ R[u], +(27) +where R[u] denotes the algebra of polynomials in the single variable u +with coefficients in R. +We momentarily specify to functionals c on the space of analytic a’s +that can be represented as polynomials in the (infinitely many) vari- +ables {zk}k≥0. This leads us to consider the algebra R[zk] of polynomials +in the variables zk with coefficients in R. The monomials +zβ := +� +k≥0 +zβ(k) +k +(28) +8where here and in the sequel k ≥ 0 stands short for k ∈ N0 + +LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES +11 +form a basis of this (infinite dimensional) linear space, where β runs +over all multi-indices9. Hence as a linear space, R[zk] can be seen as +the direct sum over the index set given by all multi-indices β, and we +think of c as being of the form +c[a] = +� +β +cβzβ[a] +for c ∈ R[zk]. +(29) +Infinitesimal u-shift. Given a shift v ∈ R, for ˜u := u − v and +˜a := a(· + v) we have ˜a(˜u) = a(u). This leads us to study the mapping +a �→ a(· + v) which provides an action/representation of the group +R ∋ v on the set R[u] ∋ a. Note that for c ∈ R[zk] and a ∈ R[u], the +function R ∋ v �→ c[a(·+v)] = � +β cβ +� +k≥0( 1 +k! +dka +du (v))β(k) is polynomial. +Thus +(D(0)c)[a] = d +dv |v=0c[a(· + v)] +(30) +is well-defined, linear in c and even a derivation10, meaning that Leib- +niz’ rule holds +(D(0)cc′) = (D(0)c)c′ + c(D(0)c′). +(31) +The latter implies that D(0) is determined by its value on the co- +ordinates zk, which by definitions (26) and (30) is given by D(0)zk += (k + 1)zk+1. Hence D(0) has to agree with the following derivation +on the algebra R[zk] +D(0) = +� +k≥0 +(k + 1)zk+1∂zk, +(32) +which is well defined since the sum is effectively finite when applied to +a monomial. +Representation of counter term. Iterating (30) we obtain by +induction in l ≥ 0 for c ∈ R[zk] and a ∈ R[u] +dl +dvl |v=0c[a(· + v)] = ((D(0))lc)[a] +and thus by Taylor’s (recall that v �→ c[a(· + v)] is polynomial) +c[a(· + v)] = +� � +l≥0 +1 +l!vl(D(0))lc +� +[a]. +(33) +9which means they associate a frequency β(k) ∈ N0 to every k ≥ 0 such that all +but finitely many β(k)’s vanish +10the index (0) is not necessary for these lecture notes, since we do not appeal +to the other derivations {D(n)}n̸=0 from [14, 15], we keep it here for consistency +with these papers + +12 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +We combine (33) with (25) to obtain the representation +h[a](v) = +� � +l≥0 +1 +l!vl(D(0))lc +� +[a]. +(34) +Hence our goal is to determine the coefficients {cβ}β in (29), which +typically will blow up as τ ↓ 0. +5. Algebrizing the solution manifold: The centered +model +The purpose of this section is to motivate the notion of a centered +model; the motivation will be in parts formal. +Parameterization of the solution manifold. If a ≡ 0 it follows +from (24) that h is a (deterministic) constant. We learned from the +discussion after Lemma 1 that – given a base point x – there is a +distinguished solution v (with v(x) = 0). Hence we may canonically +parameterize a general solution u of (21) via u = v + p, by space- +time functions p with (∂2 − ∂2 +1)p = 0. Such p are necessarily analytic. +Having realized this, it is convenient11 to free oneself from the constraint +(∂2 − ∂2 +1)p = 0, which can be done at the expense of relaxing (21) to +(∂2 − ∂2 +1)v = ξ + q +for some analytic space-time function q. +(35) +Since we think of ξ as being rough while q is infinitely smooth, this +relaxation is still constraining v. +The implicit function theorem suggests that this parameterization (lo- +cally) persists in the presence of a sufficiently small analytic nonlin- +earity a: The nonlinear manifold of all space-time functions u that +satisfy +(∂2 − ∂2 +1)u + h(u) = a(u)∂2 +1u + ξ + q +for some analytic space-time function q +(36) +is still parameterized by space-time analytic functions p. We now return +to the point of view of Section 3 of considering all nonlinearities a at +once, meaning that we consider the (still nonlinear) space of all space- +time functions that satisfy (36) for some analytic nonlinearity a. We +want to capitalize on the symmetry (23), which extends from (1) to (21) +and to (36). We do so by considering the above space of u’s modulo +constants, which we implement by focusing on increments u − u(x). +Summing up, it is reasonable to expect that the space of all space-time +functions u, modulo space-time constants, that satisfy (36) for some +analytic nonlinearity a and space-time function q (but at fixed ξ), is +parameterized by pairs (a, p) with p(x) = 0. +11otherwise, the coordinates z(2,0) and z(0,1) defined in (38) would be redundant +on p-space + +LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES +13 +Formal series representation. In line with the term-by-term ap- +proach from physics, we write the increment u(y)−u(x) as a (typically +divergent) power series +u(y) − u(x) += +� +β +Πxβ(y) +� +k≥0 +� 1 +k! +dka +duk (u(x)) +�β(k) � +n̸=0 +� 1 +n!∂np(x) +�β(n), +(37) +where β runs over all multi-indices in k ≥ 0 and n ̸= 0. Introducing +coordinates on the space of analytic space-time functions p with p(0) = +0 via12 +zn[p] = 1 +n!∂np(0) +for n ̸= 0, +(38) +(37) can be more compactly written as +u(y) = u(x) + +� +β +Πxβ(y)zβ[a(· + u(x)), p(· + x) − p(x)]. +(39) +This is reminiscent of Butcher series in the analysis of ODE discretiza- +tions. +Recall from above that for a ≡ 0 we have the explicit parameterization +u − u(x) = v + p +(40) +with the distinguished solution v of the linear equation. Hence from +setting a ≡ 0 and p ≡ 0 in (37), we learn that Πx0 = v. From keeping +a ≡ 0 but letting p vary we then deduce that for all multi-indices β ̸= 0 +which satisfy β(k) = 0 for all k ≥ 0 we must have13 +Πxβ(y) = +� +(y − x)n +provided β = en +0 +else +� +. +(41) +Hierarchy of linear equations. +The collection of coefficients +{Πxβ(y)}β from (39) is an element of the direct product with the same +index set as the direct sum R[zk, zn]. Hence the direct product inherits +the multiplication of the polynomial algebra +(ππ′) ¯β = +� +β+β′= ¯β +πβπ′ +β′, +(42) +and is denoted as the (well-defined) algebra R[[zk, zn]] of formal power +series; we denote by 1 its unit element. We claim that in terms of (39), +(36) assumes the form of +(∂2 − ∂2 +1)Πx = Π− +x +up to space-time analytic functions +(43) +12where here and in the sequel n ̸= 0 stands short for n ∈ N2 +0 − {(0, 0)} +13where we recall that β = en denotes the multi-index with β(m) = δn +m next to +β(k) = 0 + +14 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +where +Π− +x := +� +k≥0 +zkΠk +x∂2 +1Πx − +� +l≥0 +1 +l!Πl +x(D(0))lc + ξτ1, +(44) +as an identity in formal power series in zk, zn with coefficients that +are continuous space-time functions. We shall argue below that (44) +is effectively, i. e. componentwise, well-defined despite the two infinite +sums, and despite extending from c ∈ R[zk] to c ∈ R[[zk]]. Moreover, +as will become clear by (64), the β-component of (44) contains on the +r. h. s. only terms Πxβ′ for “preceding” multi-indices β′ – hence (43) +describes a hierarchy of equations. +Here comes the formal argument that relates {∂2, ∂2 +1}u, a(u), and h(u), +to {∂2, ∂2 +1}Πx[˜a, ˜p], (� +k≥0 zkΠk +x)[˜a, ˜p], and (� +l≥0 +1 +l!Πl +x(D(0))lc)[˜a, ˜p], re- +spectively. Here we have set for abbreviation ˜a = a(· + u(x)) and ˜p += p(· + x) − p(x). It is based on (39), which can be compactly written +as u(y) = u(x) + Πx[˜a, ˜p](y). Hence the statement on {∂2, ∂2 +1}u follows +immediately. Together with a(u(y)) = ˜a(u(y) −u(x)), this also implies +by (27) the desired +a(u(y)) = +� � +k≥0 +zkΠk +x(y) +� +[˜a, ˜p]. +Likewise by (24), we have h[a](u(y)) = h[˜a](u(y) − u(x)), so that by +(34), we obtain the desired +h[a](u(y)) = +� � +l≥0 +1 +l!Πl +x(y)(D(0))lc +� +[˜a, ˜p]. +Finiteness properties. The next lemma collects crucial algebraic +properties. +Lemma 2. The derivation D(0) extends from R[zk] to R[[zk]]. +Moreover, for π, π′ ∈ R[[zk, zn]], c ∈ R[[zk]], and ξ ∈ R, +π− := +� +k≥0 +zkπkπ′ − +� +l≥0 +1 +l!πl(D(0))lc + ξ1 ∈ R[[zk, zn]] +(45) +is well-defined, in the sense that the two sums are componentwise finite. +Finally, for +[β] := +� +k≥0 +kβ(k) − +� +n̸=0 +β(n) +(46) + +LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES +15 +we have the implication +πβ = π′ +β = 0 +unless +[β] ≥ 0 or β = en for some n ̸= 0 +=⇒ +π− +β = 0 +unless + + + +[β] ≥ 0 or +β = ek + en1 + · · · + enk+1 +for some k ≥ 1 and n1, . . . , nk+1 ̸= 0. + + + . +(47) +We note that for β as in the second alternative on the r. h. s. of (47), +it follows from (41) that Π− +xβ is a polynomial. Hence in view of the +modulo in (43), we learn from (47) that we may assume +Πxβ ≡ 0 +unless +[β] ≥ 0 or β = en for some n ̸= 0. +(48) +Proof of Lemma 2. We first address the extension of D(0) and note +that from (32) we may read off the matrix representation of D(0) ∈ +End(R[zk]) w. r. t. (28) given by +(D(0))γ +β = (D(0)zγ)β +(32) += +� +k≥0 +(k + 1) +� +zk+1∂zkzγ� +β +(28) += +� +k≥0 +(k + 1)γ(k) +� +1 +provided γ + ek+1 = β + ek +0 +otherwise +� +. +(49) +From this we read off that {γ|(D(0))γ +β ̸= 0} is finite for every β, which +implies that D(0) naturally extends from R[zk] to R[[zk]]. With help +of (42) the derivation property (31) can be expressed coordinate-wise, +and thus extends to R[[zk]]. +We now turn to (45), which component-wise reads +π− +β = +� +k≥0 +� +ek+β1+···+βk+1=β +πβ1 · · · πβkπ′ +βk+1 +− +� +l≥0 +1 +l! +� +β1+···+βl+1=β +πβ1 · · · πβl((D(0))lc)βl+1 + ξδ0 +β, +(50) +and claim that the two sums are effectively finite. For the first term +of the r. h. s. this is obvious since thanks to the presence of14 ek in +ek + β1 + · · · + βk+1 = β, for fixed β there are only finitely many k ≥ 0 +for which this relation can be satisfied. +In preparation for the second r. h. s. term of (50) we now establish that +((D(0))l)γ +β = 0 +unless +[β]0 = [γ]0 + l, +(51) +where we introduced the scaled length [γ]0 := � +k≥0 kγ(k) ∈ N0. The +argument for (51) proceeds by induction in l ≥ 0. It is tautological +for the base case l = 0. In order to pass from l to l + 1 we write +14γ = ek denotes the multi-index with γ(l) = δk +l next to γ(n) = 0 + +16 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +((D(0))l+1)γ +β = � +β′((D(0))l)β′ +β (D(0))γ +β′; by induction hypothesis, the first +factor vanishes unless [β]0 = [β′]0 + l. We read off (49) that the second +factor vanishes unless [β′]0 = [γ]0 + 1, so that the product vanishes +unless [β]0 = [γ]0 + (l + 1), as desired. +Equipped with (51) we now turn to the second r. h. s. term of (50) and +note that ((D(0))lc)βk+1 vanishes unless l ≤ [βk+1]0 ≤ [β]0, which shows +that also here, only finitely many l ≥ 0 contribute for fixed β. +We turn to the proof of (47). We use (50) and give the proof for every +summand separately. For the first term on the r. h. s. of (50) we obtain +by additivity of [·] that [β] = k+[β1]+· · ·+[βk+1]. Note that πβi is only +non vanishing if [βi] ≥ −1. If at least one of the β1, . . . , βk+1 satisfies +[βi] ≥ 0, we obtain therefore [β] ≥ k−k = 0. For the second r. h. s. term +in (50) we appeal to (51): Since D(0) doesn’t affect the zn components, +(51) extends from [·]0 to [·]. +Together with c ∈ R[[zk]] this yields +[βl+1] ≥ l. Hence as above [β] = [β1]+· · ·+[βl+1] ≥ −l+[βl+1] ≥ 0. +□ +Homogeneity. +We return to a heuristic discussion. +Provided we +include, like for (23), a into our considerations, the original equation +(1) has a scaling symmetry: Considering for s ∈ (0, ∞) the parabolic +space-time rescaling Sy = (sy1, s2y2), we have for any exponent α +(u, ξ, a) satisfies (1) +=⇒ +� +s−αu(S·), s2−αξ(S·), a(sα·) +� +=: (˜u, ˜ξ, ˜a) satisfies (1). +(52) +Suppose the scaling transformation ξ �→ ˜ξ preserves the law, which for +white noise is the case with α − 2 = −D +2 , i. e. α = 1 +2. Since in view of +Section 3, the counter term only depends on the law, it is natural to +postulate, in line with that section, that the solution manifold of the +renormalized problem inherits this invariance15. +It is also natural to postulate that the parameterization by the p’s +(given a base point x) is consistent with (52) in the sense that p trans- +forms as u, i. e. we have invariance under +(u, ξ, a, x, p) �→ (˜u, ˜ξ, ˜a, ˜x := S−1x, ˜p := s−αp(S·)). +We now appeal to the series expansion (37), both as it stands and +with (x, y, u, ξ, a, p) replaced by (˜x, ˜y := S−1y, ˜u, ˜ξ, ˜a, ˜p). Because of +u(y) − u(x) = sα(˜u(˜y) − ˜u(˜x)), we obtain a relation between the two +right-hand sides. It is natural to postulate that the coefficients {Π·,β}β +are individually consistent with this invariance, leading to +ΠSxβ[ξ](Sy) = s|β|Πxβ[s2−αξ(S·)](y), +(53) +15since this scale invariance in law is not consistent with the mollification ξτ this +discussion pertains to the limiting solution manifold + +LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES +17 +where the “homogeneity” |β| of the multi-index β is given by +|β| := α(1 + [β]) + |β|p, +(54) +cf. (18) and (46). We note that +|en| = |n| +(55) +so that (54) is consistent with (41). +Appealing once more to the invariance in law of ξ under (52), we obtain +from (53) +the law of s−|β|ΠSx β(Sy) does not depend on s ∈ (0, ∞). +(56) +By the invariance of the (original) solution manifold under (u, ξ) �→ +(˜u := u(·+z), ˜ξ := ξ(·+z)), which by our assumption (20) is passed on +to the renormalized solution manifold, it is natural to impose that the +parameterization is invariant under (u, ξ, x, p) �→ (˜u, ˜ξ, x + z, p(· + z)), +and that the coefficients in (39) are individually consistent with this +invariance, so that we likewise have +the law of Πx+z β(y + z) does not depend on z ∈ R2. +(57) +Specifying to x = 0, the invariance (56) implies that E +1 +p|Π0β(y)|p de- +pends on y only through +y +|y|. From the invariance (57) we thus learn +that E +1 +p |Πxβ(y)|p depends on x, y only through +y−x +|y−x|. Since +y−x +|y−x| has +compact range, this suggest that +E +1 +p |Πxβ(y)|p ≲ |y − x||β|, +which is our main result, see (60) in the next section. +The scaling invariance (52) also connects to the notion of “subcritical- +ity” which is often referred to in the realm of singular SPDEs. Loosely +speaking, it means that by zooming in on small scales, the nonlinear +term becomes negligible. Indeed, as can be seen from (52), the rescaled +nonlinearity ˜a converges to the constant a(0) in the limit s ↓ 0, i. e. the +SPDE (1) turns into a linear one. This is true iff α > 0, and provides +the reason for restricting to α > 0 in the assumption of Theorem 1, +which is the sub-critical regime for (1). +6. The main result: A stochastic estimate of the +centered model +Theorem 1. Suppose the law of ξ is invariant under (20); suppose that +it satisfies a spectral gap inequality (87) with exponent α ∈ (max{0, 1− +D +4 }, 1) \ Q. +Then given τ > 0, there exists a deterministic c ∈ R[[zk]], and for every +x ∈ R2, a random16 Πx ∈ C2[[zk, zn]], and a random Π− +x ∈ C0[[zk, zn]] +16by this we mean a formal power series in zk, zn with values in the twice con- +tinuously differentiable space-time functions + +18 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +that are related by (44) and +(∂2 − ∂2 +1)Πxβ = Π− +xβ + polynomial of degree ≤ |β| − 2, +(58) +and that satisfy (41), the population condition (48) and +cβ = 0 +unless +|β| < 2. +(59) +Moreover, we have the estimates +E +1 +p|Πxβ(y)|p ≲β,p |y − x||β|, +(60) +E +1 +p|Π− +xβt(y)|p ≲β,p ( +4√ +t)α−2( +4√ +t + |y − x|)|β|−α. +(61) +The important feature is that the constants in (60) and (61) are uniform +in τ ↓ 0. +We remark that we may pass from (61) to (60) by Lemma 1. Indeed, +because of (48) we may restrict to β with [β] ≥ 0. In this case, by our +assumption α ̸∈ Q, +[β] ≥ 0 +(54) +=⇒ +|β| ̸∈ Z, +(62) +next to |β| ≥ α. Hence we may indeed apply Lemma 1 with η = |β| +and (61) as input. The output yields a Πxβ satisfying (58) and (60). +Uniqueness and (implicit) BPHZ renormalization. The con- +struction of Πx in [15] proceeds by an inductive algorithm in β. The +ordering17 on the multi-indices is provided by +(63) +|β|≺ := |β| + λβ(0) +for fixed λ ∈ (0, α), +and we will write γ ≺ β for |γ|≺ < |β|≺. As opposed to the ordering +provided by the homogeneity, ≺ allows for the triangular structure: +(64) +Π− +xβ − cβ depends on (Πxγ, cγ) only through γ with γ ≺ β, +which can be easily checked on the component-wise level (50). More- +over, (63), as opposed to the ordering by homogeneity, is coercive: For +fixed β there are only finitely many γ with γ ≺ β, see (101), which is +important for the estimates. +We now argue that within this induction, (c, Πx, Π− +x ) is determined. +Indeed, the uniqueness statement of Lemma 1 implies that for given β, +Πxβ is determined by Π− +xβ. According to (64), Π− +xβ − cβ is determined +by the previous steps. Finally, we note that provided |β| < 2, we have +|EΠ− +xβt(x)| ≤ E|Π− +xβt(x)| +(61) +≲ ( +4√ +t)|β|−2 t↑∞ +→ 0, +(65) +17this ordering coincides with the one chosen in [13] but it slightly differs from +the one in [15], which is imposed by the restricted triangularity of dΓ∗ in Section +7; for simplicity we stick to (63) + +LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES +19 +so that cβ, because it is deterministic18 may be recovered from cβ = +− limt↑∞ E(Π− +xβ − cβ)t(x). +Hence also cβ is determined. +Fixing the +counter term by making an expectation19 vanish like in (65) corre- +sponds to what Hairer assimilates to a BPHZ renormalization. See [3, +Theorem 6.18] for the form BPHZ renormalization takes within regu- +larity structures. +Mission accomplished. Returning to the end of Section 2, we may +claim “mission accomplished”: +• On the one hand, the form of the counter terms preserve a +number of symmetries of the original solution manifold: shift +in x, reflection in x1, shift in u, and to some extend are guided +by scaling in x. +• On the other hand, in a term-by-term sense as encoded by (37), +the solution manifold of the renormalized equation stays under +control as τ ↓ 0, cf. (60) and (61). +Moreover, the constants cβ = cτ +β that determine the counter term via +(34) are (canonically) determined by the large-scale part of the estimate +(61). +As discussed in the introduction, the connection between this term-by- +term approach to the solution manifold and the solution of an actual +initial/boundary value problem is provided by the second part of reg- +ularity structures. This second part, a fixed point argument based on +a truncation of (37) to a finite sum20, is not addressed in these lecture +notes. +7. Malliavin derivative and Spectral gap (SG) +In view of the discussion at the end of the statement of Theorem 1, +the main issue is the estimate (61) of Π− +xβ. Indeed, its definition of +(44) still contains the singular product Πk +x∂2 +1Πx and the collection of +deterministic constants c that diverge as the UV regularization fades +away. Hence we seek a relation between Π− +x and Πx that is more stable +than (44); in fact, it will be a relation between the families {Π− +x }x and +{Πx}x based on symmetries under a change of the base point x. This +relation is formulated on the level of the derivative w. r. t. noise ξ, +also known as the Malliavin derivative. We start by motivating this +approach. +Heuristic discussion of a stable relation {Πx}x �→ {Π− +x }x. Let +δ denote the operation of taking the derivative of an object like Πxβ(y), +18and independent of the base point x +19in our case it is a space-time next to an ensemble average +20by restricting to homogeneities |β| < 2; in our quasi-linear case, the sum stays +infinite w. r. t. the z0-variable, but one has analyticity in that variable since 1 + z0 +plays the role of a constant elliptic coefficient + +20 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +which is a functional of ξ, in direction of an infinitesimal variation δξ +of the latter21. +Clearly, since cβ is deterministic, we have δcβ = 0. +However, applying δ to (a component of) (44) does not eliminate c +because of the specific way c enters (44), which is dictated by the +fundamental symmetry (25). However, when evaluating (44) at the +base point x itself and appealing to the built-in +(66) +Πx(x) = 0, +see (37) or (60), it collapses to +Π− +x (x) = z0∂2 +1Πx(x) − c + ξτ(x)1. +(67) +This isolates c so that it can be eliminated by applying δ: +δΠ− +x (x) = z0∂2 +1δΠx(x) + δξτ(x)1. +(68) +Clearly, (68) is impoverished in the sense that the active point coincides +with the base point. +Instead of attempting to modify the active point, the idea is to modify +the base point from x to y. Such a change of base point, which will be +rigorously introduced in Section 8, amounts to a change of coordinates +in the heuristic representation (39): +u = +� u(x) + � +β Πxβzβ[a(· + u(x)), px], +u(y) + � +β Πyβzβ[a(· + u(y)), py], +(69) +for some polynomials px, py vanishing at the origin. The form in which +the u-shift appears in (69) suggests that this change of coordinates can +be algebrized by an algebra endomorphism22 Γ∗ +yx of R[[zk, zn]] with the +properties +Πy = Γ∗ +yxΠx + Πy(x) +and +Γ∗ +yx = +� +l≥0 +1 +l!Πl +y(x)(D(0))l on R[[zk]], +(70) +see the discussion of finite u-shifts around (34). Recall that an alge- +bra endomorphism Γ∗ +yx is a linear map from R[[zk, zn]] to R[[zk, zn]] +satisfying +(71) +Γ∗ +yxππ′ = (Γ∗ +yxπ)(Γ∗ +yxπ′) +for π, π′ ∈ R[[zk, zn]]. +We claim that (70) implies +Π− +y = Γ∗ +yxΠ− +x . +(72) +21in the Gaussian case, this would be an element of the Cameron-Martin space +22in a first reading, the star should be seen as mere notation; Γ∗ +yx is actually the +algebraic dual of a linear endomorphism Γyx on the pre-dual space, see Lemma 3; +it is Γyx that can be assimilated to the object denoted by the same symbol in +regularity structures; for a concise reference see [14, Section 5.3] + +LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES +21 +Indeed, applying Γ∗ +yx to definition (44) we obtain by (71) +Γ∗ +yxΠ− +x += +� +k≥0 +(Γ∗ +yxzk)(Γ∗ +yxΠx)k∂2 +1Γ∗ +yxΠx − +� +l≥0 +1 +l!(Γ∗ +yxΠx)lΓ∗ +yx(D(0))lc + ξτ1. +We substitute Γ∗ +yxΠx according to the first item in (70), substitute +Γ∗ +yxzk = � +l≥0 +�k+l +k +� +Πl +y(x)zk+l and Γ∗ +yx(D(0))lc according to the second +item in (70) and definition (32), and finally appeal to the binomial +formula in both ensuing double sums to obtain (44) with x replaced by +y, establishing (72). +In view of the scaling (56) and the transformation (70) we expect that +the laws of s|β|−|γ|(Γ∗ +yx)γ +β and of (Γ∗ +SySx)γ +β to be identical. On the other +hand, we expect (Γ∗ +SySx)γ +β to converge to (Γ∗ +00)γ +β as s ↓ 0, and we expect +Γ∗ +00 to be the identity. This suggests strict triangularity: +(Γ∗ +yx − id)γ +β = 0 +unless +|γ| < |β|. +(73) +We claim that applying Γ∗ +yx to (68), we obtain23 +δΠ− +y (x) − (δΓ∗ +yx)Π− +x (x) += +� +k≥0 +zkΠk +y(x)∂2 +1 +� +δΠy − δΠy(x) − (δΓ∗ +yx)Πx +� +(x) + δξτ(x)1. +(74) +Since by (73), δΓ∗ +yx is strictly triangular, (74) provides an inductive +way of determining {Π− +x }x (up to expectation) in terms of {Πx}x. Here +comes the argument for (74): Applying Γ∗ +yx to the l. h. s. of (68) and +using (72) in conjunction with Leibniz’ rule w. r. t. δ, we obtain the +l. h. s. of (74). For the r. h. s. we first use the multiplicativity of Γ∗ +yx; +according to the second item in (70) and (32) we have +Γ∗ +yxz0 = +� +l≥0 +Πl +y(x)zl. +(75) +To rewrite Γ∗ +yxδΠx, we apply δ to the first identity in (70). This estab- +lishes (74). +We now argue that from an analytical point of view, (74) is not quite +adequate. Clearly, the r. h. s. of (74) still contains a potentially singular +product of Πk +y and ∂2 +1(δΠy −δΠy(x) −(δΓ∗ +yx)Πx). Here, it is crucial that +applying δ to Πy, which is a multi-linear expression in ξ, means replac- +ing one of the instances of ξ by δξ. Now as we shall explain in the next +subsection, δξ gains24 D +2 orders of regularity over ξ. However, since the +other instances of ξ remain, the regularity of δΠy is not at face value +better by D +2 orders over Πy, which is just H¨older continuous with expo- +nent α. Hence we can only expect that δΠy is locally, i. e. near a base +23of course, the r. h. s. term δΠy(x) is effectively absent due to the derivative ∂2 +1 +24however on an L2 instead of a uniform scale + +22 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +point x, described – “modelled” in the jargon of regularity structures +– to order D +2 + α in terms of Πx. The Taylor-remainder-like expression +δΠy − δΠy(x) −(δΓ∗ +yx)Πx has the potential of expressing this modeled- +ness. Hence the product of Πk +y and ∂2 +1(δΠy − δΠy(x) −(δΓ∗ +yx)Πx) has a +chance of being well-defined provided α + ( D +2 + α − 2) > 0, which gives +rise to the lower bound assumption α > 1 − D +4 in Theorem 1, which +reduces to25 α > +1 +4 for our D = 3. Since D +2 + α > 1, this only has +a chance of working provided every β-component of (δΓ∗ +yx)Πx involves +the affine function Πxe(1,0) = (· − x)1. However, this contradicts the +(strict) triangularity (73) for |β| ≤ 1. Hence δΓ∗ +yx is not rich enough to +describe all components of δΠy to the desired order near x. +In view of the preceding discussion, we are forced to loosen the pop- +ulation constraint (73). +To this purpose, we replace the directional +Malliavin derivative δΓ∗ +yx by some dΓ∗ +yx ∈ End(R[[zk, zn]]) in order to +achieve +δΠy − δΠy(x) − dΓ∗ +yxΠx = O(| · −x| +D +2 +α). +(76) +In order to preserve the identity (74) in form of +δΠ− +y (x) − dΓ∗ +yxΠ− +x (x) += +� +k≥0 +zkΠk +y(x)∂2 +1 +� +δΠy − δΠy(x) − dΓ∗ +yxΠx +� +(x) + δξτ(x)1, +(77) +we need dΓ∗ +yx to inherit the algebraic properties of δΓ∗ +yx. More precisely, +we impose that dΓ∗ +yx agrees with δΓ∗ +yx on the sub-algebra R[[zk]], +dΓ∗ +yx = δΓ∗ +yx on R[[zk]], +(78) +and that dΓ∗ +yx is in the tangent space to the manifold of algebra mor- +phisms in Γ∗ +yx, which means that for all π, π′ ∈ R[[zk, zn]] +dΓ∗ +yxππ′ = (dΓ∗ +yxπ)(Γ∗ +yxπ′) + (Γ∗ +yxπ)(dΓ∗ +yxπ′). +(79) +Here is the argument on how to pass from (78) & (79) to (77). On the +one hand, we apply δ to (44) to the effect of +δΠ− +y (x) = +� +k≥0 +zkδ +� +Πk +y(x) +� +∂2 +1Πy(x) + +� +k≥0 +zkΠk +y(x)∂2 +1δΠy(x) +− +� +l≥0 +1 +l!δ +� +Πl +y(x) +� +(D(0))lc + δξτ(x)1. +(80) +25This is the analogy of rough path construction of fractional Brownian motion. +For the case of fractional Brownian motion with Hurst parameter H, a rough path +construction can be only implemented for any H > 1 +4 by increasing the number +of iterated integrals. +However, the stochastic analysis to construct the iterated +integrals fails for fractional Brownian motion of Hurst parameter H ≤ 1 +4. See [5, +Theorem 2]. + +LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES +23 +On the other hand, we apply dΓ∗ +yx to (67) to obtain by26 (79) +dΓ∗ +yxΠ− +x (x) += (dΓ∗ +yxz0)∂2 +1Γ∗ +yxΠx(x) + (Γ∗ +yxz0)∂2 +1dΓ∗ +yxΠx(x) − dΓ∗ +yxc. +(81) +We now argue that the first r. h. s. term of (80) is identical to the one +in (81); indeed, by the first item in (70) we have ∂2 +1Γ∗ +yxΠx = ∂2 +1Πy. On +the other hand, by (78) and the second item in (70) we have +dΓ∗ +yx = +� +l≥0 +1 +l!δ +� +Πl +y(x) +� +(D(0))l +on R[[zk]]. +(82) +so that by (32) dΓ∗ +yxz0 = � +k≥0 δ(Πk +y(x))zk. Identity (82) also implies +that the third r. h. s. terms of (80) and (81) are identical. The sec- +ond r. h. s. terms of (80) and (81) combine as desired by (75). This +establishes (77). In order to use (77) inductively to define – or rather +estimate – {Π− +x }x, [15] had to come up with an ordering on multi- +indices β with respect to which dΓ∗ +yx is strictly triangular, leading to a +modification of (63). +Definition of the Malliavin derivative and SG. We have seen +that the Malliavin derivative, which we now shall rigorously define, +allows to give a more robust relation between Πx and Π− +x . Via the SG +inequality, which will be introduced here, the control of the Malliavin +derivative of a random variable F yields control of the variance of F. +Consider the Hilbert norm on (a subspace of) the space of Schwartz +distributions27 +∥δξ∥2 = +ˆ +R2 dx +� +(∂4 +1 − ∂2 +2) +1 +4(α− 1 +2)δξ +�2 = +ˆ +R2 dq +��|q|(α− 1 +2)Fδξ +��2. +(83) +Note that we encounter again A∗A = (−∂2 − ∂2 +1)(∂2 − ∂2 +1) with Fourier +symbol |q|4 = q4 +1 + q2 +2, see (4). Hence this is one of the equivalent ways +of defining the homogeneous L2(R2)-based Sobolev norm of fractional +order α − 1 +2, however of parabolic scaling, which we nevertheless still +denote by H := ˙Hα− 1 +2(R2). +We now consider “cylindrical” (nonlinear) functionals F on the space +S′(R2) of Schwartz distributions, by which one means that for some +N ∈ N, F is of the form +F[ξ] = f +� +(ξ, ζ1), · · · , (ξ, ζN) +� +with +f ∈ C∞(RN) and ζ1, · · · , ζN ∈ S(R2), +(84) +26which also implies dΓ∗ +yx1 = 0 +27we denote the argument by δξ since we think of it as an infinitesimal +perturbation. + +24 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +where we recall that (ξ, ζn) denotes the natural pairing between ξ ∈ +S′(R2) and a Schwartz function ζn ∈ S(R2). +Clearly, those func- +tion(al)s F are Fr´echet differentiable with +dF[ξ].δξ = lim +s↓0 +1 +s(F[ξ + sδξ] − F[ξ]) += +N +� +n=1 +∂nf +� +(ξ, ζ1), · · · , (ξ, ζN) +� +(δξ, ζn) = (δξ, ∂F +∂ξ [ξ]), +(85) +where ∂F +∂ξ [ξ] ∈ S(R2) is defined through +∂F +∂ξ [ξ] = +N +� +n=1 +∂nf +� +(ξ, ζ1), · · · , (ξ, ζN) +� +ζn. +We will monitor the dual norm +∥∂F +∂ξ [ξ]∥∗ := sup +δξ +(δξ, ∂F +∂ξ [ξ]) +∥δξ∥ += ∥∂F +∂ξ [ξ]∥ ˙H +1 +2 −α(R2). +(86) +Definition 1. An ensemble E of Schwartz distributions28 is said to +satisfy a SG inequality provided for all cylindrical F with E|F| < ∞ +E(F − EF)2 ≤ E∥∂F +∂ξ ∥2 +∗. +(87) +Note that the l. h. s. of (87) is the variance of F. +Inequality (87) +amounts to an L2-based Poincar´e inequality with mean value zero on +the (infinite-dimensional) space of all ξ’s. By a (parabolic) rescaling +of x, we may w. l. o. g. assume that the constant in (87) is unity. +Implicitly, we also include closability of the linear operator +cylindrical function F �→ ∂F +∂ξ ∈ {cylindrical functions} ⊗ S(R2). +(88) +This means that the closure of the graph of (88) w. r. t. the topology +of L2 and L2(H∗) is still a graph. This allows to extend the Fr´echet +derivative (88) to the Malliavin derivative +L2 ⊃ D( ∂ +∂ξ ) ∋ F �→ ∂F +∂ξ ∈ L2(H∗). +By the chain rule, we may post-process (87) to its Lp-version +E +1 +p|F − EF|p ≲p E +1 +p∥∂F +∂ξ ∥p +∗, +(89) +which is the form we use it in. A concise proof how to obtain (89) from +(87) can be found in [11, Step 2 in the proof of Lemma 3.1]. +28It does not have to be a Gaussian ensemble. + +LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES +25 +The obvious examples are Gaussian ensembles of Schwartz distributions +with +∥ · ∥ ≤ Cameron-Martin norm, +(90) +where the norm ∥ · ∥ means the Hilbert norm defined in (83), e. g. +white noise +−D +2 += α − 2 +=⇒ α = 1 +2, +free field +1 − D +2 += α − 2 +=⇒ α = 3 +2. +In other words, the SG inequality (87) holds with Gaussian ensembles +satisfying (90), see [2, Theorem 5.5.11]. +For the reader’s convenience, we sketch the simplest application of SG +from [15, Section 4.3], namely (61) for β = 0. To this aim we apply +(89) to F := (ξ, ψt(y−·)) = Πx0(y), which is of the form of (84), so that +according to (85) its Malliavin derivative is given by ∂F +∂ξ = ψt(y − ·). +In view of (86), and then appealing to (8) in conjunction with the +translation invariance and scaling of the Sobolev norm we have +∥∂F +∂ξ ∥∗ = ∥ψt(y − ·)∥ ˙H +1 +2 −α(R2) = ( +4√ +t)− D +2 − 1 +2+α∥ψt=1∥ ˙H +1 +2 −α(R2). +Noting that the exponent is α − 2 and that ψt=1 is a (deterministic) +Schwartz function we obtain from (89) +E +1 +p|Πx0(y)|p ≲ ( +4√ +t)α−2. +In view of |0| = α, this amounts to the desired (61) for β = 0. +We also remark that SG naturally complements the BPHZ-choice of +renormalization, see Section 6: +• The choice of cβ takes care of the mean EΠ− +xβt(y), while +• SG takes care of the variance of Π− +xβt(y). +Hence the main task in [15] is the estimate of E +1 +p∥ ∂F +∂ξ ∥p +∗, where F := +Π− +xβt(y), which we tackle by duality through estimating the directional +derivative +δF := (δξ, ∂F +∂ξ ) +given control of E +1 +q ∥δξ∥q. +The inductive estimate is based on (77). +Philosophically speaking, our approach is analytic rather than combi- +natorial: +analytic +combinatorial +index set: +derivatives w.r.t. a and p +Picard iteration +⇝ multi-indices on k ≥ 0, n ̸= 0 +⇝ trees with decorations +Ass. on ξ: +spectral gap inequality +cumulant bounds +Malliavin derivatives w.r.t. ξ +trees with paired nodes +⇝ estimates on E∥ ∂ +∂ξΠ− +xβ t(y)∥2 +∗ +⇝ Feynman diagrams + +26 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +For us, all combinatorics are contained in Leibniz’ rule. We also point +out that our approach may be called “top-down” rather than bottom- +up in the sense that we postulate the conditions (space-time trans- +lation, spatial reflection, shift-covariance, etc) on the counter term h +from the beginning. +A closing remark for experts in QFT: The absence of c in (77) means +that our approach does not suffer from the well-known difficulty of +“overlapping sub-divergences” in Quantum Field Theory, which is also +an issue in [4]. Our inductive approach has similarities with the one of +Epstein-Glaser, see [18, Section 3.1]. +8. The structure group and the re-expansion map +In this section we construct the endomorphism Γ∗ +yx of the algebra +R[[zk, zn]] that satisfies (70) for given Πx and Πy. In [15], the construc- +tions (and estimates) of Γ∗ +yx and Πx are actually intertwined, however +the proof of Lemma 5 has the same elements as [15, Section 5.3]. In +line with regularity structures it is convenient to adopt a more ab- +stract point of view: We start by introducing what can be assimilated +to Hairer’s structure group G, which here is a subgroup of the au- +tomorphism group of the linear space R[zk, zn], where R[zk, zn] now +plays the role of the29 (algebraic) pre-dual of R[[zk, zn]]; Γ∗ +yx will be the +transpose of a Γyx ∈ G. The elements Γ ∈ G are parameterized by +{π(n)}n ⊂ R[[zk, zn]], see Lemma 3; the group property will be estab- +lished in Lemma 4. In Lemma 5 we inductively choose {π(n) +yx }n such +that the associated Γyx satisfies (70). For a discussion of the Hopf- +and Lie-algebraic structure underlying G we refer to [14]. As opposed +to [14] and [13], we will capitalize on α < 1, which simplifies several +arguments. +Lemma 3. Given30 {π(n)}n ⊂ R[[zk, zn]] satisfying +π(n) +β += 0 +unless +|n| < |β|, +(91) +there exists a unique linear endomorphism Γ of R[zk, zn] such that Γ∗ +is an algebra endomorphism31 of R[[zk, zn]] that satisfies +Γ∗zk = +� +l≥0 +1 +l! (π(0))l(D(0))lzk +(32) += +� +l≥0 +�k+l +k +� +(π(0))lzk+l, +(92) +Γ∗zn = zn + π(n). +(93) +29canonical w. r. t. the monomial basis +30which here as opposed to earlier includes the additional (dummy) index n = 0 +we first encountered in (30) +31i. e. Γ∗ππ′ = (Γ∗π)(Γ∗π′) and Γ∗1 = 1 hold + +LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES +27 +In addition32, +(94) +(Γ∗ − id)γ +β = 0 +unless +|γ| < |β| and γ ≺ β. +We remark that the algebra endomorphism property, the mapping +property (92), and the first triangularity in (94) mimic desired proper- +ties of Γ∗ +yx, namely (71), the second item of (70), and (73), respectively. +Proof of Lemma 3. We recall that the matrix representation {Γβ +γ}β,γ of +a linear endomorphism Γ of R[zk, zn] w. r. t. the monomial basis {zβ}β +is given by +(95) +Γzβ = +� +γ +Γβ +γzγ. +The algebraic dual Γ∗, as a linear endomorphism of R[[zk, zn]], is given +by33 +(Γ∗π)β = +� +γ +(Γ∗)γ +βπγ +where +(Γ∗)γ +β := (Γ∗zγ)β = Γβ +γ. +Such a Γ∗ is an algebra endomorphism if and only if +(Γ∗)γ +β = +� +β1+···+βk=β +(Γ∗)γ1 +β1 · · · (Γ∗)γk +βk +for +γ = γ1 + · · · + γk. +(96) +This includes Γ∗1 = 1 in form of +(Γ∗)0 +β = δ0 +β +(97) +Since any multi-index γ ̸= 0 can be written as the sum of γj’s of length +one, we learn that an endomorphism Γ of R[zk, zn] with multiplica- +tive Γ∗ is determined by how Γ∗ acts on the coordinates {zk}k≥0 and +{zn}n̸=0. This establishes the uniqueness statement. +For the existence, we need to establish that the numbers {(Γ∗)γ +β}β,γ +defined through (92) & (93) in form of +(Γ∗)ek +β − δek +β = +� +l≥1 +�k+l +k +� +� +ek+l+β1+···+βl=β +π(0) +β1 · · ·π(0) +βl , +(98) +(Γ∗)en +β − δen +β = π(n) +β +(99) +and extended by (96) & (97) to all γ satisfy (for fixed β) +#{γ | (Γ∗)γ +β ̸= 0} < ∞. +(100) +Indeed, this finiteness condition allows to define Γ via (95) with Γβ +γ +:= (Γ∗)γ +β. Since thanks to (103) below in conjunction with 0 < λ, α < 1 +the ordering ≺ is coercive, by which we mean +#{γ | γ ≺ β} < ∞, +(101) +32we recall that ≺ is defined in (63) +33note that the sum is effectively finite, since there are only finitely many γ such +that Γβ +γ ̸= 0 since the monomial basis is an algebraic basis + +28 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +(100) follows once we establish the second strict triangularity in (94). +Hence, it remains to establish (94) in form of +(Γ∗)γ +β − δγ +β = 0 +unless +|γ|≺ < |β|≺ +and +|γ| < |β| +(102) +for the numbers {(Γ∗)γ +β}β,γ defined through (98) & (99) and then ex- +tended by (96). For this purpose, we note that by definition (54) in +form of +|β| − α = +� +k≥0 +kβ(k) + +� +n̸=0 +(|n| − α)β(n) +(103) +and since α ≤ 1 ≤ |n|, +| · | − α ≥ 0 +is additive +(63) +=⇒ +same for | · |≺ − α. +(104) +We first restrict to γ’s of length one in (102), and distinguish the cases +γ = en and γ = ek. +Since by (54) and (63) we have |en|≺ = |en| += |n| and |β| ≤ |β|≺, the former case follows directly via (99) from +assumption (91). We now turn to the latter case of γ = ek and to (98). +There is a contribution to the r. h. s. sum only when there exists an +l ≥ 1 and a decomposition β = ek+l + β1 + · · · + βl; this implies +|β| ≥ |ek+l| +(54) += |ek| + αl ≥ |ek| + α +(63) +=⇒ +|β|≺ ≥ |ek|≺ + (α − λ), +which yields the desired (102) because of α > λ, 0. +Finally, we need to upgrade (102) from γ’s of length one to those of +arbitrary length, which we do by induction in the length. The base +case of zero length, i. e. of γ = 0, is dealt with in (97). We carry out +the induction step with help of (96), writing a multi-index γ = γ′ + γ′′ +with γ′, γ′′ of smaller length: +(Γ∗)γ +β = +� +β′+β′′=β +(Γ∗)γ′ +β′(Γ∗)γ′′ +β′′. +(105) +We learn from the induction-hypothesis version of (102) that the sum- +mand vanishes unless +|γ′| + |γ′′| < |β′| + |β′′| and |γ′|≺ + |γ′′|≺ < |β′|≺ + |β′′|≺ +or +γ′ = β′ and γ′′ = β′′; +in the latter case the summand is equal to 1. By (104), the first al- +ternative implies |γ| < |β| and |γ|≺ < |β|≺. The second alternative +implies γ = β and then holds for exactly one summand to the desired +effect of (Γ∗)γ +β = 1. +□ +The two triangular properties (94) from Lemma 3 allow us to establish +the group property. Furthermore, a triangular dependence (106) of Γ∗ +on π(n) will play a crucial role when inductively constructing π(n) +yx in +Lemma 5. + +LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES +29 +Lemma 4. The set G of all Γ as in Lemma 3 defines a subgroup of the +automorphism group of R[zk, zn]. Moreover, +for [γ] ≥ 0, +(Γ∗)γ +β is independent of π(n) +β′ +unless +β′ ≺ β. +(106) +Remark 1. The group G is larger than the one constructed in [14], +since 1) we do not require that π(n) +β += 0 unless β satisfies (48), and +2) we do not specify the space-time shift structure of the (β = em)- +components of π(n) +β +as in [14, Proposition 5.1]. Both conditions however +are satisfied for our construction of π(n) +yxβ, see (113) and (115). +Proof of Lemma 4. We first argue that for Γ, Γ′ ∈ G we have Γ′Γ ∈ G. +More precisely, if Γ and Γ′ are associated to {π(n)}n and {π′(n)}n by +Lemma 3, respectively, we consider +�π(n) := π(n) + Γ∗π′(n). +(107) +We note that by triangularity (94) of Γ∗ w. r. t. |·|, the population prop- +erty (91) propagates from π(n), π′(n) to �π(n). Let �Γ ∈ G be associated +to {�π(n)}n; we claim that Γ′Γ = �Γ. +To this purpose, we note that (Γ′Γ)∗ = Γ∗Γ′∗ is an algebra morphism, +like �Γ∗ is. Hence by the uniqueness statement of Lemma 3, it is suffi- +cient to check that Γ∗Γ′∗ and �Γ∗ agree on the two sets of coordinates +{zk}k and {zn}n. On the latter this is easy: +�Γ∗zn +(93) += zn + �π(n) (107) += zn + π(n) + Γ∗π′(n) (93) += Γ∗(zn + π′(n)) +(93) += Γ∗Γ′∗zn. +We now turn to the zk’s, showing that the algebra endomorphisms Γ∗Γ′∗ +and �Γ∗ agree on the sub-algebra R[zk] ⊂ R[[zk, zn]]; by multiplicativity +of Γ∗ we have according to (92) for Γ′ +Γ∗Γ′∗ = +� +l′≥0 +1 +l′!(Γ∗π′(0))l′Γ∗(D(0))l′ +on R[zk]. +Since D(0) preserves R[zk], we may apply (92) for Γ and obtain by the +binomial formula: +Γ∗Γ′∗ = +� +l′≥0 +1 +l′!(Γ∗π′(0))l′ � +l≥0 +1 +l!(π(0))l(D(0))l′+l +(107) += +� +˜l≥0 +1 +˜l! +(�π(0)) +˜l(D(0)) +˜l +on R[zk], +which according to (92) agrees with �Γ∗. + +30 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +We come to the inverse of a Γ ∈ G associated to {π(n)}n. +By the +strict triangularity (94) w. r. t. the coercive ≺, cf. (101), there exists +˜π(n) ∈ R[[zk, zn]] such that +Γ∗˜π(n) = −π(n). +(108) +We now argue by induction in β w. r. t. ≺ that ˜π(n) satisfies (91). For +this, we spell (108) out as +˜π(n) +β ++ +� +γ +(Γ∗ − id)γ +β˜π(n) +γ += −π(n) +β . +If |β| ≤ |n|, the r. h. s. vanishes by (91), and by (94) the sum over +γ restricts to |γ| ≤ |β| ≤ |n|, and to γ ≺ β, so that the summand +vanishes by induction hypothesis. Thus also ˜π(n) +β +vanishes. +This allows us to argue that ˜Γ ∈ G associated to {˜π(n)}n is the inverse +of Γ. By the strict upper triangularity of Γ w. r. t. to the coercive ≺, +we already know that Γ is invertible, so that it suffices to show ˜ΓΓ = id, +which in turn follows from its transpose Γ∗˜Γ∗ = id. By the composition +rule (107) established above, Γ∗�Γ∗ is associated to {π(n) +Γ∗�π(n)}n. By +(108) we have that π(n) + Γ∗�π(n) = 0, and learn from Lemma 3 that id +is associated with 0. +We finally turn to the proof of (106). We note that β1 + · · · + βl = β +implies the componentwise βj ≤ β, which by (104) implies |βj|≺ ≤ |β|≺. +Since every γ with [γ] ≥ 0 can be written as the sum of γ’s of the form +γ = ek + en1 + · · · + enj +with +j ≤ k, +(109) +we learn from (96) that we may assume that γ is of this form. Once +more by (96) we have for these γ’s +(Γ∗)γ +β = +� +β0+···+βj=β +(Γ∗)ek +β0(Γ∗) +en1 +β1 · · · (Γ∗) +enj +βj . +From (98) & (99) we learn that this (Γ∗)γ +β is a linear combination of +π(0) +β′ +1 · · · π(0) +β′ +l (zn1 + π(n1))β1 · · · (znj + π(nj))βj, +(110) +where the multi-indices satisfy +β = ek+l + β′ +1 + · · · + β′ +l + β1 + · · · + βj. +(111) +We need to show that the product (110) contains only factors π(n) +β′ with +β′ ≺ β; w. l. o. g. we may assume l + j ≥ 1. To this purpose we apply +| · |≺ to (111); by (104) and |ek+l|≺ ≥ |ek+l| = α(1 + k + l) this implies +|β|≺ ≥ α(1 + k − j) + |β′ +1|≺ + · · · + |β′ +l|≺ + |β1|≺ + · · · + |βj|≺, which by +j ≤ k implies the desired |β′ +1|≺, . . . , |β′ +l|≺, |β1|≺, . . . , |βj|≺ < |β|≺. +□ +Finally, we show that the group G is large enough to contain the re- +expansion maps. + +LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES +31 +Lemma 5. There exists {π(n) +yx }n satisfying (91) such that the Γyx ∈ G +associated by Lemma 3 satisfies (70). +As a consequence of working with a larger group than in [14], see +Remark 1, we don’t have uniqueness of {π(n) +yx }n and thus of Γyx. We +refer the reader to [19] for a uniqueness result when working with the +smaller group. An inspection of our construction reveals transitivity in +line with [9, Definition 3.3] +Γ∗ +xyΓ∗ +yz = Γ∗ +xz +and +Γ∗ +xx = id, +see [15, Section 5.3] for the argument; it would also be a consequence +of uniqueness. +Proof of Lemma 5. We start by specifying π(n) +yxβ in the special cases of +n = 0 and of β = em for some m ̸= 0: +π(0) +yx := Πy(x), +(112) +π(n) +yxem := +� �m +n +� +(x − y)m−n +provided n < m, +0 +otherwise +� +for n ̸= 0, +(113) +where n < m means component-wise (non-strict) ordering and n ̸= m. +We note that (112) is necessary in order to bring the second item of +(70) into agreement with the form (92). We also remark that (113) +yields by (93) +(Γ∗ +yx)en +em = +� �m +n +� +(x − y)m−n +provided n ≤ m, +0 +otherwise +� +. +By the second part of (94), which implies (Γ∗ +yx)γ +0 = 0 unless γ = 0, by +(98) in form of (Γ∗ +yx)ek +em = 0, and via (96) this strengthens to +(Γ∗ +yx)γ +em = +� �m +n +� +(x − y)m−n +if γ = en with n ≤ m, +0 +otherwise +� +. +(114) +The latter is imposed upon us by taking the (β = em)-component +of the first item in (70) and plugging in (41). +The second part of +(114) implies that Γyx maps the linear span of {zm}m̸=0 into itself; +since this linear span can be identified with the space R[x1, x2]/R of +space-time polynomials (modulo constants), this can be assimilated to +Hairer’s postulate [9, Assumption 3.20]. We note that (112) and (113) +satisfy (91) because of | · | ≥ α > 0, cf. (104), and |em| = |m| > |n|, +respectively. In line with (48) and [14], we also set +π(n) +yxβ = 0 +unless +[β] ≥ 0 or β = em for some m ̸= 0. +(115) +It thus remains to construct π(n) +yxβ for n ̸= 0 and [β] ≥ 0, which we will +do by induction in β w. r. t. ≺. According to (106), we may consider +(Γ∗)γ +β as already constructed for [γ] ≥ 0. According to (64) and by the + +32 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +induction hypothesis (70), an inspection of the argument that leads +from there to (72) shows that we also have +Π− +yβ = (Γ∗ +yxΠ− +x )β. +(116) +The induction step consists in choosing {π(n) +yxβ}0<|n|<|β| such that +Πyβ = (Γ∗ +yxΠx)β + Πyβ(x) +(112) += (Γ∗ +yxΠx)β + π(0) +yxβ. +(117) +Denoting by P the projection on multi-indices γ with [γ] ≥ 0, so that +by (41) and (48) we have (id − P)Πx = � +n̸=0(· − x)nzn and thus by +(91) and (93) +(Γ∗ +yx(1 − P)Πx)β = +� +0<|n|<|β| +(· − x)nπ(n) +yxβ, +(118) +allows us to make {π(n) +yxβ}0<|n|<|β| in (117) explicit: +(Πy − Γ∗ +yxPΠx)β = +� +n:|n|<|β| +π(n) +yxβ(· − x)n. +(119) +Hence our task reads +(Πy − Γ∗ +yxPΠx)β = polynomial of degree < |β|. +(120) +According to the PDE (58), to (116), and to (118) we have +(∂2 − ∂2 +1)(Πy − Γ∗ +yxPΠx)β = polynomial of degree < |β| − 2. +(121) +In order to pass from (121) to (120), we will now appeal to the unique- +ness/Liouville statement in Lemma 1 with η = |β|, which is ̸∈ Z ac- +cording to (62) and ≥ α according to (104), and p = 1 for simplicity. +More precisely, we apply Lemma 1 to +u = (Πy − Γ∗ +yxPΠx)β − its Taylor polynomial in x of order < |β|, +which makes sense since (121) implies that (Πy − Γ∗ +yxPΠx)β is smooth, +and to f ≡ 0. Hence for the assumption (10) we need to check that +(122) +lim sup +z:|z−x|↑∞ +1 +|z − x||β|E|(Πy − Γ∗ +yxPΠx)β(z)| < ∞, +which forces us to now become semi-quantitative. +By the estimate (60) on Π, for (122) it remains to show34 +E +1 +p |(Γ∗ +yx)γ +β|p ≲β,γ,p |y − x||β|−|γ| +provided +[γ] ≥ 0. +(123) +In line with the language of [15], we split the argument for (123) into +an “algebraic argument”, where we derive (123) from +(124) +E +1 +p|π(n) +yxβ′|p ≲β′,p |x − y||β′|−|n| +for β′ ≺ β, +34which coincides with Hairer’s postulate [9, (3.2) in Definition 3.3] + +LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES +33 +and a “three-point argument”, where we derive (124) from the estimate +(60) on Π. +Here comes the argument for (123), which is modelled after the one +for (106) in Lemma 4. By H¨older’s inequality in probability and the +additivity of |·|−α, cf. (104), we may restrict to γ’s of the form (109). +We are thus lead to estimate the product (110), which now takes the +form of +π(0) +yxβ′ +1 · · · π(0) +yxβ′ +l(zn1 + π(n1) +yx )β1 · · · (znj + π(nj) +yx )βj. +(125) +Once again by H¨older’s inequality, we infer from (124) that the E +1 +p|·|p- +norm of (125) is +≲ |y − x||β′ +1| · · · |y − x||β′ +l||y − x||β1|−|n1| · · · |y − x||βj|−|nj|. +By the additivity of |·|−α, the total exponent of |y−x| can be identified +with the desired expression: +|β′ +1| + · · · + |β′ +l| + (|β1| − |n1|) + · · · + (|βj| − |nj|) +(111) += |β| − |ek+l| + (l + j)α − (|n1| + · · · + |nj|) +(109) += |β| − |γ|. +Finally, we give the “three-point argument” for the estimate (124), for +notational simplicity in case of the current multi-index β, so that we +now may use (119) and (123). By (60) and (123), the left hand side of +(119) can be estimated as follows +E +1 +p |(Πy − Γ∗ +yxPΠx)β(z)|p ≲β,p (|z − x| + |y − x|)|β|. +By the equivalence of norms on the finite-dimensional space of space- +time polynomials of degree < |β|, which by a duality argument can +be upgraded to the following estimate of annealed norms for random +polynomials +max +n: |n|<|β| |y − x||n| E +1 +p|π(n) +yxβ|p ≲ + +|z−x|≤|y−x| +dz E +1 +p�� +� +n: |n|<|β| +(z − x)nπ(n) +yxβ +��p, +we obtain (124). +□ +References +[1] H. Bahouri, J.-Y. Chemin, and R. Danchin. Fourier analysis and nonlinear +partial differential equations, volume 343. Springer, 2011. +[2] V. I. Bogachev. Gaussian measures, volume 62 of Mathematical Surveys and +Monographs. American Mathematical Society, Providence, RI, 1998. +[3] Y. Bruned, M. Hairer, and L. Zambotti. Algebraic renormalisation of regularity +structures, Invent. Math. 215(3):1039–1156, 2019. +[4] A. Chandra and M. Hairer. An analytic BPHZ theorem for Regularity Struc- +tures, arXiv:1612.08138, 2016. +[5] L. Coutin and Z. Qian. Stochastic analysis, rough path analysis and fractional +Brownian motions, Probability theory and related fields, 122(1):108–140, 2002. + +34 +FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR +[6] P. Duch. Renormalization of singular elliptic stochastic PDEs using flow equa- +tion, arXiv:2201.05031 [math.PR]. +[7] M. Gubinelli. Ramification of rough paths, J. Differential Equations 248 (2010), +no. 4, 693–721. +[8] M. Gubinelli and N. Perkowski. An introduction to singular SPDEs, Stochastic +partial differential equations and related fields, 69–99, Springer Proc. Math. +Stat., 229, Springer, Cham, 2018. +[9] M. Hairer. Regularity structures and the dynamical φ4 +3 model, arXiv:1508.05261 +[math.PR]. +[10] M. Hairer and ´E. Pardoux. A Wong-Zakai theorem for stochastic PDEs. J. +Math. Soc. Japan, 67(4):1551–1604, 2015. +[11] M. Josien and F. Otto. The annealed Calder´on-Zygmund estimate as conve- +nient tool in quantitative stochastic homogenization, J. Funct. Anal. 283 (2022), +no. 7, 74 pp. +[12] A. Kupiainen. Renormalization Group and Stochastic PDEs, Ann. Henri +Poincar´e 17, 497–535 (2016). +[13] P. Linares and F. Otto. A tree-free approach to regularity structures: the regular +case for quasi-linear equations, arXiv:2207.10627 [math.AP]. +[14] P. Linares, F. Otto and M. Tempelmayr. The structure group for quasi-linear +equations via universal enveloping algebras, arXiv:2103.04187 [math-ph]. +[15] P. Linares, F. Otto, M. Tempelmayr, and P. Tsatsoulis. A diagram-free ap- +proach to the stochastic estimates in regularity structures, arXiv:2112.10739 +[math.PR]. +[16] T. Lyons, M. Caruana and T. L´evy. Differential equations driven by rough +paths, Lecture Notes in Mathematics, 1908. Springer, Berlin, 2007. +[17] F. Otto, J. Sauer, S. Smith, and H. Weber. A priori estimates for quasi-linear +SPDEs in the full sub-critical regime, arXiv:2103.11039 [math.AP]. +[18] G. Scharf. Finite Quantum Electrodynamics: The Causal Approach, Second +version, Texts and Monographs in Physics, Springer Berlin, 1995. +[19] M. Tempelmayr. Characterizing models in regularity structures: a quasi-linear +case, to appear. +Felix Otto, Kihoon Seong, and Markus Tempelmayr +Max–Planck Institute for Mathematics in the Sciences +04103 Leipzig, Germany +felix.otto@mis.mpg.de, kihoon.seong@mis.mpg.de, +markus.tempelmayr@mis.mpg.de + diff --git a/AdAyT4oBgHgl3EQf3_pa/content/tmp_files/load_file.txt b/AdAyT4oBgHgl3EQf3_pa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1df47121aa14ec78075612b86b937cd81608692b --- /dev/null +++ b/AdAyT4oBgHgl3EQf3_pa/content/tmp_files/load_file.txt @@ -0,0 +1,843 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf,len=842 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='00778v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='PR] 2 Jan 2023 LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' These lecture notes are intended as reader’s digest of recent work on a diagram-free approach to the renormalized cen- tered model in Hairer’s regularity structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' More precisely, it is about the stochastic estimates of the centered model, based on Malli- avin calculus and a spectral gap assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We focus on a specific parabolic partial differential equation in quasi-linear form driven by (white) noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We follow a natural renormalization strategy based on preserving symmetries, and carefully introduce Hairer’s notion of a centered model, which provides the coefficients in a formal series expansion of a general solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We explain how the Malliavin derivative in conjunction with Hairer’s re-expansion map allows to reformulate this definition in a way that is stable under removing the small-scale regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' A few exemplary proofs are provided, both of analytic and of alge- braic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The working horse of the analytic arguments is an “annealed” Schauder estimate and related Liouville principle, which is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The algebra of formal power series, in variables that play the role of coordinates of the solution manifold, and its algebra morphisms are the key algebraic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Keywords: Singular SPDE, Regularity Structures, BPHZ renor- malization, Malliavin calculus, quasi-linear PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' MSC 2020: 60H17, 60L30, 60H07, 81T16, 35K59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' A singular quasi-linear SPDE 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Annealed Schauder theory 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Symmetry-motivated postulates on the form of the counter terms 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Algebrizing the counter term 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Algebrizing the solution manifold: The centered model 12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The main result: A stochastic estimate of the centered model 17 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Malliavin derivative and Spectral gap (SG) 19 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The structure group and the re-expansion map 26 1 2 FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR References 33 The theory of regularity structures by Hairer provides a systematic way to treat the small-scale divergences in singular semi-linear stochastic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Quintessential models of mathematical physics like the dynam- ical Φ4 3 model or the KPZ equation have been treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Inspired by Lyon’s theory of rough paths, this theory separates probabilistic and analytical aspects: Centered model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In a first probabilistic step, the coefficients of a local formal power series representation of a general solution of the renormalized PDE are constructed and estimated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' the co- efficients are indexed by (decorated) trees, and their stochastic estimate follows the diagrammatic approach to renormalization of quantum field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Modelled distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In a second analytical step, inspired by Gubinelli’s controlled rough path, the solution of a specific ini- tial value problem is found as a fixed point based on modulating and truncating the formal power series .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This step is purely de- terministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This automated two-pronged approach relies on an understanding of the algebraic nature of the re-expansion maps that allow to pass from one base-point to another in the local power series representation, in form of the “structure group”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The main progress of regularity struc- tures over the term-by-term treatment in the mathematical physics literature is that thanks to centering and re-expansion, the second step yields a rigorous (small data) well-posedness result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' As an introductory text to the theory of regularity structures we recommend [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In [17], motivated by the extension to a quasi-linear setting featuring a general non-linearity a(u), an alternative realization of Hairer’s reg- ularity structures was proposed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' it replaces trees with a more greedy index set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This index set of multi-indices naturally comes up when writing a general solution u as a functional of a, or rather as a func- tion of the coefficients of a in its power law expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In [17] it was established that any solution of the renormalized PDE can be locally approximated by a modelled distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This a-priori estimate was obtained under the assumption that the natural stochastic estimates on the centered model are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In [15] this program was continued: Based on scaling and other symme- tries, a canonical renormalization of the PDE and its centered model was proposed, and the centered model was stochastically constructed and estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' These notes present selected aspects of [15], providing additional motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' For a simpler setting where no renormalization and thus only purely deterministic estimates are needed, we recommend LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES 3 to also have a look at1 [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The algebraic aspects of the multi-index based regularity structures are worked out in [14], where in line with Hairer’s postulates the underlying Hopf-algebraic nature of the struc- ture group was uncovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In fact, the Hopf algebra arises from a Lie algebra generated by natural actions on the space of non-linearities a and solutions u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Other approaches to singular SPDEs include the theory of paracon- trolled distributions by Gubinelli, Imkeller, and Perkowski, we rec- ommend [8] for a first reading, and the renormalization group flow approach introduced by Kupiainen and generalized by Duch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' we rec- ommend [12] and [6] for an introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The para-controlled calculus provides an alternative to the separation into model and modelled dis- tribution, replacing localization in physical space-time by localization on the Fourier side;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' it is (typically) also indexed by trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The flow approach blends the stochastic and the deterministic step of regularity structures, and has an index set closer to multi-indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' While these alternative approaches might be more efficient in specific situations, they presumably lack the full flexibility of the two-pronged approach of regularity structures with its conceptual clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' A singular quasi-linear SPDE We are interested in nonlinear elliptic or parabolic equations with a random and thus typically rough right hand side ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Our approach is guided by moving beyond the well-studied semi-linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We con- sider a mildly quasi-linear case where the coefficients of the leading- order derivatives depend on the solution u itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' To fix ideas, we focus on the parabolic case in a single space dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' since we treat the parabolic equation in the whole space-time like an anisotropic ellip- tic equation, we denote by x1 the space-like and by x2 the time-like variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence we propose to consider (∂2 − ∂2 1)u = a(u)∂2 1u + ξ, (1) where we think of the values of a(u) to be such that the equation is parabolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We are interested in laws / ensembles of ξ where the solutions v to the linear equation (∂2 − ∂2 1)v = ξ (2) 1however,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' the setting in [13] is different in the sense that it imposes an artificial space-time periodicity: on the one hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' this allows to separate construction from estimation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' on the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' it obfuscates the quintessential scaling 4 FELIX OTTO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' KIHOON SEONG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' AND MARKUS TEMPELMAYR are (almost surely) H¨older continuous,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' where it will turn out to be convenient to express this in the “annealed” form2 of sup x̸=y 1 |y − x|αE 1 2|v(y) − v(x)|2 < ∞ (3) for some exponent α ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In view of the anisotropic nature of ∂2 − ∂2 1 and its invariance under the rescaling x1 = sˆx1 and x2 = s2ˆx2, H¨older continuity in (3) is measured w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' the Carnot-Carath´eodory distance “|y − x|” := 4� (y1 − x1)4 + (y2 − x2)2 ∼ |y1 − x1| + |y2 − x2| 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (4) By Schauder theory for ∂2 − ∂2 1, on which we shall expand on in Sub- section 2, this is the case for white noise ξ with α = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The rationale is that white noise has order of regularity −D 2 , where D is the effective dimension, which in case of (2) is D = 1 + 2 = 3 since in view of (4) the time-like variable x2 counts twice, and that (∂2 − ∂2 1)−1 increases regularity by two, leading to −D 2 + 2 = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In the range of α ∈ (0, 1), the SPDE (1) is what is called “singular”: We cannot expect that the order of regularity of u and thus a(u) is better than the one of v, which is α, and hence the order of regularity of ∂2 1u is no better than α − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Since α + (α − 2) < 0 for α < 1, the product a(u)∂2 1u cannot be classically/deterministically defined3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' As discussed at the end of Section 2, a renormalization is needed4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The same feature occurs for the (semi-linear) multiplicative heat equa- tion (∂2 −∂2 1)u = a(u)ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' in fact, our approach also applies to this semi- linear case, which already has been treated by (standard) regularity structures in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' A singular product is already present in the case when the x1-dependence is suppressed, so that the above semi-linear equation turns into the SDE du dx2 = a(u)ξ with white noise ξ in the time- like variable x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In this case, the analogue of v from (2) is Brownian motion, which is characterized by E(v(y2)−v(x2))2 = |y2−x2| and thus annealed H¨older exponent 1 2 in x2, which in view of (4) corresponds to the border-line setting α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Ito’s integral and, more recently, Lyons’ rough paths [16] and Gubinelli’s controlled rough paths [7] have been devised to tackle the issue in this SDE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 2Think of Brownian motion which satisfies E 1 2 (B(s) − B(t))2 = |s − t| 1 2 while not being H¨older continuous of exponent 1 2 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Following the jargon an- nealed/quenched from statistical mechanics models (which itself is borrowed from metallurgy), we speak of annealed norms when the inner norm is an Lp-norm w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' probability E and the outer norm is a space-time one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 3It is a classical result that the multiplication extends naturally from Cα × Cβ into D′ if and only if α + β > 0, see [1, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 4The range α > 1, while still subtle for α < 2, does not require a renormalization, see [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Annealed Schauder theory This section provides the main (linear) PDE ingredient for our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' At the same time, it will allow us to discuss (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In view of (2), we are interested in the fundamental solution of the differential operator A := ∂2 − ∂2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' It turns out to be convenient to use the more symmetric5 fundamental solution of the non-negative A∗A = (−∂2 −∂2 1)(∂2 −∂2 1) = ∂4 1 −∂2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Moreover, it will be more transparent to “disintegrate” the latter fundamental solution, by which we mean writing it as ´ ∞ 0 dtψt(z), where {ψt}t>0 are the kernels of the semi- group exp(−tA∗A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Clearly, the Fourier transform is given by Fψt(q) = exp(−t(q4 1 + q2 2)) (4) = exp(−t|q|4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (5) In particular, ψt is a Schwartz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' For a Schwartz distribution f like realizations of white noise, we thus define ft(y) as the pairing of f with ψt(y − ·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' ft is a smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' On the level of these kernels, the semi-group property translates into ψs ∗ ψt = ψs+t and ˆ ψt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (6) By construction, {ψt}t satisfies the PDE ∂tψt + (∂4 1 − ∂2 2)ψt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (7) By scale invariance of (7) under x1 = sˆx1, x2 = s2ˆx2, and t = s4ˆt, we have ψt(x1, x2) = 1 ( 4√ t)D=3 ψ1( x1 4√ t, x2 ( 4√ t)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (8) Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Let 0 < α ≤ η < ∞ with η ̸∈ Z, p < ∞, and x ∈ R2 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' For a random Schwartz distribution f with E 1 p|ft(y)|p ≤ ( 4√ t)α−2( 4√ t + |y − x|)η−α for all t > 0, y ∈ R2, (9) there exists a unique random function u of the class sup y∈R2 1 |y − x|η E 1 p|u(y)|p < ∞ (10) satisfying (distributionally in R2) (∂2 − ∂2 1)u = f + (polynomial of degree ≤ η − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (11) It actually satisfies (11) without the polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Moreover, the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' of (10) is bounded by a constant only depending on α and η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 5It is symmetric under reflection not just in space but also in time 6 FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR Now white noise ξ is an example of such a random Schwartz distri- bution: Since ξt(y) is a centered Gaussian, we have E 1 p|ξt(y)|p ≲p E 1 2(ξt(y))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' By using the characterizing property of white noise in terms of its pairing with a test function ζ E(ξ, ζ)2 = ˆ ζ2, (12) we have E 1 2(ξt(y))2 = � ´ ψ2 t (y − ·) � 1 2, which by scaling (8) is equal to ( 4√ t)− D 2 ( ´ ψ2 1) 1 2 ∼ ( 4√ t)− D 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This specifies the sense in which white noise ξ has order of regularity −D 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Fixing a “base point” x, Lemma 1 thus constructs the solution of (2) distinguished by v(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Note that the output (10) takes the form of E 1 p|v(y)−v(x)|p ≲p |y−x| 1 2, which extends (3) from p = 2 to general p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence Lemma 1 provides an annealed version of a Schauder estimate, alongside a Liouville-type uniqueness result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' By construction, ´ ∞ 0 dt(−∂2 − ∂2 1)ψt is the funda- mental solution of ∂2 − ∂2 1, so that we take the convolution of it with f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' However, in order to obtain a convergent expression for t ↑ ∞, we need to pass to a Taylor remainder: u = ˆ ∞ 0 dt(id − Tη x)(−∂2 − ∂2 1)ft, (13) where Tη x is the operation of taking the Taylor polynomial of order ≤ η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' as we shall argue the additional Taylor polynomial does not affect the PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We claim that (13) is well-defined and estimated as E 1 p|u(y)|p ≲ |y − x|η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' To this purpose, we first note that E 1 p|∂nft(y)|p ≲ ( 4√ t)α−2−|n|( 4√ t + |y − x|)η−α, (14) where ∂nf := ∂n1 1 ∂n2 2 f and |n| = n1 + 2n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (15) Indeed, by the semi-group property (6) we may write ∂nft(y) = ´ dz ∂nψ t 2(y−z) f t 2(z), so that E 1 p|∂nft(y)|p ≤ ´ dz|∂nψ t 2(y−z)|E 1 p|f t 2(z)|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence by (9), (14) follows from the kernel bound ´ dz |∂nψ t 2(y − z)| ( 4√ t + |y − x|)η−α ≲ ( 4√ t)−|n|( 4√ t + |y − x|)η−α, which itself is a conse- quence of the scaling (8) and the fact that ψ 1 2 is a Schwartz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Equipped with (14), we now derive two estimates for the integrand of (13), namely for 4√ t ≥ |y − x| (“far field”) and for 4√ t ≤ |y − x| (“near field”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We write the Taylor remainder (id − Tη x)(∂2 + ∂2 1)ft(y) LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES 7 as a linear combination of6 (y − x)n∂n(∂2 + ∂2 1)ft(z) with |n| > η and at some point z intermediate to y and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' By (14) such a term is estimated by |y − x||n|( 4√ t)α−4−|n|( 4√ t + |y − x|)η−α, which in the far field is ∼ |y − x||n|( 4√ t)η−4−|n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Since the exponent on t is < −1, we obtain as desired E 1 p| ˆ ∞ |y−x|4 dt(id − Tη x)(∂2 + ∂2 1)ft(y)|p ≲ |y − x|η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' For the near-field term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' for 4√ t ≤ |y − x|, we proceed as follows: E 1 p |(id − Tη x)(∂2 + ∂2 1)ft(y)|p ≤ E 1 p|(∂2 + ∂2 1)ft(y)|p + � |n|≤η |y − x||n|E 1 p|∂n(∂2 + ∂2 1)ft(x)|p (14) ≲ ( 4√ t)α−4|y − x|η−α + � |n|≤η |y − x||n|( 4√ t)η−4−|n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Since η is not an integer, the sum restricts to |n| < η, so that all exponents on t are > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence we obtain as desired E 1 p| ˆ |y−x|4 0 dt(id − Tη x)(∂2 + ∂2 1)ft(y)|p ≲ |y − x|η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' It can be easily checked that (13) is indeed a solution of (11), even without a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' For a detailed proof we refer to [15, Proposi- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We turn to the uniqueness of u in the class (10) satisfying (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Given two such solutions u1, u2, we observe that ¯u := u1 − u2 satisfies (10) and (11) with f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In particular ∂n(∂2 − ∂2 1)¯u = 0 for |n| > η − 2, and thus from (7) we obtain ∂t∂n¯ut = 0 provided |n| > η − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Thus, ∂n¯ut is independent of t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Moreover, (10) implies that E|∂n¯ut| → 0 as t → ∞ for |n| > η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence we learn from t → 0 that ∂n¯u = 0 for |n| > η, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' ¯u is a polynomial of degree ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Since η ̸∈ Z this strengthens to ¯u is a polynomial of degree < η, and by (10) it vanishes at x to order η which yields the desired ¯u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' □ We return to the discussion of the singular product a(u)∂2 1u, in its simplest form of v∂2 1v = ∂2 1 1 2v2 − (∂1v)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' While in view of Lemma 1 the first r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' term is well-defined as a random Schwartz distribution, we now argue that the second term 6where xn := xn1 1 xn2 2 8 FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Indeed, applying ∂1 to the representation formula (13), so that the constant Taylor term drops out, we have ∂1v = ˆ ∞ 0 dt∂1(−∂2 − ∂2 1)ξt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (16) Hence for space-time white noise E(∂1v(x))2 (16) = ˆ ∞ 0 dt ˆ ∞ 0 ds E � ∂1(−∂2 − ∂2 1)ξt(x)∂1(−∂2 − ∂2 1)ξs(x) � (12) = ˆ ∞ 0 dt ˆ ∞ 0 ds ˆ R2 dy ∂1(−∂2 − ∂2 1)ψt(x − y)∂1(−∂2 − ∂2 1)ψs(x − y) (6) = ˆ ∞ 0 dt ˆ ∞ 0 ds ∂2 1(∂2 2 − ∂4 1)ψs+t(0) (8) ∼ ˆ ∞ 0 dt ˆ ∞ 0 ds 4√ t + s −D−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Note that since 1 4(−D−6) < −2 for D = 3, the double integral diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This divergence arises from t ↓ 0 and s ↓ 0, that is, from small space- time scales, and thus is called an ultra-violet (UV) divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' A quick fix is to introduce an UV cut-off, which for instance can be implemented by mollifying ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Using the semi-group convolution ξτ specifies the UV cut-off scale to be of the order of 4√τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' It is easy to check that in this case E(∂1v(x))2 ∼ ˆ ∞ τ dt ˆ ∞ τ ds 4√ t + s −D−6 ∼ ( 4√τ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The goal is to modify the equation (1) by “counter terms” such that the solution manifold stays under control as the ultra-violet cut-off τ ↓ 0, invariances of the solution manifold are preserved i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' the solu- tion manifold keeps as many symmetries as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In view of the above discussion, we expect the coefficients of the counter terms to diverge as the cut-off tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Symmetry-motivated postulates on the form of the counter terms In view of α ∈ (0, 1), u is a function while we think of all derivatives ∂nu as being only Schwartz distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence it is natural to start from the very general Ansatz that the counter term is a polynomial in {∂nu}n̸=0 with coefficients that are general (local) functions in u: (∂2 − ∂2 1)u + � β hβ(u) � n̸=0 ( 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='∂nu)β(n) = a(u)∂2 1u + ξ, (17) LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES 9 where β runs over all multi-indices7 in n ̸= 0 and n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' := (n1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=')(n2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' For simplicity of this heuristic discussion, we drop the regularization on ξ and don’t index the counter term with τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Only counter terms that have an order strictly below the order of the leading ∂2 − ∂2 1 are desirable, so that one postulates that the sum in (17) restricts to those multi-indices for which |β|p := � n̸=0 |n|β(n) < 2 where |n| := n1 + 2n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (18) This leaves only β = 0 and β = e(1,0), where the latter means β(n) = δ(1,0) n , so that (17) collapses to (∂2 − ∂2 1)u + h(u) + h′(u)∂1u = a(u)∂2 1u + ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (19) One also postulates that h and h′ depend on the noise ξ only through its law / distribution / ensemble, hence are deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Since we assume that the law is invariant under space-time translation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' is stationary, it was natural to postulate that h and h′ do not explicitly depend on x, hence are homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Reflection symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Let us now assume that the law of ξ is invariant under space-time translation y �→ y + x, space reflection y �→ (−y1, y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (20) We now argue that under this assumption, it is natural to postulate that the term h′(u)∂1u in (19) is not present, so that we are left with (∂2 − ∂2 1)u + h(u) = a(u)∂2 1u + ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (21) To this purpose, let x ∈ R2 be arbitrary yet fixed, and consider the reflection at the line {y1 = x1} given by Ry = (2x1 − y1, y2), which by pull back acts on functions as ˜u(y) := u(Ry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Since (1) features no explicit y-dependence, and only involves even powers of ∂1, which like ∂2 commute with R, we have (u, ξ) satisfies (1) =⇒ (u(R·), ξ(R·)) satisfies (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (22) Since we postulated that h and h′ depend on ξ only via its law, and since in view of the assumption (20), ˜ξ = ξ(R·) has the same law as ξ, it is natural to postulate that the symmetry (22) extends from (1) to (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Spelled out, this means that (19) implies (∂2 − ∂2 1)˜u + h(˜u) + h′(˜u)∂1˜u = a(˜u)∂2˜u + ˜ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Evaluating both identities at y = x, and taking the difference, we get for any solution of (19) that h′(u(x))∂1u(x) = h′(u(x))(−∂1u(x)), and thus h′(u(x))∂1u(x) = 0, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 7which associate to every index n a β(n) ∈ N0 such that β(n) vanishes for all but finitely many n’s 10 FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR Covariance under u-shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We now come to our most crucial pos- tulate, which restricts how the nonlinearity h depends on the non- linearity / constitutive law a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence we no longer think of a single nonlinearity a, but consider all non-linearities at once, in the spirit of rough paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This point of view reveals another invariance of (1), namely for any shift v ∈ R (u, a) satisfies (1) =⇒ (u − v, a(· + v)) satisfies (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (23) A priori, h is a function of the u-variable that has a functional de- pendence on a, as denoted by h = h[a](u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We postulate that the symmetry (23) extends from (1) to (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This is the case provided we have the following shift-covariance property h[a](u + v) = h[a(· + v)](u) for all u ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (24) This property can also be paraphrased as: Whatever algorithm one uses to construct h from a, it should not depend on the choice of origin in what is just an affine space R ∋ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Property (24) implies that the counter term is determined by a functional c = c[a] on the space of nonlinearities a: h[a](v) = c[a(· + v)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (25) Renormalization now amounts to choosing c such that the solution manifold stays under control as the UV regularization of ξ fades away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Algebrizing the counter term In this section, we algebrize the relationship between a and the counter term h given by a functional c as in (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' To this purpose, we introduce the following coordinates8 on the space of analytic functions a of the variable u: zk[a] := 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' dka duk (0) for k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (26) These are made such that by Taylor’s a(u) = � k≥0 ukzk[a] for a ∈ R[u], (27) where R[u] denotes the algebra of polynomials in the single variable u with coefficients in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We momentarily specify to functionals c on the space of analytic a’s that can be represented as polynomials in the (infinitely many) vari- ables {zk}k≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This leads us to consider the algebra R[zk] of polynomials in the variables zk with coefficients in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The monomials zβ := � k≥0 zβ(k) k (28) 8where here and in the sequel k ≥ 0 stands short for k ∈ N0 LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES 11 form a basis of this (infinite dimensional) linear space, where β runs over all multi-indices9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence as a linear space, R[zk] can be seen as the direct sum over the index set given by all multi-indices β, and we think of c as being of the form c[a] = � β cβzβ[a] for c ∈ R[zk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (29) Infinitesimal u-shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Given a shift v ∈ R, for ˜u := u − v and ˜a := a(· + v) we have ˜a(˜u) = a(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This leads us to study the mapping a �→ a(· + v) which provides an action/representation of the group R ∋ v on the set R[u] ∋ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Note that for c ∈ R[zk] and a ∈ R[u], the function R ∋ v �→ c[a(·+v)] = � β cβ � k≥0( 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' dka du (v))β(k) is polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Thus (D(0)c)[a] = d dv |v=0c[a(· + v)] (30) is well-defined, linear in c and even a derivation10, meaning that Leib- niz’ rule holds (D(0)cc′) = (D(0)c)c′ + c(D(0)c′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (31) The latter implies that D(0) is determined by its value on the co- ordinates zk, which by definitions (26) and (30) is given by D(0)zk = (k + 1)zk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence D(0) has to agree with the following derivation on the algebra R[zk] D(0) = � k≥0 (k + 1)zk+1∂zk, (32) which is well defined since the sum is effectively finite when applied to a monomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Representation of counter term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Iterating (30) we obtain by induction in l ≥ 0 for c ∈ R[zk] and a ∈ R[u] dl dvl |v=0c[a(· + v)] = ((D(0))lc)[a] and thus by Taylor’s (recall that v �→ c[a(· + v)] is polynomial) c[a(· + v)] = � � l≥0 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='vl(D(0))lc � [a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (33) 9which means they associate a frequency β(k) ∈ N0 to every k ≥ 0 such that all but finitely many β(k)’s vanish 10the index (0) is not necessary for these lecture notes, since we do not appeal to the other derivations {D(n)}n̸=0 from [14, 15], we keep it here for consistency with these papers 12 FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR We combine (33) with (25) to obtain the representation h[a](v) = � � l≥0 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='vl(D(0))lc � [a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (34) Hence our goal is to determine the coefficients {cβ}β in (29), which typically will blow up as τ ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Algebrizing the solution manifold: The centered model The purpose of this section is to motivate the notion of a centered model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' the motivation will be in parts formal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Parameterization of the solution manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' If a ≡ 0 it follows from (24) that h is a (deterministic) constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We learned from the discussion after Lemma 1 that – given a base point x – there is a distinguished solution v (with v(x) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence we may canonically parameterize a general solution u of (21) via u = v + p, by space- time functions p with (∂2 − ∂2 1)p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Such p are necessarily analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Having realized this, it is convenient11 to free oneself from the constraint (∂2 − ∂2 1)p = 0, which can be done at the expense of relaxing (21) to (∂2 − ∂2 1)v = ξ + q for some analytic space-time function q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (35) Since we think of ξ as being rough while q is infinitely smooth, this relaxation is still constraining v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The implicit function theorem suggests that this parameterization (lo- cally) persists in the presence of a sufficiently small analytic nonlin- earity a: The nonlinear manifold of all space-time functions u that satisfy (∂2 − ∂2 1)u + h(u) = a(u)∂2 1u + ξ + q for some analytic space-time function q (36) is still parameterized by space-time analytic functions p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We now return to the point of view of Section 3 of considering all nonlinearities a at once, meaning that we consider the (still nonlinear) space of all space- time functions that satisfy (36) for some analytic nonlinearity a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We want to capitalize on the symmetry (23), which extends from (1) to (21) and to (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We do so by considering the above space of u’s modulo constants, which we implement by focusing on increments u − u(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Summing up, it is reasonable to expect that the space of all space-time functions u, modulo space-time constants, that satisfy (36) for some analytic nonlinearity a and space-time function q (but at fixed ξ), is parameterized by pairs (a, p) with p(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 11otherwise, the coordinates z(2,0) and z(0,1) defined in (38) would be redundant on p-space LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES 13 Formal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In line with the term-by-term ap- proach from physics, we write the increment u(y)−u(x) as a (typically divergent) power series u(y) − u(x) = � β Πxβ(y) � k≥0 � 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' dka duk (u(x)) �β(k) � n̸=0 � 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='∂np(x) �β(n), (37) where β runs over all multi-indices in k ≥ 0 and n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Introducing coordinates on the space of analytic space-time functions p with p(0) = 0 via12 zn[p] = 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='∂np(0) for n ̸= 0, (38) (37) can be more compactly written as u(y) = u(x) + � β Πxβ(y)zβ[a(· + u(x)), p(· + x) − p(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (39) This is reminiscent of Butcher series in the analysis of ODE discretiza- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Recall from above that for a ≡ 0 we have the explicit parameterization u − u(x) = v + p (40) with the distinguished solution v of the linear equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence from setting a ≡ 0 and p ≡ 0 in (37), we learn that Πx0 = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' From keeping a ≡ 0 but letting p vary we then deduce that for all multi-indices β ̸= 0 which satisfy β(k) = 0 for all k ≥ 0 we must have13 Πxβ(y) = � (y − x)n provided β = en 0 else � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (41) Hierarchy of linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The collection of coefficients {Πxβ(y)}β from (39) is an element of the direct product with the same index set as the direct sum R[zk, zn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence the direct product inherits the multiplication of the polynomial algebra (ππ′) ¯β = � β+β′= ¯β πβπ′ β′, (42) and is denoted as the (well-defined) algebra R[[zk, zn]] of formal power series;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' we denote by 1 its unit element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We claim that in terms of (39), (36) assumes the form of (∂2 − ∂2 1)Πx = Π− x up to space-time analytic functions (43) 12where here and in the sequel n ̸= 0 stands short for n ∈ N2 0 − {(0, 0)} 13where we recall that β = en denotes the multi-index with β(m) = δn m next to β(k) = 0 14 FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR where Π− x := � k≥0 zkΠk x∂2 1Πx − � l≥0 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='Πl x(D(0))lc + ξτ1, (44) as an identity in formal power series in zk, zn with coefficients that are continuous space-time functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We shall argue below that (44) is effectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' componentwise, well-defined despite the two infinite sums, and despite extending from c ∈ R[zk] to c ∈ R[[zk]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Moreover, as will become clear by (64), the β-component of (44) contains on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' only terms Πxβ′ for “preceding” multi-indices β′ – hence (43) describes a hierarchy of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Here comes the formal argument that relates {∂2, ∂2 1}u, a(u), and h(u), to {∂2, ∂2 1}Πx[˜a, ˜p], (� k≥0 zkΠk x)[˜a, ˜p], and (� l≥0 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='Πl x(D(0))lc)[˜a, ˜p], re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Here we have set for abbreviation ˜a = a(· + u(x)) and ˜p = p(· + x) − p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' It is based on (39), which can be compactly written as u(y) = u(x) + Πx[˜a, ˜p](y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence the statement on {∂2, ∂2 1}u follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Together with a(u(y)) = ˜a(u(y) −u(x)), this also implies by (27) the desired a(u(y)) = � � k≥0 zkΠk x(y) � [˜a, ˜p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Likewise by (24), we have h[a](u(y)) = h[˜a](u(y) − u(x)), so that by (34), we obtain the desired h[a](u(y)) = � � l≥0 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='Πl x(y)(D(0))lc � [˜a, ˜p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Finiteness properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The next lemma collects crucial algebraic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The derivation D(0) extends from R[zk] to R[[zk]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Moreover, for π, π′ ∈ R[[zk, zn]], c ∈ R[[zk]], and ξ ∈ R, π− := � k≥0 zkπkπ′ − � l≥0 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='πl(D(0))lc + ξ1 ∈ R[[zk, zn]] (45) is well-defined, in the sense that the two sums are componentwise finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Finally, for [β] := � k≥0 kβ(k) − � n̸=0 β(n) (46) LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES 15 we have the implication πβ = π′ β = 0 unless [β] ≥ 0 or β = en for some n ̸= 0 =⇒ π− β = 0 unless \uf8f1 \uf8f2 \uf8f3 [β] ≥ 0 or β = ek + en1 + · · · + enk+1 for some k ≥ 1 and n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' , nk+1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (47) We note that for β as in the second alternative on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' of (47), it follows from (41) that Π− xβ is a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence in view of the modulo in (43), we learn from (47) that we may assume Πxβ ≡ 0 unless [β] ≥ 0 or β = en for some n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (48) Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We first address the extension of D(0) and note that from (32) we may read off the matrix representation of D(0) ∈ End(R[zk]) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (28) given by (D(0))γ β = (D(0)zγ)β (32) = � k≥0 (k + 1) � zk+1∂zkzγ� β (28) = � k≥0 (k + 1)γ(k) � 1 provided γ + ek+1 = β + ek 0 otherwise � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (49) From this we read off that {γ|(D(0))γ β ̸= 0} is finite for every β, which implies that D(0) naturally extends from R[zk] to R[[zk]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' With help of (42) the derivation property (31) can be expressed coordinate-wise, and thus extends to R[[zk]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We now turn to (45), which component-wise reads π− β = � k≥0 � ek+β1+···+βk+1=β πβ1 · · · πβkπ′ βk+1 − � l≥0 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' � β1+···+βl+1=β πβ1 · · · πβl((D(0))lc)βl+1 + ξδ0 β, (50) and claim that the two sums are effectively finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' For the first term of the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' this is obvious since thanks to the presence of14 ek in ek + β1 + · · · + βk+1 = β, for fixed β there are only finitely many k ≥ 0 for which this relation can be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In preparation for the second r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' term of (50) we now establish that ((D(0))l)γ β = 0 unless [β]0 = [γ]0 + l, (51) where we introduced the scaled length [γ]0 := � k≥0 kγ(k) ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The argument for (51) proceeds by induction in l ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' It is tautological for the base case l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In order to pass from l to l + 1 we write 14γ = ek denotes the multi-index with γ(l) = δk l next to γ(n) = 0 16 FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR ((D(0))l+1)γ β = � β′((D(0))l)β′ β (D(0))γ β′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' by induction hypothesis, the first factor vanishes unless [β]0 = [β′]0 + l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We read off (49) that the second factor vanishes unless [β′]0 = [γ]0 + 1, so that the product vanishes unless [β]0 = [γ]0 + (l + 1), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Equipped with (51) we now turn to the second r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' term of (50) and note that ((D(0))lc)βk+1 vanishes unless l ≤ [βk+1]0 ≤ [β]0, which shows that also here, only finitely many l ≥ 0 contribute for fixed β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We turn to the proof of (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We use (50) and give the proof for every summand separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' For the first term on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' of (50) we obtain by additivity of [·] that [β] = k+[β1]+· · ·+[βk+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Note that πβi is only non vanishing if [βi] ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' If at least one of the β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' , βk+1 satisfies [βi] ≥ 0, we obtain therefore [β] ≥ k−k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' For the second r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' term in (50) we appeal to (51): Since D(0) doesn’t affect the zn components, (51) extends from [·]0 to [·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Together with c ∈ R[[zk]] this yields [βl+1] ≥ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence as above [β] = [β1]+· · ·+[βl+1] ≥ −l+[βl+1] ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' □ Homogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We return to a heuristic discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Provided we include, like for (23), a into our considerations, the original equation (1) has a scaling symmetry: Considering for s ∈ (0, ∞) the parabolic space-time rescaling Sy = (sy1, s2y2), we have for any exponent α (u, ξ, a) satisfies (1) =⇒ � s−αu(S·), s2−αξ(S·), a(sα·) � =: (˜u, ˜ξ, ˜a) satisfies (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (52) Suppose the scaling transformation ξ �→ ˜ξ preserves the law, which for white noise is the case with α − 2 = −D 2 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' α = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Since in view of Section 3, the counter term only depends on the law, it is natural to postulate, in line with that section, that the solution manifold of the renormalized problem inherits this invariance15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' It is also natural to postulate that the parameterization by the p’s (given a base point x) is consistent with (52) in the sense that p trans- forms as u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' we have invariance under (u, ξ, a, x, p) �→ (˜u, ˜ξ, ˜a, ˜x := S−1x, ˜p := s−αp(S·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We now appeal to the series expansion (37), both as it stands and with (x, y, u, ξ, a, p) replaced by (˜x, ˜y := S−1y, ˜u, ˜ξ, ˜a, ˜p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Because of u(y) − u(x) = sα(˜u(˜y) − ˜u(˜x)), we obtain a relation between the two right-hand sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' It is natural to postulate that the coefficients {Π·,β}β are individually consistent with this invariance, leading to ΠSxβ[ξ](Sy) = s|β|Πxβ[s2−αξ(S·)](y), (53) 15since this scale invariance in law is not consistent with the mollification ξτ this discussion pertains to the limiting solution manifold LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES 17 where the “homogeneity” |β| of the multi-index β is given by |β| := α(1 + [β]) + |β|p, (54) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (18) and (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We note that |en| = |n| (55) so that (54) is consistent with (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Appealing once more to the invariance in law of ξ under (52), we obtain from (53) the law of s−|β|ΠSx β(Sy) does not depend on s ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (56) By the invariance of the (original) solution manifold under (u, ξ) �→ (˜u := u(·+z), ˜ξ := ξ(·+z)), which by our assumption (20) is passed on to the renormalized solution manifold, it is natural to impose that the parameterization is invariant under (u, ξ, x, p) �→ (˜u, ˜ξ, x + z, p(· + z)), and that the coefficients in (39) are individually consistent with this invariance, so that we likewise have the law of Πx+z β(y + z) does not depend on z ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (57) Specifying to x = 0, the invariance (56) implies that E 1 p|Π0β(y)|p de- pends on y only through y |y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' From the invariance (57) we thus learn that E 1 p |Πxβ(y)|p depends on x, y only through y−x |y−x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Since y−x |y−x| has compact range, this suggest that E 1 p |Πxβ(y)|p ≲ |y − x||β|, which is our main result, see (60) in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The scaling invariance (52) also connects to the notion of “subcritical- ity” which is often referred to in the realm of singular SPDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Loosely speaking, it means that by zooming in on small scales, the nonlinear term becomes negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Indeed, as can be seen from (52), the rescaled nonlinearity ˜a converges to the constant a(0) in the limit s ↓ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' the SPDE (1) turns into a linear one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This is true iff α > 0, and provides the reason for restricting to α > 0 in the assumption of Theorem 1, which is the sub-critical regime for (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The main result: A stochastic estimate of the centered model Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Suppose the law of ξ is invariant under (20);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' suppose that it satisfies a spectral gap inequality (87) with exponent α ∈ (max{0, 1− D 4 }, 1) \\ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Then given τ > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' there exists a deterministic c ∈ R[[zk]],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' and for every x ∈ R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' a random16 Πx ∈ C2[[zk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' zn]],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' and a random Π− x ∈ C0[[zk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' zn]] 16by this we mean a formal power series in zk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' zn with values in the twice con- tinuously differentiable space-time functions 18 FELIX OTTO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' KIHOON SEONG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' AND MARKUS TEMPELMAYR that are related by (44) and (∂2 − ∂2 1)Πxβ = Π− xβ + polynomial of degree ≤ |β| − 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (58) and that satisfy (41),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' the population condition (48) and cβ = 0 unless |β| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (59) Moreover, we have the estimates E 1 p|Πxβ(y)|p ≲β,p |y − x||β|, (60) E 1 p|Π− xβt(y)|p ≲β,p ( 4√ t)α−2( 4√ t + |y − x|)|β|−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (61) The important feature is that the constants in (60) and (61) are uniform in τ ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We remark that we may pass from (61) to (60) by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Indeed, because of (48) we may restrict to β with [β] ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In this case, by our assumption α ̸∈ Q, [β] ≥ 0 (54) =⇒ |β| ̸∈ Z, (62) next to |β| ≥ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence we may indeed apply Lemma 1 with η = |β| and (61) as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The output yields a Πxβ satisfying (58) and (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Uniqueness and (implicit) BPHZ renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The con- struction of Πx in [15] proceeds by an inductive algorithm in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The ordering17 on the multi-indices is provided by (63) |β|≺ := |β| + λβ(0) for fixed λ ∈ (0, α), and we will write γ ≺ β for |γ|≺ < |β|≺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' As opposed to the ordering provided by the homogeneity, ≺ allows for the triangular structure: (64) Π− xβ − cβ depends on (Πxγ, cγ) only through γ with γ ≺ β, which can be easily checked on the component-wise level (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' More- over, (63), as opposed to the ordering by homogeneity, is coercive: For fixed β there are only finitely many γ with γ ≺ β, see (101), which is important for the estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We now argue that within this induction, (c, Πx, Π− x ) is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Indeed, the uniqueness statement of Lemma 1 implies that for given β, Πxβ is determined by Π− xβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' According to (64), Π− xβ − cβ is determined by the previous steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Finally, we note that provided |β| < 2, we have |EΠ− xβt(x)| ≤ E|Π− xβt(x)| (61) ≲ ( 4√ t)|β|−2 t↑∞ → 0, (65) 17this ordering coincides with the one chosen in [13] but it slightly differs from the one in [15], which is imposed by the restricted triangularity of dΓ∗ in Section 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' for simplicity we stick to (63) LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES 19 so that cβ, because it is deterministic18 may be recovered from cβ = − limt↑∞ E(Π− xβ − cβ)t(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence also cβ is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Fixing the counter term by making an expectation19 vanish like in (65) corre- sponds to what Hairer assimilates to a BPHZ renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' See [3, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='18] for the form BPHZ renormalization takes within regu- larity structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Mission accomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Returning to the end of Section 2, we may claim “mission accomplished”: On the one hand, the form of the counter terms preserve a number of symmetries of the original solution manifold: shift in x, reflection in x1, shift in u, and to some extend are guided by scaling in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' On the other hand, in a term-by-term sense as encoded by (37), the solution manifold of the renormalized equation stays under control as τ ↓ 0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (60) and (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Moreover, the constants cβ = cτ β that determine the counter term via (34) are (canonically) determined by the large-scale part of the estimate (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' As discussed in the introduction, the connection between this term-by- term approach to the solution manifold and the solution of an actual initial/boundary value problem is provided by the second part of reg- ularity structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This second part, a fixed point argument based on a truncation of (37) to a finite sum20, is not addressed in these lecture notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Malliavin derivative and Spectral gap (SG) In view of the discussion at the end of the statement of Theorem 1, the main issue is the estimate (61) of Π− xβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Indeed, its definition of (44) still contains the singular product Πk x∂2 1Πx and the collection of deterministic constants c that diverge as the UV regularization fades away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence we seek a relation between Π− x and Πx that is more stable than (44);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' in fact, it will be a relation between the families {Π− x }x and {Πx}x based on symmetries under a change of the base point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This relation is formulated on the level of the derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' noise ξ, also known as the Malliavin derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We start by motivating this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Heuristic discussion of a stable relation {Πx}x �→ {Π− x }x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Let δ denote the operation of taking the derivative of an object like Πxβ(y), 18and independent of the base point x 19in our case it is a space-time next to an ensemble average 20by restricting to homogeneities |β| < 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' in our quasi-linear case, the sum stays infinite w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' the z0-variable, but one has analyticity in that variable since 1 + z0 plays the role of a constant elliptic coefficient 20 FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR which is a functional of ξ, in direction of an infinitesimal variation δξ of the latter21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Clearly, since cβ is deterministic, we have δcβ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' However, applying δ to (a component of) (44) does not eliminate c because of the specific way c enters (44), which is dictated by the fundamental symmetry (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' However, when evaluating (44) at the base point x itself and appealing to the built-in (66) Πx(x) = 0, see (37) or (60), it collapses to Π− x (x) = z0∂2 1Πx(x) − c + ξτ(x)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (67) This isolates c so that it can be eliminated by applying δ: δΠ− x (x) = z0∂2 1δΠx(x) + δξτ(x)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (68) Clearly, (68) is impoverished in the sense that the active point coincides with the base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Instead of attempting to modify the active point, the idea is to modify the base point from x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Such a change of base point, which will be rigorously introduced in Section 8, amounts to a change of coordinates in the heuristic representation (39): u = � u(x) + � β Πxβzβ[a(· + u(x)), px], u(y) + � β Πyβzβ[a(· + u(y)), py], (69) for some polynomials px, py vanishing at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The form in which the u-shift appears in (69) suggests that this change of coordinates can be algebrized by an algebra endomorphism22 Γ∗ yx of R[[zk, zn]] with the properties Πy = Γ∗ yxΠx + Πy(x) and Γ∗ yx = � l≥0 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='Πl y(x)(D(0))l on R[[zk]], (70) see the discussion of finite u-shifts around (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Recall that an alge- bra endomorphism Γ∗ yx is a linear map from R[[zk, zn]] to R[[zk, zn]] satisfying (71) Γ∗ yxππ′ = (Γ∗ yxπ)(Γ∗ yxπ′) for π, π′ ∈ R[[zk, zn]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We claim that (70) implies Π− y = Γ∗ yxΠ− x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (72) 21in the Gaussian case, this would be an element of the Cameron-Martin space 22in a first reading, the star should be seen as mere notation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Γ∗ yx is actually the algebraic dual of a linear endomorphism Γyx on the pre-dual space, see Lemma 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' it is Γyx that can be assimilated to the object denoted by the same symbol in regularity structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' for a concise reference see [14, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='3] LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES 21 Indeed, applying Γ∗ yx to definition (44) we obtain by (71) Γ∗ yxΠ− x = � k≥0 (Γ∗ yxzk)(Γ∗ yxΠx)k∂2 1Γ∗ yxΠx − � l≥0 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (Γ∗ yxΠx)lΓ∗ yx(D(0))lc + ξτ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We substitute Γ∗ yxΠx according to the first item in (70), substitute Γ∗ yxzk = � l≥0 �k+l k � Πl y(x)zk+l and Γ∗ yx(D(0))lc according to the second item in (70) and definition (32), and finally appeal to the binomial formula in both ensuing double sums to obtain (44) with x replaced by y, establishing (72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In view of the scaling (56) and the transformation (70) we expect that the laws of s|β|−|γ|(Γ∗ yx)γ β and of (Γ∗ SySx)γ β to be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' On the other hand, we expect (Γ∗ SySx)γ β to converge to (Γ∗ 00)γ β as s ↓ 0, and we expect Γ∗ 00 to be the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This suggests strict triangularity: (Γ∗ yx − id)γ β = 0 unless |γ| < |β|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (73) We claim that applying Γ∗ yx to (68), we obtain23 δΠ− y (x) − (δΓ∗ yx)Π− x (x) = � k≥0 zkΠk y(x)∂2 1 � δΠy − δΠy(x) − (δΓ∗ yx)Πx � (x) + δξτ(x)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (74) Since by (73), δΓ∗ yx is strictly triangular, (74) provides an inductive way of determining {Π− x }x (up to expectation) in terms of {Πx}x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Here comes the argument for (74): Applying Γ∗ yx to the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' of (68) and using (72) in conjunction with Leibniz’ rule w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' δ, we obtain the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' of (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' For the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' we first use the multiplicativity of Γ∗ yx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' according to the second item in (70) and (32) we have Γ∗ yxz0 = � l≥0 Πl y(x)zl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (75) To rewrite Γ∗ yxδΠx, we apply δ to the first identity in (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This estab- lishes (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We now argue that from an analytical point of view, (74) is not quite adequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Clearly, the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' of (74) still contains a potentially singular product of Πk y and ∂2 1(δΠy −δΠy(x) −(δΓ∗ yx)Πx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Here, it is crucial that applying δ to Πy, which is a multi-linear expression in ξ, means replac- ing one of the instances of ξ by δξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Now as we shall explain in the next subsection, δξ gains24 D 2 orders of regularity over ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' However, since the other instances of ξ remain, the regularity of δΠy is not at face value better by D 2 orders over Πy, which is just H¨older continuous with expo- nent α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence we can only expect that δΠy is locally, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' near a base 23of course, the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' term δΠy(x) is effectively absent due to the derivative ∂2 1 24however on an L2 instead of a uniform scale 22 FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR point x, described – “modelled” in the jargon of regularity structures – to order D 2 + α in terms of Πx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The Taylor-remainder-like expression δΠy − δΠy(x) −(δΓ∗ yx)Πx has the potential of expressing this modeled- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence the product of Πk y and ∂2 1(δΠy − δΠy(x) −(δΓ∗ yx)Πx) has a chance of being well-defined provided α + ( D 2 + α − 2) > 0, which gives rise to the lower bound assumption α > 1 − D 4 in Theorem 1, which reduces to25 α > 1 4 for our D = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Since D 2 + α > 1, this only has a chance of working provided every β-component of (δΓ∗ yx)Πx involves the affine function Πxe(1,0) = (· − x)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' However, this contradicts the (strict) triangularity (73) for |β| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence δΓ∗ yx is not rich enough to describe all components of δΠy to the desired order near x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In view of the preceding discussion, we are forced to loosen the pop- ulation constraint (73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' To this purpose, we replace the directional Malliavin derivative δΓ∗ yx by some dΓ∗ yx ∈ End(R[[zk, zn]]) in order to achieve δΠy − δΠy(x) − dΓ∗ yxΠx = O(| · −x| D 2 +α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (76) In order to preserve the identity (74) in form of δΠ− y (x) − dΓ∗ yxΠ− x (x) = � k≥0 zkΠk y(x)∂2 1 � δΠy − δΠy(x) − dΓ∗ yxΠx � (x) + δξτ(x)1, (77) we need dΓ∗ yx to inherit the algebraic properties of δΓ∗ yx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' More precisely, we impose that dΓ∗ yx agrees with δΓ∗ yx on the sub-algebra R[[zk]], dΓ∗ yx = δΓ∗ yx on R[[zk]], (78) and that dΓ∗ yx is in the tangent space to the manifold of algebra mor- phisms in Γ∗ yx, which means that for all π, π′ ∈ R[[zk, zn]] dΓ∗ yxππ′ = (dΓ∗ yxπ)(Γ∗ yxπ′) + (Γ∗ yxπ)(dΓ∗ yxπ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (79) Here is the argument on how to pass from (78) & (79) to (77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' On the one hand, we apply δ to (44) to the effect of δΠ− y (x) = � k≥0 zkδ � Πk y(x) � ∂2 1Πy(x) + � k≥0 zkΠk y(x)∂2 1δΠy(x) − � l≥0 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='δ � Πl y(x) � (D(0))lc + δξτ(x)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (80) 25This is the analogy of rough path construction of fractional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' For the case of fractional Brownian motion with Hurst parameter H, a rough path construction can be only implemented for any H > 1 4 by increasing the number of iterated integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' However, the stochastic analysis to construct the iterated integrals fails for fractional Brownian motion of Hurst parameter H ≤ 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' See [5, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES 23 On the other hand, we apply dΓ∗ yx to (67) to obtain by26 (79) dΓ∗ yxΠ− x (x) = (dΓ∗ yxz0)∂2 1Γ∗ yxΠx(x) + (Γ∗ yxz0)∂2 1dΓ∗ yxΠx(x) − dΓ∗ yxc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (81) We now argue that the first r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' term of (80) is identical to the one in (81);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' indeed, by the first item in (70) we have ∂2 1Γ∗ yxΠx = ∂2 1Πy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' On the other hand, by (78) and the second item in (70) we have dΓ∗ yx = � l≥0 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='δ � Πl y(x) � (D(0))l on R[[zk]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (82) so that by (32) dΓ∗ yxz0 = � k≥0 δ(Πk y(x))zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Identity (82) also implies that the third r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' terms of (80) and (81) are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The sec- ond r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' terms of (80) and (81) combine as desired by (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This establishes (77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In order to use (77) inductively to define – or rather estimate – {Π− x }x, [15] had to come up with an ordering on multi- indices β with respect to which dΓ∗ yx is strictly triangular, leading to a modification of (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Definition of the Malliavin derivative and SG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We have seen that the Malliavin derivative, which we now shall rigorously define, allows to give a more robust relation between Πx and Π− x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Via the SG inequality, which will be introduced here, the control of the Malliavin derivative of a random variable F yields control of the variance of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Consider the Hilbert norm on (a subspace of) the space of Schwartz distributions27 ∥δξ∥2 = ˆ R2 dx � (∂4 1 − ∂2 2) 1 4(α− 1 2)δξ �2 = ˆ R2 dq ��|q|(α− 1 2)Fδξ ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (83) Note that we encounter again A∗A = (−∂2 − ∂2 1)(∂2 − ∂2 1) with Fourier symbol |q|4 = q4 1 + q2 2, see (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence this is one of the equivalent ways of defining the homogeneous L2(R2)-based Sobolev norm of fractional order α − 1 2, however of parabolic scaling, which we nevertheless still denote by H := ˙Hα− 1 2(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We now consider “cylindrical” (nonlinear) functionals F on the space S′(R2) of Schwartz distributions, by which one means that for some N ∈ N, F is of the form F[ξ] = f � (ξ, ζ1), · · · , (ξ, ζN) � with f ∈ C∞(RN) and ζ1, · · · , ζN ∈ S(R2), (84) 26which also implies dΓ∗ yx1 = 0 27we denote the argument by δξ since we think of it as an infinitesimal perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 24 FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR where we recall that (ξ, ζn) denotes the natural pairing between ξ ∈ S′(R2) and a Schwartz function ζn ∈ S(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Clearly, those func- tion(al)s F are Fr´echet differentiable with dF[ξ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='δξ = lim s↓0 1 s(F[ξ + sδξ] − F[ξ]) = N � n=1 ∂nf � (ξ, ζ1), · · · , (ξ, ζN) � (δξ, ζn) = (δξ, ∂F ∂ξ [ξ]), (85) where ∂F ∂ξ [ξ] ∈ S(R2) is defined through ∂F ∂ξ [ξ] = N � n=1 ∂nf � (ξ, ζ1), · · · , (ξ, ζN) � ζn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We will monitor the dual norm ∥∂F ∂ξ [ξ]∥∗ := sup δξ (δξ, ∂F ∂ξ [ξ]) ∥δξ∥ = ∥∂F ∂ξ [ξ]∥ ˙H 1 2 −α(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (86) Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' An ensemble E of Schwartz distributions28 is said to satisfy a SG inequality provided for all cylindrical F with E|F| < ∞ E(F − EF)2 ≤ E∥∂F ∂ξ ∥2 ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (87) Note that the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' of (87) is the variance of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Inequality (87) amounts to an L2-based Poincar´e inequality with mean value zero on the (infinite-dimensional) space of all ξ’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' By a (parabolic) rescaling of x, we may w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' assume that the constant in (87) is unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Implicitly, we also include closability of the linear operator cylindrical function F �→ ∂F ∂ξ ∈ {cylindrical functions} ⊗ S(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (88) This means that the closure of the graph of (88) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' the topology of L2 and L2(H∗) is still a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This allows to extend the Fr´echet derivative (88) to the Malliavin derivative L2 ⊃ D( ∂ ∂ξ ) ∋ F �→ ∂F ∂ξ ∈ L2(H∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' By the chain rule, we may post-process (87) to its Lp-version E 1 p|F − EF|p ≲p E 1 p∥∂F ∂ξ ∥p ∗, (89) which is the form we use it in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' A concise proof how to obtain (89) from (87) can be found in [11, Step 2 in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 28It does not have to be a Gaussian ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES 25 The obvious examples are Gaussian ensembles of Schwartz distributions with ∥ · ∥ ≤ Cameron-Martin norm, (90) where the norm ∥ · ∥ means the Hilbert norm defined in (83), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' white noise −D 2 = α − 2 =⇒ α = 1 2, free field 1 − D 2 = α − 2 =⇒ α = 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In other words, the SG inequality (87) holds with Gaussian ensembles satisfying (90), see [2, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' For the reader’s convenience, we sketch the simplest application of SG from [15, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='3], namely (61) for β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' To this aim we apply (89) to F := (ξ, ψt(y−·)) = Πx0(y), which is of the form of (84), so that according to (85) its Malliavin derivative is given by ∂F ∂ξ = ψt(y − ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In view of (86), and then appealing to (8) in conjunction with the translation invariance and scaling of the Sobolev norm we have ∥∂F ∂ξ ∥∗ = ∥ψt(y − ·)∥ ˙H 1 2 −α(R2) = ( 4√ t)− D 2 − 1 2+α∥ψt=1∥ ˙H 1 2 −α(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Noting that the exponent is α − 2 and that ψt=1 is a (deterministic) Schwartz function we obtain from (89) E 1 p|Πx0(y)|p ≲ ( 4√ t)α−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In view of |0| = α, this amounts to the desired (61) for β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We also remark that SG naturally complements the BPHZ-choice of renormalization, see Section 6: The choice of cβ takes care of the mean EΠ− xβt(y), while SG takes care of the variance of Π− xβt(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence the main task in [15] is the estimate of E 1 p∥ ∂F ∂ξ ∥p ∗, where F := Π− xβt(y), which we tackle by duality through estimating the directional derivative δF := (δξ, ∂F ∂ξ ) given control of E 1 q ∥δξ∥q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The inductive estimate is based on (77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Philosophically speaking, our approach is analytic rather than combi- natorial: analytic combinatorial index set: derivatives w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' a and p Picard iteration ⇝ multi-indices on k ≥ 0, n ̸= 0 ⇝ trees with decorations Ass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' on ξ: spectral gap inequality cumulant bounds Malliavin derivatives w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' ξ trees with paired nodes ⇝ estimates on E∥ ∂ ∂ξΠ− xβ t(y)∥2 ∗ ⇝ Feynman diagrams 26 FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR For us, all combinatorics are contained in Leibniz’ rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We also point out that our approach may be called “top-down” rather than bottom- up in the sense that we postulate the conditions (space-time trans- lation, spatial reflection, shift-covariance, etc) on the counter term h from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' A closing remark for experts in QFT: The absence of c in (77) means that our approach does not suffer from the well-known difficulty of “overlapping sub-divergences” in Quantum Field Theory, which is also an issue in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Our inductive approach has similarities with the one of Epstein-Glaser, see [18, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The structure group and the re-expansion map In this section we construct the endomorphism Γ∗ yx of the algebra R[[zk, zn]] that satisfies (70) for given Πx and Πy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In [15], the construc- tions (and estimates) of Γ∗ yx and Πx are actually intertwined, however the proof of Lemma 5 has the same elements as [15, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In line with regularity structures it is convenient to adopt a more ab- stract point of view: We start by introducing what can be assimilated to Hairer’s structure group G, which here is a subgroup of the au- tomorphism group of the linear space R[zk, zn], where R[zk, zn] now plays the role of the29 (algebraic) pre-dual of R[[zk, zn]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Γ∗ yx will be the transpose of a Γyx ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The elements Γ ∈ G are parameterized by {π(n)}n ⊂ R[[zk, zn]], see Lemma 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' the group property will be estab- lished in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In Lemma 5 we inductively choose {π(n) yx }n such that the associated Γyx satisfies (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' For a discussion of the Hopf- and Lie-algebraic structure underlying G we refer to [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' As opposed to [14] and [13], we will capitalize on α < 1, which simplifies several arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Given30 {π(n)}n ⊂ R[[zk, zn]] satisfying π(n) β = 0 unless |n| < |β|, (91) there exists a unique linear endomorphism Γ of R[zk, zn] such that Γ∗ is an algebra endomorphism31 of R[[zk, zn]] that satisfies Γ∗zk = � l≥0 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (π(0))l(D(0))lzk (32) = � l≥0 �k+l k � (π(0))lzk+l, (92) Γ∗zn = zn + π(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (93) 29canonical w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' the monomial basis 30which here as opposed to earlier includes the additional (dummy) index n = 0 we first encountered in (30) 31i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Γ∗ππ′ = (Γ∗π)(Γ∗π′) and Γ∗1 = 1 hold LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES 27 In addition32, (94) (Γ∗ − id)γ β = 0 unless |γ| < |β| and γ ≺ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We remark that the algebra endomorphism property, the mapping property (92), and the first triangularity in (94) mimic desired proper- ties of Γ∗ yx, namely (71), the second item of (70), and (73), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We recall that the matrix representation {Γβ γ}β,γ of a linear endomorphism Γ of R[zk, zn] w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' the monomial basis {zβ}β is given by (95) Γzβ = � γ Γβ γzγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The algebraic dual Γ∗, as a linear endomorphism of R[[zk, zn]], is given by33 (Γ∗π)β = � γ (Γ∗)γ βπγ where (Γ∗)γ β := (Γ∗zγ)β = Γβ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Such a Γ∗ is an algebra endomorphism if and only if (Γ∗)γ β = � β1+···+βk=β (Γ∗)γ1 β1 · · · (Γ∗)γk βk for γ = γ1 + · · · + γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (96) This includes Γ∗1 = 1 in form of (Γ∗)0 β = δ0 β (97) Since any multi-index γ ̸= 0 can be written as the sum of γj’s of length one, we learn that an endomorphism Γ of R[zk, zn] with multiplica- tive Γ∗ is determined by how Γ∗ acts on the coordinates {zk}k≥0 and {zn}n̸=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This establishes the uniqueness statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' For the existence, we need to establish that the numbers {(Γ∗)γ β}β,γ defined through (92) & (93) in form of (Γ∗)ek β − δek β = � l≥1 �k+l k � � ek+l+β1+···+βl=β π(0) β1 · · ·π(0) βl , (98) (Γ∗)en β − δen β = π(n) β (99) and extended by (96) & (97) to all γ satisfy (for fixed β) #{γ | (Γ∗)γ β ̸= 0} < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (100) Indeed, this finiteness condition allows to define Γ via (95) with Γβ γ := (Γ∗)γ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Since thanks to (103) below in conjunction with 0 < λ, α < 1 the ordering ≺ is coercive, by which we mean #{γ | γ ≺ β} < ∞, (101) 32we recall that ≺ is defined in (63) 33note that the sum is effectively finite, since there are only finitely many γ such that Γβ γ ̸= 0 since the monomial basis is an algebraic basis 28 FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR (100) follows once we establish the second strict triangularity in (94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence, it remains to establish (94) in form of (Γ∗)γ β − δγ β = 0 unless |γ|≺ < |β|≺ and |γ| < |β| (102) for the numbers {(Γ∗)γ β}β,γ defined through (98) & (99) and then ex- tended by (96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' For this purpose, we note that by definition (54) in form of |β| − α = � k≥0 kβ(k) + � n̸=0 (|n| − α)β(n) (103) and since α ≤ 1 ≤ |n|, | · | − α ≥ 0 is additive (63) =⇒ same for | · |≺ − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (104) We first restrict to γ’s of length one in (102), and distinguish the cases γ = en and γ = ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Since by (54) and (63) we have |en|≺ = |en| = |n| and |β| ≤ |β|≺, the former case follows directly via (99) from assumption (91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We now turn to the latter case of γ = ek and to (98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' There is a contribution to the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' sum only when there exists an l ≥ 1 and a decomposition β = ek+l + β1 + · · · + βl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' this implies |β| ≥ |ek+l| (54) = |ek| + αl ≥ |ek| + α (63) =⇒ |β|≺ ≥ |ek|≺ + (α − λ), which yields the desired (102) because of α > λ, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Finally, we need to upgrade (102) from γ’s of length one to those of arbitrary length, which we do by induction in the length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The base case of zero length, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' of γ = 0, is dealt with in (97).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We carry out the induction step with help of (96), writing a multi-index γ = γ′ + γ′′ with γ′, γ′′ of smaller length: (Γ∗)γ β = � β′+β′′=β (Γ∗)γ′ β′(Γ∗)γ′′ β′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (105) We learn from the induction-hypothesis version of (102) that the sum- mand vanishes unless |γ′| + |γ′′| < |β′| + |β′′| and |γ′|≺ + |γ′′|≺ < |β′|≺ + |β′′|≺ or γ′ = β′ and γ′′ = β′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' in the latter case the summand is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' By (104), the first al- ternative implies |γ| < |β| and |γ|≺ < |β|≺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The second alternative implies γ = β and then holds for exactly one summand to the desired effect of (Γ∗)γ β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' □ The two triangular properties (94) from Lemma 3 allow us to establish the group property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Furthermore, a triangular dependence (106) of Γ∗ on π(n) will play a crucial role when inductively constructing π(n) yx in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES 29 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The set G of all Γ as in Lemma 3 defines a subgroup of the automorphism group of R[zk, zn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Moreover, for [γ] ≥ 0, (Γ∗)γ β is independent of π(n) β′ unless β′ ≺ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (106) Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The group G is larger than the one constructed in [14], since 1) we do not require that π(n) β = 0 unless β satisfies (48), and 2) we do not specify the space-time shift structure of the (β = em)- components of π(n) β as in [14, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Both conditions however are satisfied for our construction of π(n) yxβ, see (113) and (115).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We first argue that for Γ, Γ′ ∈ G we have Γ′Γ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' More precisely, if Γ and Γ′ are associated to {π(n)}n and {π′(n)}n by Lemma 3, respectively, we consider �π(n) := π(n) + Γ∗π′(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (107) We note that by triangularity (94) of Γ∗ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' |·|, the population prop- erty (91) propagates from π(n), π′(n) to �π(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Let �Γ ∈ G be associated to {�π(n)}n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' we claim that Γ′Γ = �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' To this purpose, we note that (Γ′Γ)∗ = Γ∗Γ′∗ is an algebra morphism, like �Γ∗ is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence by the uniqueness statement of Lemma 3, it is suffi- cient to check that Γ∗Γ′∗ and �Γ∗ agree on the two sets of coordinates {zk}k and {zn}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' On the latter this is easy: �Γ∗zn (93) = zn + �π(n) (107) = zn + π(n) + Γ∗π′(n) (93) = Γ∗(zn + π′(n)) (93) = Γ∗Γ′∗zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We now turn to the zk’s, showing that the algebra endomorphisms Γ∗Γ′∗ and �Γ∗ agree on the sub-algebra R[zk] ⊂ R[[zk, zn]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' by multiplicativity of Γ∗ we have according to (92) for Γ′ Γ∗Γ′∗ = � l′≥0 1 l′!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (Γ∗π′(0))l′Γ∗(D(0))l′ on R[zk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Since D(0) preserves R[zk], we may apply (92) for Γ and obtain by the binomial formula: Γ∗Γ′∗ = � l′≥0 1 l′!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (Γ∗π′(0))l′ � l≥0 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (π(0))l(D(0))l′+l (107) = � ˜l≥0 1 ˜l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (�π(0)) ˜l(D(0)) ˜l on R[zk], which according to (92) agrees with �Γ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' 30 FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR We come to the inverse of a Γ ∈ G associated to {π(n)}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' By the strict triangularity (94) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' the coercive ≺, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (101), there exists ˜π(n) ∈ R[[zk, zn]] such that Γ∗˜π(n) = −π(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (108) We now argue by induction in β w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' ≺ that ˜π(n) satisfies (91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' For this, we spell (108) out as ˜π(n) β + � γ (Γ∗ − id)γ β˜π(n) γ = −π(n) β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' If |β| ≤ |n|, the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' vanishes by (91), and by (94) the sum over γ restricts to |γ| ≤ |β| ≤ |n|, and to γ ≺ β, so that the summand vanishes by induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Thus also ˜π(n) β vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' This allows us to argue that ˜Γ ∈ G associated to {˜π(n)}n is the inverse of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' By the strict upper triangularity of Γ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' to the coercive ≺, we already know that Γ is invertible, so that it suffices to show ˜ΓΓ = id, which in turn follows from its transpose Γ∗˜Γ∗ = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' By the composition rule (107) established above, Γ∗�Γ∗ is associated to {π(n) +Γ∗�π(n)}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' By (108) we have that π(n) + Γ∗�π(n) = 0, and learn from Lemma 3 that id is associated with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We finally turn to the proof of (106).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We note that β1 + · · · + βl = β implies the componentwise βj ≤ β, which by (104) implies |βj|≺ ≤ |β|≺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Since every γ with [γ] ≥ 0 can be written as the sum of γ’s of the form γ = ek + en1 + · · · + enj with j ≤ k, (109) we learn from (96) that we may assume that γ is of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Once more by (96) we have for these γ’s (Γ∗)γ β = � β0+···+βj=β (Γ∗)ek β0(Γ∗) en1 β1 · · · (Γ∗) enj βj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' From (98) & (99) we learn that this (Γ∗)γ β is a linear combination of π(0) β′ 1 · · · π(0) β′ l (zn1 + π(n1))β1 · · · (znj + π(nj))βj, (110) where the multi-indices satisfy β = ek+l + β′ 1 + · · · + β′ l + β1 + · · · + βj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (111) We need to show that the product (110) contains only factors π(n) β′ with β′ ≺ β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' we may assume l + j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' To this purpose we apply | · |≺ to (111);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' by (104) and |ek+l|≺ ≥ |ek+l| = α(1 + k + l) this implies |β|≺ ≥ α(1 + k − j) + |β′ 1|≺ + · · · + |β′ l|≺ + |β1|≺ + · · · + |βj|≺, which by j ≤ k implies the desired |β′ 1|≺, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' , |β′ l|≺, |β1|≺, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' , |βj|≺ < |β|≺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' □ Finally, we show that the group G is large enough to contain the re- expansion maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES 31 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' There exists {π(n) yx }n satisfying (91) such that the Γyx ∈ G associated by Lemma 3 satisfies (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' As a consequence of working with a larger group than in [14], see Remark 1, we don’t have uniqueness of {π(n) yx }n and thus of Γyx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We refer the reader to [19] for a uniqueness result when working with the smaller group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' An inspection of our construction reveals transitivity in line with [9, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='3] Γ∗ xyΓ∗ yz = Γ∗ xz and Γ∗ xx = id, see [15, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='3] for the argument;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' it would also be a consequence of uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We start by specifying π(n) yxβ in the special cases of n = 0 and of β = em for some m ̸= 0: π(0) yx := Πy(x), (112) π(n) yxem := � �m n � (x − y)m−n provided n < m, 0 otherwise � for n ̸= 0, (113) where n < m means component-wise (non-strict) ordering and n ̸= m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We note that (112) is necessary in order to bring the second item of (70) into agreement with the form (92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We also remark that (113) yields by (93) (Γ∗ yx)en em = � �m n � (x − y)m−n provided n ≤ m, 0 otherwise � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' By the second part of (94), which implies (Γ∗ yx)γ 0 = 0 unless γ = 0, by (98) in form of (Γ∗ yx)ek em = 0, and via (96) this strengthens to (Γ∗ yx)γ em = � �m n � (x − y)m−n if γ = en with n ≤ m, 0 otherwise � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (114) The latter is imposed upon us by taking the (β = em)-component of the first item in (70) and plugging in (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' The second part of (114) implies that Γyx maps the linear span of {zm}m̸=0 into itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' since this linear span can be identified with the space R[x1, x2]/R of space-time polynomials (modulo constants), this can be assimilated to Hairer’s postulate [9, Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We note that (112) and (113) satisfy (91) because of | · | ≥ α > 0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (104), and |em| = |m| > |n|, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' In line with (48) and [14], we also set π(n) yxβ = 0 unless [β] ≥ 0 or β = em for some m ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (115) It thus remains to construct π(n) yxβ for n ̸= 0 and [β] ≥ 0, which we will do by induction in β w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' ≺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' According to (106), we may consider (Γ∗)γ β as already constructed for [γ] ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' According to (64) and by the 32 FELIX OTTO, KIHOON SEONG, AND MARKUS TEMPELMAYR induction hypothesis (70), an inspection of the argument that leads from there to (72) shows that we also have Π− yβ = (Γ∗ yxΠ− x )β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (116) The induction step consists in choosing {π(n) yxβ}0<|n|<|β| such that Πyβ = (Γ∗ yxΠx)β + Πyβ(x) (112) = (Γ∗ yxΠx)β + π(0) yxβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (117) Denoting by P the projection on multi-indices γ with [γ] ≥ 0, so that by (41) and (48) we have (id − P)Πx = � n̸=0(· − x)nzn and thus by (91) and (93) (Γ∗ yx(1 − P)Πx)β = � 0<|n|<|β| (· − x)nπ(n) yxβ, (118) allows us to make {π(n) yxβ}0<|n|<|β| in (117) explicit: (Πy − Γ∗ yxPΠx)β = � n:|n|<|β| π(n) yxβ(· − x)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (119) Hence our task reads (Πy − Γ∗ yxPΠx)β = polynomial of degree < |β|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (120) According to the PDE (58), to (116), and to (118) we have (∂2 − ∂2 1)(Πy − Γ∗ yxPΠx)β = polynomial of degree < |β| − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (121) In order to pass from (121) to (120), we will now appeal to the unique- ness/Liouville statement in Lemma 1 with η = |β|, which is ̸∈ Z ac- cording to (62) and ≥ α according to (104), and p = 1 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' More precisely, we apply Lemma 1 to u = (Πy − Γ∗ yxPΠx)β − its Taylor polynomial in x of order < |β|, which makes sense since (121) implies that (Πy − Γ∗ yxPΠx)β is smooth, and to f ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Hence for the assumption (10) we need to check that (122) lim sup z:|z−x|↑∞ 1 |z − x||β|E|(Πy − Γ∗ yxPΠx)β(z)| < ∞, which forces us to now become semi-quantitative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' By the estimate (60) on Π, for (122) it remains to show34 E 1 p |(Γ∗ yx)γ β|p ≲β,γ,p |y − x||β|−|γ| provided [γ] ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (123) In line with the language of [15], we split the argument for (123) into an “algebraic argument”, where we derive (123) from (124) E 1 p|π(n) yxβ′|p ≲β′,p |x − y||β′|−|n| for β′ ≺ β, 34which coincides with Hairer’s postulate [9, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='2) in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content='3] LECTURE NOTES ON TREE-FREE REGULARITY STRUCTURES 33 and a “three-point argument”, where we derive (124) from the estimate (60) on Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Here comes the argument for (123), which is modelled after the one for (106) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' By H¨older’s inequality in probability and the additivity of |·|−α, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (104), we may restrict to γ’s of the form (109).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' We are thus lead to estimate the product (110), which now takes the form of π(0) yxβ′ 1 · · · π(0) yxβ′ l(zn1 + π(n1) yx )β1 · · · (znj + π(nj) yx )βj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' (125) Once again by H¨older’s inequality, we infer from (124) that the E 1 p|·|p- norm of (125) is ≲ |y − x||β′ 1| · · · |y − x||β′ l||y − x||β1|−|n1| · · · |y − x||βj|−|nj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' By the additivity of |·|−α, the total exponent of |y−x| can be identified with the desired expression: |β′ 1| + · · · + |β′ l| + (|β1| − |n1|) + · · · + (|βj| − |nj|) (111) = |β| − |ek+l| + (l + j)α − (|n1| + · · · + |nj|) (109) = |β| − |γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' Finally, we give the “three-point argument” for the estimate (124), for notational simplicity in case of the current multi-index β, so that we now may use (119) and (123).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' By (60) and (123), the left hand side of (119) can be estimated as follows E 1 p |(Πy − Γ∗ yxPΠx)β(z)|p ≲β,p (|z − x| + |y − x|)|β|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQf3_pa/content/2301.00778v1.pdf'} +page_content=' By the equivalence of norms on the finite-dimensional space of space- time polynomials of degree < |β|, which by a duality argument can be upgraded to the following estimate of annealed norms for random polynomials max n: |n|<|β| |y − x||n| E 1 p|π(n) yxβ|p ≲ |z−x|≤|y−x| dz E 1 p�� � n: 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a/B9E5T4oBgHgl3EQfTg8R/content/tmp_files/2301.05536v1.pdf.txt b/B9E5T4oBgHgl3EQfTg8R/content/tmp_files/2301.05536v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fbf4f433b27aa874c4c55eb774baf8319a077926 --- /dev/null +++ b/B9E5T4oBgHgl3EQfTg8R/content/tmp_files/2301.05536v1.pdf.txt @@ -0,0 +1,2756 @@ +1 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +An Electromagnetic-Information-Theory Based +Model for Efficient Characterization of MIMO +Systems in Complex Space + +Ruifeng Li, Da Li, Member, IEEE, Jinyan Ma, Zhaoyang Feng, Ling Zhang, Member, IEEE, Shurun Tan, Member, +IEEE, Wei E. I. Sha, Senior Member, IEEE, Hongsheng Chen, Fellow, IEEE, and Er-Ping Li, Fellow, IEEE +Abstract—It is the pursuit of a multiple-input-multiple-output +(MIMO) system to approach and even break the limit of channel +capacity. However, it is always a big challenge to efficiently +characterize the MIMO systems in complex space and get better +propagation performance than the conventional MIMO systems +considering only free space, which is important for guiding the +power and phase allocation of antenna units. In this manuscript, +an Electromagnetic-Information-Theory (EMIT) based model is +developed for efficient characterization of MIMO systems in +complex space. The group-T-matrix-based multiple scattering fast +algorithm, the +mode-decomposition-based +characterization +method, and their joint theoretical framework in complex space +are discussed. Firstly, key informatics parameters in free +electromagnetic space based on a dyadic Green’s function are +derived. Next, a novel group-T-matrix-based multiple scattering +fast algorithm is developed to describe a representative +inhomogeneous electromagnetic space. All the analytical results +are validated by simulations. In addition, the complete form of the +EMIT-based model is proposed to derive the informatics +parameters frequently used in electromagnetic propagation, +through integrating the mode analysis method with the dyadic +Green’s function matrix. Finally, as a proof-or-concept, +microwave anechoic chamber measurements of a cylindrical array +is performed, demonstrating the effectiveness of the EMIT-based +model. Meanwhile, a case of image transmission with limited +power is presented to illustrate how to use this EMIT-based model +to guide the power and phase allocation of antenna units for real +MIMO applications. + +Index Terms—multiple-input-multiple-output (MIMO) system, +complex space, group T matrix, mode analysis, electromagnetic +information theory (EMIT). +I. INTRODUCTION +YPICALLY, for antenna design, it is promising to +maximize the channel capacity via a multiple-input- +multiple-output (MIMO) system to approach the limit of +channel capacity during propagation. Thus, Under the demand +for high accuracy and low latency nowadays, the basic research +on efficient characterization of MIMO systems is very +important [1]. On this basis, we can carry out further work such +as the optimization solutions for the power and phase allocation +of antenna units. +Previous works for MIMO characterization can be roughly +clarified into two categories: electromagnetic (EM) methods + +This project is supported in part by Natural Science Foundation of China +(NSFC), Grant No. 62071424, 62201499 and 62027805. (Corresponding +Author: Da Li, li-da@zju.edu.cn) +and information theory. The former mainly focuses on the +radio-frequency (RF) front-end design by solving Maxwell’s +equations under different boundary conditions, consisting of the +descriptions of the complex electromagnetic space [2], [3], +while the latter mainly analyzes the channel properties under +different probability models by using Shannon information +theory [4], [5]. The above two frameworks are faced with a +major challenge in practical application: how to efficiently +model MIMO systems in complex space to achieve better +propagation performance than MIMO analysis that only +consider free space. +For EM methods, the core step of intelligent designs +nowadays is reconstructing the MIMO systems’ radiation +patterns [6], [7], [8]. To consider the effect of the EM +propagation space, full-wave numerical algorithms have been +used to incorporate the RF front-end design and environment +perception into the EM framework, consuming a lot of time [9]. +To greatly reduce the calculation time of modeling EM space, +some studies have proposed to use approximate methods like +ray tracing (RT) [10], [11], [12]. However, this is often not +acceptable due to lack of high accuracy. Besides, the T-matrix +can be easily used to characterize efficient MIMO in complex +EM space, via combining multiple scattering equations [13], +[14]. However, practical wireless communication often focuses +on some informatic parameters (such as channel capacity), +while the EM-only framework is incapable of efficiently +extracting the informatic parameters in the complex EM space. +For information theory, the common statistic model, such as +the Rayleigh fading model, is a mathematical tool based on the +assumption of rich scattering [15]. When it evolves to cluster +models like geometry-based stochastic models (GBSMs), the +EM space is equivalent to the clusters with different shapes or +distributions for convenient characterization [16], [17]. +However, the accuracy of those models will be reduced due to +the EM properties of the MIMO system. For example, the work +in [18] complements numerical methods to make up for the +problem of using only Fresnel approximation in airborne +antenna design. Moreover, the main idea of the emerging +intelligent reflective surface (IRS) is to lay out the controllable + + +The authors are with ZJU-UIUC Institute, Zhejiang Provincial Key +Laboratory of Advanced Microelectronic Intelligent Systems and Appli- +cations, and the College of Information Science and Electronic Engineering, +Zhejiang University, Hangzhou 310027, China. +T +This article has been accepted for publication in IEEE Transactions on Antennas and Propagation. This is the author's version which has not been fully edited and +content may change prior to final publication. Citation information: DOI 10.1109/TAP.2023.3235015 +© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.��See https://www.ieee.org/publications/rights/index.html for more information. +Authorized licensed use limited to: Zhejiang University. Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore. Restrictions apply. + +2 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + + +Fig. 1. System model of MIMO analysis in a complex space for indoor +communication. + +surfaces in free or complex EM space [19], [20], [21], [22], +[23]. Due to the lack of efficient MIMO characterization, this +technology is still in the trial stage. This suggests that many +basic assumptions of the information-only framework need to +be reconsidered. +Nowadays, the electromagnetic information theory (EMIT) +for the MIMO characterization attracts more attentions, which +is expected to solve the challenges mentioned above [24], [25], +[26], [27]. Researchers point out that with the wide layout of +the antenna array (e.g., Internet of vehicles), it is expected to +eliminate the step of channel estimation with the help of rich +environmental information [28]. Some works have been done +from the perspective of EM fields to study the degree of +freedom of MIMO systems [29], [30]. There are also +mathematical methods to model the source region and field +region as two sets of orthogonal bases in Hilbert space, and then +construct some characteristic parameters of the MIMO system +[31], [32]. To integrate the RF front-end design in the EMIT +framework, the surface currents of antenna elements are +modeled as the point sources with orthogonal bases. For +example, a model was established to build a channel matrix +from the angle of coordinate transformation and orthogonal +decomposition of EM plane wave expansion, applied in +holographic MIMO system [33], [34]. Additionally, the work +in [35] contains the idea of deriving the channel limit of a +MIMO system by the EM field method. Nevertheless, the above +research works on EMIT mainly focus on free space or +revealing the parameter mapping between two theories; +efficient characterization algorithms and clear EM information +analysis methods for complex EM space are still unexplored. +In this paper, we develop an EMIT-based model to conduct +the efficient characterization for MIMO systems in complex +EM space. The proposed EMIT-based model uses the group T +matrix algorithm and dyadic Green’s function-based mode +analysis method, filling the research gap of efficient +characterization algorithms and clear EM information analyses. +The main contributions of this paper are described as follows. +1) The key parameters of the MIMO systems are extracted +through the dyadic Green’s function and matrix mode +analysis. The information characteristics of the MIMO +systems are described by the EM method, revealing +some important conclusions and deducing the key +informatic parameters and valuable conclusions of +information theory by means of EM methods. +2) A fast algorithm based on the group T matrix is +developed to model the complex EM space. Since the +algorithm has semi-analytical characteristics and the +classical T matrix can be stored, which provides a faster +calculation compared with the traditional full-wave +algorithm. In contrast to the RT and pilot-based methods +for channel estimation, our EM algorithm can be easily +integrated into EMIT due to its higher accuracy and +efficiency. In other words, the benefit of our proposed +method is generated from the fast characteristics of the +group T matrix and the EM analysis of the channel +matrix (without the help of statistics). +3) The efficient EMIT-based model is proposed to +characterize the MIMO systems in complex space. As a +proof-of-concept, a microwave anechoic chamber +measurement of a cylindrical array is taken as an +example, demonstrating the effectiveness of the EMIT- +based model for the MIMO mode analysis. Meanwhile, +a case of image transmission with limited power is +presented to illustrate how to guide the MIMO feeding +based on the model, bringing a new insight into +extracting information parameters using the basis of +computational electromagnetism. +This article is organized as follows. The key informatics +parameters based on the dyadic Green’s function are derived in +Section II. Then, the proposed EMIT-based model is analyzed +in Section III. Experimental verification and an image +transmission case were conducted in Section IV. Finally, the +conclusion is drawn in Section V. +II. SYSTEM MODEL AND KEY PARAMETERS +As shown in Fig. 1, consider a typical MIMO system +including the transmitting and receiving array for indoor +communication, whose overall communication performance +will be affected by the propagation distance and the properties +of complex space. In this section, the EM propagation space is +designated as a free space for extracting key parameters of a +MIMO system. More complex EM space is characterized in the +next section. +To combine the coupling operator +TR +G + and the channel +matrix  , a series of isotropic point sources are placed in the +transmission volume and the receiving volume, with the +position vectors +Tr + and +Rr + respectively. The EM wave +received is defined as +outR +ψ +, thus the Helmholtz wave equation +is given by + +2 +0 +0 +outR +outR +incT +k +i + +− += +ψ +ψ +J +, +(1) +where +incT +J + is the transmitted source and k is the wave vector. +To solve this equation, the dyadic Green’s function G operator +based on the impulse function idea is introduced: + + +2 +exp[ +] +( +, +) +4 +R +T +R +T +R +T +ik +k + +− + + + += ++ + + +− + + +r +r +G r r +I +r +r +, +(2) +This article has been accepted for publication in IEEE Transactions on Antennas and Propagation. This is the author's version which has not been fully edited and +content may change prior to final publication. Citation information: DOI 10.1109/TAP.2023.3235015 +© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.��See https://www.ieee.org/publications/rights/index.html for more information. +Authorized licensed use limited to: Zhejiang University. Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore. Restrictions apply. + +Coupling operatorGr +Channel matrix H +Receiving array +Electromagnetic +characteristic +Information +characteristic +Complex Space +Feeding& +Beamforming3 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +where I is the unit tensor. Since the dyadic Green’s function +tensor G contains the scalar Green’s functions: + + +Gxx +Gxy +Gxz +Gyx +Gyy +Gyz +Gzx +Gzy +Gzz + + + + +=  + + + + + +G +. +(3) + +When it comes to the two independent single-polarization +situation at far field, the coupling operator is able to be +simplified into the scalar Green’s function without loss of +accuracy. Therefore, the element +ij +h in the channel matrix  +will changed to the following form in the case of a single +polarized source: + + +0( ) +exp[ +] +4 +Ri +Tj +ij +ij +Ri +Tj +ik +h +g + +− += += +− +r +r +r +r +, +(4) +where +0 +g is the scalar Green’s function, that is, the special form +of G in the case of single polarization. It is worth mentioning +that the channel matrix  at this time is not normalized, so it +contains the path loss. +Assume that the number of transmitting source points is +T +N , +and the number of receiving field points is +R +N . Therefore, the +transmitting source can be expressed as a +*1 +T +N + matrix, the +receiving electric field as a +*1 +R +N + matrix, and the coupling +operator G of the EM space as a +* +R +T +N +N + matrix. By +introducing the Dirac notation, the MIMO propagation relation +of free space is expressed as: + + +outR +incT += +ψ +G J +. +(5) + +To normalize the channel matrix, we defined the normalized +coupling operator +TR +G + as +TR + += +G +G , where  is a +normalization factor, making +2 +TR +T +R +F +N N + + + += + + + + +G +. +  + + denotes +the expectation and +F means the Frobenius norm. The +physical meaning of this normalization is that every sub- +channel should have a unity average channel gain. +According to the Hermitian nature of +† +TR +TR +G +G +, singular +value decomposition (SVD) of the coupling operator could +conduct mode analysis of MIMO EM propagation, where † +denotes the conjugate transpose: + + +† +TR +R +T += +G +U SV , +(6) + +where +T +V ( +R +U ) is a +* +T +T +N +N ( +* +R +R +N +N +) matrix, and each +column represents the EM eigenvector of the transmitting +sources (receiving fields ). Because of the unitary nature of the +SVD eigenmatrix, it is known that each column is strictly +orthogonal, which is called the EM space mode. The weight of +each pattern is determined by the corresponding element in the +diagonal matrix S. The combinations of those orthogonal modes +form two Hilbert spaces, and therefore the coupling operator +TR +G + builds a mapping between the transmitting Hilbert space +and the receiving Hilbert space, which is an important property +in the subsequent discussion. +As we all know, the upper limit of information transmission +per bandwidth in MIMO systems is also limited by Shannon’s +formula [36]: + + +† +2 +2 +2 +1 +log +det +log +1 +TR +TR +t +i +i +C +n N +n + +  + + + + + + + + +=  ++ + + + + + + + + + + + + + + + + + + += ++ + + + + + +I +G +G +, +(7) +where I is the identity matrix and +i + are the singular values +of( +) +1/ +TR +T +N +G +. Apparently, +2 +i + is the decisive parameter of +key information-carrying capacity in the MIMO system at a +given SNR +/ n + +. Besides, we can drop the expectation   + + in +(7) and no longer need to make a special distinction for large- +scale and small-scale path loss and fading, because the +amplitude and phase changes of the electric field have been +included in the operator +TR +G +. +It is seen from (6) and (7) that the singular value of EM +propagation space determines the number and weight of +independent modes, which establishes a corresponding +relationship with the number of independently available +channels and path loss of wireless communication. We give the +key informatics parameters of a MIMO system by referring to +the effective rank idea of existing work [21]: + + +min( +, +) +1 +exp( +ln( +)) +R +T +N +N +eff +i +i +i +C + + += + + += +− + +, +(8) + +where +eff +C + represents +the +EM +effective +capacity, +/ ( +) +i +i +i + + + + = + + represents the normalized singular values of +TR +G +. Hence, (8) establishes the mapping relationship between +the dyadic Green’s function matrix and typical informatics +parameters, which is an important tool for the MIMO mode +analysis. +To +understand +how +this +approach +works, +both +mathematically and physically, we set up an +* +N +N MIMO +system with the same EM space properties, as shown in Fig. 1. +In fact, in practical engineering applications, the mutual +coupling is concerned not because it affects the EM equivalent +capacity, but because it affects the radiation efficiency and +signal-to-noise ratio of the antennas. The transmitting and +receiving +antennas +are +modeled +as +isotropic +point +sources/receivers (delta function basis), which is a widely used +assumption in EM information theory. From the EM +perspective, the antennas can also be modeled as continuous +surface (equivalent) currents by +This article has been accepted for publication in IEEE Transactions on Antennas and Propagation. This is the author's version which has not been fully edited and +content may change prior to final publication. Citation information: DOI 10.1109/TAP.2023.3235015 +© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.��See https://www.ieee.org/publications/rights/index.html for more information. +Authorized licensed use limited to: Zhejiang University. Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore. Restrictions apply. + +4 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + + +Fig. 2. EM effective capability with the change of number of sources in four +communication distances. + + +Fig. 3. EM effective capability with the change of aperture sizes in four +communication distances. + + +Fig. 4. The attenuation of EM effective capability with the change of +communication distance. + + +Fig. 5. Schematic diagram of MIMO mode analysis of complex space. The +material, position, quantity and shape of the scatterers can be set arbitrarily in +our algorithm. + +the rooftop or Rao–Wilton–Glisson (RWG) basis, as frequently +utilized in the methods of moments. Different basis +representations of the currents, related to different antenna +designs, will not influence the estimations of the effective +degree of freedom limit. Since we want to focus the analysis in +this work on solving dyadic Green's function and extracting +informatics parameters in a complex space, we choose the +model carefully to avoid mutual coupling. To construct the +basic framework of EMIT, three key parameters (the number of +sources, communication distance, and the size of antenna +aperture) are considered to illustrate the relationship between +RF front-end devices’ design and the effective capability of the +MIMO system. +In Fig. 2, the relationship between the EM effective capacity +and the number of sources is presented at a given aperture +( 6 *6 + + ), showing clearly that with the increase of N , the EM +effective capacity under different communication distances will +increase with the same slope, but it converges to the channel +capacity. In this case, considering that the change of the total +power of the transmitting array will lead to different channel +capacities, we fixed the total transmitting power at +0P , +satisfying +0 +1 +T +N +i +i +P +P += += + +. +In addition, to illustrate the physical nature of the +convergence, Fig. 3 shows the EM effective capacity +corresponding to different aperture sizes with enough point +sources (30*30). Obviously, the size of the aperture plays a +determinant role in the information capacity of MIMO systems. +It suggests that the trend of antenna miniaturization is the +weakening of maximum carrying information, which cannot be +solved by multi-antenna technology. Besides, in Fig. 4, we plot +the curve of EM effective capacity changing with the +communication distance, revealing the characteristics of +wireless +communication-energy +attenuated +with +the +propagation distance from the perspective of dyadic Green’s +function. Besides, in Fig. 3 and Fig. 4, the variables we focus +on are the aperture and distance respectively, so the number of +point sources is a constant, and the total power +0P always +remains a constant. +It is worth mentioning that, due to the basis function +decomposition method (such as Rao-Wilton-Glisson (RWG) +basis in MoM) commonly used in computational +This article has been accepted for publication in IEEE Transactions on Antennas and Propagation. This is the author's version which has not been fully edited and +content may change prior to final publication. Citation information: DOI 10.1109/TAP.2023.3235015 +© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.��See https://www.ieee.org/publications/rights/index.html for more information. +Authorized licensed use limited to: Zhejiang University. Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore. Restrictions apply. + +fwith differentnumber of sources +15.1 +Distance=1*lambda +EM effective capability +Distance=6*lambda +Distance=11*lambda +Distance=16*lambda +10 +15 +20 +25 +30 +Number of sourcesCefr with different aperture +Distance=1*lambda +350 +Distance=6*lambda +Distance=11*lambda +EM effective capability +Distance=16*lambda +250 +200 +150 +00 +50 +Size of aperture ()Cofr with different communication distance +200 +180 +EM effective capability +40 +Communication distance （2)Transmitting array +Modeprofiles +Scatterers +PML +Receiving array +Complexspace5 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + + +Fig. 6. Illustration of the group-T-matrix-based algorithm. The distribution of +the field around the scatterer is decomposed, and the steady-state coefficient +matching is carried out based on the cylindrical wave expansion without +meshwork and time-domain iteration. + +electromagnetics, the specific RF front-end structure can be +decomposed into the sum of point sources through grid +partitioning, and the multi-channel coupling effect will be +considered in the coefficient term of the operator +TR +G +. +Therefore, when using the above method to perform theoretical +modeling of EMIT, the coupling can be characterized by adding +a coefficient term to the operator +TR +G +, and the specific physical +dimensions of the RF front-end can be numerically quantified +by base function equivalence. +Essentially, changing the RF front-end or the channel will +affect the value of +eff +C + in (8) by affecting the distribution of +i + on the ith channel. In other words, +eff +C + and the distribution +of +i + are the inherent property of the communication system. + However, the core assumption of this part is based on free +EM space, and the specific form of coupling operator +TR +G + will +change when numerous scatterers are introduced. The next +section will demonstrate the fast algorithms for characterizing +the complex EM complex space. +III. PROPOSED EMIT-BASED MODEL FOR EM COMPLEX SPACE +In some typical wireless communication scenarios, objects in +complex scattering environments are usually represented by +some types of scatterers for convenient EM calculations, among +which one of the commonly-used classical models is the +cylindrical array, as described in Fig.5. For example, a vehicle- +to-vehicle channel is equivalent to a scattering cluster in the +internet of vehicles channel modeling [16]. Due to the poor +accuracy and long response time of traditional channel +measurement schemes, this section proposes an EMIT-based +model for efficient MIMO characterization in this typical +scattering complex space based on the group T matrix. +A. Algorithm Description +N cylindrical scatterers in MIMO EM propagation space are +considered, which are centered at +( +1,2,..., +) +pr +p +N += + and are +with radius +( +1,2,..., +) +p +a +p +N += +. These parameters can be easily +substituted to simulate different distributions and different +shapes of scatterers. For the description of the RF front-end, we +use a +*1 +s +N + dipole antenna array, coordinate +( +1,2,..., +) +sr s +N += +, +as a convenient MIMO model. The overall algorithm +framework is shown in Fig. 6. To be clear, we focus on the +scenarios where the transceivers and receivers are in the same +horizontal plane (such as vehicle-to-vehicle communication +and indoor point-to-point communication). In this case, we can +regard the scatterer as a cluster of cylindrical scatterers, so +conducting cylindrical wave expansion is reasonable and +convenient. This benefits the convenience of calculation and the +simplicity of the model. +To take the coupling between scatterers into account, we take +the th +q + scatterer as the analysis object and decompose the total +external field +ex +q + + around it into the sum of the incident field +inc +q + + and the scattering field +s +p + of the rest scatterers: + + +1 +. +N +ex +inc +s +q +q +p +p +p q + + + += + += ++ + +(9) + +For solving the scattered fields, the electric field is expanded +as a vector cylindrical wave harmonic function: + + +( ) +( ( +)) +ex +q +q +n +n +n I +Rg +k + + += +− + +q +r +r +, +(10) + +where k is the wave vector, +( ) +q +nI + is the cylindrical wave +coefficient, which is the unknown core quantity for solving the +field distribution. +In addition, the specific mathematical form of cylindrical +wave expansion in (10) is given as follows: + + +(1) +( ( +)) +( +)exp( +) +( ( +)) +( +)exp( +) +n +n +n +n +k +H +k +in +Rg +k +J +k +in + + + + +− += +− +− += +− +p +p +p +p +rr +p +p +rr +r +r +r +r +r +r +r +r +, +(11) + +where r represents the coordinate vector of the field point,  +p +rr +represents the angle between the vectors r and +pr , +nJ is the +Bessel function of order n , +(1) +n +H + is the Hankel function of +order n , and Rg means regularization. Later, we will use the +symbol i to represent the imaginary unit. +For the mode matching, we perform the same cylindrical +wave expansion for the incident field +inc +q + + determined by the +MIMO RF front-end (here is the +*1 +s +N + dipole array), +obtaining: + + +(1) +0 +1 +( +). +4 +s +N +inc +q +s +i H +k + += += +− + +s +r +r + +(12) + +To obtain the same expansion form as (10), the vector +addition theorem is used to further expand (12) to obtain: + +This article has been accepted for publication in IEEE Transactions on Antennas and Propagation. This is the author's version which has not been fully edited and +content may change prior to final publication. Citation information: DOI 10.1109/TAP.2023.3235015 +© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.��See https://www.ieee.org/publications/rights/index.html for more information. +Authorized licensed use limited to: Zhejiang University. Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore. Restrictions apply. + +2 +y +q +x +b +pl +TpI +TPV +yip2 +AV +V +pl +pN +yot +rer +pl +pN +x +x +pl +x +Z +PId +p2 +pN +Scatterer 1 +Scatterer 2 +Scatterer N +Zo ++X06 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + + +Fig. 7. Normalized electric field distribution on the validation plane. (a) (c) (e): +FDTD solver for scatterers distribution of 1*1, 1*5 and 4*1 respectively. (b) +(d) (f): proposed EMIT-based model for scatterers distribution of 1*1, 1*5 and +4*1 respectively. + + +( +) +( +) +(1) +1 +exp +4 +( ( +)). +s +q +N +inc +q +n +n +s +n +i +H +k +in +Rg +k + + + += += +− +− + +− +  +s +s +r r +s +r +r +r +r + +(13) + +In (13), as the RF front-end information is part of prior +knowledge, the expansion coefficient of +inc +q + + is determined, +which is convenient for solving +( ) +q +nI + in (10). Next, we write the +scattering field +s +p + of the +th +p +scatterer as follows: + + +( +) +( ) +0 +n +s +TR +TR +p +dS +i +dS + + + + + + + += + +− + + + + + + +p +p +G +J r +G +M r +, (14) + +where +TR +G + is dyadic Green’s function illustrated in (2),  is +the angular frequency, ( ) +p +J r + and +( ) +p +M r + are the current +density and magnetic current density at the +th +p +scatterer, +respectively. The EM variation in the complex space is +described by the action of the coupling operator +TR +G + on ( ) +p +J r + +and +( ) +p +M r +. When ( ) +p +J r + and +( ) +p +M r + do not exist, (9) will +then degenerate into the free space case shown in section II. +In order to solve the +s +p + + described in (14), we use the +consistent mathematical form of +ex +q + + on different scatterers to +expand +s +p + into cylindrical waveform by using (10), and the +transformation relationship is shown in the red curve in Fig. 6. +Thus, (14) can be rewritten as follows based on the group-T- +matrix: + + +( ) +( ) +( ( +)) +s +p +p +p +m +m +m +mT +I +Rg +k + + += +− + +s +r +r +, +(15) + +where +( +) +p +T + is the group-T-matrix representing the relationship +between the incident field and scattering field of the +th +p +clustered scatterer, and its characteristics are only related to the +shape and material of the current scatterer. Assuming the +internal wave vector of the scatterer is +p +k , the general form of +group-T-matrix in the cylindrical coordinate system can be +obtained by using analytical methods: + + +( ) +(1) +(1) +( +) +( +) +( +) +( +) +( +) +( +) +( +) +p +m +p +p +m +p +p +m +p +p +m +m +p +m +p +p +m +p +p +m +p +p +k J +k a +J +k a +kJ +ka +T +kH +ka +J +k a +H +ka +k J +k a + + +− += + +− +. (16) + +The T-matrix of any shape objects can be solved by +numerical methods such as the method of moments (MoM) +according to (14). +The basic purpose of this section is to illustrate the +algorithm’s efficiency, and thus we consider the model of +dipole array with TM polarized waves incident on a perfect +electric conductor (PEC). In this case, (16) evolves into: + + + +( ) +(1) +( +) . +( +) +m +p +p +m +m +p +J +ka +T +H +ka += − + +(17) + +Substitute (17) into (15) to obtain the field distribution with +( ) +p +m +I + as the only variable. The matrix equation of the unknown +coefficient +( ) +p +m +I + can be obtained by combining (9), (10), (13), +and (15): + + + = +Z I +V . +(18) + +Here, in order to solve the coefficient +( ) +p +m +I +, the equations +with different scatterers are written in matrix form, and the +order of the Bessel function is truncated with the truncation +number +max +N +. Therefore, Z is a square matrix of dimension +max +(2 +1) +N +N ++ +, while V is a ( +) +max +2 +1 +1 +N +N ++ + + vector. The +specific form is: + + ( +) +( +) +( +) +( +) +( +) +1 +, +1 +( ) +(1) +1, +exp +, +p q +q +N +n +p +N +m +p +m +n m +p +q +r r +p +q +T +H +k r +r +i n +m +p +q + +−  ++ +−  ++ +− +− += + +=  +− +− +− + + +Z +,(19) + + ( +) +( +) +( +) +(1) +1 +1 +exp +4 +s +s q +N +n +s +q +r r +q +N +n +s +i +H +k r +r +in +−  ++ += += − +− +− + +V +. +(20) + + +TABLE I +This article has been accepted for publication in IEEE Transactions on Antennas and Propagation. This is the author's version which has not been fully edited and +content may change prior to final publication. Citation information: DOI 10.1109/TAP.2023.3235015 +© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.��See https://www.ieee.org/publications/rights/index.html for more information. +Authorized licensed use limited to: Zhejiang University. Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore. Restrictions apply. + +0.5 +m +0.53 +0.5 +- +0 +0 +0.5 +1 +1.5 +0 +0.5 +1 +1.5 +X (m) +x (m) +(a) +(b) +) +0.5 +() +0.5 +0 +0 +0.5 +1 +1.5 +0 +0.5 +1 +1.5 +x (m) +x (m) +(c) +(d) +3008 +(u) +0.5 +u) +0.5 +0.5 +0 +0.5 +1 +1.5 +0 +0.5 +1 +1.5 +x(m) +X (m) +(e) +(f)7 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +COMPARISON OF CPU TIME AND RMS ERROR BETWEEN FDTD AND +PROPOSED EMIT-BASED MODEL FOR THE CHARACTERISTIC OF COMPLEX +SPACE + + +In (19) and (20), Z is determined by the properties of the +complex space, and V is determined by the properties of the +RF front-end. A joint solution can semi-analytically describe +the evolution of the MIMO coupling operator +EIT +G +. +Therefore, due to the change of EM space coupling operator, +(5) will be rewritten as: + + +EIT +outR +incT += +ψ +G +J +. +(21) + +The subsequent analysis only needs to be carried out in the +same +way +as +(6)-(8) +to +complete +efficient +MIMO +characterization in complex space. It should be noted that, when +facing the time-varying channel scenario, it will be very +convenient to rewrite the coupling operator +EIT +G + into the form +based on the time-domain Green's function. + +B. Numerical Results +To verify the accuracy and efficiency of the proposed EMIT- +based model, numerical calculations of some specific scenarios +are carried out and compared with full-wave simulation results. +Fig. 7 presents the field distributions of three simple arrays. +The effectiveness of the EMIT-based model is verified by +comparing the full-wave FDTD algorithm (a, c, e) with the +proposed algorithm (b, d, f). We consider an EM space of 1 +m*1.5 m, where the MIMO system is modeled as a 3*1 dipole +array with an aperture of 0.75 m. The scatterer array element is +modeled as a metal cylinder with a height of 0.25 m and a radius +of 0.015 m, and the boundary is set as PEC. Due to the dense +mesh division of the full-wave algorithm, its application is +severely limited. However, the proposed semi-analytic +algorithm based on group-T-matrix is suitable for various +frequencies because it does not need mesh division. To obtain +the comparison results, we first define the operating frequency +at 915 MHz in this section. +Fig. 7 illustrates that the proposed EMIT-based model has +achieved good results and can accurately describe the complex +space. In order to further demonstrate its efficiency, the +scatterer distribution was adjusted, and the solving time and +error of the EMIT-based model and FDTD were calculated by +analyzing the field intensity curve at the RF back-end, as shown + +Fig. 8. The total normalized electric field distribution on the validation plane +corresponding to the proposed complex space obtained by EMIT-based model. + +in Table I. It is worth noting that the 10*15 distribution cannot +fully explain the difference between the two algorithms, +because the large number of FDTD meshes converge extremely +fast due to the inability of the electric field to propagate +effectively, and the solutions are often mediocre at this time. +Therefore, we further consider the case of a random array, that +is, randomly removing 60 scatterers from the 10*15 scatterer +distribution. Besides, we clarify that the running time of our +proposed algorithm mainly depends on the number of scatters. +Therefore, the proposed EMIT-based model has higher +computational efficiency than full-wave algorithms like FDTD, +which provides great convenience for the description of +complex space. But generally, the complexity of the real-world +environment increases with the communication distance. In this +case, an efficient way to leverage the EMIT-based method is to +build a common clustering model database. Compared with +pilot-based methods, it also has good efficiency under the +condition of a complete database. For example, a vehicle-to- +vehicle channel is equivalent to a scattering cluster in the +internet-of-vehicles channel modeling [16]. +C. Mode Analysis Step of the EMIT-Based Model +After efficient characterization of the complex space is +verified, the EMIT-based model performs a mode analysis of +the above characterization results to obtain theoretical +interpretations to guide the design of wireless communications. +Consider an actual information transmission scenario where +the RF front-end is a single-polarized dipole antenna array +operating at 2.5GHz (equivalent using an ideal line source +operating at 2.5Ghz), with a scale of 10*1 (designed to make +the MIMO feature more obvious), and the complex space is +simplified to a 4*5 metallic cylindrical scatterer cluster. +According to the quick algorithm in the previous section, the +electric field distribution on the validation plane is shown in +Fig. 8. By substituting the solved coupling operator +EIT +G + into +(6), the EM effective capacity +eff +C + of this model in wireless +communication is known as 5.2, which means that the actual +effective number of available channels is 5. However, Fig. 3 +shows that dyadic Green’s function operators +TR +G + (coupling +This article has been accepted for publication in IEEE Transactions on Antennas and Propagation. This is the author's version which has not been fully edited and +content may change prior to final publication. Citation information: DOI 10.1109/TAP.2023.3235015 +© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.��See https://www.ieee.org/publications/rights/index.html for more information. +Authorized licensed use limited to: Zhejiang University. Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore. Restrictions apply. + +(m) +0.5 +0.5 +0.5 +1.5 +x (m)8 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + + +Fig. 9. Each mode’s normalized electric field diagram on the validation plane +obtained by EMIT-based model. (a-e) Available information transfer modes; +(d-j) Unavailable higher-order information transfer modes. + +operators in free space) will bring channel gains far beyond 5.2. +Therefore, the EMIT-based model provides a convenient tool +to quantitatively explain the influence of the complex +environment on wireless communication quality. The +information at the receiving end in Fig. 8 can help us get the +operator +EIT +G +. Through the SVD mentioned above, 10 modes +at the transmitting end can be decomposed, and the EM +responses of these 10 modes in the complex space are shown in +Fig. 9. It is clearly found that the first five modes successfully +send signals to the receiver effectively in different coupling + +Fig. 10. The distribution of normalized singular values of different number of +sources. + + +Fig. 11. Simple MIMO propagation system in complex space. + +paths. However, the coupling paths of higher-order modes +bypass the receiver’s acceptance range and become unavailable +modes in wireless communication. +To define the concepts of “available” and “unavailable” more +clearly, we show the distribution of modes’ singular values for +the different number of channels in Fig. 10. A formal definition +is given as follows: if all modes are numbered according to the +normalized singular value in a descending order like Fig. 10, +then the available modes are defined as those whose index is +less than the EM effective capacity +eff +C +, and the other modes +are defined as the unavailable modes. Since the power resources +in an actual wireless communication system are limited, the +mode weight corresponding to each channel number is +normalized here. Obviously, for a MIMO system, there will be +an evident truncation of the modes’ singular value distribution, +and modes below the truncation usually become “unavailable”. +The number of “available” modes will be strictly determined by +(8) after the coupling operator +EIT +G + is obtained by the EMIT- +based model. +Obviously, the more channels available, the more +information that can be transmitted, and the greater the channel +capacity of the corresponding EM space. However, +This article has been accepted for publication in IEEE Transactions on Antennas and Propagation. This is the author's version which has not been fully edited and +content may change prior to final publication. Citation information: DOI 10.1109/TAP.2023.3235015 +© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.��See https://www.ieee.org/publications/rights/index.html for more information. +Authorized licensed use limited to: Zhejiang University. Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore. Restrictions apply. + +Model +Mode2 +目 +0.5 +(m +0.5 +0.5 +A +0 +0.5 +1 +1.5 +0 +0.5 +1.5 +x(m) +x (m) +(a) +(q)Mode3 +Mode4 +0.5 +0.5 +0.5 +U +0.5 +1 +1.5 +0 +0.5 +x (m) +1.5 +x (m) +(c) +(d)Mode5 +Mode6 +m +0.5 +0.5 +0.5 +1.5 +0 +0.5 +1.5 +x (m) +x (m) +(e) +(f)Mode7 +Mode8 +0.5 +0.5 +0.5 +1 +1.5 +0 +0.5 +1.5 +x (m) +x (m) +(g) +(h)Mode9 +Mode10 +0.5 +(m +0.5 +0 +0 +0 +0.5 +1.5 +0 +0.5 +1 +1.5 +x (m) +x (m) +(0) +()Singular value distribution +0.9 +0.8 +0.7 +0.5 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +6 +N +mode index +0 +numberofsourceTransmitting Array +Receiving Array +Scattering Region +VNA9 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + + +Fig. 12. The normalized amplitude and phase of +21 +S + in 7*7 MIMO system. (a) +Simulation results in EMIT-based model; (b) Measurement results; (c) The +error of the two above. + +communication resources of the RF front-end are often limited, +so it is important to allocate resources properly to achieve better +information transmission efficiency. In the next section, power +distribution is taken as the background problem to discuss the +guidance significance of the EMIT-based model for real +wireless communication in a complex environment. +IV. EXPERIMENTAL ANALYSIS +It is worth noting that the above discussion on the application +of the EMIT-based model is carried out by simulation. In order +to fully explain the effectiveness of the EMIT-based model and +the application method under the background of wireless +communication, we carried out experimental exploration with +the aid of a 3*1 MIMO system. +The experiment construction is shown in Fig. 11, where the +system is surrounded by the absorption boundary covered with +absorbing materials, and cylindrical metal scatterers with a +height of 0.25 m and a radius of 0.015 m are uniformly +distributed in the EM space with 4*5 arrays. The transmitting +sources and the receiving fields were replaced by dipole +antennas with a center frequency of 2.5 GHz and a gain of 2dBi. +The vector network analyzer (VNA) is used to measure the +21 +S + +between transmitting and receiving dipoles through the coaxial +feed. Since the measurement of channel matrix elements is +concerned with the single excitation properties of MIMO, we +replace the actual MIMO system by changing the spatial +position of the antenna in the transmitting aperture (shown as + +Fig. 13. The field distribution of three orthogonal modes at the receiver aperture. +The locations of the three sources are indicated by black dotted lines on the +diagram. + +the dotted white lines). Therefore, in the experimental design, +we selected the weak coupling scenario with the antenna +spacing as half-wavelength, and then measured the antenna +excitation separately to avoid the impact of coupling on the +verification of our EMIT-based model. +For the same scene, we performed effective characterization +with the EMIT-based model, and the characterization results are +shown in Fig. 12. Fig. 12 (a) and (b) respectively represent the +amplitude and phase comparison results between the simulation +results +of +EMIT-based model +and +the +experimental +measurement +values. +The +experimental +results +fully +demonstrate the effectiveness of the EMIT-based model in +complex space characterization. The purpose of conducting 7*7 +channel measurement in our experiment is to verify the EMIT- +based model more convincingly. However, the following is +mainly to illustrate how the EMIT-based model guides the RF +front-end signal transmission. Therefore, to simplify the +demonstration process, we select three groups of data evenly +spaced to form a new 3*3 MIMO system. In fact, the selection +of 3*3 channel positions is arbitrary. However, in this paper, to +make the mode orthogonality more significant and avoid the +influence of mode crosstalk on the transmitting strategy, three +positions with relatively small mode crosstalk are selected, and +the crosstalk matrix CT is as follows: + + +15 +15 +15 +15 +1 +0.1857 +4.03*10 +0.1857 +1 +6.51*10 +4.03*10 +6.51*10 +1 +− +− +− +− + + + + +=  + + + + + +CT +. +(22) + +Consider the following problems in an actual wireless +communication scenario: 3 * 3 MIMO system needs to conduct +data transmission in disorder EM space (with the standard +deviation of noise  ), the maximum transmitted power of RF +front-end is +0P . Under this constraint, since it is a very +important subject to consider the optimal power distribution, +which is related to whether the upper bound for capacity can be +achieved, the coupling operator +EIT +G + obtained by the EMIT- +This article has been accepted for publication in IEEE Transactions on Antennas and Propagation. This is the author's version which has not been fully edited and +content may change prior to final publication. Citation information: DOI 10.1109/TAP.2023.3235015 +© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.��See https://www.ieee.org/publications/rights/index.html for more information. +Authorized licensed use limited to: Zhejiang University. Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore. Restrictions apply. + +Amplitude +Phase +180 +0.5 +0 +-180 +(a) +Amplitude +Phase +180 +0.5 +0 +0 +-180 +(b) +Error +Error +0.5 +0 +0 +-180 +(c)0.16 +Port 1 +Port 2 +Port 3 +0.14 +- +Mode 1 +- +- +Mode 2 +0.12 +Mode 3 +0.1 +0.08 +0.06 +0.04 +0.02 +0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Receiving Position (m)10 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + + +Fig. 14. Image transmission case based on EMIT-based model. (a) Original +image with 128*128 pixels. (b) Optimized power distribution. (c)-(e) +Conducting single-mode transmission using mode 1, mode 2 and mode 3 +respectively. + +based model is used to endow the shape of the transmitted +signal to obtain the best quality of information transmission. +Rewrite (5-6) to obtain a new coupling equation based on the +scenario we’re considering: + + += +† +† +EIT +EIT +U +Y +SV +X , +(23) + +where +EIT +U + and +EIT +V + are determined by +EIT +G + to guide the +signal processing of transmitter and receiver, respectively. X +and Y represent the signal form of transmitter and receiver, +respectively. The singular value matrix S is disassembled to +obtain the received signal evaluation function f : + +3 +1 +m +m +m +f +V + + += +=  +. +(24) + +In (24), X is decomposed as a bitstream of information X +(here we use simple binary phase-shift keying (BPSK) +modulation) multiplied by the excitation coefficient  , and +the unitary matrix +EIT +V + is decomposed into three-mode vectors +( +1,2,3) +m +V +m = +. Since there are three sources in our MIMO +system, both +m +V + and + + here have three elements. +( +1,2,3) +m m + += + are the diagonal elements of the singular value +matrix S , representing the influence of each mode on the +receiving end. To have a clearer understanding of +m + +, we +depicted the electric field distribution at the receiving end with +the help of the EMIT-based model, as shown in Fig. 13. The +calculated proportions of the three modes are 67.55%, 23.19%, +and 9.25%, respectively. Mode 1 with the strongest proportion +just contributes its crest to the receiving end, while mode 3 with +the weakest proportion just contributes its trough to the +receiving end. This provides a clear perspective for signal +waveform design from the EM point of view, and reveals that +EM space is not the only factor determining mode contribution, +and EM space characteristics and RF front-end characteristics +should be considered together. +According to the crosstalk matrix calculated in (22), we treat +these three modes as orthogonal. Therefore, +m +V + becomes a +set of orthogonal basis in a Hilbert space, meeting +† +0 +m +n +V +V += + +and +† +1 +m +m +V +V += . Hence,  can be written as an orthogonal +basis expansion: + + +3 +1 +m +m +m +V + + += +=  +, +(25) + +where +m + + represent the corresponding weight of each basis +vector, which determines the power distribution on the +transmitting source. Therefore, the constraint of constant total +power +0P can be equivalent to that the excitation vector is +located on a fixed circle in the Hilbert space, and the received +signal evaluation function f is the sum of the weighted +projections of the excitation vector on the three basis functions: + + +0 +find : +( +1,2,3) +max : +. .: +m +m +m +f +s t +P + + + += + + += + + +. +(26) + +The optimization problem can be easily solved by using +Cauchy inequality. By substituting (8), the information transfer +function can reach the maximum value only when +/ +m +m + + is a +constant for different m . This is similar to the “water-filling” +algorithm in channel estimation, while the core difference is +that the key informatics parameters in this paper are deduced by +an effective EM algorithm. In addition, +† +TR +TR +G +G + or +† +EIT +EIT +G +G + +have the same physical meaning as the transmit signal +covariance matrix, and the main difference is that +† +TR +TR +G +G + or +† +EIT +EIT +G +G + is calculated by the EM methods based on dyadic +Green’s function. +It is seen from the mode analysis based on the EMIT-based +model that only under the guidance of a specific power +allocation strategy, MIMO information transmission can +achieve the effect of receiving power equal to transmitting +power times path loss. To fully illustrate the guiding +significance of the EMIT-based model for power distribution +(essentially waveform design), Fig. 14 shows an image +transmission case. BPSK is used to discretize every pixel in the +picture into an 8-bit data stream for transmission, and noise  +is joined to EM space. Obviously, under the premise of not +processing channel noise, the RF front-end working strategy +based on the EMIT-based model is much better than other +transmission modes. Therefore, the EMIT-based model can not +only efficiently represent the complex space, but also make +more valuable guidance for wireless communication. + +V. CONCLUSION +In this article, an EMIT-based model is presented to simulate +the performance of MIMO systems in complex EM complex +space effectively. Firstly, the EM expression of the information +coupling operator is given in the free space, and two key +informatics parameters, EM effective capacity and path loss, +This article has been accepted for publication in IEEE Transactions on Antennas and Propagation. This is the author's version which has not been fully edited and +content may change prior to final publication. Citation information: DOI 10.1109/TAP.2023.3235015 +© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.��See https://www.ieee.org/publications/rights/index.html for more information. +Authorized licensed use limited to: Zhejiang University. Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore. Restrictions apply. + +(a) +(b) +(c) +(p) +(e)11 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +are extracted from the EM perspective. It is proved that the +MIMO antennas’ aperture is the critical factor in the EM +effective capacity of the MIMO system. Secondly, the basic +principle of the EM representation method in complex space is +given, and several typical scenarios are analyzed, which proves +the accuracy and efficiency of the EMIT-based model proposed. +The results show that the EMIT-based model can reliably +analyze the electromagnetic space about 10% of the time +compared to the full-wave simulation. Finally, the MIMO +performance in real propagation scenarios is calculated using +the EMIT-based model. The experimental results verify that the +channel matrix calculated is in good agreement with the +measured ones. Based on this, it is pointed out how the EMIT- +based model can effectively guide the MIMO design and +feeding in a power distribution question. +The ultimate goal of the proposed EMIT-based model is to +advance the development of EMIT and demonstrate a new idea +of extracting information parameters to the antenna & +propagation community using the basis of computational +electromagnetism. Currently, it is suitable for cluster models +with arbitrary distribution, size, and material, providing an +efficient and reliable method for guiding the power and phase +allocation of antenna units in scattering complex space. 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Personal use is permitted, but republication/redistribution requires IEEE permission.��See https://www.ieee.org/publications/rights/index.html for more information. +Authorized licensed use limited to: Zhejiang University. Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore. Restrictions apply. + +12 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +[32] M. Lee, M. A. Neifeld, and A. Ashok, "Capacity of electromagnetic +communication modes in a noise-limited optical system," Applied optics +(2004), vol. 55, no. 6, p. 1333, 2016-01-01. 2016. +[33] S. S. A. Yuan, Z. He, X. Chen, C. Huang, and W. E. I. Sha, +"Electromagnetic Effective Degree of Freedom of an MIMO System in +Free Space," IEEE Antennas and Wireless Propagation Letters, vol. 21, +no. 3, pp. 446-450. 2022. +[34] T. K. Sarkar and M. Salazar-Palma, "MIMO: Does It Make Sense From an +Electromagnetic Perspective and Illustrated Using Computational +Electromagnetics?" 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He is currently pursuing +the Ph.D. degree at the College of Information Science +and Electronic Engineering, Zhejiang University. +His +current +research +interests +include +the +electromagnetic +information +theory +for +wireless +communication, and efficient calculation methods +applied in MIMO antennas. + + +Da Li received the B.S. degree in 2014, and the Ph.D. +degree in 2019, from Zhejiang University, Hangzhou, +China, both in electrical engineering. From 2017 to +2018, he worked at Nanyang Technological University, +Singapore, as a Project Researcher. From 2019 to 2021, +he joined Science and Technology on Antenna and +Microwave Laboratory, Nanjing, China, as a Research +Fellow. He is currently an assistant professor at Zhejiang +University. His research interests include machine +learning, antennas, matesurfaces, and electromagnetic compatibility. Dr. Li +has authored or coauthored more than 40 refereed papers and served as +Reviewers for 6 technical journals and TPC Members of 3 IEEE conferences. +He was also a recipient of the Outstanding Young Scientist Award at 2022 +Asia-Pacific International Symposium on Electromagnetic Compatibility. + +Jinyan Ma received the B.S. degree in engineering from +Zhejiang University, Hangzhou, China, in 2021. He is +currently working toward the Ph.D. degree in electronics +science and technology with the College of Information +Science and Electronic Engineering, Zhejiang University, +Hangzhou, China. +His +current +research +interests +include +the +electromagnetic information theory and efficient +electromagnetic calculation methods. + + +Zhaoyang Feng received the B.Sc degree from North +China Electric Power University, Beijing, China, in 2017. +He is currently working toward the Ph.D. degree in the +College +of +Information +Science +and +Electronic +Engineering, Zhejiang University, Hangzhou, Zhejiang. +His current research interests include electromagnetic +compatibility, +computational +electromagnetics and +multiple scattering theory + + + +Ling Zhang (Member, IEEE) received the B.S. degree in +electrical engineering from Huazhong University of +Science and Technology, Wuhan, China, in 2015, and the +M.S. and Ph.D. degrees from Missouri S&T, Rolla, MO, +USA, in 2017 and 2021, respectively, both in electrical +engineering. He was with Cisco as a student intern from +Aug. 2016 to Aug. 2017. He joined Zhejiang University, +Hangzhou, China as a research fellow in 2021. He has +authored and co-authored more than 30 journal and +conference papers. His research interests include machine learning, power +integrity, electromagnetic interference, radio-frequency interference, and signal +integrity. +Dr. Zhang was an Organizing Committee, Special Session Chair, Workshop +Session Chair, and Poster Session Chair in APEMC 2022. He has given invited +presentations at the IBIS Summit at 2021 IEEE Virtual Symposium on +EMC+SIPI, and the 2021 Virtual Asian IBIS Summit China. He was the +recipient of the Honorable Mention Paper in APEMC 2022, the Best Paper +Award in DesignCon 2019, and the Student Paper Finalist Award in ACES +Symposium in 2021. He was also the recipient of the Outstanding Young +Scientist Reward in APEMC 2022. + +Shurun Tan (S’14-M’17) received the B.E. degree in +information +engineering +and +M.Sc. +degree +in +electromagnetic field and microwave techniques from +the Southeast University, Nanjing, China, in 2009 and +2012, respectively, and the Ph.D. degree in electrical +engineering from the University of Michigan, Ann +Arbor, MI, USA, in Dec. 2016. +Dr. Tan is an assistant professor in the Zhejiang +University / University of Illinois at Urbana-Champaign +Institute located at the International Campus of Zhejiang University, Haining, +China. He is also affiliated with the State Key Laboratory of Modern Optical +Instrumentation, and the College of Information Science and Electronic +Engineering, Zhejiang University, Hangzhou, China. He is also an adjunct +assistant professor in the Department of Electrical and Computer Engineering, +University of Illinois at Urbana-Champaign, Urbana, USA. From Dec. 2010 to +Nov. 2011, he was a Visiting Student with the Department of Electrical and +Computer Engineering, the University of Houston, Houston, TX, USA. From +Sep. 2012 to Dec. 2014, he was a PhD candidate with the Department of +Electrical Engineering, the University of Washington, Seattle, WA, USA. From +Jan. 2015 to Dec. 2018, he had been affiliated with the Radiation Laboratory, +and the Department of Electrical Engineering and Computer Science, the +University of Michigan, Ann Arbor, first as a PhD candidate, and then as a +postdoctoral research fellow since Jan. 2017. +Dr. Tan is working on electromagnetic theory, computational and applied +electromagnetics. His research interests include electromagnetic scattering of +random media and periodic structures, microwave remote sensing, +electromagnetic information systems with electromagnetic wave-functional +devices, electromagnetic integrity in high-speed and high-density electronic +integration, electromagnetic environment and reliability of complex electronic +systems, etc. +This article has been accepted for publication in IEEE Transactions on Antennas and Propagation. This is the author's version which has not been fully edited and +content may change prior to final publication. Citation information: DOI 10.1109/TAP.2023.3235015 +© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.��See https://www.ieee.org/publications/rights/index.html for more information. +Authorized licensed use limited to: Zhejiang University. Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore. Restrictions apply. + +13 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + + + +Wei E. I. Sha (M’09-SM’17) received the B.S. and Ph.D. +degrees in Electronic Engineering at Anhui University, +Hefei, China, in 2003 and 2008, respectively. From Jul. +2008 to Jul. 2017, he was a Postdoctoral Research +Fellow and then a Research Assistant Professor in the +Department of Electrical and Electronic Engineering at +the University of Hong Kong, Hong Kong. From Mar. +2018 to Mar. 2019, he worked at University College +London as a Marie Skłodowska-Curie Individual Fellow. +From Oct. 2017, he joined the College of Information Science & Electronic +Engineering at Zhejiang University, Hangzhou, China, where he is currently a +tenure-tracked Assistant Professor. +Dr. Sha has authored or coauthored 180 refereed journal papers, 150 +conference publications (including 5 keynote talks and 1 short course), 9 book +chapters, and 2 books. His Google Scholar citation is 8193 with h-index of 45. +He is a senior member of IEEE and a member of OSA. He served as Reviewers +for 60 technical journals and Technical Program Committee Members of 10 +IEEE conferences. He also served as Associate Editors of IEEE Journal on +Multiscale and Multiphysics Computational Techniques, IEEE Open Journal of +Antennas and Propagation, and IEEE Access. In 2015, he was awarded Second +Prize of Science and Technology from Anhui Province Government, China. In +2007, he was awarded the Thousand Talents Program for Distinguished Young +Scholars of China. He was the recipient of ACES Technical Achievement +Award 2022 and PIERS Young Scientist Award 2021. Dr. Sha also received 6 +Best Student Paper Prizes and one Young Scientist Award with his students. +His research interests include theoretical and computational research in +electromagnetics and optics, focusing on the multiphysics and interdisciplinary +research. His research involves fundamental and applied aspects in +computational and applied electromagnetics, nonlinear and quantum +electromagnetics, micro- and nano-optics, optoelectronic device simulation, +and multiphysics modeling. + + + +Hongsheng Chen received the B.S. and Ph.D.degrees in +electrical engineering from Zhejiang University (ZJU), +Hangzhou, China, in 2000 and 2005, respectively. +In 2005, he became an Assistant Professor with ZJU, +where he was an Associate Professor in 2007 and a Full +Professor in 2011. +He was a Visiting Scientist from 2006 to 2008 and a +Visiting Professor from 2013 to 2014 with the Research +Laboratory of Electronics, Massachusetts Institute of +Technology, Cambridge, MA, USA. He is currently a Chang Jiang Scholar +Distinguished Professor with the Electromagnetics Academy, ZJU. He has +coauthored more than 200 international refereed joumal papers. His works have +been highlighted by many scientific magazines and public media, including +Nature, Scientific American, MIT Technology Review, Physorg, and so on. His +current research interests include metamaterials, invisibility cloaking, +transformation optics, graphene, and theoretical and numerical methods of +electromagnetics. +Dr. Chen serves as a Regular Reviewer for many international journals on +electromagnetics, physics, optics, and electrical engineering. He serves as a +Topical Editor for the Journal of Optics and the Editorial Board for Nature’s +Scientific Reports and Progress in Electromagnetics Research. He was a +recipient of the National Excellent Doctoral Dissertation Award in China in +2008, the Zhejiang Provincial Outstanding Youth Foundation in 2008, the +National Youth Top-Notch Talent Support Program in China in 2012, the New +Century Excellent Talents in University of China in 2012, the National Science +Foundation for Excellent Young Scholars of China in 2013, and the National +Science Foundation for Distinguished Young Scholars of China in 2016. His +research work on an invisibility cloak was selected in Science Development +Report as one of the representative achievements of Chinese Scientists in 2007. + +Er-Ping Li (S’91, M’92, SM’01, F’08) is currently a +Qiushi-Distinguished Professor with Department of +Information Science and Electronic Engineering, +Zhejiang University, China; served as Founding Dean +for Institute of Zhejiang University - University of +Illinois at Urbana-Champaign in 2016. From 1993, he has +served as a Research Fellow, Associate Professor, +Professor and Principal Scientist and Senior Director at +the Singapore Research Institute and University. Dr Li +authored or co-authored over 400 papers published in the referred international +journals, authored two books published by John-Wiley-IEEE Press and +Cambridge University Press. He holds and has filed a number of patents at the +US patent office. His research interests include electrical modeling and design +of micro/nano-scale integrated circuits, 3D electronic package integration. +Dr. Li is a Fellow of IEEE, and a Fellow of USA Electromagnetics Academy, +a Fellow of Singapore Academy of Engineering. He is the recipient of IEEE +EMC Technical Achievement Award in 2006, Singapore IES Prestigious +Engineering Achievement Award and Changjiang Chair Professorship Award +in 2007, 2015 IEEE Richard Stoddard Award on EMC, 2021 IEEE EMC +Laurence G. Cumming Award and Zhejiang Natural Science 1st Class Award. +He served as an Associate Editor for the IEEE MICROWAVE AND +WIRELESS COMPONENTS LETTERS from 2006-2008 and for IEEE +TRANSACTIOSN on EMC from 2006-2021, Guest Editor for 2006 and 2010 +IEEE TRANSACTIOSN on EMC Special Issues, Guest Editor for 2010 IEEE +TRANSACTIONS on MTT APMC Special Issue. He is currently an Associate +Editor for the IEEE TRANSACTIONS ON SIGNAL and POWER +INTEGRITY and Deputy Editor in Chief of Electromagnetics Science. He has +been a General Chair and Technical Chair, for many international conferences. +He was the President for 2006 International Zurich Symposium on EMC, the +Founding General Chair for Asia-Pacific EMC Symposium, General Chair for +2008, 2012, 2016, 2018, 2022 APEMC, and 2010 IEEE Symposium on +Electrical Design for Advanced Packaging Systems. He has been invited to give +120 invited talks and plenary speeches at various international conferences and +forums. + + +This article has been accepted for publication in IEEE Transactions on Antennas and Propagation. This is the author's version which has not been fully edited and +content may change prior to final publication. Citation information: DOI 10.1109/TAP.2023.3235015 +© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.��See https://www.ieee.org/publications/rights/index.html for more information. +Authorized licensed use limited to: Zhejiang University. Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore. Restrictions apply. + diff --git a/B9E5T4oBgHgl3EQfTg8R/content/tmp_files/load_file.txt b/B9E5T4oBgHgl3EQfTg8R/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2516f4f261c81efe7e83152c1322199074e8415c --- /dev/null +++ b/B9E5T4oBgHgl3EQfTg8R/content/tmp_files/load_file.txt @@ -0,0 +1,1147 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf,len=1146 +page_content='1 > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < An Electromagnetic-Information-Theory Based Model for Efficient Characterization of MIMO Systems in Complex Space Ruifeng Li, Da Li, Member, IEEE, Jinyan Ma, Zhaoyang Feng, Ling Zhang, Member, IEEE, Shurun Tan, Member, IEEE, Wei E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Sha, Senior Member, IEEE, Hongsheng Chen, Fellow, IEEE, and Er-Ping Li, Fellow, IEEE Abstract—It is the pursuit of a multiple-input-multiple-output (MIMO) system to approach and even break the limit of channel capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' However, it is always a big challenge to efficiently characterize the MIMO systems in complex space and get better propagation performance than the conventional MIMO systems considering only free space, which is important for guiding the power and phase allocation of antenna units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In this manuscript, an Electromagnetic-Information-Theory (EMIT) based model is developed for efficient characterization of MIMO systems in complex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The group-T-matrix-based multiple scattering fast algorithm, the mode-decomposition-based characterization method, and their joint theoretical framework in complex space are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Firstly, key informatics parameters in free electromagnetic space based on a dyadic Green’s function are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Next, a novel group-T-matrix-based multiple scattering fast algorithm is developed to describe a representative inhomogeneous electromagnetic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' All the analytical results are validated by simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In addition, the complete form of the EMIT-based model is proposed to derive the informatics parameters frequently used in electromagnetic propagation, through integrating the mode analysis method with the dyadic Green’s function matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Finally, as a proof-or-concept, microwave anechoic chamber measurements of a cylindrical array is performed, demonstrating the effectiveness of the EMIT-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Meanwhile, a case of image transmission with limited power is presented to illustrate how to use this EMIT-based model to guide the power and phase allocation of antenna units for real MIMO applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Index Terms—multiple-input-multiple-output (MIMO) system, complex space, group T matrix, mode analysis, electromagnetic information theory (EMIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' INTRODUCTION YPICALLY, for antenna design, it is promising to maximize the channel capacity via a multiple-input- multiple-output (MIMO) system to approach the limit of channel capacity during propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Thus, Under the demand for high accuracy and low latency nowadays, the basic research on efficient characterization of MIMO systems is very important [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' On this basis, we can carry out further work such as the optimization solutions for the power and phase allocation of antenna units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Previous works for MIMO characterization can be roughly clarified into two categories: electromagnetic (EM) methods This project is supported in part by Natural Science Foundation of China (NSFC), Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 62071424, 62201499 and 62027805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (Corresponding Author: Da Li, li-da@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='cn) and information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The former mainly focuses on the radio-frequency (RF) front-end design by solving Maxwell’s equations under different boundary conditions, consisting of the descriptions of the complex electromagnetic space [2], [3], while the latter mainly analyzes the channel properties under different probability models by using Shannon information theory [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The above two frameworks are faced with a major challenge in practical application: how to efficiently model MIMO systems in complex space to achieve better propagation performance than MIMO analysis that only consider free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' For EM methods, the core step of intelligent designs nowadays is reconstructing the MIMO systems’ radiation patterns [6], [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' To consider the effect of the EM propagation space, full-wave numerical algorithms have been used to incorporate the RF front-end design and environment perception into the EM framework, consuming a lot of time [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' To greatly reduce the calculation time of modeling EM space, some studies have proposed to use approximate methods like ray tracing (RT) [10], [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' However, this is often not acceptable due to lack of high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Besides, the T-matrix can be easily used to characterize efficient MIMO in complex EM space, via combining multiple scattering equations [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' However, practical wireless communication often focuses on some informatic parameters (such as channel capacity), while the EM-only framework is incapable of efficiently extracting the informatic parameters in the complex EM space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' For information theory, the common statistic model, such as the Rayleigh fading model, is a mathematical tool based on the assumption of rich scattering [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' When it evolves to cluster models like geometry-based stochastic models (GBSMs), the EM space is equivalent to the clusters with different shapes or distributions for convenient characterization [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' However, the accuracy of those models will be reduced due to the EM properties of the MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' For example, the work in [18] complements numerical methods to make up for the problem of using only Fresnel approximation in airborne antenna design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Moreover, the main idea of the emerging intelligent reflective surface (IRS) is to lay out the controllable The authors are with ZJU-UIUC Institute, Zhejiang Provincial Key Laboratory of Advanced Microelectronic Intelligent Systems and Appli- cations, and the College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' T This article has been accepted for publication in IEEE Transactions on Antennas and Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=" This is the author's version which has not been fully edited and content may change prior to final publication." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='1109/TAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='3235015 © 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Personal use is permitted, but republication/redistribution requires IEEE permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='��See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='org/publications/rights/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='html for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Authorized licensed use limited to: Zhejiang University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2 > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' System model of MIMO analysis in a complex space for indoor communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' surfaces in free or complex EM space [19], [20], [21], [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Due to the lack of efficient MIMO characterization, this technology is still in the trial stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' This suggests that many basic assumptions of the information-only framework need to be reconsidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Nowadays, the electromagnetic information theory (EMIT) for the MIMO characterization attracts more attentions, which is expected to solve the challenges mentioned above [24], [25], [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Researchers point out that with the wide layout of the antenna array (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=', Internet of vehicles), it is expected to eliminate the step of channel estimation with the help of rich environmental information [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Some works have been done from the perspective of EM fields to study the degree of freedom of MIMO systems [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' There are also mathematical methods to model the source region and field region as two sets of orthogonal bases in Hilbert space, and then construct some characteristic parameters of the MIMO system [31], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' To integrate the RF front-end design in the EMIT framework, the surface currents of antenna elements are modeled as the point sources with orthogonal bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' For example, a model was established to build a channel matrix from the angle of coordinate transformation and orthogonal decomposition of EM plane wave expansion, applied in holographic MIMO system [33], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Additionally, the work in [35] contains the idea of deriving the channel limit of a MIMO system by the EM field method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Nevertheless, the above research works on EMIT mainly focus on free space or revealing the parameter mapping between two theories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' efficient characterization algorithms and clear EM information analysis methods for complex EM space are still unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In this paper, we develop an EMIT-based model to conduct the efficient characterization for MIMO systems in complex EM space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The proposed EMIT-based model uses the group T matrix algorithm and dyadic Green’s function-based mode analysis method, filling the research gap of efficient characterization algorithms and clear EM information analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The main contributions of this paper are described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 1) The key parameters of the MIMO systems are extracted through the dyadic Green’s function and matrix mode analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The information characteristics of the MIMO systems are described by the EM method, revealing some important conclusions and deducing the key informatic parameters and valuable conclusions of information theory by means of EM methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2) A fast algorithm based on the group T matrix is developed to model the complex EM space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Since the algorithm has semi-analytical characteristics and the classical T matrix can be stored, which provides a faster calculation compared with the traditional full-wave algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In contrast to the RT and pilot-based methods for channel estimation, our EM algorithm can be easily integrated into EMIT due to its higher accuracy and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In other words, the benefit of our proposed method is generated from the fast characteristics of the group T matrix and the EM analysis of the channel matrix (without the help of statistics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 3) The efficient EMIT-based model is proposed to characterize the MIMO systems in complex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' As a proof-of-concept, a microwave anechoic chamber measurement of a cylindrical array is taken as an example, demonstrating the effectiveness of the EMIT- based model for the MIMO mode analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Meanwhile, a case of image transmission with limited power is presented to illustrate how to guide the MIMO feeding based on the model, bringing a new insight into extracting information parameters using the basis of computational electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' This article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The key informatics parameters based on the dyadic Green’s function are derived in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Then, the proposed EMIT-based model is analyzed in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Experimental verification and an image transmission case were conducted in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Finally, the conclusion is drawn in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' SYSTEM MODEL AND KEY PARAMETERS As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 1, consider a typical MIMO system including the transmitting and receiving array for indoor communication, whose overall communication performance will be affected by the propagation distance and the properties of complex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In this section, the EM propagation space is designated as a free space for extracting key parameters of a MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' More complex EM space is characterized in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' To combine the coupling operator TR G and the channel matrix \uf048 , a series of isotropic point sources are placed in the transmission volume and the receiving volume, with the position vectors Tr and Rr respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The EM wave received is defined as outR ψ , thus the Helmholtz wave equation is given by 2 0 0 outR outR incT k i\uf077\uf06d \uf0d1\uf0b4\uf0d1\uf0b4 − = ψ ψ J , (1) where incT J is the transmitted source and k is the wave vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' To solve this equation, the dyadic Green’s function G operator based on the impulse function idea is introduced: 2 exp[ ] ( , ) 4 R T R T R T ik k \uf070 − \uf0e6 \uf0f6 \uf0d1\uf0d1 = + \uf0e7 \uf0f7 − \uf0e8 \uf0f8 r r G r r I r r , (2) This article has been accepted for publication in IEEE Transactions on Antennas and Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=" This is the author's version which has not been fully edited and content may change prior to final publication." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='1109/TAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='3235015 © 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Personal use is permitted, but republication/redistribution requires IEEE permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='��See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='org/publications/rights/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='html for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Authorized licensed use limited to: Zhejiang University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Coupling operatorGr Channel matrix H Receiving array Electromagnetic characteristic Information characteristic Complex Space Feeding& Beamforming3 > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < where I is the unit tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Since the dyadic Green’s function tensor G contains the scalar Green’s functions: Gxx Gxy Gxz Gyx Gyy Gyz Gzx Gzy Gzz \uf0e9 \uf0f9 \uf0ea \uf0fa = \uf0ea \uf0fa \uf0ea \uf0fa \uf0eb \uf0fb G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (3) When it comes to the two independent single-polarization situation at far field, the coupling operator is able to be simplified into the scalar Green’s function without loss of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Therefore, the element ij h in the channel matrix \uf048 will changed to the following form in the case of a single polarized source: 0( ) exp[ ] 4 Ri Tj ij ij Ri Tj ik h g \uf070 − = = − r r r r , (4) where 0 g is the scalar Green’s function, that is, the special form of G in the case of single polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' It is worth mentioning that the channel matrix \uf048 at this time is not normalized, so it contains the path loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Assume that the number of transmitting source points is T N , and the number of receiving field points is R N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Therefore, the transmitting source can be expressed as a 1 T N matrix, the receiving electric field as a 1 R N matrix, and the coupling operator G of the EM space as a R T N N matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' By introducing the Dirac notation, the MIMO propagation relation of free space is expressed as: outR incT = ψ G J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (5) To normalize the channel matrix, we defined the normalized coupling operator TR G as TR \uf061 = G G , where \uf061 is a normalization factor, making 2 TR T R F N N \uf0e9 \uf0f9 \uf045 = \uf0ea \uf0fa \uf0eb \uf0fb G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' \uf05b \uf05d \uf045 denotes the expectation and F means the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The physical meaning of this normalization is that every sub- channel should have a unity average channel gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' According to the Hermitian nature of † TR TR G G , singular value decomposition (SVD) of the coupling operator could conduct mode analysis of MIMO EM propagation, where † denotes the conjugate transpose: † TR R T = G U SV , (6) where T V ( R U ) is a T T N N ( R R N N ) matrix, and each column represents the EM eigenvector of the transmitting sources (receiving fields ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Because of the unitary nature of the SVD eigenmatrix, it is known that each column is strictly orthogonal, which is called the EM space mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The weight of each pattern is determined by the corresponding element in the diagonal matrix S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The combinations of those orthogonal modes form two Hilbert spaces, and therefore the coupling operator TR G builds a mapping between the transmitting Hilbert space and the receiving Hilbert space, which is an important property in the subsequent discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' As we all know, the upper limit of information transmission per bandwidth in MIMO systems is also limited by Shannon’s formula [36]: † 2 2 2 1 log det log 1 TR TR t i i C n N n \uf072 \uf072 \uf073 \uf0ec \uf0fc \uf0e9 \uf0f9 \uf0e6 \uf0f6 \uf0ef \uf0ef = \uf045 + \uf0ea \uf0fa \uf0ed \uf0fd \uf0e7 \uf0f7 \uf0ea \uf0fa \uf0e8 \uf0f8 \uf0ef \uf0ef \uf0eb \uf0fb \uf0ee \uf0fe \uf0e6 \uf0f6 = + \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e5 I G G , (7) where I is the identity matrix and i \uf073 are the singular values of( ) 1/ TR T N G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Apparently, 2 i \uf073 is the decisive parameter of key information-carrying capacity in the MIMO system at a given SNR / n \uf072 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Besides, we can drop the expectation \uf05b \uf05d \uf045 in (7) and no longer need to make a special distinction for large- scale and small-scale path loss and fading, because the amplitude and phase changes of the electric field have been included in the operator TR G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' It is seen from (6) and (7) that the singular value of EM propagation space determines the number and weight of independent modes, which establishes a corresponding relationship with the number of independently available channels and path loss of wireless communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' We give the key informatics parameters of a MIMO system by referring to the effective rank idea of existing work [21]: min( , ) 1 exp( ln( )) R T N N eff i i i C \uf073 \uf073 = \uf0a2 \uf0a2 = − \uf0d5 , (8) where eff C represents the EM effective capacity, / ( ) i i i \uf073 \uf073 \uf073 \uf0a2 = \uf0e5 represents the normalized singular values of TR G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Hence, (8) establishes the mapping relationship between the dyadic Green’s function matrix and typical informatics parameters, which is an important tool for the MIMO mode analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' To understand how this approach works, both mathematically and physically, we set up an N N MIMO system with the same EM space properties, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In fact, in practical engineering applications, the mutual coupling is concerned not because it affects the EM equivalent capacity, but because it affects the radiation efficiency and signal-to-noise ratio of the antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The transmitting and receiving antennas are modeled as isotropic point sources/receivers (delta function basis), which is a widely used assumption in EM information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' From the EM perspective, the antennas can also be modeled as continuous surface (equivalent) currents by This article has been accepted for publication in IEEE Transactions on Antennas and Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=" This is the author's version which has not been fully edited and content may change prior to final publication." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='1109/TAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='3235015 © 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Personal use is permitted, but republication/redistribution requires IEEE permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='��See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='org/publications/rights/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='html for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Authorized licensed use limited to: Zhejiang University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 4 > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' EM effective capability with the change of number of sources in four communication distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' EM effective capability with the change of aperture sizes in four communication distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The attenuation of EM effective capability with the change of communication distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Schematic diagram of MIMO mode analysis of complex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The material, position, quantity and shape of the scatterers can be set arbitrarily in our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' the rooftop or Rao–Wilton–Glisson (RWG) basis, as frequently utilized in the methods of moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Different basis representations of the currents, related to different antenna designs, will not influence the estimations of the effective degree of freedom limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=" Since we want to focus the analysis in this work on solving dyadic Green's function and extracting informatics parameters in a complex space, we choose the model carefully to avoid mutual coupling." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' To construct the basic framework of EMIT, three key parameters (the number of sources, communication distance, and the size of antenna aperture) are considered to illustrate the relationship between RF front-end devices’ design and the effective capability of the MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2, the relationship between the EM effective capacity and the number of sources is presented at a given aperture ( 6 *6 \uf06c \uf06c ), showing clearly that with the increase of N , the EM effective capacity under different communication distances will increase with the same slope, but it converges to the channel capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In this case, considering that the change of the total power of the transmitting array will lead to different channel capacities, we fixed the total transmitting power at 0P , satisfying 0 1 T N i i P P = = \uf0e5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In addition, to illustrate the physical nature of the convergence, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 3 shows the EM effective capacity corresponding to different aperture sizes with enough point sources (30*30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Obviously, the size of the aperture plays a determinant role in the information capacity of MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' It suggests that the trend of antenna miniaturization is the weakening of maximum carrying information, which cannot be solved by multi-antenna technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Besides, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 4, we plot the curve of EM effective capacity changing with the communication distance, revealing the characteristics of wireless communication-energy attenuated with the propagation distance from the perspective of dyadic Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Besides, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 4, the variables we focus on are the aperture and distance respectively, so the number of point sources is a constant, and the total power 0P always remains a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' It is worth mentioning that, due to the basis function decomposition method (such as Rao-Wilton-Glisson (RWG) basis in MoM) commonly used in computational This article has been accepted for publication in IEEE Transactions on Antennas and Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=" This is the author's version which has not been fully edited and content may change prior to final publication." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='1109/TAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='3235015 © 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Personal use is permitted, but republication/redistribution requires IEEE permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='��See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='org/publications/rights/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='html for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Authorized licensed use limited to: Zhejiang University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' fwith differentnumber of sources 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Distance=1*lambda ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='EM effective capability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Distance=6*lambda ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Distance=11*lambda ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Distance=16*lambda ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Number of sourcesCefr with different aperture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Distance=1*lambda ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Distance=6*lambda ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Distance=11*lambda ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='EM effective capability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Distance=16*lambda ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Size of aperture ()Cofr with different communication distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='180 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='EM effective capability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Communication distance （2)Transmitting array ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Modeprofiles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Scatterers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='PML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Receiving array ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Complexspace5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Illustration of the group-T-matrix-based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The distribution of the field around the scatterer is decomposed, and the steady-state coefficient matching is carried out based on the cylindrical wave expansion without meshwork and time-domain iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' electromagnetics, the specific RF front-end structure can be decomposed into the sum of point sources through grid partitioning, and the multi-channel coupling effect will be considered in the coefficient term of the operator TR G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Therefore, when using the above method to perform theoretical modeling of EMIT, the coupling can be characterized by adding a coefficient term to the operator TR G , and the specific physical dimensions of the RF front-end can be numerically quantified by base function equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Essentially, changing the RF front-end or the channel will affect the value of eff C in (8) by affecting the distribution of i \uf073 on the ith channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In other words, eff C and the distribution of i \uf073 are the inherent property of the communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' However, the core assumption of this part is based on free EM space, and the specific form of coupling operator TR G will change when numerous scatterers are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The next section will demonstrate the fast algorithms for characterizing the complex EM complex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' PROPOSED EMIT-BASED MODEL FOR EM COMPLEX SPACE In some typical wireless communication scenarios, objects in complex scattering environments are usually represented by some types of scatterers for convenient EM calculations, among which one of the commonly-used classical models is the cylindrical array, as described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' For example, a vehicle- to-vehicle channel is equivalent to a scattering cluster in the internet of vehicles channel modeling [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Due to the poor accuracy and long response time of traditional channel measurement schemes, this section proposes an EMIT-based model for efficient MIMO characterization in this typical scattering complex space based on the group T matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Algorithm Description N cylindrical scatterers in MIMO EM propagation space are considered, which are centered at ( 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=', ) pr p N = and are with radius ( 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=', ) p a p N = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' These parameters can be easily substituted to simulate different distributions and different shapes of scatterers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' For the description of the RF front-end, we use a 1 s N dipole antenna array, coordinate ( 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=', ) sr s N = , as a convenient MIMO model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The overall algorithm framework is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' To be clear, we focus on the scenarios where the transceivers and receivers are in the same horizontal plane (such as vehicle-to-vehicle communication and indoor point-to-point communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In this case, we can regard the scatterer as a cluster of cylindrical scatterers, so conducting cylindrical wave expansion is reasonable and convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' This benefits the convenience of calculation and the simplicity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' To take the coupling between scatterers into account, we take the th q scatterer as the analysis object and decompose the total external field ex q \uf079 around it into the sum of the incident field inc q \uf079 and the scattering field s p \uf079 of the rest scatterers: 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' N ex inc s q q p p p q \uf079 \uf079 \uf079 = \uf0b9 = +\uf0e5 (9) For solving the scattered fields, the electric field is expanded as a vector cylindrical wave harmonic function: ( ) ( ( )) ex q q n n n I Rg k \uf079 \uf079 = − \uf0e5 q r r , (10) where k is the wave vector, ( ) q nI is the cylindrical wave coefficient, which is the unknown core quantity for solving the field distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' the specific mathematical form of cylindrical wave expansion in (10) is given as follows: (1) ( ( )) ( )exp( ) ( ( )) ( )exp( ) n n n n k H k in Rg k J k in \uf079 \uf066 \uf079 \uf066 − = − − = − p p p p rr p p rr r r r r r r r r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (11) where r represents the coordinate vector of the field point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' \uf066 p rr represents the angle between the vectors r and pr ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' nJ is the Bessel function of order n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (1) n H is the Hankel function of order n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' and Rg means regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Later, we will use the symbol i to represent the imaginary unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' For the mode matching, we perform the same cylindrical wave expansion for the incident field inc q \uf079 determined by the MIMO RF front-end (here is the 1 s N dipole array), obtaining: (1) 0 1 ( ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 4 s N inc q s i H k \uf079 = = − \uf0e5 s r r (12) To obtain the same expansion form as (10), the vector addition theorem is used to further expand (12) to obtain: This article has been accepted for publication in IEEE Transactions on Antennas and Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=" This is the author's version which has not been fully edited and content may change prior to final publication." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='1109/TAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='3235015 © 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Personal use is permitted, but republication/redistribution requires IEEE permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='��See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='org/publications/rights/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='html for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Authorized licensed use limited to: Zhejiang University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2 y q x b pl TpI TPV yip2 AV V pl pN yot rer pl pN x x pl x Z PId p2 pN Scatterer 1 Scatterer 2 Scatterer N Zo +X06 > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Normalized electric field distribution on the validation plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (a) (c) (e): FDTD solver for scatterers distribution of 1*1, 1*5 and 4*1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (b) (d) (f): proposed EMIT-based model for scatterers distribution of 1*1, 1*5 and 4*1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' ( ) ( ) (1) 1 exp 4 ( ( )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' s q N inc q n n s n i H k in Rg k \uf079 \uf066 \uf079 = = − − \uf0b4 − \uf0e5 \uf0e5 s s r r s r r r r (13) In (13), as the RF front-end information is part of prior knowledge, the expansion coefficient of inc q \uf079 is determined, which is convenient for solving ( ) q nI in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Next, we write the scattering field s p \uf079 of the th p scatterer as follows: ( ) ( ) 0 n s TR TR p dS i dS \uf079 \uf077\uf06d \uf0e9 \uf0f9 \uf0a2 \uf0a2 \uf0a2 = \uf0d7 −\uf0d1\uf0b4 \uf0d7 \uf0ea \uf0fa \uf0eb \uf0fb \uf0f2 p p G J r G M r , (14) where TR G is dyadic Green’s function illustrated in (2), \uf077 is the angular frequency, ( )\uf0a2 p J r and ( )\uf0a2 p M r are the current density and magnetic current density at the th p scatterer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The EM variation in the complex space is described by the action of the coupling operator TR G on ( )\uf0a2 p J r and ( )\uf0a2 p M r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' When ( )\uf0a2 p J r and ( )\uf0a2 p M r do not exist, (9) will then degenerate into the free space case shown in section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In order to solve the s p \uf079 described in (14), we use the consistent mathematical form of ex q \uf079 on different scatterers to expand s p \uf079 into cylindrical waveform by using (10), and the transformation relationship is shown in the red curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Thus, (14) can be rewritten as follows based on the group-T- matrix: ( ) ( ) ( ( )) s p p p m m m mT I Rg k \uf079 \uf079 = − \uf0e5 s r r , (15) where ( ) p T is the group-T-matrix representing the relationship between the incident field and scattering field of the th p clustered scatterer, and its characteristics are only related to the shape and material of the current scatterer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Assuming the internal wave vector of the scatterer is p k , the general form of group-T-matrix in the cylindrical coordinate system can be obtained by using analytical methods: ( ) (1) (1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) p m p p m p p m p p m m p m p p m p p m p p k J k a J k a kJ ka T kH ka J k a H ka k J k a \uf0a2 \uf0a2 − = \uf0a2 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (16) The T-matrix of any shape objects can be solved by numerical methods such as the method of moments (MoM) according to (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The basic purpose of this section is to illustrate the algorithm’s efficiency, and thus we consider the model of dipole array with TM polarized waves incident on a perfect electric conductor (PEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In this case, (16) evolves into: ( ) (1) ( ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' ( ) m p p m m p J ka T H ka = − (17) Substitute (17) into (15) to obtain the field distribution with ( ) p m I as the only variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The matrix equation of the unknown coefficient ( ) p m I can be obtained by combining (9), (10), (13), and (15): \uf0d7 = Z I V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (18) Here, in order to solve the coefficient ( ) p m I , the equations with different scatterers are written in matrix form, and the order of the Bessel function is truncated with the truncation number max N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Therefore, Z is a square matrix of dimension max (2 1) N N + , while V is a ( ) max 2 1 1 N N + \uf0b4 vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The specific form is: \uf05b \uf05d( ) ( ) ( ) ( ) ( ) 1 , 1 ( ) (1) 1, exp , p q q N n p N m p m n m p q r r p q T H k r r i n m p q \uf066 − \uf0b4 + − \uf0b4 + − − = \uf0ec\uf0ef = \uf0ed − − − \uf0b9 \uf0ef\uf0ee Z ,(19) \uf05b \uf05d( ) ( ) ( ) (1) 1 1 exp 4 s s q N n s q r r q N n s i H k r r in\uf066 − \uf0b4 + = = − − − \uf0e5 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (20) TABLE I This article has been accepted for publication in IEEE Transactions on Antennas and Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=" This is the author's version which has not been fully edited and content may change prior to final publication." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='1109/TAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='3235015 © 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Personal use is permitted, but republication/redistribution requires IEEE permission.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0 0.' 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1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 x (m) x (m) (c) (d) 3008 (u) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 u) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 x(m) X (m) (e) (f)7 > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < COMPARISON OF CPU TIME AND RMS ERROR BETWEEN FDTD AND PROPOSED EMIT-BASED MODEL FOR THE CHARACTERISTIC OF COMPLEX SPACE In (19) and (20), Z is determined by the properties of the complex space, and V is determined by the properties of the RF front-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' A joint solution can semi-analytically describe the evolution of the MIMO coupling operator EIT G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Therefore, due to the change of EM space coupling operator, (5) will be rewritten as: EIT outR incT = ψ G J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (21) The subsequent analysis only needs to be carried out in the same way as (6)-(8) to complete efficient MIMO characterization in complex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=" It should be noted that, when facing the time-varying channel scenario, it will be very convenient to rewrite the coupling operator EIT G into the form based on the time-domain Green's function." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Numerical Results To verify the accuracy and efficiency of the proposed EMIT- based model, numerical calculations of some specific scenarios are carried out and compared with full-wave simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 7 presents the field distributions of three simple arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The effectiveness of the EMIT-based model is verified by comparing the full-wave FDTD algorithm (a, c, e) with the proposed algorithm (b, d, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' We consider an EM space of 1 m*1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 m, where the MIMO system is modeled as a 3*1 dipole array with an aperture of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='75 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The scatterer array element is modeled as a metal cylinder with a height of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='25 m and a radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='015 m, and the boundary is set as PEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Due to the dense mesh division of the full-wave algorithm, its application is severely limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' However, the proposed semi-analytic algorithm based on group-T-matrix is suitable for various frequencies because it does not need mesh division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' To obtain the comparison results, we first define the operating frequency at 915 MHz in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 7 illustrates that the proposed EMIT-based model has achieved good results and can accurately describe the complex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In order to further demonstrate its efficiency, the scatterer distribution was adjusted, and the solving time and error of the EMIT-based model and FDTD were calculated by analyzing the field intensity curve at the RF back-end, as shown Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The total normalized electric field distribution on the validation plane corresponding to the proposed complex space obtained by EMIT-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' It is worth noting that the 10*15 distribution cannot fully explain the difference between the two algorithms, because the large number of FDTD meshes converge extremely fast due to the inability of the electric field to propagate effectively, and the solutions are often mediocre at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Therefore, we further consider the case of a random array, that is, randomly removing 60 scatterers from the 10*15 scatterer distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Besides, we clarify that the running time of our proposed algorithm mainly depends on the number of scatters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Therefore, the proposed EMIT-based model has higher computational efficiency than full-wave algorithms like FDTD, which provides great convenience for the description of complex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' But generally, the complexity of the real-world environment increases with the communication distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In this case, an efficient way to leverage the EMIT-based method is to build a common clustering model database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Compared with pilot-based methods, it also has good efficiency under the condition of a complete database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' For example, a vehicle-to- vehicle channel is equivalent to a scattering cluster in the internet-of-vehicles channel modeling [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Mode Analysis Step of the EMIT-Based Model After efficient characterization of the complex space is verified, the EMIT-based model performs a mode analysis of the above characterization results to obtain theoretical interpretations to guide the design of wireless communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Consider an actual information transmission scenario where the RF front-end is a single-polarized dipole antenna array operating at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5GHz (equivalent using an ideal line source operating at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5Ghz), with a scale of 10*1 (designed to make the MIMO feature more obvious), and the complex space is simplified to a 4*5 metallic cylindrical scatterer cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' According to the quick algorithm in the previous section, the electric field distribution on the validation plane is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' By substituting the solved coupling operator EIT G into (6), the EM effective capacity eff C of this model in wireless communication is known as 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='2, which means that the actual effective number of available channels is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' However, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 3 shows that dyadic Green’s function operators TR G (coupling This article has been accepted for publication in IEEE Transactions on Antennas and Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=" This is the author's version which has not been fully edited and content may change prior to final publication." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='1109/TAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='3235015 © 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Personal use is permitted, but republication/redistribution requires IEEE permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='��See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='org/publications/rights/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='html for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Authorized licensed use limited to: Zhejiang University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 x (m)8 > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Each mode’s normalized electric field diagram on the validation plane obtained by EMIT-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (a-e) Available information transfer modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (d-j) Unavailable higher-order information transfer modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' operators in free space) will bring channel gains far beyond 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Therefore, the EMIT-based model provides a convenient tool to quantitatively explain the influence of the complex environment on wireless communication quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The information at the receiving end in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 8 can help us get the operator EIT G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Through the SVD mentioned above, 10 modes at the transmitting end can be decomposed, and the EM responses of these 10 modes in the complex space are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' It is clearly found that the first five modes successfully send signals to the receiver effectively in different coupling Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The distribution of normalized singular values of different number of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Simple MIMO propagation system in complex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' However, the coupling paths of higher-order modes bypass the receiver’s acceptance range and become unavailable modes in wireless communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' To define the concepts of “available” and “unavailable” more clearly, we show the distribution of modes’ singular values for the different number of channels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' A formal definition is given as follows: if all modes are numbered according to the normalized singular value in a descending order like Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 10, then the available modes are defined as those whose index is less than the EM effective capacity eff C , and the other modes are defined as the unavailable modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Since the power resources in an actual wireless communication system are limited, the mode weight corresponding to each channel number is normalized here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Obviously, for a MIMO system, there will be an evident truncation of the modes’ singular value distribution, and modes below the truncation usually become “unavailable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The number of “available” modes will be strictly determined by (8) after the coupling operator EIT G is obtained by the EMIT- based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Obviously, the more channels available, the more information that can be transmitted, and the greater the channel capacity of the corresponding EM space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' However, This article has been accepted for publication in IEEE Transactions on Antennas and Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=" This is the author's version which has not been fully edited and content may change prior to final publication." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='1109/TAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='2023.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 A 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 x(m) x (m) (a) (q)Mode3 Mode4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 x (m) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 x (m) (c) (d)Mode5 Mode6 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 x (m) x (m) (e) (f)Mode7 Mode8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 x (m) x (m) (g) (h)Mode9 Mode10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 (m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 x (m) x (m) (0) ()Singular value distribution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='1 6 N mode index 0 numberofsourceTransmitting Array Receiving Array Scattering Region VNA9 > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The normalized amplitude and phase of 21 S in 7*7 MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (a) Simulation results in EMIT-based model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (b) Measurement results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (c) The error of the two above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' communication resources of the RF front-end are often limited, so it is important to allocate resources properly to achieve better information transmission efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In the next section, power distribution is taken as the background problem to discuss the guidance significance of the EMIT-based model for real wireless communication in a complex environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' EXPERIMENTAL ANALYSIS It is worth noting that the above discussion on the application of the EMIT-based model is carried out by simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In order to fully explain the effectiveness of the EMIT-based model and the application method under the background of wireless communication, we carried out experimental exploration with the aid of a 3*1 MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The experiment construction is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 11, where the system is surrounded by the absorption boundary covered with absorbing materials, and cylindrical metal scatterers with a height of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='25 m and a radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='015 m are uniformly distributed in the EM space with 4*5 arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The transmitting sources and the receiving fields were replaced by dipole antennas with a center frequency of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 GHz and a gain of 2dBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The vector network analyzer (VNA) is used to measure the 21 S between transmitting and receiving dipoles through the coaxial feed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Since the measurement of channel matrix elements is concerned with the single excitation properties of MIMO, we replace the actual MIMO system by changing the spatial position of the antenna in the transmitting aperture (shown as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The field distribution of three orthogonal modes at the receiver aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The locations of the three sources are indicated by black dotted lines on the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' the dotted white lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Therefore, in the experimental design, we selected the weak coupling scenario with the antenna spacing as half-wavelength, and then measured the antenna excitation separately to avoid the impact of coupling on the verification of our EMIT-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' For the same scene, we performed effective characterization with the EMIT-based model, and the characterization results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 12 (a) and (b) respectively represent the amplitude and phase comparison results between the simulation results of EMIT-based model and the experimental measurement values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The experimental results fully demonstrate the effectiveness of the EMIT-based model in complex space characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The purpose of conducting 7*7 channel measurement in our experiment is to verify the EMIT- based model more convincingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' However, the following is mainly to illustrate how the EMIT-based model guides the RF front-end signal transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Therefore, to simplify the demonstration process, we select three groups of data evenly spaced to form a new 3*3 MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In fact, the selection of 3*3 channel positions is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' However, in this paper, to make the mode orthogonality more significant and avoid the influence of mode crosstalk on the transmitting strategy, three positions with relatively small mode crosstalk are selected, and the crosstalk matrix CT is as follows: 15 15 15 15 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='1857 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='03*10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='1857 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='51*10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='03*10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='51*10 1 − − − − \uf0e6 \uf0f6 \uf0e7 \uf0f7 = \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 CT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (22) Consider the following problems in an actual wireless communication scenario: 3 * 3 MIMO system needs to conduct data transmission in disorder EM space (with the standard deviation of noise \uf063 ), the maximum transmitted power of RF front-end is 0P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Under this constraint, since it is a very important subject to consider the optimal power distribution, which is related to whether the upper bound for capacity can be achieved, the coupling operator EIT G obtained by the EMIT- This article has been accepted for publication in IEEE Transactions on Antennas and Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=" This is the author's version which has not been fully edited and content may change prior to final publication." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='1109/TAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='3235015 © 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Personal use is permitted, but republication/redistribution requires IEEE permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='��See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='org/publications/rights/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='html for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Authorized licensed use limited to: Zhejiang University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Amplitude Phase 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0 180 (a) Amplitude Phase 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0 0 180 (b) Error Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0 0 180 (c)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='16 Port 1 Port 2 Port 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='14 Mode 1 Mode 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='12 Mode 3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='9 Receiving Position (m)10 > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Image transmission case based on EMIT-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (a) Original image with 128*128 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (b) Optimized power distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (c)-(e) Conducting single-mode transmission using mode 1, mode 2 and mode 3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' based model is used to endow the shape of the transmitted signal to obtain the best quality of information transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Rewrite (5-6) to obtain a new coupling equation based on the scenario we’re considering: = † † EIT EIT U Y SV X , (23) where EIT U and EIT V are determined by EIT G to guide the signal processing of transmitter and receiver, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' X and Y represent the signal form of transmitter and receiver, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The singular value matrix S is disassembled to obtain the received signal evaluation function f : 3 1 m m m f V \uf073 \uf061 = = \uf0e5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (24) In (24), X is decomposed as a bitstream of information X (here we use simple binary phase-shift keying (BPSK) modulation) multiplied by the excitation coefficient \uf061 , and the unitary matrix EIT V is decomposed into three-mode vectors ( 1,2,3) m V m = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Since there are three sources in our MIMO system, both m V and \uf061 here have three elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' ( 1,2,3) m m \uf073 = are the diagonal elements of the singular value matrix S , representing the influence of each mode on the receiving end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' To have a clearer understanding of m \uf073 , we depicted the electric field distribution at the receiving end with the help of the EMIT-based model, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The calculated proportions of the three modes are 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='55%, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='19%, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='25%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Mode 1 with the strongest proportion just contributes its crest to the receiving end, while mode 3 with the weakest proportion just contributes its trough to the receiving end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' This provides a clear perspective for signal waveform design from the EM point of view, and reveals that EM space is not the only factor determining mode contribution, and EM space characteristics and RF front-end characteristics should be considered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' According to the crosstalk matrix calculated in (22), we treat these three modes as orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Therefore, m V becomes a set of orthogonal basis in a Hilbert space, meeting † 0 m n V V = and † 1 m m V V = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Hence, \uf061 can be written as an orthogonal basis expansion: 3 1 m m m V \uf061 \uf06c = = \uf0e5 , (25) where m \uf06c represent the corresponding weight of each basis vector, which determines the power distribution on the transmitting source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Therefore, the constraint of constant total power 0P can be equivalent to that the excitation vector is located on a fixed circle in the Hilbert space, and the received signal evaluation function f is the sum of the weighted projections of the excitation vector on the three basis functions: 0 find : ( 1,2,3) max : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' : m m m f s t P \uf06c \uf06c \uf0ec = \uf0ef\uf0ed \uf0ef = \uf0ee \uf0e5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (26) The optimization problem can be easily solved by using Cauchy inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' By substituting (8), the information transfer function can reach the maximum value only when / m m \uf06c \uf073 is a constant for different m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' This is similar to the “water-filling” algorithm in channel estimation, while the core difference is that the key informatics parameters in this paper are deduced by an effective EM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In addition, † TR TR G G or † EIT EIT G G have the same physical meaning as the transmit signal covariance matrix, and the main difference is that † TR TR G G or † EIT EIT G G is calculated by the EM methods based on dyadic Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' It is seen from the mode analysis based on the EMIT-based model that only under the guidance of a specific power allocation strategy, MIMO information transmission can achieve the effect of receiving power equal to transmitting power times path loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' To fully illustrate the guiding significance of the EMIT-based model for power distribution (essentially waveform design), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 14 shows an image transmission case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' BPSK is used to discretize every pixel in the picture into an 8-bit data stream for transmission, and noise \uf063 is joined to EM space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Obviously, under the premise of not processing channel noise, the RF front-end working strategy based on the EMIT-based model is much better than other transmission modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Therefore, the EMIT-based model can not only efficiently represent the complex space, but also make more valuable guidance for wireless communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' CONCLUSION In this article, an EMIT-based model is presented to simulate the performance of MIMO systems in complex EM complex space effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Firstly, the EM expression of the information coupling operator is given in the free space, and two key informatics parameters, EM effective capacity and path loss, This article has been accepted for publication in IEEE Transactions on Antennas and Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=" This is the author's version which has not been fully edited and content may change prior to final publication." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='1109/TAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='3235015 © 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Personal use is permitted, but republication/redistribution requires IEEE permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='��See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='org/publications/rights/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='html for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Authorized licensed use limited to: Zhejiang University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' (a) (b) (c) (p) (e)11 > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < are extracted from the EM perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' It is proved that the MIMO antennas’ aperture is the critical factor in the EM effective capacity of the MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Secondly, the basic principle of the EM representation method in complex space is given, and several typical scenarios are analyzed, which proves the accuracy and efficiency of the EMIT-based model proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The results show that the EMIT-based model can reliably analyze the electromagnetic space about 10% of the time compared to the full-wave simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Finally, the MIMO performance in real propagation scenarios is calculated using the EMIT-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The experimental results verify that the channel matrix calculated is in good agreement with the measured ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Based on this, it is pointed out how the EMIT- based model can effectively guide the MIMO design and feeding in a power distribution question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' The ultimate goal of the proposed EMIT-based model is to advance the development of EMIT and demonstrate a new idea of extracting information parameters to the antenna & propagation community using the basis of computational electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' This article has been accepted for publication in IEEE Transactions on Antennas and Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=" This is the author's version which has not been fully edited and content may change prior to final publication." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='1109/TAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='3235015 © 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Personal use is permitted, but republication/redistribution requires IEEE permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='��See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='org/publications/rights/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='html for more information.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Neifeld, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Ashok, "Capacity of electromagnetic communication modes in a noise-limited optical system," Applied optics (2004), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 55, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 1333, 2016-01-01.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 446-450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' [34] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Sarkar and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Salazar-Palma, "MIMO: Does It Make Sense From an Electromagnetic Perspective and Illustrated Using Computational Electromagnetics?' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Kühmayer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Ferise, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Rotter, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Davy, "Anti- reflection structure for perfect transmission through complex media," Nature, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 607, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 7918, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 281-286, 2022-07-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' [36] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Chen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Kildal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Carlsson, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Yang, "MRC Diversity and MIMO Capacity Evaluations of Multi-Port Antennas Using Reverberation Chamber and Anechoic Chamber," IEEE Transactions on Antennas and Propagation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 61, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 917-926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Ruifeng Li received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' degree in engineering from University of Electronic Science and Technology of China, Chengdu, China, in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He is currently pursuing the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' degree at the College of Information Science and Electronic Engineering, Zhejiang University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' His current research interests include the electromagnetic information theory for wireless communication, and efficient calculation methods applied in MIMO antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Da Li received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' degree in 2014, and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' degree in 2019, from Zhejiang University, Hangzhou, China, both in electrical engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' From 2017 to 2018, he worked at Nanyang Technological University, Singapore, as a Project Researcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' From 2019 to 2021, he joined Science and Technology on Antenna and Microwave Laboratory, Nanjing, China, as a Research Fellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He is currently an assistant professor at Zhejiang University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' His research interests include machine learning, antennas, matesurfaces, and electromagnetic compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Li has authored or coauthored more than 40 refereed papers and served as Reviewers for 6 technical journals and TPC Members of 3 IEEE conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He was also a recipient of the Outstanding Young Scientist Award at 2022 Asia-Pacific International Symposium on Electromagnetic Compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Jinyan Ma received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' degree in engineering from Zhejiang University, Hangzhou, China, in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He is currently working toward the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' degree in electronics science and technology with the College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' His current research interests include the electromagnetic information theory and efficient electromagnetic calculation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Zhaoyang Feng received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Sc degree from North China Electric Power University, Beijing, China, in 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He is currently working toward the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' degree in the College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, Zhejiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' His current research interests include electromagnetic compatibility, computational electromagnetics and multiple scattering theory Ling Zhang (Member, IEEE) received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' degree in electrical engineering from Huazhong University of Science and Technology, Wuhan, China, in 2015, and the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' degrees from Missouri S&T, Rolla, MO, USA, in 2017 and 2021, respectively, both in electrical engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He was with Cisco as a student intern from Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2016 to Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He joined Zhejiang University, Hangzhou, China as a research fellow in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He has authored and co-authored more than 30 journal and conference papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' His research interests include machine learning, power integrity, electromagnetic interference, radio-frequency interference, and signal integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Zhang was an Organizing Committee, Special Session Chair, Workshop Session Chair, and Poster Session Chair in APEMC 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He has given invited presentations at the IBIS Summit at 2021 IEEE Virtual Symposium on EMC+SIPI, and the 2021 Virtual Asian IBIS Summit China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He was the recipient of the Honorable Mention Paper in APEMC 2022, the Best Paper Award in DesignCon 2019, and the Student Paper Finalist Award in ACES Symposium in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He was also the recipient of the Outstanding Young Scientist Reward in APEMC 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Shurun Tan (S’14-M’17) received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' degree in information engineering and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' degree in electromagnetic field and microwave techniques from the Southeast University, Nanjing, China, in 2009 and 2012, respectively, and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' degree in electrical engineering from the University of Michigan, Ann Arbor, MI, USA, in Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Tan is an assistant professor in the Zhejiang University / University of Illinois at Urbana-Champaign Institute located at the International Campus of Zhejiang University, Haining, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He is also affiliated with the State Key Laboratory of Modern Optical Instrumentation, and the College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He is also an adjunct assistant professor in the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' From Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2010 to Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2011, he was a Visiting Student with the Department of Electrical and Computer Engineering, the University of Houston, Houston, TX, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' From Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2012 to Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2014, he was a PhD candidate with the Department of Electrical Engineering, the University of Washington, Seattle, WA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' From Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2015 to Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2018, he had been affiliated with the Radiation Laboratory, and the Department of Electrical Engineering and Computer Science, the University of Michigan, Ann Arbor, first as a PhD candidate, and then as a postdoctoral research fellow since Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Tan is working on electromagnetic theory, computational and applied electromagnetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' His research interests include electromagnetic scattering of random media and periodic structures, microwave remote sensing, electromagnetic information systems with electromagnetic wave-functional devices, electromagnetic integrity in high-speed and high-density electronic integration, electromagnetic environment and reliability of complex electronic systems, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' This article has been accepted for publication in IEEE Transactions on Antennas and Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=" This is the author's version which has not been fully edited and content may change prior to final publication." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='1109/TAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='3235015 © 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Personal use is permitted, but republication/redistribution requires IEEE permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='��See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='org/publications/rights/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='html for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Authorized licensed use limited to: Zhejiang University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Downloaded on January 13,2023 at 13:17:46 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 13 > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < Wei E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Sha (M’09-SM’17) received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' degrees in Electronic Engineering at Anhui University, Hefei, China, in 2003 and 2008, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' From Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2008 to Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2017, he was a Postdoctoral Research Fellow and then a Research Assistant Professor in the Department of Electrical and Electronic Engineering at the University of Hong Kong, Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' From Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2018 to Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2019, he worked at University College London as a Marie Skłodowska-Curie Individual Fellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' From Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' 2017, he joined the College of Information Science & Electronic Engineering at Zhejiang University, Hangzhou, China, where he is currently a tenure-tracked Assistant Professor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Sha has authored or coauthored 180 refereed journal papers, 150 conference publications (including 5 keynote talks and 1 short course), 9 book chapters, and 2 books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' His Google Scholar citation is 8193 with h-index of 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He is a senior member of IEEE and a member of OSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He served as Reviewers for 60 technical journals and Technical Program Committee Members of 10 IEEE conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He also served as Associate Editors of IEEE Journal on Multiscale and Multiphysics Computational Techniques, IEEE Open Journal of Antennas and Propagation, and IEEE Access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In 2015, he was awarded Second Prize of Science and Technology from Anhui Province Government, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In 2007, he was awarded the Thousand Talents Program for Distinguished Young Scholars of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He was the recipient of ACES Technical Achievement Award 2022 and PIERS Young Scientist Award 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Sha also received 6 Best Student Paper Prizes and one Young Scientist Award with his students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' His research interests include theoretical and computational research in electromagnetics and optics, focusing on the multiphysics and interdisciplinary research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' His research involves fundamental and applied aspects in computational and applied electromagnetics, nonlinear and quantum electromagnetics, micro- and nano-optics, optoelectronic device simulation, and multiphysics modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Hongsheng Chen received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content='degrees in electrical engineering from Zhejiang University (ZJU), Hangzhou, China, in 2000 and 2005, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' In 2005, he became an Assistant Professor with ZJU, where he was an Associate Professor in 2007 and a Full Professor in 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He was a Visiting Scientist from 2006 to 2008 and a Visiting Professor from 2013 to 2014 with the Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He is currently a Chang Jiang Scholar Distinguished Professor with the Electromagnetics Academy, ZJU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He has coauthored more than 200 international refereed joumal papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' His works have been highlighted by many scientific magazines and public media, including Nature, Scientific American, MIT Technology Review, Physorg, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' His current research interests include metamaterials, invisibility cloaking, transformation optics, graphene, and theoretical and numerical methods of electromagnetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Chen serves as a Regular Reviewer for many international journals on electromagnetics, physics, optics, and electrical engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He serves as a Topical Editor for the Journal of Optics and the Editorial Board for Nature’s Scientific Reports and Progress in Electromagnetics Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He was a recipient of the National Excellent Doctoral Dissertation Award in China in 2008, the Zhejiang Provincial Outstanding Youth Foundation in 2008, the National Youth Top-Notch Talent Support Program in China in 2012, the New Century Excellent Talents in University of China in 2012, the National Science Foundation for Excellent Young Scholars of China in 2013, and the National Science Foundation for Distinguished Young Scholars of China in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' His research work on an invisibility cloak was selected in Science Development Report as one of the representative achievements of Chinese Scientists in 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Er-Ping Li (S’91, M’92, SM’01, F’08) is currently a Qiushi-Distinguished Professor with Department of Information Science and Electronic Engineering, Zhejiang University, China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' served as Founding Dean for Institute of Zhejiang University - University of Illinois at Urbana-Champaign in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' From 1993, he has served as a Research Fellow, Associate Professor, Professor and Principal Scientist and Senior Director at the Singapore Research Institute and University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Dr Li authored or co-authored over 400 papers published in the referred international journals, authored two books published by John-Wiley-IEEE Press and Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He holds and has filed a number of patents at the US patent office.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' His research interests include electrical modeling and design of micro/nano-scale integrated circuits, 3D electronic package integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Li is a Fellow of IEEE, and a Fellow of USA Electromagnetics Academy, a Fellow of Singapore Academy of Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He is the recipient of IEEE EMC Technical Achievement Award in 2006, Singapore IES Prestigious Engineering Achievement Award and Changjiang Chair Professorship Award in 2007, 2015 IEEE Richard Stoddard Award on EMC, 2021 IEEE EMC Laurence G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Cumming Award and Zhejiang Natural Science 1st Class Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He served as an Associate Editor for the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS from 2006-2008 and for IEEE TRANSACTIOSN on EMC from 2006-2021, Guest Editor for 2006 and 2010 IEEE TRANSACTIOSN on EMC Special Issues, Guest Editor for 2010 IEEE TRANSACTIONS on MTT APMC Special Issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He is currently an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL and POWER INTEGRITY and Deputy Editor in Chief of Electromagnetics Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He has been a General Chair and Technical Chair, for many international conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He was the President for 2006 International Zurich Symposium on EMC, the Founding General Chair for Asia-Pacific EMC Symposium, General Chair for 2008, 2012, 2016, 2018, 2022 APEMC, and 2010 IEEE Symposium on Electrical Design for Advanced Packaging Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' He has been invited to give 120 invited talks and plenary speeches at various international conferences and forums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' This article has been accepted for publication in IEEE Transactions on Antennas and Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=" This is the author's version which has not been fully edited and content may change prior to final publication." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf'} +page_content=' Citation 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Traditionally, learning-based control approaches, e.g., +imitation learning, have been used to tackle deformable material +manipulation. Such approaches lack generality and often suffer +critical failure from a simple switch of material, geometric, and/or +environmental (e.g., friction) properties. In this article, we ad- +dress a fundamental but difficult step of robotic origami: forming +a predefined fold in paper with only a single manipulator. A data- +driven framework combining physically-accurate simulation and +machine learning is used to train deep neural network models +capable of predicting the external forces induced on the paper +given a grasp position. We frame the problem using scaling +analysis, resulting in a control framework robust against material +and geometric changes. Path planning is carried out over the +generated manifold to produce robot manipulation trajectories +optimized to prevent sliding. Furthermore, the inference speed +of the trained model enables the incorporation of real-time +visual feedback to achieve closed-loop sensorimotor control. Real- +world experiments demonstrate that our framework can greatly +improve robotic manipulation performance compared against +natural paper folding strategies, even when manipulating paper +objects of various materials and shapes. +Index Terms—robotic manipulation, deformable material ma- +nipulation, deep neural networks, data-driven models, closed-loop +sensorimotor control +I. INTRODUCTION +From shoelaces to clothes, we encounter flexible slender +structures throughout our everyday lives. These structures are of- +ten characterized by their ability to undergo large deformations +when subjected even to moderate forces, such as gravity. People +possess an incredible innate understanding of the dynamics of +such deformable objects; e.g., we can use gravity to perfectly +manipulate a shirt over our heads. Instilling such intuition +into robots remains an important research problem and has +the potential to breed numerous applications with considerable +economic and humanitarian potential. Some examples include +preparing deformable products in the food industry [1], [2], +assisting in the medical field [3]–[5], and providing caregiving +assistance to elderly and disabled communities, including with +respect to dressing [6]–[10] and feeding [11], [12]. However, the +The authors are with the University of California, Los Angeles (UCLA), +CA 90095, USA. +1Dezhong Tong and M. Khalid Jawed are with the UCLA Department of +Mechanical & Aerospace Engineering (email: tltl960308@g.ucla.edu; +khalidjm@seas.ucla.edu). +2Andrew Choi and Demetri Terzopoulos are with the UCLA Computer +Science Department (email: asjchoi@cs.ucla.edu; dt@cs.ucla.edu). +3Jungseock Joo is with the UCLA Department of Communication and is +currently working at NVIDIA Corporation (email: jjoo@comm.ucla.edu). +∗ Equal contribution. +† Corresponding author. +Position & +material +parameters +Learned model +Force manifold +Optimization +algorithm +Optimal +path +Intuitive trajectory +(circular curve) +Gripper +Paper +Substrate +Initial free end +Initial free end +Obvious sliding +Minimal sliding +Initial state +Folded result +Intuitive + manipulation +Our optimal + manipulation +(a) +(b) +Fig. 1. Half valley folding for A4 paper with (a) intuitive manipulation and (b) +our designed optimal manipulation. An intuitive manipulation scheme such as +tracing a semi-circle experiences significant sliding due to the bending stiffness +of the paper, resulting in a poor fold. By contrast, our optimal manipulation +approach achieves an excellent fold by taking into consideration the paper’s +deformation and thus minimizing sliding. +robotic manipulation of deformable objects is highly nontrivial +as a robot must be able to take into account future deformations +of the manipulated object to complete manipulation tasks +successfully. +Prior research has focused primarily on manipulating either +cloth [13]–[18] or ropes [12], [19]–[25] and as a result, the +robotic manipulation of many other deformable objects still +lacks robust solutions. In this article, we address a particularly +difficult deformable manipulation task — folding paper. Paper +is similar to cloth but typically has a much larger bending +stiffness and a slippery surface. Therefore, compared with +folding garments and fabrics, more delicate and insightful +manipulations are required for folding sheets of paper. +A. Our Approach +We propose a framework that combines physically accurate +simulation, scaling analysis, and machine learning to generate +folding trajectories optimized to prevent sliding. With scaling +analysis, we make the problem non-dimensional, resulting +in both dimensionality reduction and generality. We then +train neural networks, whose outputs are referred to as neural +force manifolds (NFM), to continuously approximate a scaled +force manifold sampled purely from simulation. Compared to +numerical models that require the entire geometric configuration +arXiv:2301.01968v1 [cs.RO] 5 Jan 2023 + +2 +of the paper, NFMs map the external forces of the paper given +only the grasp position. Therefore, we can generate trajectories +optimized to minimize forces (and thus minimize sliding) by +applying path planning algorithms in near real-time. We show +that our approach is capable of folding paper on extremely +slick surfaces with little-to-no sliding (Fig. 1(b)). +Our main contributions are as follows: (1) we formulate a +solution to the folding problem in a physically robust manner +using scaling analysis, resulting in complete generality with +respect to material, geometric, and environmental properties; +(2) we train a neural network with non-dimensional simulation +data forming a fast and accurate model that can generate a +descriptive force manifold for trajectory optimization; (3) we +utilize the high inference speed of our trained model with a +perception system to construct a robust and efficient closed- +loop sensorimotor control algorithm for the folding task, and +finally (4) we demonstrate full sim2real realization through +an extensive robotic case study featuring 210 experiments +across paper sheets of various materials and shapes. While +several previous works have trained their policies purely from +simulation data [7], [19], [26]–[28], these works lacked real +world validation. To our knowledge, our framework is the first +to provide optimal folding trajectories with complete generality. +We release supplementary videos as well as all source code +and CAD files as open source at https://github.com/ +StructuresComp/deep-robotic-paper-folding. +B. Overview +The remainder of the article is organized as follows: We +begin with a review of related work in Sec. II. A brief +description of the folding problem is presented in Sec. III. +The formulation of a reduced-order physics-based model +is discussed in Sec. IV, where we formulate the folding +problem using scaling analysis. In Sec. V, we formulate our +learning framework as well as algorithms for optimal path +planning. Next, in Sec. VI, we introduce our robotic system +as well as formulate our closed-loop visual feedback pipeline. +Experimental results for a robot case study and analysis of the +results are given in Sec. VII. Finally, we provide concluding +remarks and discuss the potential of future research avenues +in Sec. VIII. +II. RELATED WORK +The majority of prior works tackling the folding problem +can be roughly divided into four categories: mechanical +design-based solutions, vision-based solutions, learning-based +solutions, and model-based solutions. +Mechanical design-based approaches typically involve solv- +ing the folding problem through highly specialized manipulators +or end effectors. Early approaches involve specialized punches +and dies for sheet metal bending [29], More recently, highly +specialized manipulators for robotic origami folding have +also been developed [30]. Such methods can reliably produce +repeatable folding but are often limited to a highly specific +fold, geometry, and/or material. +Vision-based approaches involve folding deformable mate- +rials by generating folding motions purely from visual input. +These approaches are usually common for folding clothes [14], +[16], [31] as they are extremely soft, which results in the +easy predictability of their deformation state given a particular +action. Such approaches can be effective and rather simple to +implement, but do not transfer well to paper folding as paper +possesses a much higher stiffness when compared to fabric +and will attempt to restore its natural, undeformed state if not +properly handled. +Learning-based approaches involve the robot learning how +to fold through training data. The most popular has been to +learn control policies from human demonstrations, also known +as learning from demonstrations (LfD). Prior research has +demonstrated flattening and folding towels [32], [33]. Teleop +demonstrations are a popular avenue for training policies +and have been used to learn how to manipulate deformable +linear objects (DLOs) [34] as well as folding fabric [35]. +To eliminate the need for expensive human-labeled data, +researchers have also focused on tackling the sim2real problem +for robotic folding, where reinforcement learning has been +used to train robots to fold fabrics and cloths completely from +simulation [26], [28], [36]. More recently, Zheng et al. [37] used +reinforcement learning to train a robot to flip pages in a binder +through tactile feedback. Pure learning-based methods have +shown promising performance, but only for specific tasks whose +state distribution matches the training data. Such methods tend +to generalize quite poorly; e.g., when the material or geometric +properties change drastically. +Model-based approaches, where the model can either be +known or learned, often use model predictive control to +manipulate the deformable object. They involve learning +the natural dynamics of deformable objects through random +perturbations [38]. These models are generally fast, but they can +be inaccurate when experiencing new states. Known models are +often formulated to be as physically accurate as possible. They +can be referred to as physics-based (as opposed to simulated). +Their physical accuracy allows for the direct application of +their predictive capabilities in the real world. Examples are +published for rectangular cloth folding [39], strip folding [40], +and garment folding [41]. Still, known models are usually +quite expensive to run and must often face a trade-off between +accuracy and efficiency. +Despite the large quantity of prior research focusing on +2D deformable object manipulation, the majority of these +efforts have limited their scope to soft materials such as towels +and cloth. Such materials are highly compliant and often do +not exhibit complicated nonlinear deformations, thus allowing +for solutions lacking physical insight. We instead tackle the +scenario of folding papers of various stiffnesses with a single +manipulator. Because of its relatively high bending stiffness +and slippery surface, paper is significantly more difficult to +manipulate since large deformations will cause sliding of the +paper on the substrate. Such an example can be observed in +Fig. 1(a), where intuitive folding trajectories that may work +on towels and cloth fail for paper due to undesired sliding. +However, a few works have attempted to solve the paper fold- +ing problem. For example, Elbrechter et al. [42] demonstrated +paper folding using visual tracking and real-time physics-based +modeling with impressive results, but they required expensive + +3 +end effectors (two Shadow Dexterous Hands), one end effector +to hold the paper down while folding at all times, and the +paper to have AR tags for visual tracking. Similarly, Namiki et +al. [43] also achieved paper folding through dynamic motion +primitives and used physics-based simulations to estimate the +deformation of the paper sheet, also requiring highly specialized +manipulators and an end effector to hold the paper down +while folding. By contrast, our method can fold papers reliably +without any need for holding down the paper during the folding +operation and requires only an extremely simple 3D printed +gripper. Other approaches have also attempted to fold with a +single manipulator while minimizing sliding [36], [40], but +these methods focused on fabrics whose ends were taped down +to the substrate. +III. PROBLEM STATEMENT +This article studies a simple but challenging task in robotic +folding: creating a predefined crease on a sheet of paper +of typical geometry (e.g., rectangular, diamond, etc.) as is +illustrated in Fig. 2. Only one end of the paper is manipulated +while the other end is left free. Thus, extra fixtures are +unnecessary and the folding task can be completed by a single +manipulator, which simplifies the workspace, but slippage +of the paper against the substrate must be mitigated during +manipulation, which is a challenge. +The task can be divided into two sub-tasks and three states. +The first sub-task is manipulating one end of the paper from +the initial flat state (Fig. 2(a)) to the folding state (Fig. 2(b)), +with the goal that the manipulated edge or point should overlap +precisely with the crease target line or point C as shown in +the figure. With the manipulated edge of the paper at the +origin, the manipulator moves in the x direction. Since the +manipulated paper usually has relatively high bending stiffness, +large nonlinear elastic deformations are induced in the folding +state. In the second sub-task, the paper must be permanently +deformed to form the desired crease at C/2, thus achieving +the final folded state (Fig. 2(c)). +IV. PHYSICS-BASED MODEL AND ANALYSIS +We next present the numerical framework for studying the +underlying physics of the paper folding process. First, we +analyze the main deformations of the manipulated paper and +prove that a 2D model is sufficient to learn the behaviors of +the manipulated paper so long as the sheet is symmetrical. +Second, we briefly introduce a physically accurate numerical +model based on prior work in computer graphics [44]. Third, +we formulate a generalized strategy for paper folding using +scaling analysis. +A. Reduced-Order Model Representation +Paper is a unique deformable object. Unlike cloth, its +surface is developable [45]; i.e., the surface can bend but not +stretch. Furthermore, shear deformations are not of particular +importance as the geometry of the manipulated paper is +symmetrical. Therefore, the primary nonlinear deformation +when folding paper in our scenario is bending deformation. We +Initial state +Folding state +Folded state +Manipulated end +Manipulated + node +Desired crease +Desired crease +Free end +Free node +Target line +Target + node +Rectangular paper +Symmetrical paper (square) +(a) +(b) +(c) +x +z +y +o +0.5C +C +Fig. 2. Folding sheets of paper. The manipulation process involves (a) the +initial state, where the paper lies flat on the substrate, followed by (b) the +folding state, where the manipulated end is moved to the “crease target” line +C, and finally (c) the folded state, which involves forming the desired crease +on the paper. +postulate that the nonlinear behaviors of paper arise primarily +from a balance of bending and gravitational energies: ϵb ∼ ϵg. +To further understand the energy balance of the manipulated +paper, we analyze an arbitrary piece in the paper, as shown in +Fig. 3(b). The bending energy of this piece can be written as +ϵb = 1 +2kbκ2l, +(1) +where l is its undeformed length of the piece, κ is its curvature, +and its bending stiffness is +kb = 1 +12Ewh3, +(2) +where w is its undeformed width, h is its thickness, and E is +its Young’s modulus. The gravitational energy of the piece is +ϵg = ρwhlgH, +(3) +where ρ is its volume density and H is its vertical height above +the rigid substrate. +From the above equations, we obtain a characteristic length +called the gravito-bending length, which encapsulates the +influence of bending and gravity: +Lgb = +� Eh2 +24ρg +� 1 +3 +∼ +� h +κ2 +� 1 +3 +. +(4) +The length is in units of meters, and we can observe that +it scales proportionally to the ratio of thickness to curvature +squared, which are the key quantities describing the deformed +configuration of the manipulated paper. Note that the formula- +tion of Lgb contains only one geometric parameter, the paper +thickness h, which means that other geometric quantities (i.e., +length l and width w) have no influence on the deformed +configuration. + +4 +g +(a) +(b) +(c) +(d) +Mesh S +H +Rigid +substrate +q0 +q1 +qi-1 +qi-1 +qi +qi +ti-1 +ti +qi+1 +qi+1 +qN +l +h +w +Fig. 3. +(a) Schematic of a paper during the folding state. (b) Bending +deformations of a small piece in the paper. (c) Reduced-order discrete model +(planer rod) representation of our paper. (d) Notations in the discrete model. +Additionally, due to the symmetrical geometry of the paper, +curvature κ should be identical for all regions at the same height +H. Therefore, we can simply use the centerline of the paper, +as shown in Fig. 3(a), to express the paper’s configuration. We +model this centerline as a 2D planar rod since deformations are +limited to the x, z plane. We implement a discrete-differential- +geometry (DDG)-based numerical simulation to simulate the 2D +planar rod. We present the details of this numerical framework +in the next section. +B. Discrete Differential Geometry Numerical Model +Following pioneering work on physics-based modeling and +simulation of deformable curves, surfaces, and solids [46]– +[48], the computer graphics community has shown impressive +results using DDG-based simulation frameworks. For example, +the Discrete Elastic Rods (DER) [44] framework has shown +efficient and physically accurate simulation of deformable linear +objects in various scenarios including knot tying [49], helix +bifurcations [50], coiling of rods [51], and flagella buckling [52]. +Given this success, we use DER to model the centerline of the +paper as a 2D planar rod undergoing bending deformations. +As shown in Fig. 3(c), the discrete model is comprised of +N + 1 nodes, qi (0 ≤ i ≤ N). Each node, qi, represents two +degrees of freedom (DOF): position along the x and the z axes. +This results in a 2N + 2-sized DOF vector representing the +configuration of the sheet, q = [q0, q1, ..., qN]T , where T is +the transpose operator. Initially, all the nodes of the paper are +located in a line along the x-axis in the paper’s undeformed +state. As the robotic manipulator imposes boundary conditions +on the end node qN, portions of the paper deform against the +substrate as shown in Fig. 4(a). We compute the DOFs as a +function of time q(t) by integrating the equations of motion +(EOM) at each DOF. +Before describing the EOM, we first outline the elastic +energies of the rod as a function of q. Kirchhoff’s rod theory +tells us that the elastic energies of a rod can be divided into +stretching Es, bending Eb, and twisting Et energies. First, The +stretching elastic energy is +Es = 1 +2ks +N−1 +� +i=0 +� +1 − ∥qi+1 − qi∥ +∆l +�2 +∆l, +(5) +where ks = EA is the stretching stiffness; E is Young’s +modulus; A = wh is the cross-sectional area, and ∆l is the +undeformed length of each edge (segment between two nodes). +The bending energy is +Eb = 1 +2kb +N−1 +� +i=2 +� +2 tan φi +2 − 2 tan φ0 +i +2 +�2 1 +∆l, +(6) +where kb = Ewh3 +12 +is the bending stiffness; w and h are the +width and thickness respectively; φi is the “turning angle” at a +node as shown in Fig. 3(d), and φ0 +i is the undeformed turning +angle (0 for paper). Finally, since we limit our system to a 2D +plane, we can forgo twisting energies entirely. The total elastic +energy is then simply Eel = Es + Eb. +Indeed, a ratio ks/kb ∼ w/h2 >> 1 indicates that stretching +strains will be minimal which matches our intuition as paper is +usually easy to bend but not stretch. Therefore, the stretching +energy item in (5) acts as a constraint to prevent obvious +stretching for the modeled planar rod. +We can now construct our EOM as a simple force balance +P(q) ≡ M¨q + ∂Eel +∂q − Fext = 0, +(7) +where M is the diagonal lumped mass matrix; ˙( ) represents +derivatives with respect to time; − ∂Eel +∂q +is the elastic force +vector, and Fext is the external forces applied on the paper. +Note that (7) can be solved using Newton’s method, allowing +for full simulation of the 2D planar rod under manipulation. +C. Generalized Solution and Scaling Analysis +As mentioned in Sec. III, the core of the folding task is to +manipulate the end qN to the target position C starting from +an initially flat state shown in Fig. 4(a). To do so, we analyze +the physical system in order to achieve a solution capable of +minimizing sliding during manipulation. +We first denote several quantities to describe the deformed +configuration of the paper. Here, we introduce a point qC, +which is the node that connects the suspended (z > 0) and +unsuspended regions (z = 0) of the paper. We focus primarily +on the suspended region as deformations occur solely in this +region. An origin o is defined for our 2D plane which is located +at the initial manipulated end qN as shown in Fig. 4(a). For +the manipulated end, the robot end-effector imposes a position +qN = (x, z) and an orientation angle α to control the pose of +the manipulated end as shown in Fig. 4(a). On the connective +node qC, the tangent is always along the x-director. Here, we +impose a constraint that the curvature at the manipulated end is +always zero so that sharp bending deformations are prevented, +which is crucial to preventing permanent deformations during + +5 +(a) +(b) +q0 +qC +qN +(qN +') +Norm. x coord, x +x +z +s +ls=4.10 +Norm. z coord. z +c +Fig. 4. (a) Side view of a symmetrical paper during folding with coordinate +frame and relevant notations. (b) Sampled λ forces for a particular ¯ls of 4.10. +This showcases one of the sampled “partial” force manifolds that we use train +our neural network on. +the folding process. With these definitions, we can now modify +(7) with the following constraints: +P(q) = 0, +s.t. +qN = (x, z), +dqC +ds += (−1, 0), +MN = 0, +ls ≡ +� qN +qC +ds = qC · ˆx, +(8) +where MN is the external moment applied on the manipulated +end; s is the arc length of the paper’s centerline, and ls is the +arc length of the suspended region (from qC to qN). +We can solve (8) with the numerical framework presented in +Sec. IV-B resulting in a unique DOF vector q. Note that when +q is determined, we can then obtain the external forces from +the substrate along the paper Fsubstrate = Fx + Fz, orientation +angle α of the manipulated end, and the suspended length +ls. Recall that through (4), Young’s modulus E, thickness h, +and density ρ were determined to be the main material and +geometric properties of the paper. Therefore, we can outline +the following physical relationship relating all our quantities: +λ = ∥Fx∥ +∥Fz∥, +(λ, α, ls) = f (E, h, ρ, x, z) , +(9) +where f is an unknown relationship. It is then trivial to see +that to prevent sliding the relationship +λ ≤ µs +(10) +must be satisfied, where µs is the static friction coefficient +between the paper and the substrate. Therefore, a trajectory +that minimizes sliding is one that minimizes λ along its path. +One glaring problem remains in that the relation f must be +known to generate any sort of trajectory. In the absence of an +analytical solution, the numerical framework from Sec. IV-B +can be used to exhaustively find mappings between the inputs +and outputs of f. However, generating tuples in this fashion +requires solving the high-dimensional problem in (8). Such a +method would be horribly inefficient and would make real-time +operation infeasible. Instead, we opt to obtain an analytical +approximation of f by fitting a neural network on simulation +data. Currently, this approach has several shortcomings. For +one, directly learning f is difficult given that (9) currently +depends on five parameters as input, resulting in a high +dimensional relationship. Furthermore, since the formulation +directly depends on intrinsic parameters of the paper (E, ρ, +and h), an enormously exhaustive range of simulations must +be run to gather enough data to accurately learn f. +To solve all the aforementioned shortcomings, we reduce +the dimensionality of the problem by applying scaling analysis. +According to Buckingham π theorem, we construct five +dimensionless groups: ¯x = x/Lgb; ¯z = z/Lgb; ¯ls = ls/Lgb; +α, and λ = Ft/Fn, where Lgb is the gravito-bending length +from (4). This results in a non-dimensionalized formulation of +(9) which is expressed as +(λ, α, ¯ls) = F (¯x, ¯z) . +(11) +Note that the mapping relationship F is now irrelevant to +quantities with units, e.g., material and geometric properties +of the paper. As the dimensionality of our problem has been +reduced significantly, we can now express λ as a function of +just two parameters ¯x, ¯z. Therefore, training a neural network +to model F is now trivial as non-dimensionalized simulation +data from a single type of paper can be used. Furthermore, +the low dimensionality of F allows us easily visualize the +λ landscape along a non-dimensional 2D-plane. In the next +section, we will now go over the steps to model F. +V. DEEP LEARNING AND OPTIMIZATION +A. Data Generation +In order to learn the force manifold, we solve (8) for several +sampled (x, z) points. An example of the partial force manifold +produced from this sampling can be observed for a single +suspended length in Fig. 4(b). For a specific (x, z) location, +we apply incremental rotations along the y-axis and find the +optimal rotation angle α that results in MN = 0 on the +manipulated end. For a particular configuration (x, z, α), we +then record the suspended length ls as well as the tangential +and normal forces experienced on the clamped end. This +leads to a training dataset D consisting of six element tuples +(Ft, Fn, α, ls, x, z). We then non-dimensionalize this dataset +to the form (λ, α, ¯ls, ¯x, ¯z). A total of 95796 training samples +were used within a normalized suspended length of ¯ls ≤ 6.84, +which adequately includes the workspace of most papers. +B. Learning Force and Optimal Grasp Orientation +We can now train on our dataset D to obtain a generalized +neural network modeling F: +(λ, α, ¯ls) = FNN(¯x, ¯z). +(12) +To obtain the above function, a simple fully-connected feed- +forward nonlinear regression network is trained with 4 hidden +layers, each containing 392 nodes. Aside from the final output +layer, each layer is followed by a rectified linear unit (ReLU) +activation. In addition, we preprocess all inputs through the +standardization +x′ = x − ¯xD +σD +, +(13) + +3 +5 +4 +2 +3 +2 +1 +1 +0 +3 +4 +5 +6 +76 +(a) +(b) +(d) +(c) +Fig. 5. (a) Visualization of the trained neural network’s non-dimensionalized λ force manifold M and (b) α manifold. An extremely low ¯δ discretization is used +to showcase smoothness. For the force manifold, we observe two distinctive local minima canyons. Note that regions outside the workspace W are physically +inaccurate but are of no consequence to us as they are ignored. For the α manifold, we observe continuous smooth interpolation all throughout which is key +for producing feasible trajectories. Both manifolds showcase the used trajectories in the experiments for folding paper in half for Lgb ∈ [0.048, 0.060, 0.132]. +(c) Showcases the three trajectories in (a) and (b) scaled back to real space. These are the actual trajectories used by the robot. (d) Arbitrary trajectories for +various Lgb with identical start and goal states are shown to highlight the effect of the material property on our control policy. +where x is the original input, ¯xD is the mean of the dataset +D, and σD is the standard deviation of D. +We use an initial 80-20 train-val split on the dataset D with +a batch size of 128. Mean absolute error (MAE) is used as +the error. We alternate between stochastic gradient descent +(SGD) and Adam whenever training stalls. Furthermore, we +gradually increase the batch size up to 4096 and train on the +entire dataset once MAE reaches < 0.001. Using this scheme, +we achieve an MAE of < 0.0005. +C. Constructing the Neural Force Manifold +The neural force manifold (i.e. λ outputs of FNN for the +workspace set) is discretized into a rectangular grid consisting +of ¯δ × ¯δ blocks, where ¯δ = δ/Lgb. For each of the blocks, +we obtain and store a single λ value using the midpoint of +the block. This results in a discretized neural force manifold +M represented as a m × n matrix. For the purposes of path +planning, we add two components to our manifold. First, we +do not allow exploration into any region not belonging to +our dataset distribution (¯ls > 6.84). We do so by defining a +workspace W as all (¯x, ¯z) pairs within the concave hull of +the input portion of the dataset D. Secondly, we also exclude +regions within a certain ¯ls threshold. This is done as positions +with small suspended lengths and large α angles may result +in high curvatures that could cause collision with our gripper +and/or plastic deformation, both of which we wish to avoid. +We denote this region as the penalty region Ls. A visualization +of M with the workspace W and penalty boundary Ls regions +can be seen in Fig. 5(a). The α values corresponding to the +manifold are also shown in Fig. 5(b). +D. Path Planning over the Neural Force Manifold +Given the discretized manifold M, we can now generate +optimal trajectories through traditional path planning algorithms. +We define an optimal trajectory τ ∗ as one that gets to the goal +state while minimizing the sum of λ: +τ ∗ = arg min +τ∈T +i=L−1 +� +i=0 +λi, +(14) +where L is the length of the trajectory and T is the set of all +valid trajectories from the desired start to goal state. We define +a valid trajectory as one that is contained within the acceptable +region +(xi, zi) ∈ W \ Ls ∀ (xi, zi) ∈ τ, +and whose consecutive states are adjacent grid locations. Given +the discretization of the NFM, we can treat M as a graph +whose edge weights consist of λ. Therefore, we can use uniform +cost search to obtain τ ∗. The pseudocode of the path planning +algorithm can be seen in Alg. 1. +VI. ROBOTIC SYSTEM +A. Dual Manipulator Setup +For our experiments, we use two Rethink Robotics’ Sawyer +manipulators as shown in Fig. 7. One arm has an elongated +gripper designed for folding, while the other arm has a spring +compliant roller for creasing and an Intel Realsense D435 +camera for vision feedback. The elongated gripper has rubber +attached to the insides of the fingers for tight gripping. + +6 +6 + Start state +15 +Goal state +4 +5 +5 +10 +Trajectory, T +Workspace, W +Is penalty, Ls +4 +3 +0 +I23 +I23 +2 +2 +2 +1 +1 +0 + +0 +0 +0 +2 +10 +6 +10 +0 +8 +0 +0.08 +0.08 +Lg6=0.065 +Lgb=0.082 +Lgb =0.103 +0.06 +三0.06 +Lg6=0.118 +0.129 +2 +2 +0.04 +0.04 +0.02 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +α [m] +α[m]7 +Fig. 6. Example of our perception system with a top down view of the folding procedure. (a) Showcases the the intuitive baseline results while (b) showcases +our open-loop algorithm for Lgb = 0.048 and C = 0.25m. Similar to Fig. 2, the solid green line indicates the desired end effector position while the +dashed blue line indicates the crease location. We observe that the intuitive baseline has considerable sliding while our open-loop algorithm has near-perfect +performance for this case. +1 +2 +3 +4 +5 +6 +Fig. 7. Experimental apparatus: Two robot manipulators, one for folding (1) +and the other for creasing (3). An elongated gripper (2) is used for grabbing +the manipulated end of the folding paper. A roller (5) with compliant springs +(6) is used for forming the crease. An Intel RealSense D435 camera (4) is +attached to the creasing arm offer vision feedback during the folding procedure. +All gripper attachments were 3D printed. +B. Perception System +For our perception, we take an eye-in-hand approach by +attaching an Intel Realsense D435 to the roller arm. We do not +use the depth component of the camera as we align the camera +to be pointing down along the world z-axis and the distance +from the camera to the table is known. To detect the pose of +the paper, we use simple color detection to segment the paper +and then use Shi-Tomasi corner detection [53] to obtain the +position of the bottom edge. An example of the top-down view +as well as detected poses produced by the camera can be seen +in Fig. 6. +C. Vision-feedback Control +Although we minimize λ with our proposed framework, +sliding could still happen due to a substrate’s low friction +surface and/or jittering of the robot’s end-effector. Notice that +the generated optimal trajectory τ ∗ from Sec. V-D assumes +that the origin o of our coordinate system shown in Fig. 4(a) +is fixed. We can define the origin as o = q0 − Lˆx where +Algorithm 1: Uniform Cost Search +Input: ¯xs, ¯zs, ¯xg, ¯zg, M +Output: τ ∗ +1 Func UCS(¯xs, ¯zs, ¯xg, ¯zg, M): +2 +W ← valid workspace of M +3 +Ls ← ls penalty region +4 +h ← initialize min heap priority queue +5 +c ← initialize empty list +6 +ns ← node with location (¯xs, ¯zs) and cost 0 +7 +ng ← node with location (¯xg, ¯zg) and cost 0 +8 +h.push(ns) +9 +while len(h) > 0 do +10 +ni ← h.pop() +11 +if ni == ng then +12 +τ ∗ ← path from start to goal +13 +break +14 +c.append(ni) +15 +for (¯xj, ¯zj) ∈ neighbors of ni do +16 +if (¯xj, ¯zj) /∈ W \ Ls then +17 +continue +18 +nj ← node with location (¯xj, ¯zj) and cost +λj from M +19 +if nj ∈ c then +20 +continue +21 +if nj ∈ h and cost of nj is higher then +22 +continue +23 +h.push(nj) +24 +τ ∗ ← perform trajectory smoothing on τ ∗ +25 +return τ ∗ +L is the total length of the paper. Any amount of sliding +indicates that q0 is moving along the x-axis and therefore, the +origin o also moves an identical amount. When this occurs, our +position within the manifold during traversal deviates from the +optimal trajectory. Furthermore, without adaptive replanning, +the amount of sliding ∆x will directly result in ∆x amount +of error when creasing. To circumvent this, we introduce a +vision-feedback approach that mitigates the effects of sliding. +We perform vision-feedback at N evenly spaced out intervals + +(a)8 +Fig. 8. An overview of our folding pipeline. The top row showcases offline +proponents while the bottom row shows online. On the offline side, we use our +trained neural network to generate the necessary force manifold for planning. +Then, given an input tuple (xs, zs, xg, zg, Lgb), we generate an end-to-end +trajectory using uniform cost search. This end-to-end trajectory is then split +up into partial trajectories that are carried out by the robot. At the conclusion +of each partial trajectory, we measure paper sliding and replan the next partial +trajectory to rectify the error. +of the trajectory τ ∗ as shown in Fig. 8. To do so, we first split +up τ ∗ into N partial trajectories. Aside from the first partial +trajectory τ ∗ +0 , we extract the start and goal states of the other +1 ≤ i ≤ N partial trajectories resulting in a sequence of N +evenly spaced out states S = {(x1, z1, α1), ..., (xN, zN, αN)} +when accounting for overlaps. After carrying out τ ∗ +0 , we detect +the amount of sliding ∆x and incorporate this error by updating +the start state and non-dimensionalizing as +¯xc +i = xi − ∆x +Lgb +. +We then replan a partial trajectory τ ∗ +i from the updated start +state (xc +i, zi) to the next state (xi+1, zi+1) in the sequence and +carry out this updated trajectory. This is repeated until reaching +the goal state. By properly accounting for sliding, we ensure +that the traversal through the NFM is as accurate as possible. +We note that this scheme allows us obtain corrected partial +trajectories in near real time once N becomes sufficiently large +as each partial trajectory’s goal state becomes increasingly +close to its start state, allowing for uniform cost search to +conclude rapidly. We direct the reader to the supplementary +videos mentioned in Sec. I which showcase the speed of the +feedback loop. +Rectifying the sliding ∆x is not the only error we must +address. Recount that we assume an optimal grasp orientation +α for each position within the manifold. When the origin of +our NFM moves, our true position does not match the intended +position, resulting in also an angular error +αc +i = FNN(¯xc +i, ¯zi), +∆α = αi − αc +i. +Algorithm 2: Closed-loop Control Pseudocode +Input: (xs, zs), (xg, zg), Lgb, δ, N, FNN +1 M ←DiscretizeManifold (FNN, δ) +2 ¯xs, ¯zs, ¯xg, ¯zg ← non-dimensionalize with Lgb +3 ¯τ ∗ ← UCS (¯xs, ¯zs, ¯xg, ¯zg, M) +4 update ¯τ ∗ with αs using FNN +5 τ ∗ ← convert ¯τ ∗ to real space with Lgb +6 τ ∗ +0 , ..., τ ∗ +N−1 ← SplitTrajectory (τ ∗, N) +7 S ← extract start and goal states +8 carry out τ ∗ +0 on robot +9 for (xi, zi, αi) and (xi+1, zi+1, αi+1) ∈ S do +10 +∆x ← detect sliding of paper +11 +xc +i ← xi − ∆x +12 +¯xc +i, ¯zi, ¯xi+1, ¯zi+1 ← non-dimensionalize with Lgb +13 +αc +i ← FNN(¯xc +i, ¯zi) +14 +∆α ← αi − αc +i +15 +¯τ ∗ +i ← UCS (¯xc +i, ¯zi, ¯xi+1, ¯zi+1, M) +16 +L ← len(¯τ ∗ +i ) +17 +αi ← obtain αs of ¯τ ∗ +i using FNN +18 +αc +i ← αi + ∆α[1, (L − 1)/L, ..., 1/L, 0]T +19 +append ¯τ ∗ +i with αc +i +20 +τ ∗ +i ← convert ¯τ ∗ to real space with Lgb +21 +carry out τ ∗ +i on robot +22 crease paper with roller +Simply applying a −∆α update to the first point in a partial +trajectory results in a large rotational jump that only exacerbate +the sliding issue. Furthermore, we postulate that so long as +sliding is not extremely large, the incorrect α at the current +position within the manifold is still fairly optimal. Therefore, +the ∆α error is incorporated into the trajectory gradually: +τ ∗ +i = UCS(¯xc +i, ¯zi, ¯xi+1, ¯zi+1, M), +αi = FNN(τ ∗ +i ), +αc +i = αi + ∆α[1, (L − 1)/L, ..., 1/L, 0]T , +where UCS stands for uniform cost search and L is the length +of the trajectory τ ∗ +i . This gradual correction ensures that we +minimize sliding while maintaining smoothness of the trajectory. +The pseudocode for our full closed-loop algorithm can be seen +in Alg. 2. +VII. EXPERIMENTS AND ANALYSIS +A. Measuring the Material Property of Paper +To use our framework, we must develop a way to accurately +measure the parameter Lgb for a particular piece of paper. +As mentioned previously, Lgb encapsulates the influence of +bending and gravity. With this in mind, we propose a simple +way to measure the parameter. +As shown in Fig. 10(a), when one end of the paper is +fixed, the paper will deform due to the coupling of bending + +OfMine +Obtain force manifold from NN +Compute optimal end-to-end path +T* = [(Cs, Zs, Qs), .., (Cg, Zg, ag) +Path Planner (UCS) +Compute corrected +Detect paper slippage +partial trajectory +Perception +Red is the true location +Ac, Aa +obtained from vision +feedback +Transform trajectory to real space +Once goal state is +Carry out partial trajectory, +reached, crease paper +then repeat for next step +Motion +Planner +Online9 +Fig. 9. Experimental results for all folding scenarios. Each column indicates a folding scenario while the the top row (a) showcases the fold length and bottom +row (b) showcases the spin error. Boxplot results are shown color coded for the intuitive baseline, open-loop control, and closed-loop control algorithms. +Medians are shown as orange lines, means are shown as turquoise circles, and the desired target value is shown as a light blue horizontal line. We note that +both our open-loop and closed-loop algorithms have significant improvements over the intuitive baseline as shown by the broken axis in (a). Our algorithms +also have significantly less variance. +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +5 +10 +15 +20 +lh +L +(a) +(b) +Norm. paper legnth, L +Cardboard paper +US Letter paper +A4 paper +Square origami paper +Fig. 10. (a) Schematic of a hanging plate. The manipulation edge is fixed +horizontally; (b) Relationship between the ratio ϵ = lh/L and normalized +total length of the paper ¯L = L/Lgb. +and gravitational energy. Therefore, the following mapping +relationship exists: +¯L = L(ϵ), +¯L = +L +Lgb +, +ϵ = lh +L , +(15) +where lh is the vertical distance from the free end to the fixed +end and L is the total length of the paper. We can obtain the +mapping relationship L(ϵ) using numerical simulations, which +is shown in Fig. 10(b). With this mapping known, simple +algebra can be performed to obtain Lgb. First, we measure the +ratio ϵ = lh/L for a particular paper to obtain its corresponding +normalized total length ¯L. Then, the value of Lgb can be +calculated simply by Lgb = L/¯L. Once we obtain Lgb, we can +now use the non-dimensionlized mapping relationship in (11) +to find the optimal path for manipulating the paper. +B. Experimental Setup +For our experiments, we tested folding on 4 distinct types +of paper: +1) A4 paper, Lgb = 0.048m, +2) US Letter paper, Lgb = 0.060m, +3) Cardboard paper (US Letter dimensions), Lgb = 0.132m, +4) Square origami paper, Lgb = 0.043m. +For the rectangular papers (1-3), we do two sets of experiments. +The first involves folding the papers to an arbitrary crease +location (C = 0.25m for A4 and C = 0.20m for US Letter and +cardboard), while the second involves folding the papers in half. +For the square origami paper, we choose an arbitrary crease +location of C = 0.30m. This results in a total of 7 folding +scenarios. For each of the scenarios, we conduct experiments +using 3 different algorithms (an intuitive baseline, our open- +loop approach, and our closed-loop approach). We complete +10 trials for each of these algorithms, resulting in a total of +210 experiments. +C. Baseline Algorithm +To showcase the benefits of our folding algorithm, we +compare our algorithm to an intuitive baseline. We can think +of an intuitive baseline algorithm as one that would work if the +opposite end of the paper were fixed to the substrate. Naturally, +such a trajectory would be one that grabs the edge of the paper +and traces the half perimeter of a circle with radius R = C/2: +dθ = π/M, +τB = {(R cos(idθ), R sin(idθ), idθ) ∀ i ∈ [0, M]}, +(16) +where M is an arbitrary number of points used as the resolution +of trajectory. We choose M = 250 for all experiments. + +Rect, Lgb=0.048 +Rect, Lgb=0.048 +Rect, Lgb=0.060 +Rect, Lgb=0.060 +Rect, Lgb=0.132 +Rect, Lgb=0.132 +Diag, Lgb = 0.043 +(a) +C=0.25 (0.13) +C=Half (0.1485) +C=0.20 (0.105) +C=Half (0.14) +C=0.20 (0.105) +C=Haif (0.14) +C=0.30 (0.155) +0 +0.130 +0.140 +0.105 +0.14 - +0.148 +T +0.15 +0.104 +0 +0 +0.100 +0.146 +0.14 +0 +工 +0 +0.095 +← 0.102 +0.135 - +0.12 - +0.13 +0.055 +0.10 +T +0.07 +工 +0.12 - +. +T +0.06 - +空 +0.12 +0.050 +0.08 +0.11 +0 +0.06 - +0 +0.100 +0.11 +0.05 +(b) +2 - +2- +5.0 +3 - +4- +2 - +2 +2.5 - +[deg] +Q +2 +1 - +1 - +0 +2 - +T +0 +0.0 +T +0- +-0 +！ +-0 +0: +2.5 +0 +0 +1 +0 +-1 +Intuitive Baseline +Open-loop Control +Closed-loop Control10 +Fig. 11. Isometric views of different folding scenarios. (a1-2) showcases C = Half folding for Lgb = 0.048 paper with the intuitive baseline and our open-loop +algorithm, respectively. (b1-2) showcases C = 0.30m diagonal folding for Lgb = 0.043 with the intuitive baseline our closed-loop algorithm, respectively. +D. Metrics +The metrics used for the experiments were the average fold +length and the spin error. The average fold length was calculated +by simply taking the average of the left and right side lengths +up until the crease. The spin error was calculated as the angle +θerr that results in the difference between the left and right +side lengths. For square papers, the fold length was defined +as the perpendicular length from the tip to the crease and the +spin error was the angular deviation from this line to the true +diagonal. +E. Parameters +The neural force manifold M was discretized using a ¯δ +corresponding to δ = 2mm depending on the material as we +found this discretization to have good compromise between +accuracy and computational speed. All rectangular papers used +a penalty region Ls defined by ¯ls < 0.958 while the square +paper used one defined by ¯ls < 1.137. This discrepancy is +due to the fact that the diagonal paper has a smaller yield +strength compared to the the rectangular paper, i.e., to prevent +extremely high curvatures, a larger suspended length ¯ls range +must be avoided. +For closed-loop control, we chose to split all trajectories into +N = 5 intervals regardless of trajectory length. Furthermore, +we use an extremely slick (i.e. low friction) table to showcase +the robustness of our method. Using an empirical method, we +measured the static coefficient of friction of our papers and the +substrate to be approximately µs = 0.12. For comparison, the +static coefficient of friction for steel on steel (both lubricated +with castor oil) is µs = 0.15. +F. Results and Analysis +All experimental results can be seen expressed as box plots +where we showcase achieved fold lengths and spin errors in +Fig. 9(a) and (b), respectively. When observing the achieved +fold lengths, we see significant improvement over the baseline +for all folding scenarios. Due to the large gap in performance, +broken axes are used to properly display the variance of the +recorded data. We note that not only do our algorithms achieve +significantly better performance on average, the variance of +our approaches is also much lower as shown by the decreased +y-axis resolution after the axis break. We attribute the high +variance of the baseline method due to the increased influence +of friction, which can often cause chaotic, unpredictable results. +In other words, truly deterministic folding can only be achieved +when sliding is nonexistent. +For a vast majority of cases, we observe a clear improvement +over the open-loop algorithm when incorporating vision- +feedback. Intuitively, we observe a trend where the performance +gap between our open-loop and closed-loop algorithms grow +as the material stiffness increases for rectangular folding. For +softer materials (Lgb = 0.048), the open-loop algorithm has +near perfect performance as shown when folding a paper in +half in Fig. 11(a2). In comparison, Fig. 11(a1) showcases the +baseline algorithm failing with significant sliding. +The sliding problem is only exacerbated by increasing the +stiffness of the material (Lgb = 0.132) where Fig. 12(a) +showcases the baseline algorithm failing to fold the cardboard +paper in half by a margin almost as long as the paper itself. +In comparison, our open-loop algorithm is capable of folding +the cardboard with significantly better results albeit with some +visual sliding as shown in Fig. 12(b). As the material stiffness +increases, the benefits of the incorporated vision-feedback +are more clearly seen as we are able to achieve near perfect + +(al) +(a2) +(b1) +(b2)11 +Fig. 12. Isometric views for folding C = Half with the stiffest paper (Lgb = 0.132). (a) showcases the intuitive baseline, which fails drastically as the +stiffness of the paper causes excessive sliding during the folding process. (b) showcases our open-loop algorithm, which has significant improvements over the +baseline with minimal sliding. Finally, (c) showcases our closed-loop algorithm, which improves upon our open-loop results and achieves near perfect folding. +folding for cardboard in Fig. 12(c). All of our findings for +rectangular folding also match the results of our diagonal +folding experiment shown in Fig. 11(b1-b2), where closed- +loop once again achieves minimal sliding when compared to +the baseline. Overall, the matching findings across all of our +experiments showcase the robustness of our formulation against +material and geometric factors. +We observe one oddity for the folding scenario of Lgb = +0.048 and C = Half where the open-loop algorithm outper- +formed our closed-loop variant. Still, we wish to point out that +this decrease in performance is only on average 1mm, which +can easily be attributed to repetitive discretization error caused +by N = 5 replanning. In fact, as we use a discretization of +δ = 2mm for the manifold, compounding rounding errors can +easily cause 1-2mm errors. With this in mind, our closed-loop +method achieves an average fold length performance within a +1-2mm tolerance across all experiments. +In terms of spin error, we found that softer materials had +the greatest error. As the frictional surface of the table is not +perfectly even, any amount of sliding will directly result in +uneven spin as shown in Fig. 11(a). As the material stiffness +increases, the spin errors became more uniform across the +methods as the influence of friction is not enough to deform +the paper. Still, we can see that our open and closed-loop +algorithms had less sliding than the baseline on average. +VIII. CONCLUSION +We have introduced a novel control strategy capable of +robustly folding sheets of paper of varying materials and +geometries with only a single manipulator. Our framework +incorporates a combination of techniques spanning several +disciplines, including physical simulation, machine learning, +scaling analysis, and path planning. The effectiveness of +our framework was showcased through extensive real world +experiments against an intuitive baseline. Furthermore, an +efficient near real-time visual-feedback algorithm was imple- +mented that further minimizes folding error. With our closed- +loop sensorimotor control algorithm successfully accomplished +challenging scenarios such as folding stiff cardboard with +repeatable accuracy. +For future work, we hope to to tackle the difficult problem +of creating arbitrary creases along sheets of paper with non- +symmetric centerlines. Such non-symmetric papers can no +longer be represented as a reduced-order model of a 2D +elastic rod, thus requiring a different formulation. 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Shi and Tomasi, “Good features to track,” in 1994 Proceedings of IEEE +Conference on Computer Vision and Pattern Recognition, pp. 593–600, +1994. + diff --git a/D9A0T4oBgHgl3EQfAv_u/content/tmp_files/load_file.txt b/D9A0T4oBgHgl3EQfAv_u/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..746687eb586e203b4822cdcb2139e5915c1320f3 --- /dev/null +++ b/D9A0T4oBgHgl3EQfAv_u/content/tmp_files/load_file.txt @@ -0,0 +1,1017 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf,len=1016 +page_content='1 Deep Learning of Force Manifolds from the Simulated Physics of Robotic Paper Folding Dezhong Tong∗,1, Andrew Choi∗,2, Demetri Terzopoulos2, Jungseock Joo3, and Mohammad Khalid Jawed†,1 Abstract—Robotic manipulation of slender objects is challeng- ing, especially when the induced deformations are large and nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Traditionally, learning-based control approaches, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=', imitation learning, have been used to tackle deformable material manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Such approaches lack generality and often suffer critical failure from a simple switch of material, geometric, and/or environmental (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=', friction) properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' In this article, we ad- dress a fundamental but difficult step of robotic origami: forming a predefined fold in paper with only a single manipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' A data- driven framework combining physically-accurate simulation and machine learning is used to train deep neural network models capable of predicting the external forces induced on the paper given a grasp position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We frame the problem using scaling analysis, resulting in a control framework robust against material and geometric changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Path planning is carried out over the generated manifold to produce robot manipulation trajectories optimized to prevent sliding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Furthermore, the inference speed of the trained model enables the incorporation of real-time visual feedback to achieve closed-loop sensorimotor control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Real- world experiments demonstrate that our framework can greatly improve robotic manipulation performance compared against natural paper folding strategies, even when manipulating paper objects of various materials and shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Index Terms—robotic manipulation, deformable material ma- nipulation, deep neural networks, data-driven models, closed-loop sensorimotor control I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' INTRODUCTION From shoelaces to clothes, we encounter flexible slender structures throughout our everyday lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' These structures are of- ten characterized by their ability to undergo large deformations when subjected even to moderate forces, such as gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' People possess an incredible innate understanding of the dynamics of such deformable objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=', we can use gravity to perfectly manipulate a shirt over our heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Instilling such intuition into robots remains an important research problem and has the potential to breed numerous applications with considerable economic and humanitarian potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Some examples include preparing deformable products in the food industry [1], [2], assisting in the medical field [3]–[5], and providing caregiving assistance to elderly and disabled communities, including with respect to dressing [6]–[10] and feeding [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' However, the The authors are with the University of California, Los Angeles (UCLA), CA 90095, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 1Dezhong Tong and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Khalid Jawed are with the UCLA Department of Mechanical & Aerospace Engineering (email: tltl960308@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' khalidjm@seas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 2Andrew Choi and Demetri Terzopoulos are with the UCLA Computer Science Department (email: asjchoi@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' dt@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 3Jungseock Joo is with the UCLA Department of Communication and is currently working at NVIDIA Corporation (email: jjoo@comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' ∗ Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' † Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Position & material parameters Learned model Force manifold Optimization algorithm Optimal path Intuitive trajectory (circular curve) Gripper Paper Substrate Initial free end Initial free end Obvious sliding Minimal sliding Initial state Folded result Intuitive manipulation Our optimal manipulation (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Half valley folding for A4 paper with (a) intuitive manipulation and (b) our designed optimal manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' An intuitive manipulation scheme such as tracing a semi-circle experiences significant sliding due to the bending stiffness of the paper, resulting in a poor fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' By contrast, our optimal manipulation approach achieves an excellent fold by taking into consideration the paper’s deformation and thus minimizing sliding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' robotic manipulation of deformable objects is highly nontrivial as a robot must be able to take into account future deformations of the manipulated object to complete manipulation tasks successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Prior research has focused primarily on manipulating either cloth [13]–[18] or ropes [12], [19]–[25] and as a result, the robotic manipulation of many other deformable objects still lacks robust solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' In this article, we address a particularly difficult deformable manipulation task — folding paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Paper is similar to cloth but typically has a much larger bending stiffness and a slippery surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Therefore, compared with folding garments and fabrics, more delicate and insightful manipulations are required for folding sheets of paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Our Approach We propose a framework that combines physically accurate simulation, scaling analysis, and machine learning to generate folding trajectories optimized to prevent sliding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' With scaling analysis, we make the problem non-dimensional, resulting in both dimensionality reduction and generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We then train neural networks, whose outputs are referred to as neural force manifolds (NFM), to continuously approximate a scaled force manifold sampled purely from simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Compared to numerical models that require the entire geometric configuration arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='01968v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='RO] 5 Jan 2023 2 of the paper, NFMs map the external forces of the paper given only the grasp position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Therefore, we can generate trajectories optimized to minimize forces (and thus minimize sliding) by applying path planning algorithms in near real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We show that our approach is capable of folding paper on extremely slick surfaces with little-to-no sliding (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Our main contributions are as follows: (1) we formulate a solution to the folding problem in a physically robust manner using scaling analysis, resulting in complete generality with respect to material, geometric, and environmental properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (2) we train a neural network with non-dimensional simulation data forming a fast and accurate model that can generate a descriptive force manifold for trajectory optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (3) we utilize the high inference speed of our trained model with a perception system to construct a robust and efficient closed- loop sensorimotor control algorithm for the folding task, and finally (4) we demonstrate full sim2real realization through an extensive robotic case study featuring 210 experiments across paper sheets of various materials and shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' While several previous works have trained their policies purely from simulation data [7], [19], [26]–[28], these works lacked real world validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' To our knowledge, our framework is the first to provide optimal folding trajectories with complete generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We release supplementary videos as well as all source code and CAD files as open source at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='com/ StructuresComp/deep-robotic-paper-folding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Overview The remainder of the article is organized as follows: We begin with a review of related work in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' A brief description of the folding problem is presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The formulation of a reduced-order physics-based model is discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' IV, where we formulate the folding problem using scaling analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' V, we formulate our learning framework as well as algorithms for optimal path planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Next, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' VI, we introduce our robotic system as well as formulate our closed-loop visual feedback pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Experimental results for a robot case study and analysis of the results are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Finally, we provide concluding remarks and discuss the potential of future research avenues in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' RELATED WORK The majority of prior works tackling the folding problem can be roughly divided into four categories: mechanical design-based solutions, vision-based solutions, learning-based solutions, and model-based solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Mechanical design-based approaches typically involve solv- ing the folding problem through highly specialized manipulators or end effectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Early approaches involve specialized punches and dies for sheet metal bending [29], More recently, highly specialized manipulators for robotic origami folding have also been developed [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Such methods can reliably produce repeatable folding but are often limited to a highly specific fold, geometry, and/or material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Vision-based approaches involve folding deformable mate- rials by generating folding motions purely from visual input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' These approaches are usually common for folding clothes [14], [16], [31] as they are extremely soft, which results in the easy predictability of their deformation state given a particular action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Such approaches can be effective and rather simple to implement, but do not transfer well to paper folding as paper possesses a much higher stiffness when compared to fabric and will attempt to restore its natural, undeformed state if not properly handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Learning-based approaches involve the robot learning how to fold through training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The most popular has been to learn control policies from human demonstrations, also known as learning from demonstrations (LfD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Prior research has demonstrated flattening and folding towels [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Teleop demonstrations are a popular avenue for training policies and have been used to learn how to manipulate deformable linear objects (DLOs) [34] as well as folding fabric [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' To eliminate the need for expensive human-labeled data, researchers have also focused on tackling the sim2real problem for robotic folding, where reinforcement learning has been used to train robots to fold fabrics and cloths completely from simulation [26], [28], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' More recently, Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' [37] used reinforcement learning to train a robot to flip pages in a binder through tactile feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Pure learning-based methods have shown promising performance, but only for specific tasks whose state distribution matches the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Such methods tend to generalize quite poorly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=', when the material or geometric properties change drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Model-based approaches, where the model can either be known or learned, often use model predictive control to manipulate the deformable object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' They involve learning the natural dynamics of deformable objects through random perturbations [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' These models are generally fast, but they can be inaccurate when experiencing new states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Known models are often formulated to be as physically accurate as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' They can be referred to as physics-based (as opposed to simulated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Their physical accuracy allows for the direct application of their predictive capabilities in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Examples are published for rectangular cloth folding [39], strip folding [40], and garment folding [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Still, known models are usually quite expensive to run and must often face a trade-off between accuracy and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Despite the large quantity of prior research focusing on 2D deformable object manipulation, the majority of these efforts have limited their scope to soft materials such as towels and cloth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Such materials are highly compliant and often do not exhibit complicated nonlinear deformations, thus allowing for solutions lacking physical insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We instead tackle the scenario of folding papers of various stiffnesses with a single manipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Because of its relatively high bending stiffness and slippery surface, paper is significantly more difficult to manipulate since large deformations will cause sliding of the paper on the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Such an example can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 1(a), where intuitive folding trajectories that may work on towels and cloth fail for paper due to undesired sliding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' However, a few works have attempted to solve the paper fold- ing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For example, Elbrechter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' [42] demonstrated paper folding using visual tracking and real-time physics-based modeling with impressive results, but they required expensive 3 end effectors (two Shadow Dexterous Hands), one end effector to hold the paper down while folding at all times, and the paper to have AR tags for visual tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Similarly, Namiki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' [43] also achieved paper folding through dynamic motion primitives and used physics-based simulations to estimate the deformation of the paper sheet, also requiring highly specialized manipulators and an end effector to hold the paper down while folding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' By contrast, our method can fold papers reliably without any need for holding down the paper during the folding operation and requires only an extremely simple 3D printed gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Other approaches have also attempted to fold with a single manipulator while minimizing sliding [36], [40], but these methods focused on fabrics whose ends were taped down to the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' PROBLEM STATEMENT This article studies a simple but challenging task in robotic folding: creating a predefined crease on a sheet of paper of typical geometry (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=', rectangular, diamond, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=') as is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Only one end of the paper is manipulated while the other end is left free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Thus, extra fixtures are unnecessary and the folding task can be completed by a single manipulator, which simplifies the workspace, but slippage of the paper against the substrate must be mitigated during manipulation, which is a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The task can be divided into two sub-tasks and three states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The first sub-task is manipulating one end of the paper from the initial flat state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 2(a)) to the folding state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 2(b)), with the goal that the manipulated edge or point should overlap precisely with the crease target line or point C as shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' With the manipulated edge of the paper at the origin, the manipulator moves in the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Since the manipulated paper usually has relatively high bending stiffness, large nonlinear elastic deformations are induced in the folding state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' In the second sub-task, the paper must be permanently deformed to form the desired crease at C/2, thus achieving the final folded state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 2(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' PHYSICS-BASED MODEL AND ANALYSIS We next present the numerical framework for studying the underlying physics of the paper folding process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' First, we analyze the main deformations of the manipulated paper and prove that a 2D model is sufficient to learn the behaviors of the manipulated paper so long as the sheet is symmetrical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Second, we briefly introduce a physically accurate numerical model based on prior work in computer graphics [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Third, we formulate a generalized strategy for paper folding using scaling analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Reduced-Order Model Representation Paper is a unique deformable object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Unlike cloth, its surface is developable [45];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=', the surface can bend but not stretch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Furthermore, shear deformations are not of particular importance as the geometry of the manipulated paper is symmetrical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Therefore, the primary nonlinear deformation when folding paper in our scenario is bending deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We Initial state Folding state Folded state Manipulated end Manipulated node Desired crease Desired crease Free end Free node Target line Target node Rectangular paper Symmetrical paper (square) (a) (b) (c) x z y o 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='5C C Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Folding sheets of paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The manipulation process involves (a) the initial state, where the paper lies flat on the substrate, followed by (b) the folding state, where the manipulated end is moved to the “crease target” line C, and finally (c) the folded state, which involves forming the desired crease on the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' postulate that the nonlinear behaviors of paper arise primarily from a balance of bending and gravitational energies: ϵb ∼ ϵg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' To further understand the energy balance of the manipulated paper, we analyze an arbitrary piece in the paper, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The bending energy of this piece can be written as ϵb = 1 2kbκ2l, (1) where l is its undeformed length of the piece, κ is its curvature, and its bending stiffness is kb = 1 12Ewh3, (2) where w is its undeformed width, h is its thickness, and E is its Young’s modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The gravitational energy of the piece is ϵg = ρwhlgH, (3) where ρ is its volume density and H is its vertical height above the rigid substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' From the above equations, we obtain a characteristic length called the gravito-bending length, which encapsulates the influence of bending and gravity: Lgb = � Eh2 24ρg � 1 3 ∼ � h κ2 � 1 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (4) The length is in units of meters, and we can observe that it scales proportionally to the ratio of thickness to curvature squared, which are the key quantities describing the deformed configuration of the manipulated paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Note that the formula- tion of Lgb contains only one geometric parameter, the paper thickness h, which means that other geometric quantities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=', length l and width w) have no influence on the deformed configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 4 g (a) (b) (c) (d) Mesh S H Rigid substrate q0 q1 qi-1 qi-1 qi qi ti-1 ti qi+1 qi+1 qN l h w Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (a) Schematic of a paper during the folding state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (b) Bending deformations of a small piece in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (c) Reduced-order discrete model (planer rod) representation of our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (d) Notations in the discrete model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Additionally, due to the symmetrical geometry of the paper, curvature κ should be identical for all regions at the same height H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Therefore, we can simply use the centerline of the paper, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 3(a), to express the paper’s configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We model this centerline as a 2D planar rod since deformations are limited to the x, z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We implement a discrete-differential- geometry (DDG)-based numerical simulation to simulate the 2D planar rod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We present the details of this numerical framework in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Discrete Differential Geometry Numerical Model Following pioneering work on physics-based modeling and simulation of deformable curves, surfaces, and solids [46]– [48], the computer graphics community has shown impressive results using DDG-based simulation frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For example, the Discrete Elastic Rods (DER) [44] framework has shown efficient and physically accurate simulation of deformable linear objects in various scenarios including knot tying [49], helix bifurcations [50], coiling of rods [51], and flagella buckling [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Given this success, we use DER to model the centerline of the paper as a 2D planar rod undergoing bending deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 3(c), the discrete model is comprised of N + 1 nodes, qi (0 ≤ i ≤ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Each node, qi, represents two degrees of freedom (DOF): position along the x and the z axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' This results in a 2N + 2-sized DOF vector representing the configuration of the sheet, q = [q0, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=', qN]T , where T is the transpose operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Initially, all the nodes of the paper are located in a line along the x-axis in the paper’s undeformed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' As the robotic manipulator imposes boundary conditions on the end node qN, portions of the paper deform against the substrate as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We compute the DOFs as a function of time q(t) by integrating the equations of motion (EOM) at each DOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Before describing the EOM, we first outline the elastic energies of the rod as a function of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Kirchhoff’s rod theory tells us that the elastic energies of a rod can be divided into stretching Es, bending Eb, and twisting Et energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' First, The stretching elastic energy is Es = 1 2ks N−1 � i=0 � 1 − ∥qi+1 − qi∥ ∆l �2 ∆l, (5) where ks = EA is the stretching stiffness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' E is Young’s modulus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' A = wh is the cross-sectional area, and ∆l is the undeformed length of each edge (segment between two nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The bending energy is Eb = 1 2kb N−1 � i=2 � 2 tan φi 2 − 2 tan φ0 i 2 �2 1 ∆l, (6) where kb = Ewh3 12 is the bending stiffness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' w and h are the width and thickness respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' φi is the “turning angle” at a node as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 3(d), and φ0 i is the undeformed turning angle (0 for paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Finally, since we limit our system to a 2D plane, we can forgo twisting energies entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The total elastic energy is then simply Eel = Es + Eb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Indeed, a ratio ks/kb ∼ w/h2 >> 1 indicates that stretching strains will be minimal which matches our intuition as paper is usually easy to bend but not stretch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Therefore, the stretching energy item in (5) acts as a constraint to prevent obvious stretching for the modeled planar rod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We can now construct our EOM as a simple force balance P(q) ≡ M¨q + ∂Eel ∂q − Fext = 0, (7) where M is the diagonal lumped mass matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' ˙( ) represents derivatives with respect to time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' − ∂Eel ∂q is the elastic force vector, and Fext is the external forces applied on the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Note that (7) can be solved using Newton’s method, allowing for full simulation of the 2D planar rod under manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Generalized Solution and Scaling Analysis As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' III, the core of the folding task is to manipulate the end qN to the target position C starting from an initially flat state shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' To do so, we analyze the physical system in order to achieve a solution capable of minimizing sliding during manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We first denote several quantities to describe the deformed configuration of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Here, we introduce a point qC, which is the node that connects the suspended (z > 0) and unsuspended regions (z = 0) of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We focus primarily on the suspended region as deformations occur solely in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' An origin o is defined for our 2D plane which is located at the initial manipulated end qN as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For the manipulated end, the robot end-effector imposes a position qN = (x, z) and an orientation angle α to control the pose of the manipulated end as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' On the connective node qC, the tangent is always along the x-director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=" Here, we impose a constraint that the curvature at the manipulated end is always zero so that sharp bending deformations are prevented, which is crucial to preventing permanent deformations during 5 (a) (b) q0 qC qN (qN ') Norm." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' x coord, x x z s ls=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='10 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' z coord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' z c Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (a) Side view of a symmetrical paper during folding with coordinate frame and relevant notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (b) Sampled λ forces for a particular ¯ls of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' This showcases one of the sampled “partial” force manifolds that we use train our neural network on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' the folding process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' With these definitions, we can now modify (7) with the following constraints: P(q) = 0, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' qN = (x, z), dqC ds = (−1, 0), MN = 0, ls ≡ � qN qC ds = qC · ˆx, (8) where MN is the external moment applied on the manipulated end;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' s is the arc length of the paper’s centerline, and ls is the arc length of the suspended region (from qC to qN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We can solve (8) with the numerical framework presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' IV-B resulting in a unique DOF vector q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Note that when q is determined, we can then obtain the external forces from the substrate along the paper Fsubstrate = Fx + Fz, orientation angle α of the manipulated end, and the suspended length ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Recall that through (4), Young’s modulus E, thickness h, and density ρ were determined to be the main material and geometric properties of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Therefore, we can outline the following physical relationship relating all our quantities: λ = ∥Fx∥ ∥Fz∥, (λ, α, ls) = f (E, h, ρ, x, z) , (9) where f is an unknown relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' It is then trivial to see that to prevent sliding the relationship λ ≤ µs (10) must be satisfied, where µs is the static friction coefficient between the paper and the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Therefore, a trajectory that minimizes sliding is one that minimizes λ along its path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' One glaring problem remains in that the relation f must be known to generate any sort of trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' In the absence of an analytical solution, the numerical framework from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' IV-B can be used to exhaustively find mappings between the inputs and outputs of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' However, generating tuples in this fashion requires solving the high-dimensional problem in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Such a method would be horribly inefficient and would make real-time operation infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Instead, we opt to obtain an analytical approximation of f by fitting a neural network on simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Currently, this approach has several shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For one, directly learning f is difficult given that (9) currently depends on five parameters as input, resulting in a high dimensional relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Furthermore, since the formulation directly depends on intrinsic parameters of the paper (E, ρ, and h), an enormously exhaustive range of simulations must be run to gather enough data to accurately learn f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' To solve all the aforementioned shortcomings, we reduce the dimensionality of the problem by applying scaling analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' According to Buckingham π theorem, we construct five dimensionless groups: ¯x = x/Lgb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' ¯z = z/Lgb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' ¯ls = ls/Lgb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' α, and λ = Ft/Fn, where Lgb is the gravito-bending length from (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' This results in a non-dimensionalized formulation of (9) which is expressed as (λ, α, ¯ls) = F (¯x, ¯z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (11) Note that the mapping relationship F is now irrelevant to quantities with units, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=', material and geometric properties of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' As the dimensionality of our problem has been reduced significantly, we can now express λ as a function of just two parameters ¯x, ¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Therefore, training a neural network to model F is now trivial as non-dimensionalized simulation data from a single type of paper can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Furthermore, the low dimensionality of F allows us easily visualize the λ landscape along a non-dimensional 2D-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' In the next section, we will now go over the steps to model F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' DEEP LEARNING AND OPTIMIZATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Data Generation In order to learn the force manifold, we solve (8) for several sampled (x, z) points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' An example of the partial force manifold produced from this sampling can be observed for a single suspended length in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For a specific (x, z) location, we apply incremental rotations along the y-axis and find the optimal rotation angle α that results in MN = 0 on the manipulated end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For a particular configuration (x, z, α), we then record the suspended length ls as well as the tangential and normal forces experienced on the clamped end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' This leads to a training dataset D consisting of six element tuples (Ft, Fn, α, ls, x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We then non-dimensionalize this dataset to the form (λ, α, ¯ls, ¯x, ¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' A total of 95796 training samples were used within a normalized suspended length of ¯ls ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='84, which adequately includes the workspace of most papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Learning Force and Optimal Grasp Orientation We can now train on our dataset D to obtain a generalized neural network modeling F: (λ, α, ¯ls) = FNN(¯x, ¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (12) To obtain the above function, a simple fully-connected feed- forward nonlinear regression network is trained with 4 hidden layers, each containing 392 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Aside from the final output layer, each layer is followed by a rectified linear unit (ReLU) activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' In addition, we preprocess all inputs through the standardization x′ = x − ¯xD σD , (13) 3 5 4 2 3 2 1 1 0 3 4 5 6 76 (a) (b) (d) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (a) Visualization of the trained neural network’s non-dimensionalized λ force manifold M and (b) α manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' An extremely low ¯δ discretization is used to showcase smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For the force manifold, we observe two distinctive local minima canyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Note that regions outside the workspace W are physically inaccurate but are of no consequence to us as they are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For the α manifold, we observe continuous smooth interpolation all throughout which is key for producing feasible trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Both manifolds showcase the used trajectories in the experiments for folding paper in half for Lgb ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='048, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='060, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (c) Showcases the three trajectories in (a) and (b) scaled back to real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' These are the actual trajectories used by the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (d) Arbitrary trajectories for various Lgb with identical start and goal states are shown to highlight the effect of the material property on our control policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' where x is the original input, ¯xD is the mean of the dataset D, and σD is the standard deviation of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We use an initial 80-20 train-val split on the dataset D with a batch size of 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Mean absolute error (MAE) is used as the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We alternate between stochastic gradient descent (SGD) and Adam whenever training stalls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Furthermore, we gradually increase the batch size up to 4096 and train on the entire dataset once MAE reaches < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Using this scheme, we achieve an MAE of < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='0005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Constructing the Neural Force Manifold The neural force manifold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' λ outputs of FNN for the workspace set) is discretized into a rectangular grid consisting of ¯δ × ¯δ blocks, where ¯δ = δ/Lgb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For each of the blocks, we obtain and store a single λ value using the midpoint of the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' This results in a discretized neural force manifold M represented as a m × n matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For the purposes of path planning, we add two components to our manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' First, we do not allow exploration into any region not belonging to our dataset distribution (¯ls > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We do so by defining a workspace W as all (¯x, ¯z) pairs within the concave hull of the input portion of the dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Secondly, we also exclude regions within a certain ¯ls threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' This is done as positions with small suspended lengths and large α angles may result in high curvatures that could cause collision with our gripper and/or plastic deformation, both of which we wish to avoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We denote this region as the penalty region Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' A visualization of M with the workspace W and penalty boundary Ls regions can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The α values corresponding to the manifold are also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Path Planning over the Neural Force Manifold Given the discretized manifold M, we can now generate optimal trajectories through traditional path planning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We define an optimal trajectory τ ∗ as one that gets to the goal state while minimizing the sum of λ: τ ∗ = arg min τ∈T i=L−1 � i=0 λi, (14) where L is the length of the trajectory and T is the set of all valid trajectories from the desired start to goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We define a valid trajectory as one that is contained within the acceptable region (xi, zi) ∈ W \\ Ls ∀ (xi, zi) ∈ τ, and whose consecutive states are adjacent grid locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Given the discretization of the NFM, we can treat M as a graph whose edge weights consist of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Therefore, we can use uniform cost search to obtain τ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The pseudocode of the path planning algorithm can be seen in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' ROBOTIC SYSTEM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Dual Manipulator Setup For our experiments, we use two Rethink Robotics’ Sawyer manipulators as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' One arm has an elongated gripper designed for folding, while the other arm has a spring compliant roller for creasing and an Intel Realsense D435 camera for vision feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The elongated gripper has rubber attached to the insides of the fingers for tight gripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 6 6 Start state 15 Goal state 4 5 5 10 Trajectory, T Workspace, W Is penalty, Ls 4 3 0 I23 I23 2 2 2 1 1 0 + 0 0 0 2 10 6 10 0 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='08 Lg6=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='065 Lgb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='082 Lgb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='06 三0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='06 Lg6=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='129 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='25 α [m] α[m]7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Example of our perception system with a top down view of the folding procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (a) Showcases the the intuitive baseline results while (b) showcases our open-loop algorithm for Lgb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='048 and C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='25m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 2, the solid green line indicates the desired end effector position while the dashed blue line indicates the crease location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We observe that the intuitive baseline has considerable sliding while our open-loop algorithm has near-perfect performance for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 1 2 3 4 5 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Experimental apparatus: Two robot manipulators, one for folding (1) and the other for creasing (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' An elongated gripper (2) is used for grabbing the manipulated end of the folding paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' A roller (5) with compliant springs (6) is used for forming the crease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' An Intel RealSense D435 camera (4) is attached to the creasing arm offer vision feedback during the folding procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' All gripper attachments were 3D printed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Perception System For our perception, we take an eye-in-hand approach by attaching an Intel Realsense D435 to the roller arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We do not use the depth component of the camera as we align the camera to be pointing down along the world z-axis and the distance from the camera to the table is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' To detect the pose of the paper, we use simple color detection to segment the paper and then use Shi-Tomasi corner detection [53] to obtain the position of the bottom edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' An example of the top-down view as well as detected poses produced by the camera can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Vision-feedback Control Although we minimize λ with our proposed framework, sliding could still happen due to a substrate’s low friction surface and/or jittering of the robot’s end-effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Notice that the generated optimal trajectory τ ∗ from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' V-D assumes that the origin o of our coordinate system shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 4(a) is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We can define the origin as o = q0 − Lˆx where Algorithm 1: Uniform Cost Search Input: ¯xs, ¯zs, ¯xg, ¯zg, M Output: τ ∗ 1 Func UCS(¯xs, ¯zs, ¯xg, ¯zg, M): 2 W ← valid workspace of M 3 Ls ← ls penalty region 4 h ← initialize min heap priority queue 5 c ← initialize empty list 6 ns ← node with location (¯xs, ¯zs) and cost 0 7 ng ← node with location (¯xg, ¯zg) and cost 0 8 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='push(ns) 9 while len(h) > 0 do 10 ni ← h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='pop() 11 if ni == ng then 12 τ ∗ ← path from start to goal 13 break 14 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='append(ni) 15 for (¯xj, ¯zj) ∈ neighbors of ni do 16 if (¯xj, ¯zj) /∈ W \\ Ls then 17 continue 18 nj ← node with location (¯xj, ¯zj) and cost λj from M 19 if nj ∈ c then 20 continue 21 if nj ∈ h and cost of nj is higher then 22 continue 23 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='push(nj) 24 τ ∗ ← perform trajectory smoothing on τ ∗ 25 return τ ∗ L is the total length of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Any amount of sliding indicates that q0 is moving along the x-axis and therefore, the origin o also moves an identical amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' When this occurs, our position within the manifold during traversal deviates from the optimal trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Furthermore, without adaptive replanning, the amount of sliding ∆x will directly result in ∆x amount of error when creasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' To circumvent this, we introduce a vision-feedback approach that mitigates the effects of sliding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We perform vision-feedback at N evenly spaced out intervals (a)8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' An overview of our folding pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The top row showcases offline proponents while the bottom row shows online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' On the offline side, we use our trained neural network to generate the necessary force manifold for planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Then, given an input tuple (xs, zs, xg, zg, Lgb), we generate an end-to-end trajectory using uniform cost search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' This end-to-end trajectory is then split up into partial trajectories that are carried out by the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' At the conclusion of each partial trajectory, we measure paper sliding and replan the next partial trajectory to rectify the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' of the trajectory τ ∗ as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' To do so, we first split up τ ∗ into N partial trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Aside from the first partial trajectory τ ∗ 0 , we extract the start and goal states of the other 1 ≤ i ≤ N partial trajectories resulting in a sequence of N evenly spaced out states S = {(x1, z1, α1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=', (xN, zN, αN)} when accounting for overlaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' After carrying out τ ∗ 0 , we detect the amount of sliding ∆x and incorporate this error by updating the start state and non-dimensionalizing as ¯xc i = xi − ∆x Lgb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We then replan a partial trajectory τ ∗ i from the updated start state (xc i, zi) to the next state (xi+1, zi+1) in the sequence and carry out this updated trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' This is repeated until reaching the goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' By properly accounting for sliding, we ensure that the traversal through the NFM is as accurate as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We note that this scheme allows us obtain corrected partial trajectories in near real time once N becomes sufficiently large as each partial trajectory’s goal state becomes increasingly close to its start state, allowing for uniform cost search to conclude rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We direct the reader to the supplementary videos mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' I which showcase the speed of the feedback loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Rectifying the sliding ∆x is not the only error we must address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Recount that we assume an optimal grasp orientation α for each position within the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' When the origin of our NFM moves, our true position does not match the intended position, resulting in also an angular error αc i = FNN(¯xc i, ¯zi), ∆α = αi − αc i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Algorithm 2: Closed-loop Control Pseudocode Input: (xs, zs), (xg, zg), Lgb, δ, N, FNN 1 M ←DiscretizeManifold (FNN, δ) 2 ¯xs, ¯zs, ¯xg, ¯zg ← non-dimensionalize with Lgb 3 ¯τ ∗ ← UCS (¯xs, ¯zs, ¯xg, ¯zg, M) 4 update ¯τ ∗ with αs using FNN 5 τ ∗ ← convert ¯τ ∗ to real space with Lgb 6 τ ∗ 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=', τ ∗ N−1 ← SplitTrajectory (τ ∗, N) 7 S ← extract start and goal states 8 carry out τ ∗ 0 on robot 9 for (xi, zi, αi) and (xi+1, zi+1, αi+1) ∈ S do 10 ∆x ← detect sliding of paper 11 xc i ← xi − ∆x 12 ¯xc i, ¯zi, ¯xi+1, ¯zi+1 ← non-dimensionalize with Lgb 13 αc i ← FNN(¯xc i, ¯zi) 14 ∆α ← αi − αc i 15 ¯τ ∗ i ← UCS (¯xc i, ¯zi, ¯xi+1, ¯zi+1, M) 16 L ← len(¯τ ∗ i ) 17 αi ← obtain αs of ¯τ ∗ i using FNN 18 αc i ← αi + ∆α[1, (L − 1)/L, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=', 1/L, 0]T 19 append ¯τ ∗ i with αc i 20 τ ∗ i ← convert ¯τ ∗ to real space with Lgb 21 carry out τ ∗ i on robot 22 crease paper with roller Simply applying a −∆α update to the first point in a partial trajectory results in a large rotational jump that only exacerbate the sliding issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Furthermore, we postulate that so long as sliding is not extremely large, the incorrect α at the current position within the manifold is still fairly optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Therefore, the ∆α error is incorporated into the trajectory gradually: τ ∗ i = UCS(¯xc i, ¯zi, ¯xi+1, ¯zi+1, M), αi = FNN(τ ∗ i ), αc i = αi + ∆α[1, (L − 1)/L, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=', 1/L, 0]T , where UCS stands for uniform cost search and L is the length of the trajectory τ ∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' This gradual correction ensures that we minimize sliding while maintaining smoothness of the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The pseudocode for our full closed-loop algorithm can be seen in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' EXPERIMENTS AND ANALYSIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Measuring the Material Property of Paper To use our framework, we must develop a way to accurately measure the parameter Lgb for a particular piece of paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' As mentioned previously, Lgb encapsulates the influence of bending and gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' With this in mind, we propose a simple way to measure the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 10(a), when one end of the paper is fixed, the paper will deform due to the coupling of bending OfMine Obtain force manifold from NN Compute optimal end-to-end path T* = [(Cs, Zs, Qs), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='., (Cg, Zg, ag) Path Planner (UCS) Compute corrected Detect paper slippage partial trajectory Perception Red is the true location Ac, Aa obtained from vision feedback Transform trajectory to real space Once goal state is Carry out partial trajectory, reached, crease paper then repeat for next step Motion Planner Online9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Experimental results for all folding scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Each column indicates a folding scenario while the the top row (a) showcases the fold length and bottom row (b) showcases the spin error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Boxplot results are shown color coded for the intuitive baseline, open-loop control, and closed-loop control algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Medians are shown as orange lines, means are shown as turquoise circles, and the desired target value is shown as a light blue horizontal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We note that both our open-loop and closed-loop algorithms have significant improvements over the intuitive baseline as shown by the broken axis in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Our algorithms also have significantly less variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='8 1 0 5 10 15 20 lh L (a) (b) Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' paper legnth, L Cardboard paper US Letter paper A4 paper Square origami paper Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (a) Schematic of a hanging plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The manipulation edge is fixed horizontally;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (b) Relationship between the ratio ϵ = lh/L and normalized total length of the paper ¯L = L/Lgb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' and gravitational energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Therefore, the following mapping relationship exists: ¯L = L(ϵ), ¯L = L Lgb , ϵ = lh L , (15) where lh is the vertical distance from the free end to the fixed end and L is the total length of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We can obtain the mapping relationship L(ϵ) using numerical simulations, which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 10(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' With this mapping known, simple algebra can be performed to obtain Lgb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' First, we measure the ratio ϵ = lh/L for a particular paper to obtain its corresponding normalized total length ¯L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Then, the value of Lgb can be calculated simply by Lgb = L/¯L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Once we obtain Lgb, we can now use the non-dimensionlized mapping relationship in (11) to find the optimal path for manipulating the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Experimental Setup For our experiments, we tested folding on 4 distinct types of paper: 1) A4 paper, Lgb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='048m, 2) US Letter paper, Lgb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='060m, 3) Cardboard paper (US Letter dimensions), Lgb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='132m, 4) Square origami paper, Lgb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='043m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For the rectangular papers (1-3), we do two sets of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The first involves folding the papers to an arbitrary crease location (C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='25m for A4 and C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='20m for US Letter and cardboard), while the second involves folding the papers in half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For the square origami paper, we choose an arbitrary crease location of C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='30m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' This results in a total of 7 folding scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For each of the scenarios, we conduct experiments using 3 different algorithms (an intuitive baseline, our open- loop approach, and our closed-loop approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We complete 10 trials for each of these algorithms, resulting in a total of 210 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Baseline Algorithm To showcase the benefits of our folding algorithm, we compare our algorithm to an intuitive baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We can think of an intuitive baseline algorithm as one that would work if the opposite end of the paper were fixed to the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Naturally, such a trajectory would be one that grabs the edge of the paper and traces the half perimeter of a circle with radius R = C/2: dθ = π/M, τB = {(R cos(idθ), R sin(idθ), idθ) ∀ i ∈ [0, M]}, (16) where M is an arbitrary number of points used as the resolution of trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We choose M = 250 for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Rect, Lgb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='048 Rect, Lgb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='048 Rect, Lgb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='060 Rect, Lgb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='060 Rect, Lgb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='132 Rect, Lgb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='132 Diag, Lgb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='043 (a) C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='25 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='13) C=Half (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='1485) C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='20 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='105) C=Half (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='14) C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='20 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='105) C=Haif (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='14) C=0.' metadata={'source': 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+page_content=' 0 0: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='5 0 0 1 0 1 Intuitive Baseline Open-loop Control Closed-loop Control10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Isometric views of different folding scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (a1-2) showcases C = Half folding for Lgb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='048 paper with the intuitive baseline and our open-loop algorithm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (b1-2) showcases C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='30m diagonal folding for Lgb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='043 with the intuitive baseline our closed-loop algorithm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Metrics The metrics used for the experiments were the average fold length and the spin error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The average fold length was calculated by simply taking the average of the left and right side lengths up until the crease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The spin error was calculated as the angle θerr that results in the difference between the left and right side lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For square papers, the fold length was defined as the perpendicular length from the tip to the crease and the spin error was the angular deviation from this line to the true diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Parameters The neural force manifold M was discretized using a ¯δ corresponding to δ = 2mm depending on the material as we found this discretization to have good compromise between accuracy and computational speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' All rectangular papers used a penalty region Ls defined by ¯ls < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='958 while the square paper used one defined by ¯ls < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' This discrepancy is due to the fact that the diagonal paper has a smaller yield strength compared to the the rectangular paper, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=', to prevent extremely high curvatures, a larger suspended length ¯ls range must be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For closed-loop control, we chose to split all trajectories into N = 5 intervals regardless of trajectory length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Furthermore, we use an extremely slick (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' low friction) table to showcase the robustness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Using an empirical method, we measured the static coefficient of friction of our papers and the substrate to be approximately µs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For comparison, the static coefficient of friction for steel on steel (both lubricated with castor oil) is µs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Results and Analysis All experimental results can be seen expressed as box plots where we showcase achieved fold lengths and spin errors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 9(a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' When observing the achieved fold lengths, we see significant improvement over the baseline for all folding scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Due to the large gap in performance, broken axes are used to properly display the variance of the recorded data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We note that not only do our algorithms achieve significantly better performance on average, the variance of our approaches is also much lower as shown by the decreased y-axis resolution after the axis break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We attribute the high variance of the baseline method due to the increased influence of friction, which can often cause chaotic, unpredictable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' In other words, truly deterministic folding can only be achieved when sliding is nonexistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For a vast majority of cases, we observe a clear improvement over the open-loop algorithm when incorporating vision- feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Intuitively, we observe a trend where the performance gap between our open-loop and closed-loop algorithms grow as the material stiffness increases for rectangular folding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For softer materials (Lgb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='048), the open-loop algorithm has near perfect performance as shown when folding a paper in half in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 11(a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' In comparison, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 11(a1) showcases the baseline algorithm failing with significant sliding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The sliding problem is only exacerbated by increasing the stiffness of the material (Lgb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='132) where Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 12(a) showcases the baseline algorithm failing to fold the cardboard paper in half by a margin almost as long as the paper itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' In comparison, our open-loop algorithm is capable of folding the cardboard with significantly better results albeit with some visual sliding as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 12(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' As the material stiffness increases, the benefits of the incorporated vision-feedback are more clearly seen as we are able to achieve near perfect (al) (a2) (b1) (b2)11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Isometric views for folding C = Half with the stiffest paper (Lgb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='132).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (a) showcases the intuitive baseline, which fails drastically as the stiffness of the paper causes excessive sliding during the folding process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' (b) showcases our open-loop algorithm, which has significant improvements over the baseline with minimal sliding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Finally, (c) showcases our closed-loop algorithm, which improves upon our open-loop results and achieves near perfect folding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' folding for cardboard in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 12(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' All of our findings for rectangular folding also match the results of our diagonal folding experiment shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 11(b1-b2), where closed- loop once again achieves minimal sliding when compared to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Overall, the matching findings across all of our experiments showcase the robustness of our formulation against material and geometric factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' We observe one oddity for the folding scenario of Lgb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content='048 and C = Half where the open-loop algorithm outper- formed our closed-loop variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Still, we wish to point out that this decrease in performance is only on average 1mm, which can easily be attributed to repetitive discretization error caused by N = 5 replanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' In fact, as we use a discretization of δ = 2mm for the manifold, compounding rounding errors can easily cause 1-2mm errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' With this in mind, our closed-loop method achieves an average fold length performance within a 1-2mm tolerance across all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' In terms of spin error, we found that softer materials had the greatest error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' As the frictional surface of the table is not perfectly even, any amount of sliding will directly result in uneven spin as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' 11(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' As the material stiffness increases, the spin errors became more uniform across the methods as the influence of friction is not enough to deform the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Still, we can see that our open and closed-loop algorithms had less sliding than the baseline on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' CONCLUSION We have introduced a novel control strategy capable of robustly folding sheets of paper of varying materials and geometries with only a single manipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Our framework incorporates a combination of techniques spanning several disciplines, including physical simulation, machine learning, scaling analysis, and path planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' The effectiveness of our framework was showcased through extensive real world experiments against an intuitive baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Furthermore, an efficient near real-time visual-feedback algorithm was imple- mented that further minimizes folding error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' With our closed- loop sensorimotor control algorithm successfully accomplished challenging scenarios such as folding stiff cardboard with repeatable accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' For future work, we hope to to tackle the difficult problem of creating arbitrary creases along sheets of paper with non- symmetric centerlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Such non-symmetric papers can no longer be represented as a reduced-order model of a 2D elastic rod, thus requiring a different formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9A0T4oBgHgl3EQfAv_u/content/2301.01968v1.pdf'} +page_content=' Additionally, folding along regions of 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Finding the global minimizer of polynomials is an impor- +tant topic in almost all fields in applied mathematics, statistics, and +engineering, such as signal processing, machine learning, and data sci- +ence, etc. In this paper, we investigate the possibility of the backward- +differential-flow-like algorithm which starts from the minimum of con- +vexification version of the polynomial. +We apply the heat evolution +convexification approach through Gaussian filtering p(x, t) = p(x) ∗ +gt(x) with variance t > 0, which is actually an accumulation version +of Steklov’s regularization. This heat equation plays a multiscale anal- +ysis framework in mathematics, image processing and computer vision. +We generalize the fingerprint theory which was proposed in the theory +of computer vision by A.L. Yuille and T. Poggio in 1980s, in particu- +lar their fingerprint trajectory equation, to characterize the evolution +of minimizers across the scale (time) t. +On the other hand, we pro- +pose the ”seesaw” polynomials p(x|s) by replacing the coefficient of x of +p(x) with an arbitrary real parameter s, and we find a seesaw differen- +tial equation to characterize the evolution of global minimizer x∗(s) of +p(x|s) while varying s. Essentially, both the fingerprints FP2 and FP3 +of p(x), consisting of the zeros of ∂2p(x,t) +∂x2 +and ∂3p(x,t) +∂x3 +, respectively, are +independent of seesaw coefficient s, upon which we define the Confine- +ment Zone and Escape Zone. Meanwhile, varying s will monotonically +condition the location of global minimizer of p(x|s), and all these loca- +tion form the Attainable Zone. Based on these concepts, we prove that +the global minimizer x∗ of p(x) can be inversely evolved from the global +minimizer of its convexification polynomial p(x, t0) if and only if x∗ +is included in the Escape Zone. In particular, we give detailed analy- +sis for quartic and six degree polynomials. For quartic polynomial, we +proved that the Attainable Zone is completely contained in the Escape +Zone, thus heat evolution approach must converge to global minimizer, +and we even find a simpler Euler’s algorithm which must converge to +the global minimizer, without heat evolution. For six and higher degree +polynomials, we illustrate that the Attainable Zone might intersect with +Confinement Zone, which leads to the failure of immediate backward +differential flow like algorithm. In this case, we show that how to attain +the global minimizer through our seesaw differential equation. +Date: December 31, 2022. +2010 Mathematics Subject Classification. 35Q90,46N10,35Q93,90C26. +Key words and phrases. convex optimization, non-convex optimization, heat equation, +maximum principle, multiscale Gaussian filter, computer vision, quartic polynomial. +1 +arXiv:2301.00326v1 [math.OC] 1 Jan 2023 + +2 +QIAO WANG +1. Background and Motivations +Global optimization of real polynomials is an important non-convex op- +timization problem (cf. +[1] and references there in), and produces many +excellent theories in past decades. Among them, N. Z. Shor [2] first trans- +formed univariate polynomial optimization to convex problem through qua- +dratic optimization in 1987, which can offer approximate solution to this +global optimization. After that, N.Z. Shor further studied its relationship +with Hilbert’s 17th problem [3]. Also in 1987, V. N. Nefnov [4] proposed +an algorithm by computing the roots of algebraic equation for finding the +minimizer. +In 2014, J. Zhu, S. Zhao and G. Liu [6] proposed a backward differential +flow formulation, comes from Kuhn-Tucker equation of constrained opti- +mization, to find out the global minimizer of polynomials. They consider +the problem for sufficient smooth function p(x), +min p(x) +s.t. x ∈ D := {x ∈ Rn| ∥x∥ < a} +(1.1) +by introducing a set +G = {ρ > 0| [∇2p(x) + ρI] > 0, ∀x ∈ D}, +(1.2) +and an initial pair (�ρ, �x) ∈ G × D satisfying +∇p(�x) + �ρ�x = 0. +(1.3) +Then they proved that the back differential flow �x(ρ), defined near �ρ, +d�x +dρ+[∇2p(�x) + ρI]−1�x = 0, +�x(�ρ) = �x +(1.4) +will lead to the solution of (1.1). +The above work is under the condition that all global minimizers of this +polynomial occur only in a known ball, thus the unconstrained optimization +problem may be reduced to a constrained optimization problem. However, +O. Arikan, R.S. Burachik and C.Y. Kaya [7] pointed out in 2015 that the +method in [6] may not converge to global minimizer by a counter-example +of quartic polynomial +p(x) = x4 − 8x3 − 18x2 + 56x. +(1.5) +Furthermore, they [8] proposed a Steklov regularization and trajectory method +to this optimization for univariate polynomials in 2019. +Then in 2020, +R.S. Burachik and C.Y. Kaya [9] generalized it to the multi-variable case. +In these works, the quartic polynomial optimization plays an interesting +role as toy examples. In addition, the six degree polynomials may fail to +attain the global minimizers, which are illustrated by several examples and +counter-examples in [8]. + +HEAT EVOLUTION +3 +Actually, the Steklov regularization [8] +µ(x, t) = 1 +2t +� x+t +x−t +f(τ) dτ +(1.6) +is a low-pass filter, in the viewpoint of signal processing, since we may write +µ(x, t) = 1 +2t +� x+t +x−t +f(τ) dτ = f(x) ∗ 1[−t,t](x) +(1.7) +where +1[−t,t](x) = +� +1 +2t, +x ∈ [−t, t] +0, +x /∈ [−t, t] +(1.8) +from which one may obtain µx(x, t), µxt(x, t) and µxx(x, t) (where the sub- +script means partial derivative). However, µt(x, t) is not explicitly in this +regime, since we can merely represent a differential equation +2µt + tµtt = tµx. +(1.9) +Obviously it brings some inconvenience in analyzing the evolution of local +minimizers. Therefore, we require an approach which can balance between +the simple differential equation and filters, as well as offer convexification +for polynomials. Fortunately, the heat conduct equation +∂p +∂t = 1 +2 +∂2p +∂x2 +(1.10) +with initial condition +p(x, 0) = p(x) +(1.11) +is a nice framework to implement the convexification and the similar op- +timization algorithm. In addition, the analysis for evolution of all critical +points becomes more analytically tractable. +Remark 1. It should be pointed that the initial problem of heat equation +(1.10) is equivalent to Gaussian filter, which will be explained in Subsection +2.1. But on the other hand, the accumulation of Steklov regularization will +lead to Gaussian distribution, since that +1[−t,t] ∗ 1[−t,t] ∗ · · · ∗ 1[−t,t] +� +�� +� +n +→ N(0, 2nt3 +3 +), +(1.12) +for n large enough. Thus replacing Steklov regularity with heat evolution, +i.e., the Gaussian filtering, is very natural in this paper. +Our interest in this paper is to explore the method of optimizing the even +degree polynomial +min +x p(x) = xn + +n +� +j=1 +cjxn−j. +(1.13) + +4 +QIAO WANG +Different from Kuhn-Tucker’s equation based backward differential flow in +[6], we propose in this paper a constructive way, through evolving the poly- +nomial by heat conduct (Gaussian filtering) to build a backward-differential- +flow-like algorithm, in which we can even explicitly express the differential +equation to attain the minimizer. However, the algorithm converges to the +global minimizer for quartic polynomial, and partially success for higher +degree polynomials. This phenomena was actually observed in [8] with ex- +amples for Steklov regularization. +In this paper, we explain this convexification derived trajectory algorithm, +i.e., a backward-differential-flow-like algorithm, by building the convexifica- +tion of heat evolution to polynomials, and in particular, we build the suffi- +cient and necessary condition (Theorem 8) under which the algorithm must +attain the global minimizer. Our analysis is based on the Yuille-Poggio’s +fingerprints theory and their trajectory differential equation in the theory +of computer vision [12][13] which were built in 1980s. In addition, to attain +the global minimizer when the previous algorithm fails, we build a new tra- +jectory differential equation (Theorem 5) which characterizes the minimizer +moving from the global minimizer of ”Seesaw”1 polynomial p(x|s) to that +of original polynomial p(x). +Before ending this introduction, we slightly sketch the motivation in our +contributions. +The elegant framework of multiscale Gaussian filter is equivalent to the +model of heat conduct equation. +Applying this theory, any even degree +monic polynomials p(x) will become convex by p(x, t) = gt(x) ∗ p(x) for t +large enough2, where gt stands for Gaussian filter with variance t. Moreover, +for quartic polynomial p(x), the global minimizer xmin will continuously +evolve along t > 0 such that it remains global minimizer xt +min of p(x, t) at +each scale t ≥ 0. Therefore, reversely and continuously evolving from any +global minimizer xt +min of p(x, t) to xmin of p(x) is guaranteed. +A natural question is, whether the global minimizer xmin of a higher de- +gree polynomial also evolves continuously to global minimizer of scaled ver- +sion p(x, t), like the quartic polynomial case? Unfortunately this extremely +expected property doesn’t hold in general for polynomials whose degree is +more than 4. We will illustrate it by a counter-example on 6-degree polyno- +mial. Furthermore, we give a condition which is both sufficient and necessary +for the convergence to global minimizer. +The multi-scale Gaussian filter and equivalent heat conduct equation is +a standard content in the theory of PDEs, signal processing and so on. +In particular in the field of computer vision, it brought us many powerful +1Here the ”seesaw” polynomial of a polynomial p(x), say p(x) = x6 −2x4 +3x3 +4x2 + +5x + 6, is p(x|s) = x6 − 2x4 + 3x3 + 4x2 + sx + 6, in which 5x + 6 is replaced by sx + 6, +where s ∈ R can be conditioned such that sx performs like a seesaw. +2I definitely believe that this very simple fact should have been already established. +But I have not gotten any references, limited to my scope of reading. + +HEAT EVOLUTION +5 +Notations +Definition +Index +Zt,k +real zeros of ∂kp(x,t) +∂xk +(3.1) +µ(x, t) +Steklov regularity of p(x) +(1.7) +p(x, t) +heat evolution of p(x) +(2.1) +FPk(p) +k-th fingerprints, (k = 2 is +Yuille-Poggio’s fingerprint) +(3.4) +Yuille-Poggio’s +fingerprint +trajectory equation +(3.8) +FlowY P (p) +Yuille-Poggio’s flow +(3.10) +Q(p) + S(p) +Quadric and higher plus See- +saw decomposition +(3.15)(3.16)(3.17) +S(p, s) +seesaw term +(3.18) +seesaw differential equation +(3.25) +AZ(p) +attainable zone +(3.24) +Ω(p) and Ωc(p) +confinement zone and escape +zone +Definition 5 +Table 1. List of notations and symbols +theoretical tools since 1950s (cf. +[10][11]). +Among them, the fingerprint +theory proposed in 1980s (cf. [12][13]) plays a kernel role for many years. +In this paper, we apply the ideas of fingerprint from computer vision, and +define three fingerprints of scaled polynomials p(x, t) across scale t. The +first fingerprint FP1 characterizes all the local extremals of p(x, t) for each +t, and the second one, FP2, characterizes the stationary points of p(x, t) +at each t, which indicate the domain of convexity of p(x, t) during the time +evolution. Furthermore, FP3 indicates the evolution of curves in FP2. All +these powerful fingerprints tools offer us insightful understandings to the +evolution of both local and global extremals of the polynomials, from which +we proposed a sufficient and necessary condition for attaining the global +minimizer by the backward trajectory algorithm. +For the sake of simplicity, we list all the symbol and notations in this +paper as below. +2. Heat Evolution and Convexification of polynomials +2.1. Heat evolution of p(x). Consider the heat conduct equation (1.10) +with initial condition (1.11), the general solution of (1.10) is +p(x, t) = p(x) ∗ gt(x), +(2.1) +in which gt(x) stands for the Gaussian filter +gt(x) = +1 +√ +2πte− x2 +2t , +t ≥ 0. +(2.2) +In signal processing and computer vision, this time variable t is also called +scale (of Gaussian filtering) or artificial time. Notice that any differential + +6 +QIAO WANG +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +10 +12 +-2000 +0 +2000 +4000 +6000 +8000 +10000 +12000 +14000 +Figure 1. The heat evolution of quartic polynomial p(x) = +x4−8x3−18x2+56x illustrated in x ∈ [−8, 12]. See Example +3 for more details. +-4 +-2 +0 +2 +4 +6 +8 +-1000 +-500 +0 +500 +1000 +1500 +2000 +2500 +Figure 2. The partial enlarged view of p(x) = x4 − 8x3 − +18x2 + 56x. See Example 3 for more details. +operator D is commutative with convolution operator ∗, i.e., +D(f ∗ g) = f ∗ Dg = Df ∗ g. +(2.3) +For polynomials p(x, t), the heat equation (1.10) can be enhanced to +∂kp +∂tk = 1 +2k +∂2kp +∂x2k , +(k = 1, 2, · · · ) +(2.4) + +HEAT EVOLUTION +7 +by differentiating both sides of (1.10) w.r.t t, since the smoothness is guar- +anteed. Then performing Taylor’s expansion for p(x, t) about t will yield +p(x, t) = p(x, 0) + t · ∂p +∂t +���� +t=0 ++ t2 +2 +∂2p +∂t2 +���� +t=0 ++ · · · . +(2.5) +If using the heat equation derived (2.4), we may rewrite (2.5) as +p(x, t) = p(x, 0) + t +2 · ∂2p +∂x2 +���� +t=0 ++ t2 +8 +∂4p +∂x4 +���� +t=0 ++ · · · . +(2.6) +The convexification of even degree polynomials by heat evolution is char- +acterized by following Theorem3. +Theorem 1. For each even degree monic polynomial p(x), there exists an +specified T ∗ = T ∗(p) such that the heat convolution p(x, t) is convex w.r.t x +at any t > T ∗. +We require the following basic results existing in many standard text- +books. +Lemma 1. The Gaussian density gt(x) defined in (2.2) satisfies the follow- +ing equations: +(1) The moment formula +� +∞ +−∞ +xmgt(x) dx = +� +t +m +2 (m − 1)!!, +(m even) +0, +(m odd) +(2.7) +(2) The convolution formula +xm ∗ gt(x) = xm + m(m − 1)txm−2 + · · · + rm(x, t), +(2.8) +where +rm(x, t) = +� +m!! t +m +2 , +(m even) +(m − 1)!! t +m−1 +2 x, +(m odd). +(2.9) +Proof. The equation (2.7) can be verified immediately, from which we have +xm ∗ gt(x) += +� +(x − y)mgt(y) dy += +m +� +k=0 +�m +k +� +xk(−1)m−k +� +ym−kgt(y) dy += +m +� +k=0 +m−k is even +�m +k +� +(m − k)!! t +m−k +2 xk += +xm + m(m − 1)txm−2 + · · · + rm(x, t), +(2.10) +where the last term rm(x, t) is presented at (2.9). +□ +Now we prove the Theorem 1. +3Once again, I believe that this convexity result must be known in some literature. + +8 +QIAO WANG +Proof of Theorem 1. In what follows, the subscription k in Pk(x) and Qk(x) +stands for the degree of polynomials. +Let’s consider even degree monic +polynomial +P2m(x) = x2m + P2m−1(x). +(2.11) +Observing the expansion +P2m(x) ∗ gt(x) = x2m ∗ gt(x) + P2m−1(x) ∗ gt(x), +(2.12) +according to (2.8) and (2.9), we may write +P2m(x) ∗ gt(x) = P2m(x) + β(x, t), +(2.13) +in which +β(x, t) = (2m)!! tm + +m−1 +� +k=1 +tm−kQ2k(x). +(2.14) +Using the heat evolution, we have +1 +2 +∂2p(x, t) +∂x2 += ∂p(x, t) +∂t += ∂β +∂t . +(2.15) +In our case, +∂β +∂t = m(2m)!! tm−1 + +m−1 +� +k=1 +(m − k)tm−k−1Q2k(x). +(2.16) +Clearly, all these leading terms of Q2n(x) are contributed by x2m(x) ∗ +gt(x) − x2m, and must be positive. In more detail, +the coefficient of leading term of Q2n(x) = +�2m +2n +� +(2m − 2n)!! > 0 (2.17) +which implies that there exists bounded constants K, such that +Q2n(x) > K > −∞, +(n = 2, 3, · · · , 2m − 2) +(2.18) +So that we have +∂β +∂t > m(2m)!! tm−1 + K(tm−2 + tm−3 + · · · + 1). +(2.19) +Therefore, there exists a T ∗ > 0, such that for all t > T ∗, we have ∂β +∂t > 0. +Thus the convexity is guaranteed by heat evolution. +□ +2.2. Comparison principle. The most important mechanism in heat evo- +lution is the comparison principle, from which we understand that usually +a local minimizer will merge to a local maximizer during the evolution, like +the ”annihilation” action between the pair of minimizer and maximizer. +Theorem 2 (Comparison principle). Assume that x∗ be a critical point of +p(x, t∗), then for t > t∗, the heat evolution of the critical point satisfies +p(x∗(t), t) ≥ p(x∗, t∗), if x∗ is local minimum; +(2.20) +p(x∗(t), t) ≤ p(x∗, t∗), if x∗ is local maximum. +(2.21) + +HEAT EVOLUTION +9 +Proof. Without loss of generality, we set t∗ = 0, due to that the heat operator +U t : f(x) �→ gt(x) ∗ f(x) forms a semi-group (Lie group). Let x = x(t) be +one of the integral curves of critical points of p(x, t) w.r.t x, then from +dp(x(t), t) +dt +=∂p(x, t) +∂x +˙x(t) + ∂p(x, t) +∂t +=0 + ∂p(x, t) +∂t +=1 +2 +∂2p(x, t) +∂x2 +, +thus we can get the required result. Notice that the last equality comes from +heat conduct equation. +□ +Remark 2. If we consider the domain (x, t) ∈ [x∗ − ϵ, x∗ + ϵ] × [0, T) near +each critical point x∗, we can show this result by maximum principle for +parabolic operator +∂ +∂t − 1 +2 +∂2 +∂x2 (cf. [16][17]). +This comparison principle reveals that the (local) minimizer and (local) +maximizer might merge pair-wisely during the evolution. Ideally, there ex- +ists n−1 critical points for a n degree polynomial (here n is even). Thus we +hope the global minimizer will not merge with any local maximizer during +the heat evolution. However, it might fail in some cases, and we will analyze +this mechanism in details. +3. Global minimizer and scale space fingerprint +3.1. Fingerprints of scale space. The scale space fingerprint was intro- +duced by A.L. Yuille and T.A. Poggio in 1980s (cf. [12] [13] etc.), which +plays an important role in computer vision. +To capture the information of a signal or image p(x), the multi-scale +version p(x, t) = p(x) ∗ gt(x), which comes from heat conduct equation, is +applied, in which the variance t ≥ 0 of Gaussian filter is also called artificial +time. +Consider that all the polynomials in our situation are of real coefficients, +for the sake of simplicity, we need to generalize Yuille-Poggio’s definition of +fingerprints of multi-scale zero-crossings to more general case as below. +Definition 1. Denote the set of real zeros of k-th derivative of polynomial +p(x, t) as +Zt,k(p) := +� +xi(t) ∈ R; +∂kp(xi(t), t) +∂xk += 0, i = 1, 2, · · · . +� +, +(3.1) +and denote the sets +FP+ +k (p) := +� +(x, t); ∂kp(x, t) +∂xk +> 0, t ≥ 0, +� +, +FP− +k (p) := +� +(x, t); ∂kp(x, t) +∂xk +< 0, t ≥ 0, +� +. +(3.2) + +10 +QIAO WANG +then the k-th order fingerprints of polynomial p(x) are defined as +FPk(p) := FP+ +k (p) +� +FP− +k (p). +(3.3) +In above notations S represents the topological closure of set S. In our +case, this topological closure is very simple thus we may characterize FPk +by algebraic equations +FPk(p) = +� +(x, t); ∂kp(x, t) +∂xk += 0 +� +, +(3.4) +due to the sufficient smoothness of all polynomials. +Remark 3. When k = 2, the fingerprint FP2 of so-called zero-crossings, as +well as the equation of zero-crossing contour, are introduced by A.L. Yuille +and T.A. Poggio [13]. Here, we generalize their fingerprints from FP2 to +more general FPk (k ≥ 2) in this paper. In other words, if we consider +P(x) whose derivative is P ′(x) = p(x), then FP1(p) = FP2(P). That is to +say, our framework of FPk is essentially a generalization of Yuille-Poggio’s +fingerprints in the theory of computer vision. +According to this notation, FP1 is the fingerprint of extremal values +(critical points), and FP2 the zero-crossings (convexity)4, of polynomial +p(x), respectively. Essentially, as in the theory of computer vision, we can +get more information from FP+ +2 and FP− +2 . In this paper, we generalize the +classic concept FP1 and FP2 to general FPk, in particular, FP3 is included +such that our main results can be represented on these three fingerprints. +We further consider the dynamics of the elements in FP1, i.e., the tra- +jectories. Our main interest is to obtain the curves x = x(t) which obey the +equation +∂p(x(t), t) +∂x += 0, +(3.5) +as well as initial conditions +x(0) = xi ∈ Z0,1(p), +(i = 1, 2, · · · ) +(3.6) +where xi (i = 1, 2, · · · ) are the critical points of p(x). To solve these curves, +an ODE by varying t as follows is introduced by A.L. Yuille and T.A. Poggio +in [13], +0 = d +dt +�∂p(x(t), t) +∂x +� += ∂2p(x, t) +∂x2 +dx(t) +dt ++ ∂2p(x, t) +∂x∂t +. +(3.7) +Therefore, we may characterize the fingerprint which contains all the maxi- +mums at different t > 0 by rewriting (3.7) as +dx(t) +dt += − +∂2p(x,t) +∂x∂t +∂2p(x,t) +∂x2 += − +∂3p(x,t) +∂x3 +2 · ∂2p(x,t) +∂x2 +, +(3.8) +4Although there exists certain gap between the rigorous meaning and the definition +here, we omit it in this paper. + +HEAT EVOLUTION +11 +-0.6 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +x +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +t +(a) FP1 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +x +0 +0.005 +0.01 +0.015 +0.02 +0.025 +0.03 +0.035 +0.04 +0.045 +t +(b) FP2 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +x +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +t +(c) FP3 +Figure 3. The fingerprints in Example 1, separately illustrated. + +12 +QIAO WANG +Figure 4. The joint illustration of fingerprints FP1, FP2 +and FP3 of previous figures about Example 1. +as well as suitable initial conditions5 +x(0) ∈ Z0,1(p). +(3.9) +In this paper, we call this ODE (3.8) the Yuille-Poggio equation, since it was +first proposed in (3.3) of A.L. Yuille and T.A. Poggio’s seminal work [13]. +On the other hand, we also call this equation (3.8) the trajectory equa- +tion, since the reversely evolution algorithm will backward evolute along +this curve, provided the initial value is given. Given any initial position, +one may obtain a trajectory by this equation. In particular, when the initial +condition is located at the critical points of p(x), the trajectories form the +fingerprint FP1(p). +We may further generalize FP1(p) to Yuille-Poggio’s flow. +Definition 2. For any h ∈ R, the integral curve generated by Yuille-Poggio +equation (3.8) associated with initial value x(0) = h is called a Yuille- +Poggio’s curve. All these Yuille-Poggio’s curves consist the set +FlowY P (p) := +� +(x(t), t); +dx(t) +dt += − +∂2p(x,t) +∂x2 +2∂p3(x,t) +∂x3 +, +x(0) = h, +∀h ∈ R, +� +, +(3.10) +and we call it the Yuille-Poggio’s flow generated by polynomial p(x). +Clearly, the fingerprint curve in the fingerprint FP1(p) is a special Yuille- +Poggio’s curve whose initial value x(0) is restricted to Z0,1(p), i.e., satisfies +p′(x(0)) = 0. Thus we have +5If p(x) is n-degree polynomial, there exists at most n − 1 distinct initial conditions. + +0.05 +0.0450.04 +0.035 +0.03 +七 +0.025 +0.02 +0.015 +0.01 +0.005 +0 +-0.6 +-0.4 +-0.2 +0 +0.2 +X0.4HEAT EVOLUTION +13 +Theorem 3. The fingerprint FP1 can be represented as +FP1(p) = +� +(x, t); +dx(t) +dt += − +∂2p(x,t) +∂x2 +2 ∂p3(x,t) +∂x3 +, +x(0) ∈ Z0,1 +� +. +(3.11) +And +FP1(p) ⊂ FlowY P (p). +(3.12) +Notice that the singularity occurs at which the denominator +∂2p(x, t) +∂x2 += 0. +(3.13) +Example 1. We illustrate the fingerprints of six degree polynomial +p(x) = x6 − 0.3726x4 + 0.0574x3 + 0.0306x2 − 0.0084x +in Fig.3 and Fig.4. We point out that the global minimizer (”*” in Fig.3(a)) +does not evolute to infinity, which means the convex convolution for this p(x) +will not converge to its global minimizer. +The differential equation (3.8) characterizes the trajectory of FP1(p), the +evolution of critical points of p(x) in scale space, which inspires an backward +differential flow algorithm, which is actually Euler’s algorithm along the +trajectory described by Yuille-Poggio’s equation. That is, to solve the global +minimizer of p(x), we first build its convex version p(x, t0) for certain t0 > 0 +large enough. +According to Theorem 1 at Section 2, this t0 > 0 exists. +Suppose that x∗(t0) be the global minimizer of convex polynomial p(x, t0), +we inversely evolute it to x∗(0) according to the trajectory equation (3.8) +from t = t0 to t = 0. We expect that x∗(0) be the global minimizer of p(x). +However, this strategy may fail since in some cases the reversely evolution +might result in a local minimizer of p(x). +In this paper, we will analyze the mechanism according to Yuille-Poggio’s +flow and derived zones, and further build a new trajectory differential equa- +tion to attain the true global minimizer from connected global minimizer of +its ”Seesaw” polynomial. +3.2. Q-S (”Quadric and higher plus Seesaw”) decomposition. As +we point out, that the heat conduct based backward-differential-flow-like +algorithm is not guaranteed to converge to theoretically global minimizer. +This is similar to a six degree polynomial counter-example of Steklov’s regu- +larization approach in [8]. In this paper, we explain how the convexification +method converge to global minimizer, and why it may fail in some cases. +Furthermore, to recover from the failed cases, we propose a ”Quadric plus +Seesaw” decomposition (Q-S decomposition), then build a new ordinary dif- +ferential equation that describes the evolution of global minimizer on account +of varying S(x) according to this Q-S decomposition. + +14 +QIAO WANG +Definition 3 (”Quadric and higher plus Seesaw” decomposition). For any +polynomial +p(x) = xn + +n−1 +� +k=0 +ckxk, +(3.14) +we define its Q-S decomposition +p(x) = Q(p) + S(p), +(3.15) +in which +Q(p) = xn + +n−1 +� +k=2 +ckxk +(3.16) +stands for the ”Quadric and higher terms”, and +S(p) = c1x + c0 +(3.17) +stands for the ”Seesaw terms”. We further define the generalized Seesaw +term +S(p, s) = sx + c0, +s ∈ R. +(3.18) +Instead of studying p(x) = Q(p) + S(p), we will consider its ”Seesaw” +family of polynomials Q(p) + S(p, s). We have +Lemma 2. Every Seesaw term S(p, s) is invariant under heat evolution, +i.e., +S(p, s) ∗ gt(x) = S(p, s), +∀s ∈ R. +(3.19) +Proof. Applying Lemma 1 will lead to above result immediately. +□ +Actually, this Lemma 2 leads to an insight on multi-scale decomposition +of p(x) by +p(x, t) = p(x)∗gt(x) = Q(p)∗gt(x)+S(p)∗gt(x) = Q(P)∗gt(x)+S(p), (3.20) +upon which we see that the fingerprints FP2 and FP3 of p(x) is essential of +Q(p) but independent of S(p). Instead, the fingerprint FP1 of p(x) concerns +both Q(p) and S(p). That is +Theorem 4. For any polynomial p(x), all of its Seesaw polynomial +p(x|s) = Q(p) + S(p, s) = +n +� +k=2 +ckxk + sx + c0 +(3.21) +satisfy the following equality, +FPk(p(x|s)) = FPk(p(x)), +k ≥ 2. +(3.22) +Meanwhile, +FP1(p(x|s)) ̸= FP1(p(x)). +(3.23) +For these seesaw polynomials, we define + +HEAT EVOLUTION +15 +Definition 4. For the even degree polynomial p(x), we denote by x∗(s) the +global minimizer of seesaw polynomials p(x|s) = Q(p) + S(p, s) for each s, +and call it the seesaw minimizer. For given p(x), the set of global minimizers +of p(x|s) by varying s ∈ R is called attainable zone given Q(p), i.e., +AZ(p) = {x∗ ∈ R; ∃s ∈ R, x∗ is the global minimizer of p(x|s)} . +(3.24) +We first focus on those cases that the global minimizer can not be obtained +from heat evolution from convexificated version p(x, t) of polynomial p(x). +If case is this, we investigate the global minimizer of Q-S form Q(p)+S(p, s) +where S(p, s) = sx + c0. Notice that c0 is always a dumb parameter since it +doesn’t affect the location of the global minimizer. +Theorem 5. [seesaw differential equation of minimizers moving of seesaw +polynomials] The global minimizers x∗(s) (and any critical points) of seesaw +polynomials p(x|s) = Q(p) + S(p, s), i.e., the x∗(s) ∈ AZ(p), must satisfy +the seesaw differential equation +dx +ds = − +1 +p′′(x). +(3.25) +Proof. For each s ∈ R, the global minimizer of p(x|s) w.r.t. x satisfies +0 = p′(x|s) = +� +� +n +� +j=2 +cjxj +� +� +′ ++ s = 0. +(3.26) +Then differentiating both sides w.r.t. s will lead to +0 = p′′(x) dx +ds + 1 = 0, +(3.27) +which produces the required result. +□ +Corollary 1. The global minimizer x∗(s) of seesaw polynomial p(x|s) is +monotonically decreasing as s increasing, i.e., +s ≥ s′ =⇒ x∗(s) ≤ x∗(s′). +(3.28) +Proof. It follows from (3.25) that dx∗(s) +ds +< 0 since that p′′(x) > 0 when x∗(s) +is the global minimizer of p(x∗(s)|s). +□ +These results will help us in some situations, may start from the true +global minimizer of a suitable p(x|s) as initial value, then move it from x∗(s) +to required location x∗(c1), and finally obtain the true global minimizer of +p(x|c1). +The following Theorem explains the ”Seesaw” properties of p(x|s). +Theorem 6. For any even degree monic polynomial p(x), let x(s) be the +(global or local) minimizers of seesaw polynomials p(x|s), then they satisfy +the differential equation +dp(x(s)|s) +ds += x(s), +(3.29) + +16 +QIAO WANG +and +d2p(x(s)|s) +ds2 += − +1 +p′′(x) < 0. +(3.30) +Proof. This differential equation can be verified immediately, +dp(x(s)|s) +ds += ∂p(x(s)|s) +∂x +· dx(s) +ds ++ x(s) = x(s). +(3.31) +The reminder is a simple application of previous Theorem 5, and the function +p(x(s)|x) is concave with respect to s. +□ +Remark 4. The (3.29) doesn’t distinct the global and local minimizers for +these x(s). +That is, if x(s0) is the global minimizer of seesaw polyno- +mial p(x|s0), the connected minimizer x(s1) might be the local minimizer +of p(x|s1). Thus we must identify the interval on which x(s) generated from +equation (3.29) with initial x(s0) is global or local. +3.3. Confinement zone and escape zone. In our following analysis, we +will give basic framework of FP2 +� FP3, essentially dependent on Q(x), and +varying initial condition of trajectory ODE to partition R into Confinement +Zone and Escape Zone, as well as varying S(p) to obtain Attainable Zone +for given Q(p). +It should be stress that in our study, all the fingerprints are about poly- +nomials, thus we have some obvious properties. +Lemma 3. For any polynomial p(x) and its heat evolution p(x, t), if +(x′, t′) ∈ FPi +� +FPi+1, +(3.32) +then x′ must be a real double root of polynomial equation ∂ip(x,t) +∂xi += 0, and a +real root of polynomial equation ∂i+1p(x,t) +∂xi+1 += 0. +Lemma 4. For n-th (n is even) order polynomial p(x), the set FP2 +� FP3 +contains at most n +2 − 1 points (xi, ti), where i = 1, 2, · · · , n +2 − 1. +Definition 5. Let c ∈ R, if the Yuille-Poggio’s curve from (c, 0) will not +intersect with any Yuille-Poggio’s curve from (c′, 0) ̸= (c, 0), we call this c +is in Escape Zone. Otherwise, we say it is in the Confinement Zone, which +is denoted by Ω. Accordingly, the Escape Zone is denoted by Ωc. +Theorem 7 (Characterization of confinement zone and escape zone). The +confinement zone Ω is +Ω := +n +2 −1 +� +i=1 +[XLL +i +, XRR +i +]. +(3.33) +where +XLL +i += +lim +L(�xL +i ,�t)→(xi,ti) +xLL +i +, +(3.34) +XRR +i += +lim +L(�xL +i ,�t)→(xi,ti) +xRR +i +, +(3.35) + +HEAT EVOLUTION +17 +Figure 5. The illustration of Yuille-Poggio’s flow as well as +FP2 and FP3. + +18 +QIAO WANG +in which the limitation means the point (�xL +i , �t) (or (�xR +i , �t), resp.) moves +to the destination (xi, ti) along the local FP2 fingerprint curve fpi +2(L) (or +fpi +2(R), resp.). Here (�xL +i , �t) (or (�xR +i , �t), resp.) is the end of Yuille-Poggio +curve connected to (xLL +i +, 0) (or (xRR +i +, 0), resp.). +Proof. We first prove that +Ω := +n +2 −1 +� +i=1 +� +[XLL +i +, XLR +i +] +� +[XRL +i +, XRR +i +] +� +, +(3.36) +in which we add two notations, +XLR +i += +lim +L(�xL +i ,�t)→(xi,ti) +xLR +i += K, +(3.37) +XRL +i += +lim +L(�xL +i ,�t)→(xi,ti) +xRL +i += K. +(3.38) +Here K stands for the intersection point (K, 0) between the curve in FP3(p) +and straight line t = 0. +Let’s show that the right hand side of (3.36) is well defined. As illustrated +at Fig.5, connecting to each (xi, ti) ∈ FP2 +� FP3, there exists a pair of +curves in FP2, corresponding to (xi + 0, ti − 0) and (xi − 0, ti − 0), denoted +by fpi +2(R) and fpi +2(L) respectively. +For any point (�xR +i , �t) ∈ fpi +2(R), when (�xR +i , �t) ̸= (xi, ti), there are a pair of +trajectories satisfying (3.8) which contains the point (�xR +i , �t). We may denote +their ends at t = 0 as (xRL +i +, 0) and (xRR +i +, 0) respectively. Here we assume +that xRL +i +≤ xRR +i +. +Similarly, for any point (�xL +i , �t) ∈ fpi +2(L), when (�xL +i , �t) ̸= (xi, ti), there are +a pair of trajectories satisfying (3.8) which contains the point (�xL +i , �t). We +denote their ends at t = 0 as (xLL +i +, 0) and (xLR +i +, 0) respectively. Here we +assume that xLL +i +≤ xLR +i +. +Now we may write that +xLL +i +≤ xLR +i +< K < xRL +i +≤ xRR +i +(3.39) +due to that the Yuille-Poggio curve can not intersect with FP3(p) otherwise +it will bring singularities, according to the denominator of the right hand +side of Yuille-Poggio equation (3.8). +Furthermore, the limitation process in (3.34) etc. remains monotonicity. +That is, moving from (�xL +i , �t) to (�xL+ +i +, �t+) and finally to (xi, ti), we may +observe that +− ∞ < �xLL+ +i +< �xLL +i +< �xLR +i +< �xLR+ +i +< K. +(3.40) +This implies that all the limitation (3.34) and so on are well-defined. Finally, +we note that ∀h ∈ (K − ϵ, K), there must exist a Yuille-Poggio curve starts +from (h, 0), and for ∀ϵ > 0, for any point (x′, t′) ∈ fpi +2(L), that satisfy +t′ < t, x′ > xi and ∥(x′, t′) − (xi, ti)∥2 < ϵ, there must exist a Yuille-Poggio +curve pass the point (x′, t′). That is to say, the Yuille-Poggio curves near + +HEAT EVOLUTION +19 +the FP3 curve connecting (K, 0) and (xi, ti), are dense. Here for the sake +of simplicity, we omit the topology and differential dynamics description. +Now the set Ω in (3.33) is well defined. We observe that any Yuille-Poggio +curve starting from (h, 0) for h ∈ Ω occurs if and only if there exists another +Yuille-Poggio curve, starting from (h′, 0) for some h′ ∈ Ω. In particular, +these two curves meet at fpi +2(L) or fpi +2(R). Thus the current Ω in (3.36) is +agreed with that in Definition 5. +□ +Clearly, we further have +Theorem 8. Let p(x) be any even order polynomial with positive leading +coefficient. Assume that xt0 +min be the global minimizer of convex polynomial +(sufficient scaled version) p(x, t) = p(x) ∗ gt(x) of p(x) at t = t0, and the +end of the trajectory by (3.8) at t = 0 is x∗. Then the global minimizer x∗ of +p(x) can be inversely involved from the global minimizer of its convexification +version p(x, t0), if and only if x∗ is in the Escape Zone Ωc. +Proof. If x∗ ∈ Ω, then the maximum of t coordinate of all the corresponding +Yuille-Poggio curves is bounded, thus all these Yuille-Poggio curves can not +connect to the point in Rx × Rt with large t > 0. +□ +Remark 5. Although the explicit representation of XLL +i +, XLR +i +, XRL +i +, XRR +i +is expected, it is not available in algebraic form since that when the degree of +polynomial is no less than 6, the curve in FP1 will involve algebraic equation +at least 5 degree. Thus we intend to give some numerical methods to give +these values. +Remark 6. The methodology of analysis declared here for convexification +by heat conduct equation, i.e., the Gaussian filtering, still works for the case +of Steklov regularization. +4. Case study of Quartic polynomials +In what follows, we will get explicit representation for the fingerprints of +quartic polynomials, and explain their geometric properties, such that we +can build the algorithm for solving the global minimizer of quartic polyno- +mials. +4.1. The structure of fingerprints. For the quartic polynomial p(x), we +see that +p(x, t) = p(x) + (6x2 + 3ax + b) · t + 3t2 += x4 + ax3 + (b + 6t)x2 + (c + 3at)x + (d + bt + 3t2). +(4.1) + +20 +QIAO WANG +Continue to differentiate both sides of (4.1) w.r.t x, the information of +∂p(x,t) +∂x +across time t may be represented as +∂p(x, t) +∂x += ∂p(x) +∂x ++ (12x + 3a) · t += (4x3 + 3ax2 + 2bx + c) + (12x + 3a) · t += 4x3 + 3ax2 + (2b + 12t)x + (c + 3at) += 0. +(4.2) +Similarily, we have +∂2p(x, t) +∂x2 += ∂2p(x) +∂x2 ++ 12t = (12x2 + 6ax + 2b) + 12t = 0, +(4.3) +and +∂3p(x, t) +∂x3 += 24x + 6a = 0. +(4.4) +These form the description of Fingerprints FP1, FP2 and FP3, respectively. +4.1.1. The structure of fingerprint FP1. Based on (4.2), the fingerprint FP1 +is characterized by following time-varying cubic equation +x3 + 3a +4 x2 + b + 6t +2 +x + c + 3at +4 += 0, +(4.5) +Now, if xi is a real root of (4.5) at t = 0, then it leads to the trajectory +described by the differential equation (3.8). For more details, we have +Lemma 5. For quartic polynomial p(x), the local extremal values points +xt +i (i = 1, 2, 3) of p(x, t) w.r.t x at scale t satisfy the trajectory differential +equation +dx(t) +dt += − +12x + 3a +12x2 + 6ax + 2b + 12t, +(4.6) +with following (at most three) initial conditions, +xi(0) = xi, +(i = 1, 2, 3). +(4.7) +Here xi is the local extremal of p(x). +Proof. Inserting (4.3) and (4.4) into (3.8) will lead to required results. +□ +The equation (4.5) possesses (at most) three real roots at t = 0, corre- +sponds to (at most) three trajectories, which form the Fingerprint FP1. +However, on the viewpoint of differential algebra (see, e.g. [15]), actually +the solution of differential equation (4.6) is real algebraic curve, i.e., a poly- +nomial F(x, t) about x(t) and t which satisfy F(x, t) = 0. +In our case, +the polynomial equation (4.5) describes this algebraic curve, thus we may +immediately apply the algebraic representation of FP1: +FP1 = +� +(x, t); t = −4x3 + 3ax2 + 2bx + c +12x + 3a +, +x ̸= −a +4, and t ≥ 0 +� +. (4.8) + +HEAT EVOLUTION +21 +According to Subsection A.2, to get the information of the roots of (4.5), +we need its discriminant, +∆(t) = +�a3 − 4ab + 8c +64 +�2 ++ +�−3a2 + 8b +48 ++ t +�3 +, +(4.9) +which will be explained in details in (4.12). +Lemma 6. The discriminant ∆(t) of equation (4.5) is monotonically in- +creasing to infinity. Its unique zero is +tu = a2 +16 − b +6 − 1 +16(a3 − 4ab + 8c) +2 +3 . +(4.10) +Proof. Using (A.5), we write +f(t) = b +2 − 3a2 +16 + 3t, +g(t) = a3 +32 − ab +8 + c +4. +(4.11) +Now the time-varying discriminant +∆(t) =[g(t)]2 +4 ++ [f(t)]3 +27 +, += +�a3 − 4ab + 8c +64 +�2 ++ +�−3a2 + 8b +48 ++ t +�3 +, +(4.12) +which means that ∆(t) increases monotonically w.r.t. t. Immediately, (4.12) +leads to (4.10). +□ +Theorem 9 (The ”1+2” structure of FP1). Let tu, defined in (4.10), be the +zero of discriminant ∆(t). If tu < 0, then FP1 contains only one trajectory +x(t) described by equation (4.6), which evolutes as t → +∞. If tu ≥ 0, then +during t ∈ [0, tu] the Fingerprint FP1 contains three distinct trajectories +described by (4.6), one of which continues to evolute to +∞ during t > tu, +and the other two trajectories will start from t = 0 but merge (stop) when +t = tu at the point (x(tu), tu). Here, +x(tu) = +�a3 − 4ab + 8c +64 +�1/3 +− a +4. +(4.13) +Proof. According to Lemma 6, we know that if tu < 0, then ∆(t) > 0 for +all t ≥ 0, which means the equation (4.5) has only one root at each t ≥ 0. +When tu ≥ 0, then ∆(t) < 0 (= 0, > 0, respectively) while t ∈ [0, tu) +(t = tu, t > tu, respectively), and the equation (4.5) has three distinct real +roots (one real and a pair of double real roots, or one real root, respectively). +In particular, we consider the critical case t = tu at which ∆(t) = 0. If case +is this, the equation (4.5) at t = tu possesses one real root and a real double + +22 +QIAO WANG +root. It follows from (A.8) that the real double root is +x(tu) = +�g(t) +2 +� 1 +3 +− 1 +3 · 3a +4 . +(4.14) +Substituting (4.11) into this formula will produces (4.13). +□ +4.1.2. The structure of FP2 and FP3. The structure of FP3 is very simple +for quartic polynomial, since from (4.4) we may write +FP3 = +� +(x, t); x = −a +4, t ≥ 0 +� +. +(4.15) +To analyze the structure of FP2, we have a Lemma as below. +Lemma 7. Denote +t∗ = a2 +16 − b +6, +(4.16) +then the polynomial p(x, t) defined in (4.1) is convex about x at each t > +max(t∗, 0). Furthermore, this t∗ can not be improved. +Proof. Consider the lower bound of (4.3) at t = 0, +∂2p(x) +∂x2 += 12x2 + 6ax + 2b += 12 +� +x + a +4 +�2 +− 3a2 +4 ++ 2b +≥ −3a2 +4 ++ 2b = −12t∗. +(4.17) +Combining it with (4.3), we would have +∂2p(x, t) +∂x2 +≥ −3a2 +4 ++ 2b + 12t = 12(t − t∗), +(4.18) +which implies the required results. +On the other hand, at any fixed t′ < t∗, notice that at x = − a +4, we have +∂2p(x, t′) +∂x2 += 12x2 + 6ax + 2b + 12t′ += 12 +� +x + a +4 +�2 +− 12(t∗ − t′) += −12(t∗ − t′) < 0, +(4.19) +which is not convex at this x = − a +4, such that t∗ is the optimal, and can not +be improved. +□ +Theorem 10 (The structure of FP2). For the structure of FP2 of quartic +polynomial p(x), +(a) when t∗ < 0, the fringerprint FP2 is empty. + +HEAT EVOLUTION +23 +(b) when t∗ = 0, the +FP2 = +� +(x, t) = (−a +4, 0) +� +has only single element; +(c) when t∗ > 0, the fingerprint FP2 consists of two curves: the left one +is +xL(t) = −a +4 − +√ +t∗ − t, +(t∗ ≥ t ≥ 0), +(4.20) +and the right one is +xR(t) = −a +4 + +√ +t∗ − t, +(t∗ ≥ t ≥ 0). +(4.21) +Specifically, these two curves must meets up at t = t∗, i.e., at the +point +(xL(t∗), t∗) = (xR(t∗), t∗) = +� +−a +4, t∗� +. +(4.22) +Proof. (a) comes from the fact that for every t ≥ 0, all the +∂2p +∂x2 > 0. That +is, FP+ +2 = {(x, t); x ∈ R, t ∈ [0, +∞)}, but FP− +2 = ∅. (b) is an immediate +result, and (c) is from the quadratic equation (4.18). +□ +4.1.3. The intersection between fingerprints. According to Lemma 3, we +may summarize the intersection of fingerprints. +Theorem 11. For the monic quartic polynomials p(x) = x4+ax3+bx2+cx, +the two intersection sets +FP2 +� +FP3 = +� +(−a +4, t∗) +� +, +(4.23) +and +FP1 +� +FP2 = {(x(tu), tu)}, +(4.24) +in which t∗ is defined in (4.16), tu and x(tu) are defined in (4.10) and (4.13) +respectively. +Remark 7 (Three phase of time evolution). In general settings, the evo- +lution of polynomial p(x) can be categorized into three phases according to +0 ≤ tu ≤ t∗. At first phase, t evolutes from 0 to tu, and FP1 contains three +distinct trajectories. Two of them will merge at t = tu. +Then at the second phase, tu < t < t∗, the FP1 contains only one trajec- +tory, but p(x, t) is not convex. +Finally, at the third phase, t > t∗, the FP1 contains only one trajectory, +and p(x, t) is convex. +4.2. Confinement zone. Now we compute the confinement zone of the +quartic polynomial p(x). We have +Theorem 12. The confinement zone of quartic polynomial p(x) is +� +−a +4 − +√ +3t∗, −a +4 + +√ +3t∗ +� +, +(4.25) +where t∗ is defined in (4.16). + +24 +QIAO WANG +Proof. Perform Q-S decomposition for quartic polynomial p(x), +p(x, t) = Q(x, t) + S(p), +(4.26) +where S(p) = cx + d. Clearly, we have +FPi(p) = FPi(Q), +i = 2, 3. +(4.27) +Thus we may vary W(p), i.e., vary c, to form a pair of trajectories such +that they can expand the scope as large as possible in R, which forms the +confinement zone. Re-write (4.10) as +tu(c) = t∗ − 1 +16(a3 − 4ab + 8c) +2 +3 . +(4.28) +We see that we should vary c such that tu(c) = t∗, i.e., +a3 − 4ab + 8c = 0 =⇒ c = ab +2 − a3 +8 . +(4.29) +Substituting this c into the trajectory algebraic curve equation (4.8) and +setting t = 0, we get the equation +4x3 + 3ax2 + 2bx + ab +2 − a3 +8 = 0. +(4.30) +The three roots of this equation are +x1 = −a +4, x2,3 = −a +4 ± +√ +3t∗, +(4.31) +which produces two pair of trajectories started from t∗ but reversely evolutes +to t = 0, whose four destinations form the confinement zone +� +−a +4 − +√ +3t∗, −a +4 +� � � +−a +4, −a +4 + +√ +3t∗ +� += +� +−a +4 − +√ +3t∗, −a +4 + +√ +3t∗ +� +(4.32) +□ +Remark 8. This confinement zone of p(x) is essentially dependent of Q(x, t) +but independent of S(p). +4.3. Differential equation of critical points across scale. Denote p(x) +the quartic polynomial as (1.13). Through out this paper, we denote by +x1, x2, x3 the roots of cubic equation p′(x) = 0, i.e., +x3 + 3a +4 x2 + b +2x + c +4 = 0. +(4.33) +Clearly, the global minimizer of p(x) must be one of x1, x2, x3. Comparing +this equation to (14), we may represent a, b, c in terms of x1, x2, x3 as +� +� +� +� +� +� +� +a = − 4 +3(x1 + x2 + x3), +b =2(x1x2 + x2x3 + x3x1), +c = − 4x1x2x3. +(4.34) +Now we give the representation of t∗ and tu in terms of roots of ∂p(x,t) +∂x += 0. + +HEAT EVOLUTION +25 +Lemma 8. Let x1, x2, x3 be the roots of (4.33), t∗ is defined in (4.16), and +tu defined in (4.10), then they can be represented as +t∗ = +�x1 + x2 + x3 +3 +�2 +− x1x2 + x2x3 + x3x1 +3 +, +(4.35) +and +tu = t∗ − +�32 +27(2x1 − x2 − x3)(2x2 − x3 − x1)(2x3 − x1 − x2) +� 2 +3 +. +(4.36) +This can be verified by substituting with (4.34). +Theorem 13. The singularity of the equation (4.6) occurs only at +xtu = x(tu) = +�a3 − 4ab + 8c +64 +�1/3 +− a +4, +t = tu. +(4.37) +Proof. The singularity occurs at differential equation (4.6), which describes +the FP1, so it must satisfy (4.5). Meanwhile, the denominator of the r.h.s. +of (4.6) is actually the fingerprint of FP2, which should satisfy (4.18). Thus +we may combine these two algebraic equations to solve (x, t). +Multiplying both sides of (4.18) by x + a +4, and subtracted it from (4.5) +will produce +t = (3a2 − 8b)x − (6c − ab) +48x + 12a +, +x ̸= −a +4. +(4.38) +Substituting it into (4.18) will yield a cubic equation about x, +48x3 + 36ax2 + 9a2x + (3ab − 6c) = 0. +(4.39) +This cubic equation has only one real solution (4.37). Substituting this x +into (4.38) will show that t = tu. +□ +When − a +4 is not a critical point, this (xtu, tu) occurs only at two FP1 +integral curves of (4.6) whose initial point is a local minimum and a local +maximum, respectively. Among them, one curve corresponds to the case +˙x(tu) = +∞ and another to ˙x(tu) = −∞. Most importantly, the integral +curve starts with globally minimum will not pass this (xtu, tu), which is the +main discovery of this paper, and will be explained in details in Section 4.4. +Remark 9. If there exists a critical point x′ at t = 0 such that x′ = − a +4, then +(4.6) implies its fingerprint curve x(t) ≡ − a +4. This happens when x′ is the +local maximizer, and other two critical points x1 and x3 satisfy x1+x3 = 2x′. +If case is this, all three fingerprint curves meet up at x′ = − a +4 when t = tu, +which will be explained in details in the following sections. +Example 2. Consider the polynomial p(x) = x4 + 0.2x3 − 0.5x2 + 0.01x, +the illustration is in Fig.6. + +26 +QIAO WANG +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +x +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +t +tu +t* +(a) FPi, (i = 1, 2, 3), tu and t∗. Notice +that FP1 corresponds to c = 0.01. +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +x +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +t +(b) FP2, FP3 and trajectories of c = +−0.05. +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +x +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +t +(c) critical trajectories when c = −0.051, +which are symmetric about x = − a +4 . +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +x +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +t +(d) FP2, FP3 and trajectories of c = +−0.2. +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +x +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +t +(e) trajectories by varying c. +(f) more trajectories by varying c. +Figure 6. Illustration of Fingerprints and Trajectories in +Example 2. To observe the change of trajectories with coef- +ficient c in the polynomial, we vary it in c ∈ [−2, 2]. + +0.2 +2 < c < 2 tr +0.18 +FP2 +FP.jectories0.16 +0.14 +0.12 +0.1 +0.08 +0.06 +0.04 +T +0.02 +0 +-1 +-0.5 +0 +0.5 +X1HEAT EVOLUTION +27 +4.4. Heat evolution of critical points of quartic polynomials. Now +we investigate the evolution of the critical points, and in particular, the quart +polynomial case. Essentially, we concentrate on the case tu > 0 in which +there exist three distinct critical points x1 < x2 < x3, and they correspond- +ingly evolve to the critical points xt +1, xt +2, xt +3 when 0 < t < tu. Generally, both +x1 and x3 are local minimizers and x2 is local maximizer. Our main con- +cern is the behavior associate with heat evolution, characterized by equation +(4.6). +Our next concern is the comparison principle between two local minimums +during heat evolution. Surprisingly, we have the very expected result for heat +evolution of quartic polynomials: +Theorem 14. For monic quartic polynomial p(x), assume that tu > 0, +and denote its three critical points x1 < x2 < x3 (or x1 > x2 > x3). If +p(x1) < p(x3), then p(xt +1, t) < p(xt +2, t). +Before showing this result, we need several lemmas. +Lemma 9. Let x1, x2, x3 be the critical points of quartic polynomial p(x), +then we have +p(x3) − p(x1) = −(x3 − x1)3 · (x1 + x3 − 2x2)/3. +(4.40) +Proof. Represent a, b, c in terms of x1, x2, x3, by (4.34). Thus we obtain +p(x1) − p(x3). Factorizing it will lead to required result. +□ +This Lemma 9 implies that +Lemma 10. For any t ∈ [0, tu), p(xt +3, t) = p(xt +1, t) if and only if xt +1 + xt +3 = +2xt +2. +Consequently, we see that +Lemma 11. For any t ∈ [0, tu), xt +1 + xt +3 = 2xt +2 if and only if xt +2 = − a +4. +Proof. Notice that the coefficient of x3 in p(x, t) is invariant with t, and the +coefficient of x2 of ∂p +∂x is also invariant with t. According to Appendix A, we +have +xt +1 + xt +2 + xt +3 = −3a +4 . +(4.41) +Applying Lemma 10 will yield the result. +□ +Lemma 12. Assume that x1 < x2 < x3 are three critical points of monic +quartic polynomial p(x), if p(x1) = p(x3), then for all t ∈ (−∞, tu), we have +p(xt +1, t) = p(xt +3, t). +(4.42) +Proof. Recall (4.5), and apply (4.34), we can actually represent ∂p(x,t) +∂x += 0 +in terms of x1, x2, x3 as +x3 − (x1 + x2 + x3)x2+(x1x2 + x2x3 + x3x1 + 3t)x +−[x1x2x3 + (x1 + x2 + x3)t] = 0. +(4.43) + +28 +QIAO WANG +Now if p(x1) = p(x3), Lemma 10 tells us x3 = 2x2−x1, thus we may simplify +the above equation as +x3 − 3x2x2 + (2x1x2 + 2x2 +2 − x2 +1 + 3t)x − (2x1x2 +2 − x2 +1x2 + 3x2t) = 0, (4.44) +whose solution is +� +� +� +� +� +� +� +� +� +� +� +xt +2 = x2, +xt +1 = x2 − +� +(x1 − x2)2 − 3t, +xt +3 = x2 + +� +(x1 − x2)2 − 3t, +−∞ < t < min +�(x1 − x2)2 +3 +, tu +� +(4.45) +which shows that xt +1 + xt +3 = 2xt +2. According to Lemma 10, this leads to +p(xt +1, t) = p(xt +3, t). +□ +At present stage, we summarize all lemmas as below, +Theorem 15. Under the same assumptions as Theorem 14, and denote +xt +1 < xt +2 < xt +3 the critical points of p(x, t). Then the following statements +are equivalent: +(1) p(x1) = p(x3), +(2) p(xt +1, t) = p(xt +3, t), ∀t ∈ [0, tu); +(3) x1 + x3 = 2x2; +(4) xt +1 + xt +3 = 2xt +2, ∀t ∈ [0, tu); +(5) x2 = − a +4; +(6) xt +2 = − a +4, ∀t ∈ [0, tu). +(7) t∗ = tu. +Proof. We will prove that (1) =⇒ (3) =⇒ (5) =⇒ (6) =⇒ (4) =⇒ (2) =⇒ +(1). In addition, (4) ⇐⇒ (7). Actually, this routine is partially repeated +with previous proofs. +(1) =⇒ (3) (also (4) =⇒ (2)) comes from Lemma 10, (3) =⇒ (5) (also +(6) =⇒ (4)) from Lemma 11, (5) =⇒ (6) from differential equation (4.6). +Finally, (4) ⇐⇒ (7) comes from (4.36) in Lemma 8 as well as the condition +x1 < x2 < x3. +□ +Proof of Theorem 10. The dynamical equation (4.6) states that the evo- +lution of three critical points are continuous when t ∈ [0, tu). +Thus if +p(x1) < p(x3), we must have p(xt +1, t) < p(xt +3, t) for t ∈ [0, tu), otherwise, +there must have +p(xt′ +1 , t′) = p(xt′ +3 , t′) +for some t′ ∈ (0, tu). But, if case is this, Theorem 15 or Lemma 12 tells us +that p(x1) = p(x3) since t is reversible. This leads to conflict with assump- +tion. +□ +To intuitively explain this result, we suggest a triangle representation at +Fig.20 for each t, where the cortes of triangle consists of (xt +i, p(xt +i, t)), (i = +1, 2, 3) when t < tu. Notice that the sequence of triangles when 0 ≤ t ≤ tu + +HEAT EVOLUTION +29 +and continued curve ˙x(t) actually connected to global minimum of p(x, t) at +each t ≥ 0. +Finally, we discuss an interesting problem: if x1 < x2 < x3 are three +critical points of quartic polynomial p(x), can we judge which one of them +is global minimizer without valuating all these p(xi)? The answer is YES. +Theorem 16. Let x1 < x2 < x3 be three distinct critical points of monic +quartic polynomial p(x), then the following statements are equivalent: +(1) x3 (resp. x1) is global minimizer; +(2) x1 + x3 > 2x2 (resp. x1 + x3 < 2x2); +(3) x2 < −a/4 (resp. x2 > −a/4). +Proof. Apply Lemma 9. +□ +This Theorem inspired the following very simple Euler’s algorithm with- +out Heat Convolution for quartic polynomials. +Theorem 17. For any monic quartic polynomial p(x) with a the coefficient +of x3, the Euler’s algorithm with FIXED initial position x(0) = − a +4, +x(k+1) = x(k) − ∆x · p′(x(k)), +(4.46) +MUST converge to the global minimizer of p(x). +4.5. Algorithm. Actually, the Euler’s algorithm may work from t > tu. +Fortunately, we know at t ≥ tu, the p(x, t) has only single minimum about +x. Recall the formula (A.2), in which we see that the sum of all real roots of +fingerprint cubic equation (4.5) should be invariant under 0 ≤ t ≥ tu, thus +we know the remaining critical point xinit at t = tu can be solved since we +already know the information of (x(tu), tu) from Theorem 13. This means +we may adopt +xinit = − 3a +4 − 2x(tu) += − a +4 − 2 +�a3 − 4ab + 8c +64 +�1/3 +, +(4.47) +at t = tu as initial point, then perform Euler’s algorithm for equation (4.6), +and finally attain the global minimum of p(x). This implies the following +result. +Theorem 18. For the quartic polynomial (1.13), if tu ≤ 0, this polynomial +has only one critical point. If tu > 0, the polynomial contains three distinct +critical points, then if all the critical points satisfy x ̸= − a +4, we backward +perform the differential equation +dx(t) +dt += − +12x + 3a +12x2 + 6ax + 2b + 12t, +(4.48) +with initial condition +xtu = x(tu) = −a +4 − (a3 − 4ab + 8c)1/3 +2 +(4.49) + +30 +QIAO WANG +from t = tu > 0 to t = 0, must attain the global minimizer of (1.13) at t = 0. +Finally, if one critical point equals − a +4, then p(x) has two global minimizers, +and they are the roots of quadratic polynomial +p(x) +x + a +4 +. +(4.50) +So far, we may start with verifying that whether − a +4 is a root of cubic +polynomial ∂p(x) +∂x . The global minimizer can be obtained immediately if − a +4 +is a root. Otherwise, set x(0) = xinit, and t(0) = tu. Then motivated by +(4.6), the iteration process is as below +x(i+1) =x(i) − ∆t · +12x(i) + 3a +12x(i)2 + 6ax(i) + 2b + 12t(i) , +t(i+1) =t(i) − ∆t. +(4.51) +Here the prescribed step ∆t > 0 is small enough, and we may stop the +iteration while t(n) ≈ 0. Finally, this algorithm provides +lim +i→∞ x(i) = xmin. +(4.52) +Instead beginning with xinit, we can also start by sufficient evolution +p(x, t). This will cost more steps of iterations. +4.6. Numerical experiments. +Example 3. This counter-example [7] is proposed against ’backward differ- +ential flow’ method of [6] , in which p(x) = x4−8x3−18x2+56x. In our heat +conduct framework, we have p(x, t) = x4−8x3−(18−6t)x2+32x−(18t−3t2). +Notice that Fig. 20 demonstrates the triangle series of critical points. +Example 4. Set p(x) = x4 + 0.2114x3 − 2.6841x2 − 0.1110x + 1.2406, then +in Fig. 7 the most left curve x1(t) is the global minimizer of corresponding +p(x, t) at each t ≥ 0, and Fig.8 illustrates the fingerprint FP1. The the- +oretical minimizer is x1 = −1.2307, and our iteration algorithm provides +x1 = −1.2308. +Example 5. Consider p(x) = x4 − 4x3 − 2x2 + 12x, then we actually have +three critical points x1 = −1, x2 = 1, x3 = 3. Notice that x1+x3 = 2x2 thus +p(x1, t) = p(x3, t) and x2(t) = x2 = 1 for all x ∈ [0, tu]. One can further +verify that in this symmetric case, we must have t∗ = tu. +The detailed +explain can be referred as in Theorem 15. +4.7. Summary of quartic polynomial case. For the global minimizer of +quartic polynomial p(x), while generating its multi-scale version p(x, t) = +p(x) ∗ gt(x) on account of Gaussian filter gt(x) with variance from t = 0 to ++∞, we will see that: + +HEAT EVOLUTION +31 +Require: a, b, c, d of p(x) = x4 + ax3 + bx2 + cx + d, and ∆t, pre. +Ensure: global minimizer xmin +1: function Iteration(a, b, c, d) +2: +[tu, xinit] ← Initialize(a, b, c) +3: +4: +if tu < 0 or −a/4 is critical point then computing x +5: +6: +else +7: +t ← tu +8: +x ← xint +9: +while t > pre do +10: +r ← (12x + 3a)/(12x2 + 6ax + 2b + 12t) +11: +x ← x − ∆t · r +12: +t ← t − ∆t +13: +end while +14: +15: +end if +16: +return x +17: end function +18: function Initialize(a, b, c) +19: +h ← a3 − 4ab + 8c +20: +t∗ ← a2/16 − b/6 +21: +tu ← t∗ − h +2 +3 /16 +22: +xinit ← −a/4 − h1/3/2 +23: +return tu, xinit +24: end function +• If tu < 0, p(x) itself is not necessary convex, but it has unique critical +point. Consequently, each p(x, t) has only one critical point at any +t ≥ 0; +• If further tu ≤ t∗ < 0, p(x) must be convex. Then each p(x, t) is +convex about x at any t ≥ 0; +• If tu > 0, the polynomial p(x) has three distinct critical points x1 < +x2 < x3 when 0 ≤ t < tu; +• When t = tu, the critical point corresponding to the global minimizer +will evolve continuously from tu to t∗, and the local minimizer will +meet up with local maximum xt +2. +Even more, these two critical +points will stop evolution at t = tu; +• When tu < t < t∗, the polynomial p(x, t) has unique minimizer at +each t. +• When t ≥ t∗, the polynomial p(x, t) will become convex about x, +and possesses unique minimizer. + +32 +QIAO WANG +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +x +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +y +x1(0) +x2(0) +x3(0) +x2(tu)=x3(tu) +x1(tu) +x3(t) +x2(t) +x1(t) +Figure 7. An example of triangle series in (x, y) system, of +p(x) = x4 + 0.2114x3 − 2.6841x2 − 0.1110x + 1.2406. See +Example 4 for more details. +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +x +0 +0.5 +1 +1.5 +2 +2.5 +3 +t +Figure 8. Fingerprint FP1 in (x, t) system, of p(x) = x4 + +0.2114x3 − 2.6841x2 − 0.1110x + 1.2406. See Example 4 for +more details. +5. Case study of sixth degree polynomials +5.1. Evolution and fingerprints. Now we consider 6 degree monic poly- +nomial +p(x) = x6 + bx4 + cx3 + dx2 + ex + f, +(5.1) +For the sake of simplicity, here we already regularize the coefficient of x5 +by setting it as zero, which is a standard technique in treating the algebraic + +HEAT EVOLUTION +33 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +2.5 +3 +x +-10 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +y +x1(0) +x3(0) +x1(tu)=x2(tu)=x3(tu) +x2(0) +x(t), t> tu +Figure 9. Triangle series of p(x) = x4 − 4x3 − 2x2 + 12x, +in which there exist two global minimizers. Notice that at +x1(tu) = x2(tu) = x3(tu), the fingerprint of x2(t) is a line +segmentation, which is partial repeated by global minimizer +curve x(t) after t ≥ tu. See Example 5 for more details. +equations. This implies that the heat evolution is +p(x, t) =p(x) + t · ∂p +∂t + t2 +2 +∂2p +∂t2 + t3 +6 +∂3p +∂t3 +=p(x) + t +2 · ∂2p +∂x2 + t2 +8 +∂4p +∂x4 + t3 +48 +∂6p +∂x6 +=x6 + b(t)x4 + c(t)x3 + d(t)x2 + e(t)x + f(t), +(5.2) +in which +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +b(t) = b + 15t, +c(t) = c, +d(t) = d + 6bt + 45t2, +e(t) = e + 3ct, +f(t) = f + dt + 3bt2 + 15t3. +(5.3) +The critical points of (5.1) satisfy the 5 degree equation +0 = 1 +6 +∂p(x, t) +∂x += x5 + B(t)x3 + C(t)x2 + D(t)x + E(t), +(5.4) +in which +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +B(t) = 2b +3 + 10t, +C(t) = c +2, +D(t) = d +3 + 2bt + 15t2, +E(t) = e +6 + ct +2 . +(5.5) + +34 +QIAO WANG +-0.5 +0 +0.5 +0 +10 +20 +10-3 +t = 0 +-0.5 +0 +0.5 +-0.1 +0 +0.1 +0.2 +-0.5 +0 +0.5 +0 +1 +2 +-0.5 +0 +0.5 +-20 +0 +20 +-0.5 +0 +0.5 +0 +10 +20 +10-3t = 0.02 +-0.5 +0 +0.5 +-0.5 +0 +0.5 +-0.5 +0 +0.5 +0 +1 +2 +3 +-0.5 +0 +0.5 +-20 +0 +20 +-0.5 +0 +0.5 +0 +10 +20 +10-3t = 0.05 +-0.5 +0 +0.5 +-1 +0 +1 +-0.5 +0 +0.5 +0 +5 +-0.5 +0 +0.5 +-20 +0 +20 +Figure 10. The evolution of six degree polynomial p(x, t) +and its derivatives ∂p(x,t) +∂x +, ∂2p(x,t) +∂x2 +, ∂3p(x,t) +∂x3 +. +The solution of (5.4) for t ≥ 0 consists the fingerprint FP1. Unfortunately, +the roots of this fifth degree equation is algebraically intractable [18]. +Similarly, we may write the equation of FP2 as below, +1 +30 +∂2p +∂x2 = x4 + +�2b +5 + 6t +� +x2 + c +5x + d +15 + 2b +5 t + 3t2 = 0. +(5.6) +Then the equation of FP3 is +1 +120 +∂3p +∂x3 = x3 + +�b +5 + 3t +� +x + c +20 = 0. +(5.7) +Our interest is the set FP2 +� FP3. Geometrically, there exist two pair of +real double roots of quartic equation (5.21), based on following Lemma. +Lemma 13. If x0 be a root of both polynomial p(x) and its derivative p′(x), +then it must be at least a double root of p(x). +Different from quartic polynomials case, we have not an explicit represen- +tation for FP2 +� FP3, and numerical approach is required here for exper- +iments. Clearly, the real double root of quartic equation (5.6) must be the +common roots of both (5.6) and (5.7). In general, there exist two 0 ≤ t1 < t2 + +HEAT EVOLUTION +35 +0 +0.005 +0.01 +0.015 +0.02 +0.025 +0.03 +0.035 +0.04 +t +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +(t) +10-6 +(t) +0 +Figure 11. The discriminant ∆(t) of quartic equation is +generated from ∂2p(x,t) +∂x2 += 0 and defined in (5.8). Here the +data comes from Example 1. The two real roots of ∆(t) are +t1 = 0.002341, at which the quartic equation possesses a real +double root and two distinct real roots, and t2 = 0.034887, +at which the quartic equation possesses a real double root +and a pair of conjugate complex roots. +such that the corresponding x1 and x2 are those two real double roots. The +following Theorem 19 explains the process of numerical approach. +Theorem 19. For any six degree monic polynomial p(x), the set FP2 +� FP3 +contains a pair of elements (xi, ti), i = 1, 2, or one element (x1, t1), or +empty. Specifically, +(1) Any ti must be the zero of discriminant +∆(t) = 27648t′6 + 1728c2 +25 +t′3 − 256 +625h2t′2 − 288 +625c2ht′ − 256h3 +753 +− 27c4 +625 +(5.8) +where +t′ = t + b +15, +h = b2 − 5d. +(5.9) + +36 +QIAO WANG +0 +0.005 +0.01 +0.015 +0.02 +0.025 +0.03 +0.035 +0.04 +t +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +x(t) +0.23516 +-0.078914 +Figure 12. The function x(t) is defined in (5.10), where the +data comes from the Example 1. Here we obtain two solution +(t1, x1) = (0.0023, 0.23516), (t2, x2) = (0.03489, −0.078914), +which consist the set FP2 +� FP3. +(2) Any xi is dependent of ti by the function +x = −c · +1800 +� +t + b +15 +�2 − 4(b2 − 5d) +36000 +� +t + b +15 +�3 + 80(b2 − 5d) +� +t + b +15 +� ++ 45c2 . +(5.10) +Proof. We will give a detailed analysis in Apendix B, based on which, we +know that in general settings there exist at most two merge time t1 and t2 +from the discriminant equation ∆(t) = 0 of quartic equation (5.6) and (B.8), +This function (5.8) can be verified by (B.7) and (B.8) immediately, but +we omit the detailed computation here. We may find out its two real zeros +t1 and t2 through numerical computation, then the real double roots x1 and +x2 of (5.6) at t1 and t2 respectively, could be obtained according to following +(5.10). Thus in general settings, i.e., when p′(x) has five distinct real roots, +we will have +FP2 +� +FP3 = {(x1, t1), (x2, t2)}. +(5.11) +In degenerated cases, this intersection set might possess one point or even +null. When it is null, the polynomial p(x) is globally convex. + +HEAT EVOLUTION +37 +Here we may take Euclidean algorithm to reduce the degree of the poly- +nomials about x for (5.6) and (5.7). At first, multiplying (5.7) with x, and +subtracted both sides of (5.6) resp., we get a second degree polynomial +� +t + b +15 +� +x2 + c +20x + +� +t + b +15 +�2 +− b2 − 5d +225 += 0. +(5.12) +Again, multiplying with x for both sides of (5.12), and subtracted from both +sides of (5.7) multiplied with t + b +15, then we obtain +− c +20x2 + +� +2 +� +t + b +15 +�2 ++ b2 − 5d +225 +� +x + c +20 +� +t + b +15 +� += 0. +(5.13) +Finally, eliminating the second degree term by combining (5.13) and +(5.12) will lead to (5.10). +□ +As explained in Appendix B, we can obtain the suitable t from the dis- +criminant ∆(t) = 0 at first, then substitute it into the above (5.10), then +choose x from (5.10). +5.2. The boundary of confinement zone. To characterize the boundary +of confinement zone, we must study the singular trajectory +dx +dt = − +∂3p +∂x3 +2 ∂2p +∂x2 +, +Initial Condition : (t′, x(t′)) ∈ FP2 +� +FP3. +(5.14) +Unfortunately, this equation is singular due to its initial data, which re- +sults in +dx +dt = 0 +0. +(5.15) +However, at critical time t′, the multiplicity of common root x′ of both FP2 +and FP3 is different. Clearly, x′ is double root of FP2 and single root of +FP3, thus the reciprocal equation of (5.14) +dt +dx = −2 ∂2p +∂x2 +∂3p +∂x3 +, +Initial Condition : (t(x′), x′) ∈ FP2 +� +FP3 +(5.16) +contains only removable singularity near FP2 +� FP3. +Fig.13 demonstrates the limitation curve satisfying (3.8) and inversely +evolutes as t → +0 starting from the top points near FP2 +� FP3. + +38 +QIAO WANG +-0.6 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +x +0 +0.005 +0.01 +0.015 +0.02 +0.025 +0.03 +0.035 +0.04 +0.045 +0.05 +t +X1 +LL +X1 +LR +X1 +RL +X1 +RR +X2 +LL +X2 +RR +X2 +LR,X2 +RL +Figure 13. +Numerically, the partition of Confinement +Zone and Escape Zone associated with p(x) defined in Ex- +ample 1 is obtained through Matlab ODE packet of @ode25, +and +the +Confinement +Zone +is +[−0.5082, 0.0858] �[0.1603, 0.3267]. +5.3. Why convecification approach can not guarantee attaining the +global minimizer? Recall the regularized polynomial (5.1), which contains +the parameters {b, c, d, e, f}, and the FP1 equation (5.4) contains {b, c, d, e}. +However, the FP2 equation (5.6) and the FP3 equation (5.7), contains only +{b, c, d} and {b, c} respectively, which is independent of e. Thus FP2 +� FP3 +doesn’t contain the information of e, which actually affect the location of +global minimizer of polynomial p(x) in (5.1)6. +On the other hand, to determine the scope of the Confinement Zone, we +require the equation (3.8) which depends only on {b, c, d}. Thus we must +investigate the affection of e in the (5.1) to the global minimizer. +Thus +we may consider that for fixed b, c, d and vary e, the variation of global +minimizer of corresponding p(x), and when it is included in the Escape +6The parameter f in (5.1) doesn’t affect the location of global minimizer of six degree +polynomial. + +HEAT EVOLUTION +39 +Zone. To this end, we may define the mapping +R(e|b, c, d) = +� +1, +x∗ ∈ Escape Zone, +−1, +x∗ ∈ Confinement Zone. +(5.17) +where x∗ represents the global of polynomial p(x) in (5.1). The detailed +analysis reveals that when +e ∈ (−∞, 0.676739] +� +[−0.617543, −0.58523] +� +[−0.67115, +∞), +(5.18) +the corresponding global minimizer x∗ falls into the scape of Escape Zone. +Fig17 illustrates the curve of x∗(e|b, c, d), the global minimizer of polynomial +p(x) in (5.1) in which b, c, d remains invariant, while varying the parameter e. +Those x∗ fails to be obtained by convexification approach is demonstrated. +5.4. Comparison principle and criterion function for evolution poly- +nomials. Now we describe the comparison criterion for p(xi, t) > p(xj, t), +where xi and xj are critical points of p(x, t) at time t. Apparently, it follows +immediately from (5.1) that +p(xi, t) − p(xj, t) = (xi − xj) · Q5(xi, xj, t) +(5.19) +in which Q5 is a fifth degree polynomial. However, it is too complicated for +analysis. Instead, we can give a more concise representation for factoring +the p(xi, t) − p(xj, t). +Theorem 20. Let ξ = ξ(t) and η = η(t) be critical points of p(x, t), we +have +p(ξ, t) − p(η, t) = −(ξ − η)3 +10 +· K(ξ, η, t), +(5.20) +where the criterion function +K(ξ, η, t) = 20(ξ3 + η3) + 30(ξ2η + ξη2) ++15a(ξ2 + η2) + 20aξη ++ (10b + 150t)(ξ + η) + (5c + 50at) +=K(ξ, η, 0) + 150t(ξ + η) + 50at. +(5.21) +If both ξ and η (where we suppose that ξ ̸= η) are real critical points of p(x), +then for sixth degree monic polynomial p(x, t), +p(ξ, t) > p(η, t) ⇐⇒ (ξ − η)K(ξ, η, t) < 0. +(5.22) +Remark 10. If we transform the critical points by translation xi → xi − a +5, +we may set a = 0 in (5.21), then the criterion function (5.21) can be reduced +to +K(ξ, η, t) = 20(ξ3 + η3) + 30(ξ2η + ξη2) + (10b + 150t)(ξ + η) + 5c. (5.23) + +40 +QIAO WANG +Proof of Theorem 20. Let xi (i = 1, 2, 3, 4, 5) be the critical points of monic +sixth degree polynomial p(x, t) at t, i.e., the roots of equation 1 +6 +∂p(x,t) +∂x += 0. +Thus we may write +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +A(t) = − +� +i +xi, +B(t) = +� +i p(η, t) ⇐⇒ K(ξ, η, t) > 0. +In general settings, a monic sixth degree polynomial p(x) has five real +critical points x1 < x2 < x3 < x4 < x5, and +p(x1) < p(x2) > p(x3) < p(x4) > p(x5). +Accordingly, we may see that +K(x1, x2), K(x3, x4) < 0, +and +K(x2, x3), K(x4, x5) > 0. +Our main task is to determine the +arg +min +i∈{1,3,5} p(xi). +(5.26) + +HEAT EVOLUTION +41 +Thus we have a graphical criterion by partition the (ξ, η) plane into P/N +parts according to the sign of K(ξ, η, t = 0): +Theorem 21 (criterion of global minimizer). Let x1 < x2 < x3 < x4 < x5 +be five critical points of monic sixth degree polynomial p(x), we have the +following criterion +(1) x1 = arg minx p(x) if and only if K(x1, x3) ≤ 0 and K(x1, x5) ≤ 0; +(2) x3 = arg minx p(x) if and only if K(x1, x3) ≥ 0 and K(x3, x5) ≤ 0; +(3) x5 = arg minx p(x) if and only if K(x1, x5) ≥ 0 and K(x3, x5) ≥ 0. +In a summary, the sign of K(x1, x3), K(x1, x5) and K(x3, x5) determines +the global minimizer. +5.5. The level set of criterion surface. Our main challenge is to explain +the merge of two critical points at time evolution. That is, in general set- +tings, we intend to find out those two merge time t1 and t2 such that at +each merge time ti, there is a pair of critical points satisfy ξ(t1) = η(t1), and +another pair of points satisfy ξ(t2) = η(t2). Notice that these don’t mean +that they satisfy K(ξ, η, ti) = 0. +In general setting, we assume that all five critical points are real and +separate. Define the sets +Zt(K) ={(ξ, η); K(ξ, η, t) = 0}, +Z+ +t (K) ={(ξ, η); K(ξ, η, t) > 0}, +Z− +t (K) ={(ξ, η); K(ξ, η, t) < 0}. +(5.27) +Proposition 1. If the initial sixth degree polynomial contains five real crit- +ical points xi, then the regularization a = 0 will lead to +b ≤ 0, +(5.28) +And each xi satisfies +|xi| ≤ +√ +−b. +(5.29) +Proof. We have +0 = a2 = +�� +i +xi +�2 += +� +i +x2 +i + 2 +� +i 0, or c < 0, resp.). +Proof. This can be immediately obtained from (5.23). +□ +To understand the evolution of criterion surface, it is more natural to +change the coordinates while setting a = 0. Define +u = ξ + η, +v = η − ξ, +(5.31) + +42 +QIAO WANG +we may rewrite the criterion surface K(ξ, η, t) as +˜K(u, v, t) = 25 +2 u3 + +�15 +2 v2 + 10b + 150t +� +u + 5c. +(5.32) +Then the level set Zt(K) = 0 can be characterized by +u3 + +�3 +5v2 + 4b +5 + 12t +� +u + 2c +5 = 0. +(5.33) +Notice that ξ < η is equivalent to v > 0, and ξ = η means v = 0. Thus we +may represent the discriminant of the cubic function of u (5.32) by +∆(v, t) = +�v2 +5 + 4b +15 + 4t +�3 ++ c2 +25. +(5.34) +The solution about u in ˜K(u, v, t) = 0 contains three distinct, (or one and +a repeated pair, and single, resp.), real roots, when ∆(v, t) < 0, (or =, > 0 +resp.) Notice that the equation (5.32) is symmetric about v, we may consider +only about v ≥ 0. Clearly, the sign of ∆(v, t) is the same as that of +v2 +5 + 4b +15 + 4t + +�c +5 +� 2 +3 , +(5.35) +which is monotonically increasing with t. Therefore we have +Theorem 22 (Level Set). For ˜K(u, v, t) = 0, set +t∗ +v = −v2 +20 − b +15 − +� c +40 +� 2 +3 . +(5.36) +Then for 0 ≤ t < t∗ +v, the equation has three distinct real roots, +� +� +� +� +� +� +� +� +� +� +� +� +� +u1(v, t) = P(v, t) + Q(v, t), +u2(v, t) = −1 +2 [P(v, t) + Q(v, t)] + +√ +3 +2 i [P(v, t) − Q(v, t)] , +u3(v, t) = −1 +2 [P(v, t) + Q(v, t)] − +√ +3 +2 i [P(v, t) − Q(v, t)] , +(5.37) +in which +� +� +� +� +� +� +� +� +� +P(v, t) = +3 +� +− c +40 + +� +∆(v, t), +Q(v, t) = +3 +� +− c +40 − +� +∆(v, t). +(5.38) +If t > t∗ +v ≥ 0, i.e., ∆(v, t) > 0, it has only one real solution +u(t) = P(v, t) + Q(v, t). +(5.39) +Finally, when t = t∗ +v, i.e., ∆(v, t) = 0, it has one real root and in addition, +two repeated real roots, +u1(v, t∗ +v) = −2 3 +� c +40, +u2(v, t∗ +v) = u3(v, t∗ +v) = +3 +� c +40. +(5.40) + +HEAT EVOLUTION +43 +This Theorem indicates the structure of the level set Zt(K). +Theorem 23. Under the reduced form (a = 0), define +t∗ +max = − b +15 − +� c +40 +� 2 +3 . +(5.41) +(1) When t∗ +max < 0, the level set contains a unique continuous curve C, +which is asymptotically by a line parallel to ξ + η = 0; +(2) When t∗ +max > 0, for any fixed t ∈ [0, t∗ +max], there are two types of +the configuration of the level set. If c > 0, the oval-like part of Zt is +included in the north-east part of the plane w.r.t. the long curve C. +If c < 0, it is contained in the south-east part. +(3) When t∗ +max > 0, then for any fixed t ∈ [0, t∗ +max], then the oval-like +closed curve spanned in the scope of v2 ≤ 20(t∗ +max − t). The level set +inside this scope is characterized by (5.37). In particular, one pair +of the vortex of this oval like curve is +u = − 3 +� +− c +40, +v(t) = ± +� +20(t∗max − t). +(5.42) +and another pair of vertex at v = 0 is u1,2(t), which are the solution +of (5.37) except for the minimal one if c > 0 (or maximal one if +c < 0). +(4) When t∗ +max = 0, then at t = 0, the level set contains a curve C and +a point u = − 3�− c +40, v = 0. +5.6. Symmetry and trend of the criterion function. We consider the +case u = 0, i.e., +ξ + η = 0. +(5.43) +Clearly, we see that +K(x, y, t) = 5c, +if ξ = x + y = 0. +(5.44) +We further consider v = 0, i.e., ξ = η at zero level set, that is, +K(ξ, ξ, t) = ˜K(u = 2ξ, v = 0, t) = 0. +(5.45) +According to (5.23) we can obtain a concise cubic equation +ξ3 + +�b +5 + 3t +� +ξ + c +20 = 0. +(5.46) +Remark 12. This equation formally agrees with the equation +∂3p(x, t) +∂x3 += 0, +(a = 0), +(5.47) +since that +1 +120 +∂3p(x, t) +∂x3 += x3 + +�b +5 + 3t +� +x + c +20. +(5.48) + +44 +QIAO WANG +(a) a = 0, b = −0.3726, c = 0.0574 +(b) a = 0, b = −0.2938, c = −0.0797 +Figure 14. Criterion functions and P/N partition with pa- +rameters a = 0, b = −0.3726, c = 0.0574 (a), and a = 0, b = +−0.2938, c = −0.0797 (b). Here the green domain is positive +domain. + +150 +100 +50 +0 +-50 +-100 +-150 +-1 +-0.5 +0 +0.5 +1 +-1 +n150 +100 +50 +0: +-50 +-100 +-150 +-1 +-0.5 +0 +0.50 +nHEAT EVOLUTION +45 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +-0.6 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +(a) t = 0 +-1 +-0.5 +0 +0.5 +1 +-1 +-0.5 +0 +0.5 +1 +(b) t = 0.002 +Figure 15. The evolution of position and sign (”+” or +”−”) indicates the critical points of p(x) = x6 − 0.3726x4 + +0.0574x3 + 0.0306x2 − 0.0084x by level set of criterion func- +tion K(ξ, η, t). The ”o” at ξ = η stands for the critical points +(xi, xi) (i = 1, 2, 3, 4, 5) and their evolution or merge. Note +that all K(x1, xi, 0) < 0, then x1 is the global minimizer of +p(x) according to Theorem 21. + +46 +QIAO WANG +Notice that the structure of the solution of equation (5.46) and (5.48) is +only a special case of (5.33) as v = 0. +In a summary, the intersection of level set and the line ξ = η is just the +roots of ∂3p +∂x3 = 0. +5.7. The merge time. +5.7.1. Merge time of critical points. Now we discuss the merge phenomenon +of critical points. Two critical points meet up when ξ = η at time t, that +is, v = 0 in above equations (5.31)-(5.36). Notice that if case is this, they +must satisfy both the fingerprint equation ∂p +∂x = 0 and ∂2p +∂x2 = 0. +Now we apply the reduced form for both equations and by assuming that +a = 0, and obtain +� +� +� +� +� +� +� +x5 + +�2b +3 + 10t +� +x3 + c +2x2 + +� +15t2 + 2bt + d +3 +� +x + 3ct + e +6 += 0, +x4 + +�2b +5 + 6t +� +x2 + c +5x + +� +3t2 + 2bt +5 + d +15 +� += 0, +(5.49) +Subtracted the second equation multiplied by x from the first one, we have +�4b +15 + 4t +� +x3 + 3c +10x + +� +12t2 + 8bt +5 + 4d +15 +� +x + 3ct + e +6 += 0, +(5.50) +Then, multiplied by x − +3c +40(t+b/15) and subtracted from the second equation +multiplied by 4b +15 + 4t, we obtain a quadratic equation like +F(t)x2 + G(t)x + H(t) = 0 +(5.51) +We may first consider the merge time of fingerprints FP1 and FP2. But +the systems of fifth degree polynomial and a quadratic polynomial will not +be explicitly expressed. We apply Euclidean method to decrease the order +of x. We begin with +R1(x, t) = 1 +6 +∂p(x, t) +∂x +, +R2(x) = 1 +30 +∂2p(x, t) +∂x2 +. +(5.52) +Then we apply the Euclidean algorithm, +Ri(x, t) = (αi(t)x + βi(t))Ri+1(x, t) + Ri+2(x, t), +i = 1, 2, 3, 4, +(5.53) +where all the terms are polynomials. Finally we obtain a rational polynomial +representation +x(t) = −M2(t) +M1(t), +(5.54) +in which +M1(t) = +5 +� +i=0 +Niti, +M2(t) = +6 +� +i=0 +diti, +(5.55) + +HEAT EVOLUTION +47 +where +N5 = − 34992000c, +N4 = − 6480000e − 10368000bc, +N3 = − 1287360b2c − 1728000be + 388800cd, +N2 = − 64512b3c − 273600b2e − 23040bcd − 94770c3 + 504000de, +N1 = − 1536b4c − 21120b3e + 4416b2cd − 6156bc3 + 67200bde − 32400c2e − 31680cd2, +N0 = − 768b4e + 128b3cd + 4160b2de +− 3510bc2e − 1152bcd2 + 729c3d + 3375ce2 − 4800d2e, +(5.56) +and +D6 =466560000, +D5 =186624000b, +D4 =34214400b2 − 15552000d, +D3 =3594240b3 − 4147200db + 2041200c2, +D2 =211968b4 − 322560b2d + 537840bc2 − 648000ce − 230400d2, +D1 =6144b5 − 6144b3d + 27504b2c2 − 104400bce +− 30720bd2 + 93960c2d + 45000e2, +D0 =1024b4d − 384b3c2 − 1920b2ce − 5632b2d2 + 8424bc2d ++ 3000be2 − 2187c4 − 10800cde + 7680d3. +(5.57) +The motivation of this rational representation comes from the fact that +at the merge time t we have dx(t) +dt += ∞, which means the suitable t must be +the singularity. Thus we intend to obtain the singularity, i.e., the zeros of +M1(t). +5.7.2. merge time of inflection points. Then consider the merge time of fin- +gerprints FP2 and FP3. +Combining the equation (5.49) with the criterion equation (5.23) (setting +ξ = x) will yield a quadratic equation +�b +5 + 3t +� +x2 + 3c +20x + +� +3t2 + 2bt +5 + d +15 +� += 0, +(5.58) +with quadratic discriminant +∆2(t) = −36t3 − 36b +5 t2 − +�8b2 +25 + 4d +5 +� +t + 9c2 +400 − 4bd +75 . +(5.59) +When ∆2(t) ≥ 0, the equation possesses a pair of roots +x1,2(t) = − 3c +20 ± +� +∆2(t) +2 +� b +5 + 3t +� +. +(5.60) + +48 +QIAO WANG +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +t +-10 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +10 +x +(a) The singularity of − M2(t) +M1(t) occurs. +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +t +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +x +(b) The zeros of M1(t) +Figure 16. The singularity occurs + +HEAT EVOLUTION +49 +Therefore, combining it with (5.37) will produce the repeated root of (5.46). +We can actually continue using Euclidean’s algorithm to reduce the order +of polynomials w.r.t x, and obtain +M1(t)x + M2(t) = 0, +(5.61) +thus +x = −M1(t) +M2(t), +(5.62) +in which +M1(t) =18t3 + 18b +5 t2 + +�7b2 +25 − d +5 − 9ac +40 +� +t +− b +5 +� d +15 − b2 +25 +� ++ 3c +20 +�3c +20 − ab +10 +� +, +M2(t) =9c − 3ab +10 +t2 + 6bc − ab2 − 5ad +50 +t + 5cd + b2c +500 +− abd +150 +(5.63) +5.8. Global minimizers with varying S(x) = sx and differential +equation. Now we explain the evolution of global minimizer of +p(x) = x6 − 0.3726x4 + 0.0574x3 + 0.0376x2 + sx, +(5.64) +which is a modified version of Example 1. Fig.17 shows the evolution of +global minimizers of this seesaw polynomial w.r.t the parameter s, the coef- +ficient of x. Actually, at two points s1 and s2, there exists two distinct global +minimizers pair x1(s1) and x2(s1), and x1(s2) and x2(s2), which occurs at +p(x1(s1)) = p(x2(s1)) and p(x1(s2)) = p(x2(s2)) respectively. +We give a detailed analysis on locating the s1 and s2. +Actually, the +polynomial p(x|s) has at most five real critical points for each s. Among +them, three are local minimizers. +Assume that x1, x2, x3 are three local +minimizers of p(x), we may apply the seesaw equation to characterize their +evolution as s changes. At the first phase, we start with s = −2, and apply +dx +ds = − +1 +p′′(x), +xi = xi(s = −2), +i = 1, 2, 3. +(5.65) +We take s goes to +∞, in order to obtain these three continuous curves +begin with s = −2. +5.9. Comparison principle and criterion function for seesaw poly- +nomials. Similar to the comparison criterion of evolution polynomials, we +may compare p(xi|s) > p(xj|s), where xi and xj are critical points of p(x|s) +at seesaw parameter s. +Theorem 24. Let ξ = ξ(s) and η = η(s) be critical points of p(x|s), we +have +p(ξ|s) − p(η|s) = −(ξ − η)3 +10 +· H(ξ, η), +(5.66) + +50 +QIAO WANG +Figure 17. The global minimizers x∗(s) of six degree poly- +nomials p(x) = Q(p) + S(p), where Q(p) = x6 − 0.3726x4 + +0.0574x3 + 0.0376x2, and S(p) = sx with varying parame- +ter s. This is the modified version of Example 1. There are +two concave domains, each one starts with a vertical dashed +line and ends with a vertical straight line. +Both the blue +points and red points represent the global minimizer x∗(s), +and they always appears at convex domain. Specifically, the +blue point occurs at Escape Zone, means that at correspond- +ing s the global minimizer x∗(s) can be obtained by inversely +heat conduct algorithm. However, the red point appears at +the Confinement Zone which means that this global mini- +mizers x∗(s) can not be obtained immediately through the +inverse heat conduct algorithm. However, this red part in +confinement zone can still be accessed by solving seesaw dif- +ferential equation (3.25) with initial global minimums from +attainable zone in connected blue part. +where the criterion function +H(ξ, η) = 20(ξ3 + η3) + 30(ξ2η + ξη2) ++15a(ξ2 + η2) + 20aξη ++ 10b(ξ + η) + 5c +(5.67) + +2 +EscapeZone +1.5Concave +0.5 +S +0 +-0.5 +-1 +-1.5 +confinementZone +-2 +-1 +-0.5 +0 +0.5 +* +X1HEAT EVOLUTION +51 +-1.2 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +x +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.1 +t +* stands for global minimum +Figure 18. An example of Fingerprint of p(x) = x6 + +0.6987x5 − 1.0908x4 − 0.4216x3 + 0.2177x2 + 0.1071x. Here +∗ stands for the global minimizer, and the Euler’s method +along the critical point Fingerprint from large t will back- +ward to the true global minimizer. +If both ξ and η (where we suppose that ξ ̸= η) are real critical points of +p(x|s), +p(ξ|s) > p(η|s) ⇐⇒ (ξ − η)H(ξ, η) < 0. +(5.68) +Our interest is to find out the seesaw parameter s such that ξ(s) ̸= η(s) +and +min +x p(x|s) = p(ξ(s)|s) = p(η(s)|s). +(5.69) +That is, the seesaw polynomial p(x|s) attains a state that occurs the jump +phenomena: it possesses (at least) two global minimizers ξ(s) and η(s). +5.10. Numerical examples for 6-degree polynomials. It is extremely +expected to generalize the heat evolution algorithm to find out global min- +imizer of 6 or higher even degree polynomials. However, the Theorem 14 +can not be generalized to higher degree polynomials. Here we illustrate the +positive and negative examples. +Example 6. The fingerprint FP1 of p(x) = x6 + 0.6987x5 − 1.0908x4 − +0.4216x3 + 0.2177x2 + 0.1071x illustrated in Fig.18 shows that the global +minimizer is included in the integral curve to convex p(x, t). +Example 7. The fingerprint FP1 of p(x) = x6 − 0.8529x5 − 0.4243x4 − +0.2248x3 + 0.0916x2 − 0.0074x illustrated in Fig.19 shows that the global +minimizer is NOT included in the integral curve to convex p(x, t). + +52 +QIAO WANG +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +x +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.1 +t +* stands for global minimum +Figure 19. A counter-example of fingerprint of p(x) = x6 − +0.8529x5−0.4243x4−0.2248x3+0.0916x2−0.0074x. Here the +∗ stands for the global minimizer, but the most right curve +started from large t > 0, connected only to the local mini- +mizer. +6. Conclusion +In this paper, we investigate the possibility of finiding the global mini- +mizer of a polynomial p(x) by inversely evolution from the global minimizer +of its conxification version p(x, t) = p(x) ∗ gt(x). We propose the concepts +of confinement zone and escape zone, as well as attainable zone, of the poly- +nomial p(x). +We apply Yuille-Poggio’s fingerprint theory including the Yuille-Poggio +equation in computer vision to characterize the critical points of p(x, t), +and propose a seesaw decomposition which produces the seesaw polynomial +p(x|s). We further propose a seesaw differential equation to characterize +the change of minimizers of p(x|s). Here, the fingerprint FP2 and FP3 are +independent of seesaw parameter s, but the information of critical points of +p(x|s) are contained in Yuille-Poggio’s flow. +We showed in this paper that the global minimizer x∗ of a polynomial +p(x) can be evolved inversely from the global minimizer of its conxification +version p(x, t) = p(x) ∗ gt(x), if and only if this x∗ is in the escape zone +of polynomial p(x). When x∗ is not in the escape zone, we may apply the +seesaw equation by varying s through the global minimizer x∗(s) of p(x|s) +to obtain x∗. +However, the characterization of escape zone and attainable zone of a +polynomial p(x) is in general algebraically not tractable, according to the +Galois theory. Thus efficient numerical methods, as well as various criterions +of judge the zones are extremely expected. + +HEAT EVOLUTION +53 +Some results concerning the multivariate cases will be giving in our forth- +coming works. + +54 +QIAO WANG +-4 +-2 +0 +2 +4 +6 +8 +-900 +-800 +-700 +-600 +-500 +-400 +-300 +-200 +-100 +0 +100 +x1(t) +x3(t) +x2(t) +x1(t) and x2(t) meets up at tu +x3(t) is the unique critical point when t>tu +at tu +Figure 20. The triangle series of critical points of evolution. Here the polynomial is p(x) = x4 − 8x3 − +18x2 + 56x, which is explained in Example 3. + +HEAT EVOLUTION +55 +-4 +-2 +0 +2 +4 +6 +8 +x +0 +5 +10 +15 +scale t +convexity fingerprint +-4 +-2 +0 +2 +4 +6 +8 +x +0 +5 +10 +15 +scale t +extreme fingerprint +-4 +-2 +0 +2 +4 +6 +8 +x +0 +5 +10 +15 +scale t +mixtured fingerprints +Figure 21. Both the fingerprints FP1 and FP2 characterize the distribution of critical points and convexity +of heat evolved version of a quartic polynomial. + +56 +QIAO WANG +Appendix A. Real solutions of cubic equation +A.1. Representation by roots. Recall the classical theory of cubic alge- +braic equation (cf. [14]) +x3 + αx2 + βx + γ = 0, +(A.1) +According to Newton’s method, we have +Lemma 14. Let xi (i = 1, 2, 3) be the roots (real or complex) of polynomial +equation (A.1). We have the following propositions, +x1 + x2 + x3 = −α, +x1x2 + x2x3 + x3x1 = β, +x1x2x3 = −γ, +x2 +1 + x2 +2 + x2 +3 = α2 − 2β, +x3 +1 + x3 +2 + x3 +3 = −α3 + 3αβ − 3γ, +x4 +1 + x4 +2 + x4 +3 = α4 − 4α2β + 4α2 + 2β2. +(A.2) +When the coefficients of the equation are real, the discriminant of the +equation is +∆ = (x1 − x2)2(x2 − x3)2(x3 − x1)2, +(A.3) +which is equivalent to +∆ = g2 +4 + f3 +27, +(A.4) +in which +f = β − α2 +3 +and g = 2α3 +27 − αβ +3 + γ. +(A.5) +A.2. The real roots described by discriminant. Then the solution of +this cubic equation is as follows: +If ∆ < 0, the equation (A.1) contains three distinct real roots. +x1 = +2 +√ +3 +� +−f sin(θ) − α +3 , +x2 = − 2 +√ +3 +� +−f sin +� +θ + π +3 +� +− α +3 , +x3 = +2 +√ +3 +� +−f cos +� +θ + π +6 +� +− α +3 , +(A.6) +where +θ = 1 +3 arcsin +� +3 +√ +3g +2(√−f)3 +� +. +(A.7) +If ∆ = 0, the solutions contains a single root and two repeated roots, +x1 = −2 +�g +2 +� 1 +3 − α +3 , +x2 = x3 = +�g +2 +� 1 +3 − α +3 . +(A.8) + +HEAT EVOLUTION +57 +Finally, when ∆ > 0, the equation has only single real root +x = +� +−g +2 + +√ +∆ +� 1 +3 + +� +−g +2 − +√ +∆ +� 1 +3 . +(A.9) +Appendix B. Real double roots of quartic polynomials and the +structure of FP2 +� FP3 +We consider the regularized form of real quartic equation +x4 + βx2 + γx + δ = 0. +(B.1) +The structure of its roots is described in [19]. It possesses repeated roots if +and only if its discriminant ∆ = 0, where +∆ = 256γ3 − 128β2δ2 + 144βγ2δ − 27γ4 + 16β4δ − 4β3γ2. +(B.2) +In addition, we require another four polynomials, +P =8β, +R =8γ, +∆0 =β2 + 12δ, +D =64δ − 16β2. +(B.3) +The equation (B.1) has one double real roots and two other distinct real +roots, if and only if +∆ = 0 and P < 0 and D < 0 and ∆0 ̸= 0. +(B.4) +The equation (B.1) has one double real roots and a pair of complex roots, +if and only if +(∆ = 0 and D > 0) or (∆ = 0 and P > 0 and (D ̸= 0 or R ̸= 0)). +(B.5) +To apply the above results to the quartic equation (5.23), we may repre- +sent the variables according to +β =2b +5 + 6t, +γ =c +5, +δ = d +15 + 2b +5 t + 3t2. +(B.6) +Finally, we see that +∆(t) = +6 +� +k=0 +c6−ktk, +(B.7) + +58 +QIAO WANG +in which +c0 =27648, +c1 =55296b +5 +, +c2 =9216b2 +5 +, +c3 =4096b3 + 1728c2 +25 +, +c4 =4864b4 +625 ++ 512db2 + 1728bc2 +125 +− 256d2 +25 +, +c5 =32(b2 + 5d)(48b3 − 80db + 135c2) +9375 +, +c6 =256b4d +9375 − 32b3c2 +3125 − 512b2d2 +5625 ++ 96bc2d +625 +− 27c4 +625 + 256d3 +3375 +(B.8) +Here we should explicitly represent the conditions in (B.4) (B.5). Actually, +based on (B.6), we have +P < 0 ⇐⇒ t < − b +15, +R ̸= 0 ⇐⇒ c ̸= 0, +∆0 ̸= 0 ⇐⇒ d > b2 +5 , +D > 0 (= 0, < 0, resp.) ⇐⇒ +� +t + b +15 +�2 +− 1 +90 +� +d − b2 +5 +� +> 0 (= 0, < 0, resp.). +(B.9) +These implies that +(B.4) ⇐⇒ − b +15 − +1 +√ +90 +� +d − b2 +5 < t < − b +15, d − b2 +5 > 0, and ∆(t) = 0. +If we denote +�t = +1 +√ +90 +� +d − b2 +5 − b +15, +then we see that +• If d − b2 +5 < 0, then (B.5) ⇐⇒ ∆(t) = 0. +• If d − b2 +5 = 0, then D > 0 is equivalent to t ̸= − b +15. Thus +(B.5) ⇐⇒ t ̸= − b +15 and ∆(t) = 0. +• If d − b2 +5 > 0, we may classify it into two cases. At first case, if +d ≥ 3b2 +5 , then D > 0 is equivalent to +0 ≤ t < − b +15 − +1 +√ +90 +� +d − b2 +5 , +or t > − b +15 + +1 +√ +90 +� +d − b2 +5 . +(B.10) + +HEAT EVOLUTION +59 +Notice that +D > 0 ∨ (P > 0 ∧ (D ̸= 0 ∨ R ̸= 0)) +=(D > 0 ∨ P > 0) ∧ (D > 0 ∨ D ̸= 0 ∨ R ̸= 0) +=(D > 0 ∨ P > 0) ∧ (D ̸= 0 ∨ R ̸= 0) +(B.11) +which indicates that +(B.5) ⇐⇒ ∆(t) = 0, and t ∈ +� +0, − b +15 − +1 +√ +90 +� +d − b2 +5 +� � � +− b +15, +∞ +� +, +excluding that both D = 0 and c = 0. +But if 3b2 +5 > d > b2 +5 , then D > 0 is equivalent to +t > − b +15 + +1 +√ +90 +� +d − b2 +5 . +(B.12) +then (B.5) ⇐⇒ ∆(t) = 0 and t > − b +15 excluding that both D = 0 +and c = 0. +In a summary, we can obtain at 0 ≤ t1 ≤ t2, respectively corresponds to +a dual real roots x1 and x2 of the equation (B.1), thus +{(x1, t1), (x2, t2)} = FP2 +� +FP3, +in which t2 ≥ t1 and 0 ≤ t1 < − b +15. +References +[1] J. B. Lasserre, Global optimization with polynomials and the problem of moments, +SIAM J. Optim. Vol. 11(3), pp:796-817, 2001. +[2] N.Z. Shor, Quadratic optimization problems, Soviet J. Comput. Systems Sci., +25(1987), pp:1-11. +[3] N.Z. Shor, Nondifferentiable optimization and polynomial problems, Kluwer Aca- +demic Publishers, 1998 +[4] V.N. Nefedov, Polynomial optimization problem, U.S.S.R. Comput. Maths. Math. +Phys., Vol.27, No.3, pp.l3-21, 1987 +[5] J. Zhu and X. Zhang, On global optimizations with polynomials, Optimization Let- +ters, (2008)2: 239-249. +[6] J. Zhu, S. Zhao and G. Liu, Solution to global minimization of polynomials by back- +ward differential flow, J. Optim Theory Appl (2014)161: 828-836. +[7] O. Arikan, R.S. Burachik and C.Y. Kaya, ”Backward differential flow” may not con- +verge to a global minimizer of polynomials, J. Optim Theory Appl (2015): 167: +401-408. +[8] O. Arikan, R.S. Burachik and C.Y. Kaya, Steklov regularization and trajectory meth- +ods for univariate global optimization, J. Global Optimization, 2019. +[9] Burachik, R.S., Kaya, C.Y. Steklov convexification and a trajectory method for +global optimization of multivariate quartic polynomials. Math. Program. (2020). +https://doi.org/10.1007/s10107-020-01536-8 +[10] T. Iijima, basic theory of pattern observation, Papers of Tech. Group on Automata +and Automatic Control, IEICE, Japan, 1959 (in Japanese). +[11] T. Iijima, basic theory on normalization of pattern (in case of typical one-dimensional +pattern), Bulletin of the Electrotechnical Lab., Vol.26: 368-388, 1962. +[12] A. L. Yuille and T. Poggio, Fingerprints theorems for zero crossings, J. Opt. Soc. +Am. A, 2(5): 683-692, 1985. doi = 10.1364/JOSAA.2.000683 + +60 +QIAO WANG +[13] A. L. Yuille and T. A. Poggio, Scaling theorems for zero crossings, IEEE Trans. +on Pattern Analysis and Machine Intelligence, vol.8(1), pp. 15-25, Jan. 1986, doi: +10.1109/TPAMI.1986.4767748. +[14] E. Zeidler, Oxford users’ guide to mathematics, Oxford Univ. Press, 2013. +[15] Irving Kaplansky, an introduction to differential algebra, Hermann, Paris, 1957. +[16] Fritz John, Partial differential equations, Springer-Verlag, 4.ed., 1982. +[17] W.A. Strauss, Partial differential equations. John Wiley and Sons Inc. 1992. +[18] I.N. Stewart, Galois theory, Chapman and Hall/CRC, 4.ed., 2015. +[19] E. L. Rees, Graphical discussion of the roots of a quartic equation, The American +Mathematical Monthly, 29:2, 51-55, 1922. +[20] Birkhoff, G. and Mac Lane, S. A survey of modern algebra, 3rd ed. New York: +Macmillan, 1965. +[21] Nickalls, R.W.D, The quartic equation: invariants and Euler’s solution revealed, The +Mathematical Gazette, vol.93(526), 66-75, 2009. +[22] Louis Nirenberg, A strong maximum principle for parabolic equations, Comm. Pure. +Appl. Math., Vol.6, 167-177, 1953. +School of Information Science and Engineering, Southeast University, Nan- +jing, 210096, China +Email address: qiaowang@seu.edu.cn + diff --git a/ENAyT4oBgHgl3EQfevhN/content/tmp_files/load_file.txt b/ENAyT4oBgHgl3EQfevhN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..626a16423b99cca28f0b638c85508b1ece0b865e --- /dev/null +++ b/ENAyT4oBgHgl3EQfevhN/content/tmp_files/load_file.txt @@ -0,0 +1,1950 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf,len=1949 +page_content='YUILLE-POGGIO’S FLOW AND GLOBAL MINIMIZER OF POLYNOMIALS THROUGH CONVEXIFICATION BY HEAT EVOLUTION QIAO WANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Finding the global minimizer of polynomials is an impor- tant topic in almost all fields in applied mathematics, statistics, and engineering, such as signal processing, machine learning, and data sci- ence, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In this paper, we investigate the possibility of the backward- differential-flow-like algorithm which starts from the minimum of con- vexification version of the polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We apply the heat evolution convexification approach through Gaussian filtering p(x, t) = p(x) ∗ gt(x) with variance t > 0, which is actually an accumulation version of Steklov’s regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' This heat equation plays a multiscale anal- ysis framework in mathematics, image processing and computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We generalize the fingerprint theory which was proposed in the theory of computer vision by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Yuille and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Poggio in 1980s, in particu- lar their fingerprint trajectory equation, to characterize the evolution of minimizers across the scale (time) t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' On the other hand, we pro- pose the ”seesaw” polynomials p(x|s) by replacing the coefficient of x of p(x) with an arbitrary real parameter s, and we find a seesaw differen- tial equation to characterize the evolution of global minimizer x∗(s) of p(x|s) while varying s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Essentially, both the fingerprints FP2 and FP3 of p(x), consisting of the zeros of ∂2p(x,t) ∂x2 and ∂3p(x,t) ∂x3 , respectively, are independent of seesaw coefficient s, upon which we define the Confine- ment Zone and Escape Zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Meanwhile, varying s will monotonically condition the location of global minimizer of p(x|s), and all these loca- tion form the Attainable Zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Based on these concepts, we prove that the global minimizer x∗ of p(x) can be inversely evolved from the global minimizer of its convexification polynomial p(x, t0) if and only if x∗ is included in the Escape Zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In particular, we give detailed analy- sis for quartic and six degree polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For quartic polynomial, we proved that the Attainable Zone is completely contained in the Escape Zone, thus heat evolution approach must converge to global minimizer, and we even find a simpler Euler’s algorithm which must converge to the global minimizer, without heat evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For six and higher degree polynomials, we illustrate that the Attainable Zone might intersect with Confinement Zone, which leads to the failure of immediate backward differential flow like algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In this case, we show that how to attain the global minimizer through our seesaw differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Date: December 31, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 35Q90,46N10,35Q93,90C26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' convex optimization, non-convex optimization, heat equation, maximum principle, multiscale Gaussian filter, computer vision, quartic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='00326v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='OC] 1 Jan 2023 2 QIAO WANG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Background and Motivations Global optimization of real polynomials is an important non-convex op- timization problem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' [1] and references there in), and produces many excellent theories in past decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Among them, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Shor [2] first trans- formed univariate polynomial optimization to convex problem through qua- dratic optimization in 1987, which can offer approximate solution to this global optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' After that, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Shor further studied its relationship with Hilbert’s 17th problem [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Also in 1987, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Nefnov [4] proposed an algorithm by computing the roots of algebraic equation for finding the minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In 2014, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Zhu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Zhao and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Liu [6] proposed a backward differential flow formulation, comes from Kuhn-Tucker equation of constrained opti- mization, to find out the global minimizer of polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' They consider the problem for sufficient smooth function p(x), min p(x) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' x ∈ D := {x ∈ Rn| ∥x∥ < a} (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1) by introducing a set G = {ρ > 0| [∇2p(x) + ρI] > 0, ∀x ∈ D}, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2) and an initial pair (�ρ, �x) ∈ G × D satisfying ∇p(�x) + �ρ�x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3) Then they proved that the back differential flow �x(ρ), defined near �ρ, d�x dρ+[∇2p(�x) + ρI]−1�x = 0, �x(�ρ) = �x (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4) will lead to the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The above work is under the condition that all global minimizers of this polynomial occur only in a known ball, thus the unconstrained optimization problem may be reduced to a constrained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' However, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Arikan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Burachik and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Kaya [7] pointed out in 2015 that the method in [6] may not converge to global minimizer by a counter-example of quartic polynomial p(x) = x4 − 8x3 − 18x2 + 56x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5) Furthermore, they [8] proposed a Steklov regularization and trajectory method to this optimization for univariate polynomials in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Then in 2020, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Burachik and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Kaya [9] generalized it to the multi-variable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In these works, the quartic polynomial optimization plays an interesting role as toy examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In addition, the six degree polynomials may fail to attain the global minimizers, which are illustrated by several examples and counter-examples in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' HEAT EVOLUTION 3 Actually, the Steklov regularization [8] µ(x, t) = 1 2t � x+t x−t f(τ) dτ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) is a low-pass filter, in the viewpoint of signal processing, since we may write µ(x, t) = 1 2t � x+t x−t f(τ) dτ = f(x) ∗ 1[−t,t](x) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='7) where 1[−t,t](x) = � 1 2t, x ∈ [−t, t] 0, x /∈ [−t, t] (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) from which one may obtain µx(x, t), µxt(x, t) and µxx(x, t) (where the sub- script means partial derivative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' However, µt(x, t) is not explicitly in this regime, since we can merely represent a differential equation 2µt + tµtt = tµx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='9) Obviously it brings some inconvenience in analyzing the evolution of local minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Therefore, we require an approach which can balance between the simple differential equation and filters, as well as offer convexification for polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Fortunately, the heat conduct equation ∂p ∂t = 1 2 ∂2p ∂x2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10) with initial condition p(x, 0) = p(x) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='11) is a nice framework to implement the convexification and the similar op- timization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In addition, the analysis for evolution of all critical points becomes more analytically tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' It should be pointed that the initial problem of heat equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10) is equivalent to Gaussian filter, which will be explained in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' But on the other hand, the accumulation of Steklov regularization will lead to Gaussian distribution, since that 1[−t,t] ∗ 1[−t,t] ∗ · · · ∗ 1[−t,t] � �� � n → N(0, 2nt3 3 ), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='12) for n large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Thus replacing Steklov regularity with heat evolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', the Gaussian filtering, is very natural in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Our interest in this paper is to explore the method of optimizing the even degree polynomial min x p(x) = xn + n � j=1 cjxn−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='13) 4 QIAO WANG Different from Kuhn-Tucker’s equation based backward differential flow in [6], we propose in this paper a constructive way, through evolving the poly- nomial by heat conduct (Gaussian filtering) to build a backward-differential- flow-like algorithm, in which we can even explicitly express the differential equation to attain the minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' However, the algorithm converges to the global minimizer for quartic polynomial, and partially success for higher degree polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' This phenomena was actually observed in [8] with ex- amples for Steklov regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In this paper, we explain this convexification derived trajectory algorithm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', a backward-differential-flow-like algorithm, by building the convexifica- tion of heat evolution to polynomials, and in particular, we build the suffi- cient and necessary condition (Theorem 8) under which the algorithm must attain the global minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Our analysis is based on the Yuille-Poggio’s fingerprints theory and their trajectory differential equation in the theory of computer vision [12][13] which were built in 1980s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In addition, to attain the global minimizer when the previous algorithm fails, we build a new tra- jectory differential equation (Theorem 5) which characterizes the minimizer moving from the global minimizer of ”Seesaw”1 polynomial p(x|s) to that of original polynomial p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Before ending this introduction, we slightly sketch the motivation in our contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The elegant framework of multiscale Gaussian filter is equivalent to the model of heat conduct equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Applying this theory, any even degree monic polynomials p(x) will become convex by p(x, t) = gt(x) ∗ p(x) for t large enough2, where gt stands for Gaussian filter with variance t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Moreover, for quartic polynomial p(x), the global minimizer xmin will continuously evolve along t > 0 such that it remains global minimizer xt min of p(x, t) at each scale t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Therefore, reversely and continuously evolving from any global minimizer xt min of p(x, t) to xmin of p(x) is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' A natural question is, whether the global minimizer xmin of a higher de- gree polynomial also evolves continuously to global minimizer of scaled ver- sion p(x, t), like the quartic polynomial case?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Unfortunately this extremely expected property doesn’t hold in general for polynomials whose degree is more than 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We will illustrate it by a counter-example on 6-degree polyno- mial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Furthermore, we give a condition which is both sufficient and necessary for the convergence to global minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The multi-scale Gaussian filter and equivalent heat conduct equation is a standard content in the theory of PDEs, signal processing and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In particular in the field of computer vision, it brought us many powerful 1Here the ”seesaw” polynomial of a polynomial p(x), say p(x) = x6 −2x4 +3x3 +4x2 + 5x + 6, is p(x|s) = x6 − 2x4 + 3x3 + 4x2 + sx + 6, in which 5x + 6 is replaced by sx + 6, where s ∈ R can be conditioned such that sx performs like a seesaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 2I definitely believe that this very simple fact should have been already established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' But I have not gotten any references, limited to my scope of reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' HEAT EVOLUTION 5 Notations Definition Index Zt,k real zeros of ∂kp(x,t) ∂xk (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1) µ(x, t) Steklov regularity of p(x) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='7) p(x, t) heat evolution of p(x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1) FPk(p) k-th fingerprints, (k = 2 is Yuille-Poggio’s fingerprint) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4) Yuille-Poggio’s fingerprint trajectory equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) FlowY P (p) Yuille-Poggio’s flow (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10) Q(p) + S(p) Quadric and higher plus See- saw decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='15)(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='16)(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='17) S(p, s) seesaw term (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='18) seesaw differential equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='25) AZ(p) attainable zone (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='24) Ω(p) and Ωc(p) confinement zone and escape zone Definition 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' List of notations and symbols theoretical tools since 1950s (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' [10][11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Among them, the fingerprint theory proposed in 1980s (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' [12][13]) plays a kernel role for many years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In this paper, we apply the ideas of fingerprint from computer vision, and define three fingerprints of scaled polynomials p(x, t) across scale t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The first fingerprint FP1 characterizes all the local extremals of p(x, t) for each t, and the second one, FP2, characterizes the stationary points of p(x, t) at each t, which indicate the domain of convexity of p(x, t) during the time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Furthermore, FP3 indicates the evolution of curves in FP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' All these powerful fingerprints tools offer us insightful understandings to the evolution of both local and global extremals of the polynomials, from which we proposed a sufficient and necessary condition for attaining the global minimizer by the backward trajectory algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For the sake of simplicity, we list all the symbol and notations in this paper as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Heat Evolution and Convexification of polynomials 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Heat evolution of p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Consider the heat conduct equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10) with initial condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='11), the general solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10) is p(x, t) = p(x) ∗ gt(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1) in which gt(x) stands for the Gaussian filter gt(x) = 1 √ 2πte− x2 2t , t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2) In signal processing and computer vision, this time variable t is also called scale (of Gaussian filtering) or artificial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Notice that any differential 6 QIAO WANG 8 6 4 2 0 2 4 6 8 10 12 2000 0 2000 4000 6000 8000 10000 12000 14000 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The heat evolution of quartic polynomial p(x) = x4−8x3−18x2+56x illustrated in x ∈ [−8, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' See Example 3 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 4 2 0 2 4 6 8 1000 500 0 500 1000 1500 2000 2500 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The partial enlarged view of p(x) = x4 − 8x3 − 18x2 + 56x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' See Example 3 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' operator D is commutative with convolution operator ∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', D(f ∗ g) = f ∗ Dg = Df ∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3) For polynomials p(x, t), the heat equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10) can be enhanced to ∂kp ∂tk = 1 2k ∂2kp ∂x2k , (k = 1, 2, · · · ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4) HEAT EVOLUTION 7 by differentiating both sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='t t, since the smoothness is guar- anteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Then performing Taylor’s expansion for p(x, t) about t will yield p(x, t) = p(x, 0) + t · ∂p ∂t ���� t=0 + t2 2 ∂2p ∂t2 ���� t=0 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5) If using the heat equation derived (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4), we may rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5) as p(x, t) = p(x, 0) + t 2 · ∂2p ∂x2 ���� t=0 + t2 8 ∂4p ∂x4 ���� t=0 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) The convexification of even degree polynomials by heat evolution is char- acterized by following Theorem3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For each even degree monic polynomial p(x), there exists an specified T ∗ = T ∗(p) such that the heat convolution p(x, t) is convex w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='t x at any t > T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We require the following basic results existing in many standard text- books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The Gaussian density gt(x) defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2) satisfies the follow- ing equations: (1) The moment formula � +∞ −∞ xmgt(x) dx = � t m 2 (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', (m even) 0, (m odd) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='7) (2) The convolution formula xm ∗ gt(x) = xm + m(m − 1)txm−2 + · · · + rm(x, t), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) where rm(x, t) = � m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' t m 2 , (m even) (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' t m−1 2 x, (m odd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='7) can be verified immediately, from which we have xm ∗ gt(x) = � (x − y)mgt(y) dy = m � k=0 �m k � xk(−1)m−k � ym−kgt(y) dy = m � k=0 m−k is even �m k � (m − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' t m−k 2 xk = xm + m(m − 1)txm−2 + · · · + rm(x, t), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10) where the last term rm(x, t) is presented at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ Now we prove the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 3Once again, I believe that this convexity result must be known in some literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 8 QIAO WANG Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In what follows, the subscription k in Pk(x) and Qk(x) stands for the degree of polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Let’s consider even degree monic polynomial P2m(x) = x2m + P2m−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='11) Observing the expansion P2m(x) ∗ gt(x) = x2m ∗ gt(x) + P2m−1(x) ∗ gt(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='12) according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='9), we may write P2m(x) ∗ gt(x) = P2m(x) + β(x, t), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='13) in which β(x, t) = (2m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' tm + m−1 � k=1 tm−kQ2k(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='14) Using the heat evolution, we have 1 2 ∂2p(x, t) ∂x2 = ∂p(x, t) ∂t = ∂β ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='15) In our case, ∂β ∂t = m(2m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' tm−1 + m−1 � k=1 (m − k)tm−k−1Q2k(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='16) Clearly, all these leading terms of Q2n(x) are contributed by x2m(x) ∗ gt(x) − x2m, and must be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In more detail, the coefficient of leading term of Q2n(x) = �2m 2n � (2m − 2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' > 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='17) which implies that there exists bounded constants K, such that Q2n(x) > K > −∞, (n = 2, 3, · · · , 2m − 2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='18) So that we have ∂β ∂t > m(2m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' tm−1 + K(tm−2 + tm−3 + · · · + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='19) Therefore, there exists a T ∗ > 0, such that for all t > T ∗, we have ∂β ∂t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Thus the convexity is guaranteed by heat evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Comparison principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The most important mechanism in heat evo- lution is the comparison principle, from which we understand that usually a local minimizer will merge to a local maximizer during the evolution, like the ”annihilation” action between the pair of minimizer and maximizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Theorem 2 (Comparison principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Assume that x∗ be a critical point of p(x, t∗), then for t > t∗, the heat evolution of the critical point satisfies p(x∗(t), t) ≥ p(x∗, t∗), if x∗ is local minimum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='20) p(x∗(t), t) ≤ p(x∗, t∗), if x∗ is local maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='21) HEAT EVOLUTION 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Without loss of generality, we set t∗ = 0, due to that the heat operator U t : f(x) �→ gt(x) ∗ f(x) forms a semi-group (Lie group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Let x = x(t) be one of the integral curves of critical points of p(x, t) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='t x, then from dp(x(t), t) dt =∂p(x, t) ∂x ˙x(t) + ∂p(x, t) ∂t =0 + ∂p(x, t) ∂t =1 2 ∂2p(x, t) ∂x2 , thus we can get the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Notice that the last equality comes from heat conduct equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' If we consider the domain (x, t) ∈ [x∗ − ϵ, x∗ + ϵ] × [0, T) near each critical point x∗, we can show this result by maximum principle for parabolic operator ∂ ∂t − 1 2 ∂2 ∂x2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' [16][17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' This comparison principle reveals that the (local) minimizer and (local) maximizer might merge pair-wisely during the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Ideally, there ex- ists n−1 critical points for a n degree polynomial (here n is even).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Thus we hope the global minimizer will not merge with any local maximizer during the heat evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' However, it might fail in some cases, and we will analyze this mechanism in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Global minimizer and scale space fingerprint 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Fingerprints of scale space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The scale space fingerprint was intro- duced by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Yuille and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Poggio in 1980s (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' [12] [13] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' ), which plays an important role in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' To capture the information of a signal or image p(x), the multi-scale version p(x, t) = p(x) ∗ gt(x), which comes from heat conduct equation, is applied, in which the variance t ≥ 0 of Gaussian filter is also called artificial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Consider that all the polynomials in our situation are of real coefficients, for the sake of simplicity, we need to generalize Yuille-Poggio’s definition of fingerprints of multi-scale zero-crossings to more general case as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Denote the set of real zeros of k-th derivative of polynomial p(x, t) as Zt,k(p) := � xi(t) ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' ∂kp(xi(t), t) ∂xk = 0, i = 1, 2, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1) and denote the sets FP+ k (p) := � (x, t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' ∂kp(x, t) ∂xk > 0, t ≥ 0, � , FP− k (p) := � (x, t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' ∂kp(x, t) ∂xk < 0, t ≥ 0, � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2) 10 QIAO WANG then the k-th order fingerprints of polynomial p(x) are defined as FPk(p) := FP+ k (p) � FP− k (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3) In above notations S represents the topological closure of set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In our case, this topological closure is very simple thus we may characterize FPk by algebraic equations FPk(p) = � (x, t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' ∂kp(x, t) ∂xk = 0 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4) due to the sufficient smoothness of all polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' When k = 2, the fingerprint FP2 of so-called zero-crossings, as well as the equation of zero-crossing contour, are introduced by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Yuille and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Poggio [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Here, we generalize their fingerprints from FP2 to more general FPk (k ≥ 2) in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In other words, if we consider P(x) whose derivative is P ′(x) = p(x), then FP1(p) = FP2(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' That is to say, our framework of FPk is essentially a generalization of Yuille-Poggio’s fingerprints in the theory of computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' According to this notation, FP1 is the fingerprint of extremal values (critical points), and FP2 the zero-crossings (convexity)4, of polynomial p(x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Essentially, as in the theory of computer vision, we can get more information from FP+ 2 and FP− 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In this paper, we generalize the classic concept FP1 and FP2 to general FPk, in particular, FP3 is included such that our main results can be represented on these three fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We further consider the dynamics of the elements in FP1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', the tra- jectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Our main interest is to obtain the curves x = x(t) which obey the equation ∂p(x(t), t) ∂x = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5) as well as initial conditions x(0) = xi ∈ Z0,1(p), (i = 1, 2, · · · ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) where xi (i = 1, 2, · · · ) are the critical points of p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' To solve these curves, an ODE by varying t as follows is introduced by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Yuille and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Poggio in [13], 0 = d dt �∂p(x(t), t) ∂x � = ∂2p(x, t) ∂x2 dx(t) dt + ∂2p(x, t) ∂x∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='7) Therefore, we may characterize the fingerprint which contains all the maxi- mums at different t > 0 by rewriting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='7) as dx(t) dt = − ∂2p(x,t) ∂x∂t ∂2p(x,t) ∂x2 = − ∂3p(x,t) ∂x3 2 · ∂2p(x,t) ∂x2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) 4Although there exists certain gap between the rigorous meaning and the definition here, we omit it in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' HEAT EVOLUTION 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 x 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='06 t (a) FP1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3 x 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='045 t (b) FP2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3 x 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='06 t (c) FP3 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The fingerprints in Example 1, separately illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 12 QIAO WANG Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The joint illustration of fingerprints FP1, FP2 and FP3 of previous figures about Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' as well as suitable initial conditions5 x(0) ∈ Z0,1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='9) In this paper, we call this ODE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) the Yuille-Poggio equation, since it was first proposed in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3) of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Yuille and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Poggio’s seminal work [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' On the other hand, we also call this equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) the trajectory equa- tion, since the reversely evolution algorithm will backward evolute along this curve, provided the initial value is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Given any initial position, one may obtain a trajectory by this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In particular, when the initial condition is located at the critical points of p(x), the trajectories form the fingerprint FP1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We may further generalize FP1(p) to Yuille-Poggio’s flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For any h ∈ R, the integral curve generated by Yuille-Poggio equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) associated with initial value x(0) = h is called a Yuille- Poggio’s curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' All these Yuille-Poggio’s curves consist the set FlowY P (p) := � (x(t), t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' dx(t) dt = − ∂2p(x,t) ∂x2 2∂p3(x,t) ∂x3 , x(0) = h, ∀h ∈ R, � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10) and we call it the Yuille-Poggio’s flow generated by polynomial p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Clearly, the fingerprint curve in the fingerprint FP1(p) is a special Yuille- Poggio’s curve whose initial value x(0) is restricted to Z0,1(p), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', satisfies p′(x(0)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Thus we have 5If p(x) is n-degree polynomial, there exists at most n − 1 distinct initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='0450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='03 七 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='005 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4HEAT EVOLUTION 13 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The fingerprint FP1 can be represented as FP1(p) = � (x, t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' dx(t) dt = − ∂2p(x,t) ∂x2 2 ∂p3(x,t) ∂x3 , x(0) ∈ Z0,1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='11) And FP1(p) ⊂ FlowY P (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='12) Notice that the singularity occurs at which the denominator ∂2p(x, t) ∂x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='13) Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We illustrate the fingerprints of six degree polynomial p(x) = x6 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3726x4 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='0574x3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='0306x2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='0084x in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We point out that the global minimizer (”*” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3(a)) does not evolute to infinity, which means the convex convolution for this p(x) will not converge to its global minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The differential equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) characterizes the trajectory of FP1(p), the evolution of critical points of p(x) in scale space, which inspires an backward differential flow algorithm, which is actually Euler’s algorithm along the trajectory described by Yuille-Poggio’s equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' That is, to solve the global minimizer of p(x), we first build its convex version p(x, t0) for certain t0 > 0 large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' According to Theorem 1 at Section 2, this t0 > 0 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Suppose that x∗(t0) be the global minimizer of convex polynomial p(x, t0), we inversely evolute it to x∗(0) according to the trajectory equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) from t = t0 to t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We expect that x∗(0) be the global minimizer of p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' However, this strategy may fail since in some cases the reversely evolution might result in a local minimizer of p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In this paper, we will analyze the mechanism according to Yuille-Poggio’s flow and derived zones, and further build a new trajectory differential equa- tion to attain the true global minimizer from connected global minimizer of its ”Seesaw” polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Q-S (”Quadric and higher plus Seesaw”) decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' As we point out, that the heat conduct based backward-differential-flow-like algorithm is not guaranteed to converge to theoretically global minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' This is similar to a six degree polynomial counter-example of Steklov’s regu- larization approach in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In this paper, we explain how the convexification method converge to global minimizer, and why it may fail in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Furthermore, to recover from the failed cases, we propose a ”Quadric plus Seesaw” decomposition (Q-S decomposition), then build a new ordinary dif- ferential equation that describes the evolution of global minimizer on account of varying S(x) according to this Q-S decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 14 QIAO WANG Definition 3 (”Quadric and higher plus Seesaw” decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For any polynomial p(x) = xn + n−1 � k=0 ckxk, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='14) we define its Q-S decomposition p(x) = Q(p) + S(p), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='15) in which Q(p) = xn + n−1 � k=2 ckxk (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='16) stands for the ”Quadric and higher terms”, and S(p) = c1x + c0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='17) stands for the ”Seesaw terms”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We further define the generalized Seesaw term S(p, s) = sx + c0, s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='18) Instead of studying p(x) = Q(p) + S(p), we will consider its ”Seesaw” family of polynomials Q(p) + S(p, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We have Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Every Seesaw term S(p, s) is invariant under heat evolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', S(p, s) ∗ gt(x) = S(p, s), ∀s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='19) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Applying Lemma 1 will lead to above result immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ Actually, this Lemma 2 leads to an insight on multi-scale decomposition of p(x) by p(x, t) = p(x)∗gt(x) = Q(p)∗gt(x)+S(p)∗gt(x) = Q(P)∗gt(x)+S(p), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='20) upon which we see that the fingerprints FP2 and FP3 of p(x) is essential of Q(p) but independent of S(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Instead, the fingerprint FP1 of p(x) concerns both Q(p) and S(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' That is Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For any polynomial p(x), all of its Seesaw polynomial p(x|s) = Q(p) + S(p, s) = n � k=2 ckxk + sx + c0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='21) satisfy the following equality, FPk(p(x|s)) = FPk(p(x)), k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='22) Meanwhile, FP1(p(x|s)) ̸= FP1(p(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='23) For these seesaw polynomials, we define HEAT EVOLUTION 15 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For the even degree polynomial p(x), we denote by x∗(s) the global minimizer of seesaw polynomials p(x|s) = Q(p) + S(p, s) for each s, and call it the seesaw minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For given p(x), the set of global minimizers of p(x|s) by varying s ∈ R is called attainable zone given Q(p), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', AZ(p) = {x∗ ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' ∃s ∈ R, x∗ is the global minimizer of p(x|s)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='24) We first focus on those cases that the global minimizer can not be obtained from heat evolution from convexificated version p(x, t) of polynomial p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' If case is this, we investigate the global minimizer of Q-S form Q(p)+S(p, s) where S(p, s) = sx + c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Notice that c0 is always a dumb parameter since it doesn’t affect the location of the global minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' [seesaw differential equation of minimizers moving of seesaw polynomials] The global minimizers x∗(s) (and any critical points) of seesaw polynomials p(x|s) = Q(p) + S(p, s), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', the x∗(s) ∈ AZ(p), must satisfy the seesaw differential equation dx ds = − 1 p′′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='25) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For each s ∈ R, the global minimizer of p(x|s) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' x satisfies 0 = p′(x|s) = � � n � j=2 cjxj � � ′ + s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='26) Then differentiating both sides w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' s will lead to 0 = p′′(x) dx ds + 1 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='27) which produces the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The global minimizer x∗(s) of seesaw polynomial p(x|s) is monotonically decreasing as s increasing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', s ≥ s′ =⇒ x∗(s) ≤ x∗(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='28) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='25) that dx∗(s) ds < 0 since that p′′(x) > 0 when x∗(s) is the global minimizer of p(x∗(s)|s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ These results will help us in some situations, may start from the true global minimizer of a suitable p(x|s) as initial value, then move it from x∗(s) to required location x∗(c1), and finally obtain the true global minimizer of p(x|c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The following Theorem explains the ”Seesaw” properties of p(x|s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For any even degree monic polynomial p(x), let x(s) be the (global or local) minimizers of seesaw polynomials p(x|s), then they satisfy the differential equation dp(x(s)|s) ds = x(s), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='29) 16 QIAO WANG and d2p(x(s)|s) ds2 = − 1 p′′(x) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='30) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' This differential equation can be verified immediately, dp(x(s)|s) ds = ∂p(x(s)|s) ∂x dx(s) ds + x(s) = x(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='31) The reminder is a simple application of previous Theorem 5, and the function p(x(s)|x) is concave with respect to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='29) doesn’t distinct the global and local minimizers for these x(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' That is, if x(s0) is the global minimizer of seesaw polyno- mial p(x|s0), the connected minimizer x(s1) might be the local minimizer of p(x|s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Thus we must identify the interval on which x(s) generated from equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='29) with initial x(s0) is global or local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Confinement zone and escape zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In our following analysis, we will give basic framework of FP2 � FP3, essentially dependent on Q(x), and varying initial condition of trajectory ODE to partition R into Confinement Zone and Escape Zone, as well as varying S(p) to obtain Attainable Zone for given Q(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' It should be stress that in our study, all the fingerprints are about poly- nomials, thus we have some obvious properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For any polynomial p(x) and its heat evolution p(x, t), if (x′, t′) ∈ FPi � FPi+1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='32) then x′ must be a real double root of polynomial equation ∂ip(x,t) ∂xi = 0, and a real root of polynomial equation ∂i+1p(x,t) ∂xi+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For n-th (n is even) order polynomial p(x), the set FP2 � FP3 contains at most n 2 − 1 points (xi, ti), where i = 1, 2, · · · , n 2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Let c ∈ R, if the Yuille-Poggio’s curve from (c, 0) will not intersect with any Yuille-Poggio’s curve from (c′, 0) ̸= (c, 0), we call this c is in Escape Zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Otherwise, we say it is in the Confinement Zone, which is denoted by Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Accordingly, the Escape Zone is denoted by Ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Theorem 7 (Characterization of confinement zone and escape zone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The confinement zone Ω is Ω := n 2 −1 � i=1 [XLL i , XRR i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='33) where XLL i = lim L(�xL i ,�t)→(xi,ti) xLL i , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='34) XRR i = lim L(�xL i ,�t)→(xi,ti) xRR i , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='35) HEAT EVOLUTION 17 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The illustration of Yuille-Poggio’s flow as well as FP2 and FP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 18 QIAO WANG in which the limitation means the point (�xL i , �t) (or (�xR i , �t), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=') moves to the destination (xi, ti) along the local FP2 fingerprint curve fpi 2(L) (or fpi 2(R), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Here (�xL i , �t) (or (�xR i , �t), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=') is the end of Yuille-Poggio curve connected to (xLL i , 0) (or (xRR i , 0), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We first prove that Ω := n 2 −1 � i=1 � [XLL i , XLR i ] � [XRL i , XRR i ] � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='36) in which we add two notations, XLR i = lim L(�xL i ,�t)→(xi,ti) xLR i = K, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='37) XRL i = lim L(�xL i ,�t)→(xi,ti) xRL i = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='38) Here K stands for the intersection point (K, 0) between the curve in FP3(p) and straight line t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Let’s show that the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='36) is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' As illustrated at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5, connecting to each (xi, ti) ∈ FP2 � FP3, there exists a pair of curves in FP2, corresponding to (xi + 0, ti − 0) and (xi − 0, ti − 0), denoted by fpi 2(R) and fpi 2(L) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For any point (�xR i , �t) ∈ fpi 2(R), when (�xR i , �t) ̸= (xi, ti), there are a pair of trajectories satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) which contains the point (�xR i , �t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We may denote their ends at t = 0 as (xRL i , 0) and (xRR i , 0) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Here we assume that xRL i ≤ xRR i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Similarly, for any point (�xL i , �t) ∈ fpi 2(L), when (�xL i , �t) ̸= (xi, ti), there are a pair of trajectories satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) which contains the point (�xL i , �t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We denote their ends at t = 0 as (xLL i , 0) and (xLR i , 0) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Here we assume that xLL i ≤ xLR i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Now we may write that xLL i ≤ xLR i < K < xRL i ≤ xRR i (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='39) due to that the Yuille-Poggio curve can not intersect with FP3(p) otherwise it will bring singularities, according to the denominator of the right hand side of Yuille-Poggio equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Furthermore, the limitation process in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='34) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' remains monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' That is, moving from (�xL i , �t) to (�xL+ i , �t+) and finally to (xi, ti), we may observe that − ∞ < �xLL+ i < �xLL i < �xLR i < �xLR+ i < K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='40) This implies that all the limitation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='34) and so on are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Finally, we note that ∀h ∈ (K − ϵ, K), there must exist a Yuille-Poggio curve starts from (h, 0), and for ∀ϵ > 0, for any point (x′, t′) ∈ fpi 2(L), that satisfy t′ < t, x′ > xi and ∥(x′, t′) − (xi, ti)∥2 < ϵ, there must exist a Yuille-Poggio curve pass the point (x′, t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' That is to say, the Yuille-Poggio curves near HEAT EVOLUTION 19 the FP3 curve connecting (K, 0) and (xi, ti), are dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Here for the sake of simplicity, we omit the topology and differential dynamics description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Now the set Ω in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='33) is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We observe that any Yuille-Poggio curve starting from (h, 0) for h ∈ Ω occurs if and only if there exists another Yuille-Poggio curve, starting from (h′, 0) for some h′ ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In particular, these two curves meet at fpi 2(L) or fpi 2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Thus the current Ω in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='36) is agreed with that in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ Clearly, we further have Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Let p(x) be any even order polynomial with positive leading coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Assume that xt0 min be the global minimizer of convex polynomial (sufficient scaled version) p(x, t) = p(x) ∗ gt(x) of p(x) at t = t0, and the end of the trajectory by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) at t = 0 is x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Then the global minimizer x∗ of p(x) can be inversely involved from the global minimizer of its convexification version p(x, t0), if and only if x∗ is in the Escape Zone Ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' If x∗ ∈ Ω, then the maximum of t coordinate of all the corresponding Yuille-Poggio curves is bounded, thus all these Yuille-Poggio curves can not connect to the point in Rx × Rt with large t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Although the explicit representation of XLL i , XLR i , XRL i , XRR i is expected, it is not available in algebraic form since that when the degree of polynomial is no less than 6, the curve in FP1 will involve algebraic equation at least 5 degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Thus we intend to give some numerical methods to give these values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The methodology of analysis declared here for convexification by heat conduct equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', the Gaussian filtering, still works for the case of Steklov regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Case study of Quartic polynomials In what follows, we will get explicit representation for the fingerprints of quartic polynomials, and explain their geometric properties, such that we can build the algorithm for solving the global minimizer of quartic polyno- mials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The structure of fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For the quartic polynomial p(x), we see that p(x, t) = p(x) + (6x2 + 3ax + b) · t + 3t2 = x4 + ax3 + (b + 6t)x2 + (c + 3at)x + (d + bt + 3t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1) 20 QIAO WANG Continue to differentiate both sides of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='t x, the information of ∂p(x,t) ∂x across time t may be represented as ∂p(x, t) ∂x = ∂p(x) ∂x + (12x + 3a) · t = (4x3 + 3ax2 + 2bx + c) + (12x + 3a) · t = 4x3 + 3ax2 + (2b + 12t)x + (c + 3at) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2) Similarily, we have ∂2p(x, t) ∂x2 = ∂2p(x) ∂x2 + 12t = (12x2 + 6ax + 2b) + 12t = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3) and ∂3p(x, t) ∂x3 = 24x + 6a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4) These form the description of Fingerprints FP1, FP2 and FP3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The structure of fingerprint FP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Based on (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2), the fingerprint FP1 is characterized by following time-varying cubic equation x3 + 3a 4 x2 + b + 6t 2 x + c + 3at 4 = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5) Now, if xi is a real root of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5) at t = 0, then it leads to the trajectory described by the differential equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For more details, we have Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For quartic polynomial p(x), the local extremal values points xt i (i = 1, 2, 3) of p(x, t) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='t x at scale t satisfy the trajectory differential equation dx(t) dt = − 12x + 3a 12x2 + 6ax + 2b + 12t, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) with following (at most three) initial conditions, xi(0) = xi, (i = 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='7) Here xi is the local extremal of p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Inserting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) will lead to required results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ The equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5) possesses (at most) three real roots at t = 0, corre- sponds to (at most) three trajectories, which form the Fingerprint FP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' However, on the viewpoint of differential algebra (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' [15]), actually the solution of differential equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) is real algebraic curve, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', a poly- nomial F(x, t) about x(t) and t which satisfy F(x, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In our case, the polynomial equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5) describes this algebraic curve, thus we may immediately apply the algebraic representation of FP1: FP1 = � (x, t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' t = −4x3 + 3ax2 + 2bx + c 12x + 3a , x ̸= −a 4, and t ≥ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) HEAT EVOLUTION 21 According to Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2, to get the information of the roots of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5), we need its discriminant, ∆(t) = �a3 − 4ab + 8c 64 �2 + �−3a2 + 8b 48 + t �3 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='9) which will be explained in details in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The discriminant ∆(t) of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5) is monotonically in- creasing to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Its unique zero is tu = a2 16 − b 6 − 1 16(a3 − 4ab + 8c) 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5), we write f(t) = b 2 − 3a2 16 + 3t, g(t) = a3 32 − ab 8 + c 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='11) Now the time-varying discriminant ∆(t) =[g(t)]2 4 + [f(t)]3 27 , = �a3 − 4ab + 8c 64 �2 + �−3a2 + 8b 48 + t �3 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='12) which means that ∆(t) increases monotonically w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Immediately, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='12) leads to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ Theorem 9 (The ”1+2” structure of FP1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Let tu, defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10), be the zero of discriminant ∆(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' If tu < 0, then FP1 contains only one trajectory x(t) described by equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6), which evolutes as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' If tu ≥ 0, then during t ∈ [0, tu] the Fingerprint FP1 contains three distinct trajectories described by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6), one of which continues to evolute to +∞ during t > tu, and the other two trajectories will start from t = 0 but merge (stop) when t = tu at the point (x(tu), tu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Here, x(tu) = �a3 − 4ab + 8c 64 �1/3 − a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' According to Lemma 6, we know that if tu < 0, then ∆(t) > 0 for all t ≥ 0, which means the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5) has only one root at each t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' When tu ≥ 0, then ∆(t) < 0 (= 0, > 0, respectively) while t ∈ [0, tu) (t = tu, t > tu, respectively), and the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5) has three distinct real roots (one real and a pair of double real roots, or one real root, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In particular, we consider the critical case t = tu at which ∆(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' If case is this, the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5) at t = tu possesses one real root and a real double 22 QIAO WANG root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' It follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) that the real double root is x(tu) = �g(t) 2 � 1 3 − 1 3 · 3a 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='14) Substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='11) into this formula will produces (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The structure of FP2 and FP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The structure of FP3 is very simple for quartic polynomial, since from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4) we may write FP3 = � (x, t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' x = −a 4, t ≥ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='15) To analyze the structure of FP2, we have a Lemma as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Denote t∗ = a2 16 − b 6, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='16) then the polynomial p(x, t) defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1) is convex about x at each t > max(t∗, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Furthermore, this t∗ can not be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Consider the lower bound of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3) at t = 0, ∂2p(x) ∂x2 = 12x2 + 6ax + 2b = 12 � x + a 4 �2 − 3a2 4 + 2b ≥ −3a2 4 + 2b = −12t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='17) Combining it with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3), we would have ∂2p(x, t) ∂x2 ≥ −3a2 4 + 2b + 12t = 12(t − t∗), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='18) which implies the required results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' On the other hand, at any fixed t′ < t∗, notice that at x = − a 4, we have ∂2p(x, t′) ∂x2 = 12x2 + 6ax + 2b + 12t′ = 12 � x + a 4 �2 − 12(t∗ − t′) = −12(t∗ − t′) < 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='19) which is not convex at this x = − a 4, such that t∗ is the optimal, and can not be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ Theorem 10 (The structure of FP2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For the structure of FP2 of quartic polynomial p(x), (a) when t∗ < 0, the fringerprint FP2 is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' HEAT EVOLUTION 23 (b) when t∗ = 0, the FP2 = � (x, t) = (−a 4, 0) � has only single element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (c) when t∗ > 0, the fingerprint FP2 consists of two curves: the left one is xL(t) = −a 4 − √ t∗ − t, (t∗ ≥ t ≥ 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='20) and the right one is xR(t) = −a 4 + √ t∗ − t, (t∗ ≥ t ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='21) Specifically, these two curves must meets up at t = t∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', at the point (xL(t∗), t∗) = (xR(t∗), t∗) = � −a 4, t∗� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='22) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (a) comes from the fact that for every t ≥ 0, all the ∂2p ∂x2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' That is, FP+ 2 = {(x, t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' x ∈ R, t ∈ [0, +∞)}, but FP− 2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (b) is an immediate result, and (c) is from the quadratic equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The intersection between fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' According to Lemma 3, we may summarize the intersection of fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For the monic quartic polynomials p(x) = x4+ax3+bx2+cx, the two intersection sets FP2 � FP3 = � (−a 4, t∗) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='23) and FP1 � FP2 = {(x(tu), tu)}, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='24) in which t∗ is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='16), tu and x(tu) are defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='13) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Remark 7 (Three phase of time evolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In general settings, the evo- lution of polynomial p(x) can be categorized into three phases according to 0 ≤ tu ≤ t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' At first phase, t evolutes from 0 to tu, and FP1 contains three distinct trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Two of them will merge at t = tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Then at the second phase, tu < t < t∗, the FP1 contains only one trajec- tory, but p(x, t) is not convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Finally, at the third phase, t > t∗, the FP1 contains only one trajectory, and p(x, t) is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Confinement zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Now we compute the confinement zone of the quartic polynomial p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We have Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The confinement zone of quartic polynomial p(x) is � −a 4 − √ 3t∗, −a 4 + √ 3t∗ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='25) where t∗ is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 24 QIAO WANG Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Perform Q-S decomposition for quartic polynomial p(x), p(x, t) = Q(x, t) + S(p), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='26) where S(p) = cx + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Clearly, we have FPi(p) = FPi(Q), i = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='27) Thus we may vary W(p), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', vary c, to form a pair of trajectories such that they can expand the scope as large as possible in R, which forms the confinement zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Re-write (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10) as tu(c) = t∗ − 1 16(a3 − 4ab + 8c) 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='28) We see that we should vary c such that tu(c) = t∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', a3 − 4ab + 8c = 0 =⇒ c = ab 2 − a3 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='29) Substituting this c into the trajectory algebraic curve equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) and setting t = 0, we get the equation 4x3 + 3ax2 + 2bx + ab 2 − a3 8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='30) The three roots of this equation are x1 = −a 4, x2,3 = −a 4 ± √ 3t∗, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='31) which produces two pair of trajectories started from t∗ but reversely evolutes to t = 0, whose four destinations form the confinement zone � −a 4 − √ 3t∗, −a 4 � � � −a 4, −a 4 + √ 3t∗ � = � −a 4 − √ 3t∗, −a 4 + √ 3t∗ � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='32) □ Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' This confinement zone of p(x) is essentially dependent of Q(x, t) but independent of S(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Differential equation of critical points across scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Denote p(x) the quartic polynomial as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Through out this paper, we denote by x1, x2, x3 the roots of cubic equation p′(x) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', x3 + 3a 4 x2 + b 2x + c 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='33) Clearly, the global minimizer of p(x) must be one of x1, x2, x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Comparing this equation to (14), we may represent a, b, c in terms of x1, x2, x3 as � � � � � � � a = − 4 3(x1 + x2 + x3), b =2(x1x2 + x2x3 + x3x1), c = − 4x1x2x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='34) Now we give the representation of t∗ and tu in terms of roots of ∂p(x,t) ∂x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' HEAT EVOLUTION 25 Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Let x1, x2, x3 be the roots of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='33), t∗ is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='16), and tu defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10), then they can be represented as t∗ = �x1 + x2 + x3 3 �2 − x1x2 + x2x3 + x3x1 3 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='35) and tu = t∗ − �32 27(2x1 − x2 − x3)(2x2 − x3 − x1)(2x3 − x1 − x2) � 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='36) This can be verified by substituting with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The singularity of the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) occurs only at xtu = x(tu) = �a3 − 4ab + 8c 64 �1/3 − a 4, t = tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='37) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The singularity occurs at differential equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6), which describes the FP1, so it must satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Meanwhile, the denominator of the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) is actually the fingerprint of FP2, which should satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Thus we may combine these two algebraic equations to solve (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Multiplying both sides of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='18) by x + a 4, and subtracted it from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5) will produce t = (3a2 − 8b)x − (6c − ab) 48x + 12a , x ̸= −a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='38) Substituting it into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='18) will yield a cubic equation about x, 48x3 + 36ax2 + 9a2x + (3ab − 6c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='39) This cubic equation has only one real solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Substituting this x into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='38) will show that t = tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ When − a 4 is not a critical point, this (xtu, tu) occurs only at two FP1 integral curves of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) whose initial point is a local minimum and a local maximum, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Among them, one curve corresponds to the case ˙x(tu) = +∞ and another to ˙x(tu) = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Most importantly, the integral curve starts with globally minimum will not pass this (xtu, tu), which is the main discovery of this paper, and will be explained in details in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' If there exists a critical point x′ at t = 0 such that x′ = − a 4, then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) implies its fingerprint curve x(t) ≡ − a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' This happens when x′ is the local maximizer, and other two critical points x1 and x3 satisfy x1+x3 = 2x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' If case is this, all three fingerprint curves meet up at x′ = − a 4 when t = tu, which will be explained in details in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Consider the polynomial p(x) = x4 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2x3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5x2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='01x, the illustration is in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 26 QIAO WANG 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8 1 x 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 t tu t* (a) FPi, (i = 1, 2, 3), tu and t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Notice that FP1 corresponds to c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8 1 x 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 t (b) FP2, FP3 and trajectories of c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8 1 x 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 t (c) critical trajectories when c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='051, which are symmetric about x = − a 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8 1 x 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 t (d) FP2, FP3 and trajectories of c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8 1 x 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 t (e) trajectories by varying c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (f) more trajectories by varying c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Illustration of Fingerprints and Trajectories in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' To observe the change of trajectories with coef- ficient c in the polynomial, we vary it in c ∈ [−2, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 2 < c < 2 tr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='18 FP2 FP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='jectories0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='04 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='02 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 X1HEAT EVOLUTION 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Heat evolution of critical points of quartic polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Now we investigate the evolution of the critical points, and in particular, the quart polynomial case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Essentially, we concentrate on the case tu > 0 in which there exist three distinct critical points x1 < x2 < x3, and they correspond- ingly evolve to the critical points xt 1, xt 2, xt 3 when 0 < t < tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Generally, both x1 and x3 are local minimizers and x2 is local maximizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Our main con- cern is the behavior associate with heat evolution, characterized by equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Our next concern is the comparison principle between two local minimums during heat evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Surprisingly, we have the very expected result for heat evolution of quartic polynomials: Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For monic quartic polynomial p(x), assume that tu > 0, and denote its three critical points x1 < x2 < x3 (or x1 > x2 > x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' If p(x1) < p(x3), then p(xt 1, t) < p(xt 2, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Before showing this result, we need several lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Let x1, x2, x3 be the critical points of quartic polynomial p(x), then we have p(x3) − p(x1) = −(x3 − x1)3 · (x1 + x3 − 2x2)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='40) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Represent a, b, c in terms of x1, x2, x3, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Thus we obtain p(x1) − p(x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Factorizing it will lead to required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ This Lemma 9 implies that Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For any t ∈ [0, tu), p(xt 3, t) = p(xt 1, t) if and only if xt 1 + xt 3 = 2xt 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Consequently, we see that Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For any t ∈ [0, tu), xt 1 + xt 3 = 2xt 2 if and only if xt 2 = − a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Notice that the coefficient of x3 in p(x, t) is invariant with t, and the coefficient of x2 of ∂p ∂x is also invariant with t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' According to Appendix A, we have xt 1 + xt 2 + xt 3 = −3a 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='41) Applying Lemma 10 will yield the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Assume that x1 < x2 < x3 are three critical points of monic quartic polynomial p(x), if p(x1) = p(x3), then for all t ∈ (−∞, tu), we have p(xt 1, t) = p(xt 3, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='42) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Recall (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5), and apply (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='34), we can actually represent ∂p(x,t) ∂x = 0 in terms of x1, x2, x3 as x3 − (x1 + x2 + x3)x2+(x1x2 + x2x3 + x3x1 + 3t)x −[x1x2x3 + (x1 + x2 + x3)t] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='43) 28 QIAO WANG Now if p(x1) = p(x3), Lemma 10 tells us x3 = 2x2−x1, thus we may simplify the above equation as x3 − 3x2x2 + (2x1x2 + 2x2 2 − x2 1 + 3t)x − (2x1x2 2 − x2 1x2 + 3x2t) = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='44) whose solution is � � � � � � � � � � � xt 2 = x2, xt 1 = x2 − � (x1 − x2)2 − 3t, xt 3 = x2 + � (x1 − x2)2 − 3t, −∞ < t < min �(x1 − x2)2 3 , tu � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='45) which shows that xt 1 + xt 3 = 2xt 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' According to Lemma 10, this leads to p(xt 1, t) = p(xt 3, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ At present stage, we summarize all lemmas as below, Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Under the same assumptions as Theorem 14, and denote xt 1 < xt 2 < xt 3 the critical points of p(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Then the following statements are equivalent: (1) p(x1) = p(x3), (2) p(xt 1, t) = p(xt 3, t), ∀t ∈ [0, tu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3) x1 + x3 = 2x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4) xt 1 + xt 3 = 2xt 2, ∀t ∈ [0, tu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (5) x2 = − a 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (6) xt 2 = − a 4, ∀t ∈ [0, tu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (7) t∗ = tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We will prove that (1) =⇒ (3) =⇒ (5) =⇒ (6) =⇒ (4) =⇒ (2) =⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In addition, (4) ⇐⇒ (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Actually, this routine is partially repeated with previous proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (1) =⇒ (3) (also (4) =⇒ (2)) comes from Lemma 10, (3) =⇒ (5) (also (6) =⇒ (4)) from Lemma 11, (5) =⇒ (6) from differential equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Finally, (4) ⇐⇒ (7) comes from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='36) in Lemma 8 as well as the condition x1 < x2 < x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ Proof of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The dynamical equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) states that the evo- lution of three critical points are continuous when t ∈ [0, tu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Thus if p(x1) < p(x3), we must have p(xt 1, t) < p(xt 3, t) for t ∈ [0, tu), otherwise, there must have p(xt′ 1 , t′) = p(xt′ 3 , t′) for some t′ ∈ (0, tu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' But, if case is this, Theorem 15 or Lemma 12 tells us that p(x1) = p(x3) since t is reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' This leads to conflict with assump- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ To intuitively explain this result, we suggest a triangle representation at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='20 for each t, where the cortes of triangle consists of (xt i, p(xt i, t)), (i = 1, 2, 3) when t < tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Notice that the sequence of triangles when 0 ≤ t ≤ tu HEAT EVOLUTION 29 and continued curve ˙x(t) actually connected to global minimum of p(x, t) at each t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Finally, we discuss an interesting problem: if x1 < x2 < x3 are three critical points of quartic polynomial p(x), can we judge which one of them is global minimizer without valuating all these p(xi)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The answer is YES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Let x1 < x2 < x3 be three distinct critical points of monic quartic polynomial p(x), then the following statements are equivalent: (1) x3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' x1) is global minimizer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (2) x1 + x3 > 2x2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' x1 + x3 < 2x2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (3) x2 < −a/4 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' x2 > −a/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Apply Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ This Theorem inspired the following very simple Euler’s algorithm with- out Heat Convolution for quartic polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For any monic quartic polynomial p(x) with a the coefficient of x3, the Euler’s algorithm with FIXED initial position x(0) = − a 4, x(k+1) = x(k) − ∆x · p′(x(k)), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='46) MUST converge to the global minimizer of p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Actually, the Euler’s algorithm may work from t > tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Fortunately, we know at t ≥ tu, the p(x, t) has only single minimum about x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Recall the formula (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2), in which we see that the sum of all real roots of fingerprint cubic equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5) should be invariant under 0 ≤ t ≥ tu, thus we know the remaining critical point xinit at t = tu can be solved since we already know the information of (x(tu), tu) from Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' This means we may adopt xinit = − 3a 4 − 2x(tu) = − a 4 − 2 �a3 − 4ab + 8c 64 �1/3 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='47) at t = tu as initial point, then perform Euler’s algorithm for equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6), and finally attain the global minimum of p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' This implies the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For the quartic polynomial (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='13), if tu ≤ 0, this polynomial has only one critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' If tu > 0, the polynomial contains three distinct critical points, then if all the critical points satisfy x ̸= − a 4, we backward perform the differential equation dx(t) dt = − 12x + 3a 12x2 + 6ax + 2b + 12t, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='48) with initial condition xtu = x(tu) = −a 4 − (a3 − 4ab + 8c)1/3 2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='49) 30 QIAO WANG from t = tu > 0 to t = 0, must attain the global minimizer of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='13) at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Finally, if one critical point equals − a 4, then p(x) has two global minimizers, and they are the roots of quadratic polynomial p(x) x + a 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='50) So far, we may start with verifying that whether − a 4 is a root of cubic polynomial ∂p(x) ∂x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The global minimizer can be obtained immediately if − a 4 is a root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Otherwise, set x(0) = xinit, and t(0) = tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Then motivated by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6), the iteration process is as below x(i+1) =x(i) − ∆t · 12x(i) + 3a 12x(i)2 + 6ax(i) + 2b + 12t(i) , t(i+1) =t(i) − ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='51) Here the prescribed step ∆t > 0 is small enough, and we may stop the iteration while t(n) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Finally, this algorithm provides lim i→∞ x(i) = xmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='52) Instead beginning with xinit, we can also start by sufficient evolution p(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' This will cost more steps of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' This counter-example [7] is proposed against ’backward differ- ential flow’ method of [6] , in which p(x) = x4−8x3−18x2+56x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In our heat conduct framework, we have p(x, t) = x4−8x3−(18−6t)x2+32x−(18t−3t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Notice that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 20 demonstrates the triangle series of critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Set p(x) = x4 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2114x3 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6841x2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1110x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2406, then in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 7 the most left curve x1(t) is the global minimizer of corresponding p(x, t) at each t ≥ 0, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8 illustrates the fingerprint FP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The the- oretical minimizer is x1 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2307, and our iteration algorithm provides x1 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Consider p(x) = x4 − 4x3 − 2x2 + 12x, then we actually have three critical points x1 = −1, x2 = 1, x3 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Notice that x1+x3 = 2x2 thus p(x1, t) = p(x3, t) and x2(t) = x2 = 1 for all x ∈ [0, tu].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' One can further verify that in this symmetric case, we must have t∗ = tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The detailed explain can be referred as in Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Summary of quartic polynomial case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For the global minimizer of quartic polynomial p(x), while generating its multi-scale version p(x, t) = p(x) ∗ gt(x) on account of Gaussian filter gt(x) with variance from t = 0 to +∞, we will see that: HEAT EVOLUTION 31 Require: a, b, c, d of p(x) = x4 + ax3 + bx2 + cx + d, and ∆t, pre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Ensure: global minimizer xmin 1: function Iteration(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' d) 2: [tu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' xinit] ← Initialize(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' c) 3: 4: if tu < 0 or −a/4 is critical point then computing x 5: 6: else 7: t ← tu 8: x ← xint 9: while t > pre do 10: r ← (12x + 3a)/(12x2 + 6ax + 2b + 12t) 11: x ← x − ∆t · r 12: t ← t − ∆t 13: end while 14: 15: end if 16: return x 17: end function 18: function Initialize(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' c) 19: h ← a3 − 4ab + 8c 20: t∗ ← a2/16 − b/6 21: tu ← t∗ − h 2 3 /16 22: xinit ← −a/4 − h1/3/2 23: return tu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' xinit 24: end function If tu < 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' p(x) itself is not necessary convex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' but it has unique critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Consequently, each p(x, t) has only one critical point at any t ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' If further tu ≤ t∗ < 0, p(x) must be convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Then each p(x, t) is convex about x at any t ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' If tu > 0, the polynomial p(x) has three distinct critical points x1 < x2 < x3 when 0 ≤ t < tu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' When t = tu, the critical point corresponding to the global minimizer will evolve continuously from tu to t∗, and the local minimizer will meet up with local maximum xt 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Even more, these two critical points will stop evolution at t = tu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' When tu < t < t∗, the polynomial p(x, t) has unique minimizer at each t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' When t ≥ t∗, the polynomial p(x, t) will become convex about x, and possesses unique minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 32 QIAO WANG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 x 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 2 y x1(0) x2(0) x3(0) x2(tu)=x3(tu) x1(tu) x3(t) x2(t) x1(t) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' An example of triangle series in (x, y) system, of p(x) = x4 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2114x3 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6841x2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1110x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' See Example 4 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 1 x 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 3 t Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Fingerprint FP1 in (x, t) system, of p(x) = x4 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2114x3 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6841x2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1110x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' See Example 4 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Case study of sixth degree polynomials 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Evolution and fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Now we consider 6 degree monic poly- nomial p(x) = x6 + bx4 + cx3 + dx2 + ex + f, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1) For the sake of simplicity, here we already regularize the coefficient of x5 by setting it as zero, which is a standard technique in treating the algebraic HEAT EVOLUTION 33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 3 x 10 8 6 4 2 0 2 4 6 8 y x1(0) x3(0) x1(tu)=x2(tu)=x3(tu) x2(0) x(t), t> tu Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Triangle series of p(x) = x4 − 4x3 − 2x2 + 12x, in which there exist two global minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Notice that at x1(tu) = x2(tu) = x3(tu), the fingerprint of x2(t) is a line segmentation, which is partial repeated by global minimizer curve x(t) after t ≥ tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' See Example 5 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' This implies that the heat evolution is p(x, t) =p(x) + t · ∂p ∂t + t2 2 ∂2p ∂t2 + t3 6 ∂3p ∂t3 =p(x) + t 2 · ∂2p ∂x2 + t2 8 ∂4p ∂x4 + t3 48 ∂6p ∂x6 =x6 + b(t)x4 + c(t)x3 + d(t)x2 + e(t)x + f(t), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2) in which � � � � � � � � � � � � � � � b(t) = b + 15t, c(t) = c, d(t) = d + 6bt + 45t2, e(t) = e + 3ct, f(t) = f + dt + 3bt2 + 15t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3) The critical points of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1) satisfy the 5 degree equation 0 = 1 6 ∂p(x, t) ∂x = x5 + B(t)x3 + C(t)x2 + D(t)x + E(t), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4) in which � � � � � � � � � � � � � � � � � � � � � B(t) = 2b 3 + 10t, C(t) = c 2, D(t) = d 3 + 2bt + 15t2, E(t) = e 6 + ct 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5) 34 QIAO WANG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 10 20 10-3 t = 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 20 0 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 10 20 10-3t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 20 0 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 10 20 10-3t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 1 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 20 0 20 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The evolution of six degree polynomial p(x, t) and its derivatives ∂p(x,t) ∂x , ∂2p(x,t) ∂x2 , ∂3p(x,t) ∂x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4) for t ≥ 0 consists the fingerprint FP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Unfortunately, the roots of this fifth degree equation is algebraically intractable [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Similarly, we may write the equation of FP2 as below, 1 30 ∂2p ∂x2 = x4 + �2b 5 + 6t � x2 + c 5x + d 15 + 2b 5 t + 3t2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) Then the equation of FP3 is 1 120 ∂3p ∂x3 = x3 + �b 5 + 3t � x + c 20 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='7) Our interest is the set FP2 � FP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Geometrically, there exist two pair of real double roots of quartic equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='21), based on following Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' If x0 be a root of both polynomial p(x) and its derivative p′(x), then it must be at least a double root of p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Different from quartic polynomials case, we have not an explicit represen- tation for FP2 � FP3, and numerical approach is required here for exper- iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Clearly, the real double root of quartic equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) must be the common roots of both (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' In general, there exist two 0 ≤ t1 < t2 HEAT EVOLUTION 35 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='04 t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 2 (t) 10-6 (t) 0 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The discriminant ∆(t) of quartic equation is generated from ∂2p(x,t) ∂x2 = 0 and defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Here the data comes from Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The two real roots of ∆(t) are t1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='002341, at which the quartic equation possesses a real double root and two distinct real roots, and t2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='034887, at which the quartic equation possesses a real double root and a pair of conjugate complex roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' such that the corresponding x1 and x2 are those two real double roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The following Theorem 19 explains the process of numerical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Theorem 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' For any six degree monic polynomial p(x), the set FP2 � FP3 contains a pair of elements (xi, ti), i = 1, 2, or one element (x1, t1), or empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Specifically, (1) Any ti must be the zero of discriminant ∆(t) = 27648t′6 + 1728c2 25 t′3 − 256 625h2t′2 − 288 625c2ht′ − 256h3 753 − 27c4 625 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) where t′ = t + b 15, h = b2 − 5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='9) 36 QIAO WANG 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='04 t 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8 1 x(t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='23516 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='078914 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The function x(t) is defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10), where the data comes from the Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Here we obtain two solution (t1, x1) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='0023, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='23516), (t2, x2) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='03489, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='078914), which consist the set FP2 � FP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (2) Any xi is dependent of ti by the function x = −c · 1800 � t + b 15 �2 − 4(b2 − 5d) 36000 � t + b 15 �3 + 80(b2 − 5d) � t + b 15 � + 45c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We will give a detailed analysis in Apendix B, based on which, we know that in general settings there exist at most two merge time t1 and t2 from the discriminant equation ∆(t) = 0 of quartic equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8), This function (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) can be verified by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='7) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) immediately, but we omit the detailed computation here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' We may find out its two real zeros t1 and t2 through numerical computation, then the real double roots x1 and x2 of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) at t1 and t2 respectively, could be obtained according to following (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Thus in general settings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', when p′(x) has five distinct real roots, we will have FP2 � FP3 = {(x1, t1), (x2, t2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='11) In degenerated cases, this intersection set might possess one point or even null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' When it is null, the polynomial p(x) is globally convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' HEAT EVOLUTION 37 Here we may take Euclidean algorithm to reduce the degree of the poly- nomials about x for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' At first, multiplying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='7) with x, and subtracted both sides of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', we get a second degree polynomial � t + b 15 � x2 + c 20x + � t + b 15 �2 − b2 − 5d 225 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='12) Again, multiplying with x for both sides of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='12), and subtracted from both sides of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='7) multiplied with t + b 15, then we obtain − c 20x2 + � 2 � t + b 15 �2 + b2 − 5d 225 � x + c 20 � t + b 15 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='13) Finally, eliminating the second degree term by combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='13) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='12) will lead to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' □ As explained in Appendix B, we can obtain the suitable t from the dis- criminant ∆(t) = 0 at first, then substitute it into the above (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10), then choose x from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The boundary of confinement zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' To characterize the boundary of confinement zone, we must study the singular trajectory dx dt = − ∂3p ∂x3 2 ∂2p ∂x2 , Initial Condition : (t′, x(t′)) ∈ FP2 � FP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='14) Unfortunately, this equation is singular due to its initial data, which re- sults in dx dt = 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='15) However, at critical time t′, the multiplicity of common root x′ of both FP2 and FP3 is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Clearly, x′ is double root of FP2 and single root of FP3, thus the reciprocal equation of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='14) dt dx = −2 ∂2p ∂x2 ∂3p ∂x3 , Initial Condition : (t(x′), x′) ∈ FP2 � FP3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='16) contains only removable singularity near FP2 � FP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='13 demonstrates the limitation curve satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) and inversely evolutes as t → +0 starting from the top points near FP2 � FP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 38 QIAO WANG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4 x 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='05 t X1 LL X1 LR X1 RL X1 RR X2 LL X2 RR X2 LR,X2 RL Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Numerically, the partition of Confinement Zone and Escape Zone associated with p(x) defined in Ex- ample 1 is obtained through Matlab ODE packet of @ode25, and the Confinement Zone is [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='5082, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='0858] �[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1603, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3267].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Why convecification approach can not guarantee attaining the global minimizer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Recall the regularized polynomial (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1), which contains the parameters {b, c, d, e, f}, and the FP1 equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4) contains {b, c, d, e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' However, the FP2 equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='6) and the FP3 equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='7), contains only {b, c, d} and {b, c} respectively, which is independent of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Thus FP2 � FP3 doesn’t contain the information of e, which actually affect the location of global minimizer of polynomial p(x) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' On the other hand, to determine the scope of the Confinement Zone, we require the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='8) which depends only on {b, c, d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Thus we must investigate the affection of e in the (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1) to the global minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Thus we may consider that for fixed b, c, d and vary e, the variation of global minimizer of corresponding p(x), and when it is included in the Escape 6The parameter f in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1) doesn’t affect the location of global minimizer of six degree polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' HEAT EVOLUTION 39 Zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' To this end, we may define the mapping R(e|b, c, d) = � 1, x∗ ∈ Escape Zone, −1, x∗ ∈ Confinement Zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='17) where x∗ represents the global of polynomial p(x) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' The detailed analysis reveals that when e ∈ (−∞, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='676739] � [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='617543, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='58523] � [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='67115, +∞), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='18) the corresponding global minimizer x∗ falls into the scape of Escape Zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Fig17 illustrates the curve of x∗(e|b, c, d), the global minimizer of polynomial p(x) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1) in which b, c, d remains invariant, while varying the parameter e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Those x∗ fails to be obtained by convexification approach is demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Comparison principle and criterion function for evolution poly- nomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Now we describe the comparison criterion for p(xi, t) > p(xj, t), where xi and xj are critical points of p(x, t) at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Apparently, it follows immediately from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='1) that p(xi, t) − p(xj, t) = (xi − xj) · Q5(xi, xj, t) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='19) in which Q5 is a fifth degree polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' However, it is too complicated for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Instead, we can give a more concise representation for factoring the p(xi, t) − p(xj, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Theorem 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Let ξ = ξ(t) and η = η(t) be critical points of p(x, t), we have p(ξ, t) − p(η, t) = −(ξ − η)3 10 K(ξ, η, t), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='20) where the criterion function K(ξ, η, t) = 20(ξ3 + η3) + 30(ξ2η + ξη2) +15a(ξ2 + η2) + 20aξη + (10b + 150t)(ξ + η) + (5c + 50at) =K(ξ, η, 0) + 150t(ξ + η) + 50at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='21) If both ξ and η (where we suppose that ξ ̸= η) are real critical points of p(x), then for sixth degree monic polynomial p(x, t), p(ξ, t) > p(η, t) ⇐⇒ (ξ − η)K(ξ, η, t) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='22) Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' If we transform the critical points by translation xi → xi − a 5, we may set a = 0 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='21), then the criterion function (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='21) can be reduced to K(ξ, η, t) = 20(ξ3 + η3) + 30(ξ2η + ξη2) + (10b + 150t)(ξ + η) + 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='23) 40 QIAO WANG Proof of Theorem 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Let xi (i = 1, 2, 3, 4, 5) be the critical points of monic sixth degree polynomial p(x, t) at t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=', the roots of equation 1 6 ∂p(x,t) ∂x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfevhN/content/2301.00326v1.pdf'} +page_content=' Thus we may write � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � A(t) = − � i xi, B(t) = � i n +, x ∈ [0, 1]. +(1.1) +In [23], using a probabilistic method, Stancu generalized Bernstein’s funda- +mental functions as +wn,k,r(x) := + + + +(1 − x) pn−r,k (x) ; +0 ≤ k < r +(1 − x) pn−r,k (x) + xpn−r,k−r (x) ; +r ≤ k ≤ n − r +xpn−r,k−r (x) ; +n − r < k ≤ n +, x ∈ [0, 1], +(1.2) +where r is a non-negative integer parameter, n is any natural number such +that n > 2r, for which each pn−r,k is given by (1.1), and therefore, con- +structed and studied Bernstein-type positive linear operators as +Ln,r (f; x) := +n +� +k=0 +wn,k,r(x)f +�k +n +� +, +x ∈ [0, 1], +(1.3) +for f ∈ C[0, 1]. In doing so Stancu was guided by an article of Brass [8]. +This is further discussed by Gonska [11]. Among others, estimates in terms +of the second order modulus of smoothness are given there for continuous +functions. +It is clear that for x ∈ [0, 1] Stancu’s fundamental functions in (1.2) satisfy +wn,k,r(x) ≥ 0 and +n +� +k=0 +wn,k,r(x) = 1, +Key words and phrases. Multivariate Kantorovich operator; Multivariate averaged +modulus of smoothness; Multivariate K-functional +2010 MSC: 41A36, 41A25, 26A45 +∗This paper is an extension of a talk given in ICATA 2022. +1 + +2 +G¨ULEN BAS¸CANBAZ-TUNCA AND HEINER GONSKA +hence the operators Ln,r can be expressed as +Ln,r (f; x) := +n−r +� +k=0 +pn−r,k (x) +� +(1 − x) f +�k +n +� ++ xf +�k + r +n +�� +, +(1.4) +are defined for n ≥ r and satisfy the end point interpolation Ln,r (f; 0) = +f (0) , Ln,r (f; 1) = f (1). +It thus seems to be justified to call the Ln,r +Brass-Stancu-Bernstein (BSB) operators. +In [24] Stancu gave uniform convergence limn→∞ Ln,r (f) = f on [0, 1] for +f ∈ C[0, 1] and presented an expression for the remainder Rn,r(f; x) of the +approximation formula f(x) = Ln,r(f; x) + Rn,r(f; x) by means of second +order divided differences and also obtained an integral representation for +the remainder. Moreover, the author estimated the order of approximation +by the operators Ln,r (f) via the classical modulus of continuity. He also +studied the spectral properties of Ln,r. +In the cases r = 0 and r = 1, the operators Ln,r reduce to the classical +Bernstein operators Bn, i.e., +Bn (f; x) = +n +� +k=0 +pn,k(x)f +�k +n +� +. +What also has to be mentioned: Stancu himself in his 1983 paper observed +that ”we can optimize the error bound of the approximation of the function +f by means of Ln,rf if we take r = 0 or r = 1, when the operator Ln,r +reduces to Bernstein’s.” So there is a shortcoming. +Since Bernstein polynomials are not appropriate for approximation of +discontinuous functions (see [14, Section 1.9]), by replacing the point evalu- +ations f +� k +n +� +with the integral means over small intervals around the knots +k +n, Kantorovich [12] generalized the Bernstein operators as +Kn (f; x) = +n +� +k=0 +pn,k (x) (n + 1) +k+1 +n+1 +� +k +n+1 +f (t) dt, +x ∈ [0, 1], n ∈ N, +(1.5) +for Lebesgue integrable functions f on [0, 1]. +On p. 239 of his mathematical memoirs [13] Kantorovich writes: ”While +I was waiting for a student who was late, I was looking over vol. XIII of +Fundamenta Math. and saw in it a note from the Moscow Mathematician +Khlodovskii related to Bernstein polynomials. In it I first caught sight of +Bernstein polynomials, which he proposed in 1912 for an elementary proof +of the well known Weierstrass theorem ... I at once wondered if it is not +possible in these polynomials to change the values of the function at certain +points into the more stable average of the function in the corresponding +interval. It turned out that this was possible, and the polynomials could be +written in such a form not only for a continuous function but also for any +Lebesgue-summable function.” +Lorentz [14] proved that lim +n→∞ ∥Kn(f) − f∥p = 0, f ∈ Lp[0, 1], 1 ≤ p < ∞. +There are a lot of articles dealing with classical Kantorovich operators, +and, in particular, their degree of approximation and the importance of + +BRASS-STANCU-KANTOROVICH OPERATORS ON A HYPERCUBE +3 +second order moduli of different types. See, e.g., the work of Berens and +DeVore [5], [6], Swetits and Wood [25] and Gonska and Zhou [10]. +It is +beyond the scope of this note to further discuss this matter. +As further +work on the classical case here we only mention the 1976 work of M¨uller +[16], Maier [15], and Altomare et al. [1], see also the references therein. +Similarly to Kantorovich operators Bodur et al. [7] constructed a Kan- +torovich type modification of BSB operators as +Kn,r (f; x) := +n +� +k=0 +wn,k,r(x) + + + +(n + 1) +k+1 +n+1 +� +k +n+1 +f (t) dt + + + + , +x ∈ [0, 1], +(1.6) +for f ∈ L1 [0, 1], where r is a non-negative integer parameter, n is a natural +number such that n > 2r and wn,k,r(x) are given by (1.2). And, it was +shown that If f ∈ Lp[0, 1], 1 ≤ p < ∞, then +lim +n→∞ ∥Kn,r(f) − f∥p = 0. +In addition, it was obtained that each Kn,r is variation detracting as well +[7]. Throughout the paper, we shall call the operators Kn,r given by (1.6) +”Brass-Stancu-Kantorovich”, BSK operators. +Notice that from the definition of wn,k,r, Kn,r (f; x) can be expressed as +Kn,r (f; x) +(1.7) += +n−r +� +k=0 +pn−r,k (x) (n + 1) + +(1 − x) +k+1 +n+1 +� +k +n+1 +f (t) dt + x +k+r+1 +n+1 +� +k+r +n+1 +f (t) dt + + +and in the cases r = 0 and r = 1 they reduce to the Kantorovich operators; +Kn,0 = Kn,1 = Kn given by (1.5). Again they are defined for all n ≥ r. +MULTIVARIATE SITUATION +Some work has been done in the multivariate setting for BSB and BSK +operators. For the standard simplex this was done, e.g., by Yang, Xiong +and Cao [27] and Cao [9], For example, Cao proved that multivariate Stancu +operators preserve the properties of multivariate moduli of continuity and +obtained the rate of convergence with the help of Ditzian-Totik’s modulus +of continuity. +In this work, motivated by the work Altomare et al. [3], we deal with +a multivariate extension of the BSK operators on a d-dimensional unit hy- +percube and we study Lp -approximation by these operators. For the rate +of convergence we provide an estimate in terms of the so called first order +multivariate τ-modulus, a quantity coming from the Bulgarian school of +Approximation Theory. Also, inspired by M¨uller’s approach in [17], we give +estimates for differentiable functions and such in terms of the Lp-modulus +of smoothness, using properties of the τ-modulus. Here the work of Quak +[20], [21] was helpful. + +4 +G¨ULEN BAS¸CANBAZ-TUNCA AND HEINER GONSKA +2. Preliminaries +Consider the space Rd, d ∈ N. +Let ∥x∥∞ denote the max-norm of a +point x = (x1, . . . , xd) ∈ Rd; +∥x∥∞ := ∥x∥max = +max +i∈{1,...,d} |xi| +and let 1 denote the constant function 1 : Rd → R such that 1 (x) = 1 for +x ∈ Rd. And, for each j = 1, . . . , d, let +prj : Rd → R +stand for the jth coordinate function defined for x ∈ Rd by +prj (x) = xj. +Definition 2.1. A multi-index is a d-tuple α = (α1, . . . , αd) of non-negative +integers. Its norm (length) is the quantity +|α| = +d +� +i=1 +αi. +The differential operator Dα is defined by +Dαf = Dα1 +1 · · · Dαd +d f, +where Di, i = 1, . . . , d, is the corresponding partial derivative operator (see +[4, p. 335]). +Throughout the paper Qd := [0, 1]d, d ∈ N, will denote the d-dimensional +unit hypercube and we consider the space +Lp (Qd) = {f : Qd → R | f p-integrable on Qd} , 1 ≤ p < ∞, +with the standard norm ∥.∥p. Recall the following definition of the usual +Lp-modulus of smoothness of first order: +Definition 2.2. Let f ∈ Lp (Qd) , 1 ≤ p < ∞, h ∈ Rd and δ > 0. The +modulus of smoothness of the first order for the function f and step δ in +Lp-norm is given by +ω1 (f; δ)p = +sup +0<∥h∥∞≤δ + + + +� +Qd +|f (x + h) − f (x)|p dx + + + +1/p +if x, x + h ∈ Qd [21]. +Let M (Qd) := {f | f bounded and measurable on Qd}. Below, we present +the concept of the first order averaged modulus of smoothness. +Definition 2.3. Let f ∈ M (Qd) , h ∈ Rd and δ > 0. The multivariate +averaged modulus of smoothness, or τ-modulus, of the first order for function +f and step δ in Lp-norm is given by +τ 1 (f, δ)p := ∥ω1 (f, .; δ)∥p , 1 ≤ p < ∞, +where +ω1 (f, x; δ) = +sup +� +|f (t + h) − f (t)| : t, t + h ∈ Qd, ∥t − x∥∞ ≤ δ +2, ∥t + h − x∥∞ ≤ δ +2 +� + +BRASS-STANCU-KANTOROVICH OPERATORS ON A HYPERCUBE +5 +is the multivariate local modulus of smoothness of first order for the function +f at the point x ∈ Qd and for step δ. [21]. +For our future purposes, we need the following properties of first order +multivariate averaged modulus of smoothness: +For f ∈ M (Qd) , 1 ≤ p < ∞ and δ, λ, γ ∈ R+, there hold +τ 1) τ 1 (f, δ)p ≤ τ 1 (f, λ)p for 0 < δ ≤ λ, +τ 2) τ 1 (f, λδ)p ≤ (2 ⌊λ⌋ + 2)d+1 τ 1 (f, δ)p, where ⌊λ⌋ is the greatest inte- +ger that does not exceed λ, +τ 3) τ 1 (f, δ)p ≤ 2 � +|α|≥1 +δ|α| ∥Dαf∥p , αi = 0 or 1, if Dαf ∈ Lp (Qd) for +all multi-indices α with |α| ≥ 1 and αi = 0 or 1 (see [19] or [21]). +For a detailed knowledge concerning averaged modulus of smoothness, we +refer to the book of Sendov and Popov [22]. +Now, consider the Sobolev space W p +1 (Qd) of functions f ∈ Lp (Qd) , 1 ≤ +p < ∞, with (distributional) derivatives Dαf belong to Lp (Qd), where +|α| ≤ 1, with the seminorm +|f|W p +1 = +� +|α|=1 +∥Dαf∥p +(see [4, p. 336]). Recall that for all f ∈ Lp (Qd) the K-functional, in Lp- +norm, is defined as +K1,p (f; t) := inf +� +∥f − g∥p + t |g|W p +1 : g ∈ W p +1 (Qd) +� +(t > 0) . +(2.1) +K1,p (f; t) is equivalent with the usual first order modulus of smoothness of +f, ω1 (f; t)p; namely, there are positive constants c1 and c2 such that +c1K1,p (f; t) ≤ ω1 (f; t)p ≤ c2K1,p (f; t) +(t > 0) +(2.2) +holds for all f ∈ Lp (Qd) (see [4, Formula 4.42 in p. 341]). +The following result due to Quak [21] is an upper estimate for the Lp-norm +of the approximation error by the multivariate positive linear operators in +terms of the first order averaged modulus of smoothness. Note that this +idea was used first by Popov for the univariate case in [18]. +Theorem 2.1. Let L : M (Qd) → M (Qd) be a positive linear operator that +preserves the constants. Then for every f ∈ M (Qd) and 1 ≤ p < ∞, the +following estimate holds: +∥L(f) − f∥p ≤ Cτ1 +� +f, +2d√ +A +� +p , +where C is a positive constant and +A := sup +� +L +� +(pri ◦ ψx)2 ; x +� +: i = 1, . . . , d, x ∈ Qd +� +, +in which ψx (y) := y − x for fixed x ∈ Qd and for every y ∈ Qd and +A ≤ 1 [21]. + +6 +G¨ULEN BAS¸CANBAZ-TUNCA AND HEINER GONSKA +3. Multivariate BSK-Operators +In this section, motivated by the works of Altomare et al. [1] and Al- +tomare et al. [3], we consider the multivariate extension of BSK-operators +on Lp (Qd) and study approximation properties of these operators in Lp- +norm. We investigate the rate of the convergence in terms of the first order +τ-modulus and the usual Lp-modulus of smoothness of the first order. +Let r be a given non-negative integer. +For any n ∈ N such that n > +2r, k = (k1, . . . , kd) ∈ {0, . . . , n}d and x = (x1, . . . , xd) ∈ Qd, we set +wn,k,r(x) := +d +� +i=1 +wn,ki,r(xi), +(3.1) +where, wn,ki,r(xi) is Stancu’s fundamental function given by (1.2), written +for each i = 1, . . . , d, 0 ≤ ki ≤ n and xi ∈ [0, 1]. Thus, for x ∈ Qd, we have +wn,k,r(x) ≥ 0 and +� +k∈{0,...,n}d +wn,k,r(x) = 1. +(3.2) +For f ∈ L1 (Qd) and x = (x1, . . . , xd) ∈ Qd we consider the following +multivariate extension of the BSK-operators Kn,r given by (1.6): +Kd +n,r (f; x) = +n +� +k1,...,kd=0 +d +� +i=1 +wn,ki,r(xi) +� +Qd +f +�k1 + u1 +n + 1 , . . . , kd + ud +n + 1 +� +du1 · · · dud. +Notice that from (3.1), and denoting, as usual, any f ∈ L1 (Qd) of x = +(x1, . . . , xd) ∈ Qd by f (x) = f (x1, . . . , xd), we can express these operators +in compact form as +Kd +n,r (f; x) = +� +k∈{0,...,n}d +wn,k,r(x) +� +Qd +f +�k + u +n + 1 +� +du. +(3.3) +It is clear that multivariate BSK-operators are positive and linear and the +cases r = 0 and 1 give the multivariate Kantorovich operators on the hyper- +cube Qd, which can be captured from [1] as a special case. +Lemma 3.1. For x ∈ Qd, we have +Kd +n,r (1; x) += +1, +Kd +n,r (pri; x) += +n +n + 1xi + +1 +2 (n + 1), +Kd +n,r +� +pr2 +i ; x +� += +n2 +(n + 1)2 +� +x2 +i + +� +1 + r (r − 1) +n +� xi (1 − xi) +n +� ++ 3nxi + 1 +3 (n + 1)2 , +for i = 1, . . . , d. +Taking this lemma into consideration, by the well-known theorem of +Volkov [26], we immediately get that +Theorem 3.1. Let r be a non-negative fixed integer and f ∈ C (Qd). Then +lim +n→∞ Kd +n,r (f) = f uniformly on Qd. + +BRASS-STANCU-KANTOROVICH OPERATORS ON A HYPERCUBE +7 +Now, we need the following evaluations for the subsequent result: For +0 ≤ xi ≤ 1, i = 1, . . . , d, we have +1 +� +0 +(1 − xi) pn−r,ki (xi) dxi += +�n − r +ki +� +1 +� +0 +xki +i (1 − xi)n−r−ki+1 dxi += +n − r − ki + 1 +(n − r + 2) (n − r + 1) +when 0 ≤ ki < r and +1 +� +0 +xipn−r,ki−r (xi) dxi += +�n − r +ki − r +� +1 +� +0 +xki−r+1 +i +(1 − xi)n−ki dxi += +ki − r + 1 +(n − r + 2) (n − r + 1) +when n − r < ki ≤ n. Thus, from (1.1) and (1.2), it follows that +1 +� +0 +wn,ki,r(xi)dxi = + + + + + +n−r−ki+1 +(n−r+2)(n−r+1); +0 ≤ ki < r +n−2r+2 +(n−r+2)(n−r+1); +r ≤ ki ≤ n − r +ki−r+1 +(n−r+2)(n−r+1); +n − r < ki ≤ n +. +(3.4) +Note that we can write the following estimates +n − r − ki + 1 +≤ +n − r + 1 when 0 ≤ ki < r, +n − 2r + 2 +≤ +n − r + 1 when r ≤ ki ≤ n − r, +ki − r + 1 +≤ +n − r + 1 when n − r < ki ≤ n +(3.5) +for each i = 1, . . . , d, where in the middle term, we have used the hypothesis +n > 2r. Making use of (3.5), (3.4) and (3.1), we obtain +� +Qd +wn,k,r(x)dx = +d +� +i=1 +1 +� +0 +wn,ki,r(xi)dxi ≤ +1 +(n − r + 2)d . +(3.6) +Lp-approximation by the sequence of the multivariate Stancu-Kantorovich +operators is presented in the following theorem. +Theorem 3.2. Let r be a non-negative fixed integer and f ∈ Lp (Qd) , 1 ≤ +p < ∞. Then lim +n→∞ +��Kd +n,r(f) − f +�� +p = 0. +Proof. Since the cases r = 0 and 1 correspond to the multivariate Kan- +torovich operators (see [1] or [3]), we consider only the cases r > 1, which is +taken as fixed. From Theorem 3.1, we obtain that lim +n→∞ +��Kd +n,r(f) − f +�� +p = +0 for any f ∈ C (Qd). Since C (Qd) is dense in Lp (Qd), denoting the norm +of the operator Kd +n,r acting on Lp (Qd) onto itself by +��Kd +n,r +��, it remains to +show that there exists an Mr, where Mr is a positive constant that maybe +depends on r, such that +��Kd +n,r +�� ≤ Mr for all n > 2r. Now, as in [3, p.604], +we adopt the notation +Qn,k := +d +� +i=1 +� +ki +n + 1, ki + 1 +n + 1 +� +⊂ Qd; +� +k∈{0,...,n}d +Qn,k = Qd. + +8 +G¨ULEN BAS¸CANBAZ-TUNCA AND HEINER GONSKA +Making use of the convexity of the function ϕ (t) := |t|p , t ∈ R, 1 ≤ p < +∞ (see, e.g., [2]), and (3.2), for every f ∈ Lp (Qd) , n > 2r, and x ∈ Qd, we +obtain +���Kd +n,r (f; x) +��� +p +≤ +� +k∈{0,...,n}d +wn,k,r(x) +� +Qd +����f +�k + u +n + 1 +����� +p +du += +� +k∈{0,...,n}d +wn,k,r(x) (n + 1)d +� +Qn,k +|f (v)|p dv. +Taking (3.6) into consideration, we reach to +� +Qd +���Kd +n,r (f; x) +��� +p +dx ≤ +� +k∈{0,...,n}d +� +n + 1 +n − r + 2 +�d � +Qn,k +|f (v)|p dv. +Since sup +n>2r +� +n+1 +n−r+2 +�d += +� +2r+2 +r+3 +�d +:= Mr for r > 1, where 1 < 2r+2 +r+3 < 2, we +get +� +Qd +���Kd +n,r (f; x) +��� +p +dx ≤ Mr +� +Qd +|f (v)|p dv, +which implies that +��Kd +n,r (f) +�� +p ≤ M1/p +r +∥f∥p. Note that for the cases r = 0 +and 1; we have Mr = 1 (see [3]). Therefore, the proof is completed. +□ +4. Estimates for the rate of convergence +In [17], M¨uller studied Lp-approximation by the sequence of the Cheney- +Sharma-Kantorovich operators (CSK). The author gave an estimate for this +approximation in terms of the univariate τ-modulus and moreover, using +some properties of the τ-modulus, he also obtained upper estimates for the +Lp-norm of the error of approximation for first order differentiable functions +as well as for continuous ones. In this part, we show that similar estimates +can also be obtained for +��Kd +n,r (f) − f +�� +p in the multivariate setting. Our +first result is an application of Quak’s method in Theorem 2.1 +Theorem 4.1. Let r be a non-negative fixed integer, f ∈ M (Qd) and 1 ≤ +p < ∞. Then +���Kd +n,r (f) − f +��� +p ≤ Cτ 1 +� +f, 2d +� +3n + 1 + 3r (r − 1) +12 (n + 1)2 +� +p +(4.1) +for all n ∈ N such that n > 2r, where the positive constant C does not +depend on f. +Proof. According to Theorem 2.1; by taking ψx (y) = y − x for fixed x ∈ +Qd and for every y ∈ Qd, and defining +An,r := sup +� +Kd +n,r +� +(pri ◦ ψx)2 ; x +� +: i = 1, . . . , d, x ∈ Qd +� +, +where (pri ◦ ψx)2 = pr2 +i − 2xipri + x2 +i 1, i = 1, . . . , d, we get the following +estimate +���Kd +n,r (f) − f +��� +p ≤ Cτ1 +� +f; 2d� +An,r +� + +BRASS-STANCU-KANTOROVICH OPERATORS ON A HYPERCUBE +9 +for any f ∈ M (Qd), under the condition that An,r ≤ 1. Now, applying the +operators Kd +n,r and making use of Lemma 3.1, for every i = 1, . . . , d and +x ∈ Qd, we obtain +Kd +n,r +� +(pri ◦ ψx)2 ; x +� += +n − 1 + r (r − 1) +(n + 1)2 +xi (1 − xi) + +1 +3 (n + 1)2 +≤ +n − 1 + r (r − 1) +4 (n + 1)2 ++ +1 +3 (n + 1)2 += +3n + 1 + 3r (r − 1) +12 (n + 1)2 +for all n ∈ N such that n > 2r, where r ∈ N ∪ {0}. Therefore, since we have +n ≥ 2r + 1, we take r ≤ n−1 +2 +and obtain that An,r ≤ 3n+1+3r(r−1) +12(n+1)2 +≤ 1 is +satisfied, which completes the proof. +□ +Now, making use of the properties τ1)-τ3) of the multivariate first order +τ-modulus, we obtain +Theorem 4.2. Let r be a non-negative fixed integer, f ∈ Lp (Qd) , 1 ≤ p < +∞, and Dαf ∈ Lp (Qd) for all multi-indices α with |α| ≥ 1, αi = 0 or 1. +Then +���Kd +n,r (f) − f +��� +p ≤ 2Cr +� +|α|≥1 +� +1 +2d√n + 1 +�|α| +∥Dαf∥p , +for all n ∈ N such that n > 2r, where Cr is a positive constant depending +on r. +Proof. Since n > 2r, we immediately have n + 1 ≥ 2 (r + 1). Thus, the +term appearing inside the 2dth root in the formula (4.1) can be estimated, +respectively, for r > 1, and r = 0, 1, as +3n + 1 + 3r (r − 1) +12 (n + 1)2 += +3n + 3 + 3r (r − 1) − 2 +12(n + 1)2 += +1 +n + 1 +�1 +4 + 3r (r − 1) − 2 +12(n + 1) +� +≤ +1 +n + 1 +�1 +4 + 3r (r − 1) − 2 +24(r + 1) +� += +1 +n + 1 +�3r2 + 3r + 4 +24(r + 1) +� +and +3n + 1 +12 (n + 1)2 = +1 +n + 1 +3n + 1 +4 (3n + 3) < +1 +4 (n + 1). +Now, defining +Br := +� +3r2+3r+4 +24(r+1) ; +r > 1, +1 +4; +r = 0, 1, + +10 +G¨ULEN BAS¸CANBAZ-TUNCA AND HEINER GONSKA +and making use of the properties τ 1)-τ 3) of τ-modulus, from (4.1), we arrive +at +���Kd +n,r (f) − f +��� +p +≤ +Cτ1 +� +f, 2d +� +3n + 1 + 3r (r − 1) +12 (n + 1)2 +� +p +≤ +Cτ1 +� +f, +2d� +Br +1 +2d√n + 1 +� +p +≤ +C +� +2 +� +2d� +Br +� ++ 2 +�d+1 +τ 1 +� +f, +1 +2d√n + 1 +� +p +≤ +2Cr +� +|α|≥1 +� +1 +2d√n + 1 +�|α| +∥Dαf∥p , +where the positive constant Cr is defined as Cr := C +� +2 +� 2d√Br +� ++ 2 +�d+1 . +□ +For non-differentiable functions we have the following estimate in terms +of the first order modulus of smoothness, in Lp-norm. +Theorem 4.3. Let r be a non-negative fixed integer and f ∈ Lp (Qd) , 1 ≤ +p < ∞. +Then +���Kd +n,r (f) − f +��� +p ≤ c2Cr,pω1 +� +f; +1 +2d√n + 1 +� +p +, +where ω1 is the first order multivariate modulus of smoothness of f and Cr,p +is a constant depending on r and p. +Proof. By Theorem 3.2, since Kd +n,r is bounded, with +��Kd +n,r +�� +p ≤ M1/p +r +, for +all n ∈ N such that n > 2r, we have +��Kd +n,r (g) − g +�� +p ≤ +� +M1/p +r ++ 1 +� +∥g∥p for +g ∈ Lp (Qd). Moreover, from Theorem 4.2, we can write +���Kd +n,r (g) − g +��� +p ≤ 2Cr +� +|α|≥1 +� +1 +2d√n + 1 +�|α| +∥Dαg∥p +for those g such that Dαg ∈ Lp (Qd), for all multi-indices α with |α| ≥ 1 +and αi = 0 or 1. Hence, for f ∈ Lp (Qd), it readily follows that +���Kd +n,r (f) − f +��� +p +≤ +���Kd +n,r (f − g) − (f − g) +��� +p + +���Kd +n,r (g) − g +��� +p +≤ +� +M1/p +r ++ 1 +� + + +∥f − g∥p + 2Cr +� +|α|≥1 +� +1 +2d√n + 1 +�|α| +∥Dαg∥p + + + . + +BRASS-STANCU-KANTOROVICH OPERATORS ON A HYPERCUBE +11 +Passing to the infimum for all g ∈ W p +1 (Qd) in the last formula, since the +infimum of a superset does not exceed that of subset, we obtain +���Kd +n,r (f) − f +��� +p +≤ +� +M1/p +r ++ 1 +� +inf + + +∥f − g∥p + +2Cr +2d√n + 1 +� +|α|=1 +∥Dαg∥p : g ∈ W p +1 (Qd) + + + += +� +M1/p +r ++ 1 +� +inf +� +∥f − g∥p + +2Cr +2d√n + 1 |g|W p +1 : g ∈ W p +1 (Qd) +� += +� +M1/p +r ++ 1 +� +K1,p +� +f; +2Cr +2d√n + 1 +� +, +(4.2) +where K1,p is the K-functional given by (2.1). The proof follows from the +equivalence (2.2) of the K-functional and the first order modulus of smooth- +ness in Lp-norm and the non-decreasingness property of the modulus. In- +deed, we get +K1,p +� +f; +2Cr +2d√n + 1 +� +≤ +c2ω1 +� +f; +2Cr +2d√n + 1 +� +p +≤ +c2 (2Cr + 1) ω1 +� +f; +1 +2d√n + 1 +� +p +. +(4.3) +Combining (4.3) with (4.2) and defining Cr,p := +� +M1/p +r ++ 1 +� +(2Cr + 1), +where M1/p +r +and Cr are the same as in Theorems 3.2 and 4.2, respectively, +we obtain the desired result. +□ +References +[1] F. Altomare, M. Cappelletti Montano, V. Leonessa, On a generalization of Kan- +torovich operators on simplices and hypercubes, Adv. Pure Appl. Math. 1 (2010), no. +3, 359-385. +[2] F. Altomare, Korovkin-type Theorems and Approximation by Positive Linear Oper- +ators, Surv. Approx. Theory 5 (2010), 92-164. +[3] F. Altomare, M. Cappelletti Montano, V. Leonessa, I. Ra¸sa, A generalization of +Kantorovich operators for convex compact subsets, Banach J. Math. Anal. 11 (2017), +no. 3, 591–614. +[4] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press Inc., 1988. +[5] H. Berens, R. 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Proc. + +12 +G¨ULEN BAS¸CANBAZ-TUNCA AND HEINER GONSKA +int. conf. ded. to the memory of Vl. K. Dzyadyk, Kiev, Ukraine, May 27–31, 1999. +Pr. Inst. Mat. Nats. Akad. Nauk Ukr., Mat. Zastos. 31, 161–180 (2000). +[12] L. V. Kantorovich, Sur certains d´eveloppements suivant les polynˆomes de la forme +de S. Bernstein, I, II, C. R. Acad. Sci. URSS, (1930), 563–568, 595–600. +[13] L. V. Kantorovich, My journey in science (proposed report to the Moscow Mathe- +matical Society), Russ. Math. Surv, 42(2) (1987), 233–270. Russian original: Uspekhi +Mat. Nauk 42(2) (1987), 183–213. +[14] G. G. Lorentz, Bernstein Polynomials, University of Toronto Press, Toronto, 1953 +(2.ed., Chelsea Publishing Co., New York, 1986). +[15] V. Maier, Lp-approximation by Kantoroviˇc operators, Anal. Math., 4 (1978), 289– +295. +[16] M. W. M¨uller, Die G¨ute der Lp- Approximation durch Kantoroviˇc-polynome, Math. +Z. 151 (1976), 243–247. +[17] M. W. M¨uller, Approximation by Cheney-Sharma-Kantoroviˇc polynomials in the Lp- +metric, Rocky Mountain J. Math., 19(1) (1989), 281–291. +[18] V. A. Popov, On the quantitative Korovkin theorems in Lp, Compt. Rend. Acad. +Bulg. Sci., 35 (1982), 897–900. +[19] V. A. Popov, V. Kh. Khristov, “Averaged moduli of smoothness for functions of +several variables and function spaces generated by them”, Orthogonal series and +approximations of functions, Collection of articles. Dedicated to Academician N. N. +Luzin on the occasion of the 100th anniversary of his birth, Trudy Mat. Inst. Steklov., +164, 1983, 136–141; Proc. Steklov Inst. Math., 164 (1985), 155–160 +[20] E. Quak, Uni- und multivariate Lp-Abschatzungen des Approximationsfehlers posi- +tiver linearer Operatoren mit Hilfe des -Moduls, Ph.D. Thesis, University of Dort- +mund 1985. +[21] E. Quak, Multivariate Lp-error estimates for positive linear operators via the first- +order τ-modulus, J. Approx. Theory 56 (1989), 277–286. +[22] B. Sendov, V. A. Popov, The Averaged Moduli of Smoothness. Applications in Nu- +merical Methods and Approximation, Chichester (UK) etc.: Wiley 1988. +[23] D. D. Stancu, Quadrature formulas constructed by using certain linear positive op- +erators, Numerical Integration (Proc. Conf., Oberwolfach, 1981), ISNM 57 (1982), +241–251, Birkh¨auser Verlag, Basel. +[24] D. D. Stancu, Approximation of functions by means of a new generalized Bernstein +operator, Calcolo, 20 (1983), no. 2, 211–229. +[25] J. J. Swetits, B. Wood, Quantitative estimates for Lp approximation with positive +linear operators, J. Approx. Theory 38 (1983), 81–89. +[26] V. I. Volkov, On the convergence of sequences of linear positive operators in the space +of continuous functions of two variables (Russian), Dokl. Akad. Nauk. SSSR (N.S.), +115 (1957), 17–19. +[27] R. Yang, J. Xiong, F. Cao, Multivariate Stancu operators defined on a simplex, Appl. +Math. Comput., 138 (2003), 189–198. +Ankara University, Faculty of Science, Department of Mathematics, Str. +D¨ogol 06100, Bes¸evler, Ankara, Turkey +Email address: tunca@science.ankara.edu.tr +University of Duisburg-Essen, Faculty of Mathematics, Forsthausweg 2, +D-47057 Duisburg, Germany +Email address: heiner.gonska@uni-due.de and gonska.sibiu@gmail.com + diff --git a/FNAzT4oBgHgl3EQfG_ve/content/tmp_files/load_file.txt b/FNAzT4oBgHgl3EQfG_ve/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..62bb975700a765fab9291b98837fb7251b550966 --- /dev/null +++ b/FNAzT4oBgHgl3EQfG_ve/content/tmp_files/load_file.txt @@ -0,0 +1,532 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf,len=531 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='01039v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='CA] 3 Jan 2023 BRASS-STANCU-KANTOROVICH OPERATORS ON A HYPERCUBE∗ G¨ULEN BAS¸CANBAZ-TUNCA AND HEINER GONSKA This study is dedicated to Professor Ioan Ra¸sa on the occasion of his 70th birthday Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' We deal with multivariate Brass-Stancu-Kantorovich oper- ators depending on a non-negative integer parameter and defined on the space of all Lebesgue integrable functions on a unit hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' We prove Lp-approximation and provide estimates for the Lp-norm of the error of approximation in terms of a multivariate averaged modulus of continuity and of the corresponding Lp-modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Introduction and Historical Notes The fundamental functions of the well-known Bernstein operators are defined by pn,k(x) = � �n k � xk(1 − x)n−k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 0 ≤ k ≤ n 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' k < 0 or k > n , x ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1) In [23], using a probabilistic method, Stancu generalized Bernstein’s funda- mental functions as wn,k,r(x) := \uf8f1 \uf8f2 \uf8f3 (1 − x) pn−r,k (x) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 0 ≤ k < r (1 − x) pn−r,k (x) + xpn−r,k−r (x) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' r ≤ k ≤ n − r xpn−r,k−r (x) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' n − r < k ≤ n , x ∈ [0, 1], (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2) where r is a non-negative integer parameter, n is any natural number such that n > 2r, for which each pn−r,k is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1), and therefore, con- structed and studied Bernstein-type positive linear operators as Ln,r (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) := n � k=0 wn,k,r(x)f �k n � , x ∈ [0, 1], (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='3) for f ∈ C[0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' In doing so Stancu was guided by an article of Brass [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' This is further discussed by Gonska [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Among others, estimates in terms of the second order modulus of smoothness are given there for continuous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' It is clear that for x ∈ [0, 1] Stancu’s fundamental functions in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2) satisfy wn,k,r(x) ≥ 0 and n � k=0 wn,k,r(x) = 1, Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Multivariate Kantorovich operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Multivariate averaged modulus of smoothness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Multivariate K-functional 2010 MSC: 41A36, 41A25, 26A45 ∗This paper is an extension of a talk given in ICATA 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 1 2 G¨ULEN BAS¸CANBAZ-TUNCA AND HEINER GONSKA hence the operators Ln,r can be expressed as Ln,r (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) := n−r � k=0 pn−r,k (x) � (1 − x) f �k n � + xf �k + r n �� , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='4) are defined for n ≥ r and satisfy the end point interpolation Ln,r (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 0) = f (0) , Ln,r (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 1) = f (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' It thus seems to be justified to call the Ln,r Brass-Stancu-Bernstein (BSB) operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' In [24] Stancu gave uniform convergence limn→∞ Ln,r (f) = f on [0, 1] for f ∈ C[0, 1] and presented an expression for the remainder Rn,r(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) of the approximation formula f(x) = Ln,r(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) + Rn,r(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) by means of second order divided differences and also obtained an integral representation for the remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Moreover, the author estimated the order of approximation by the operators Ln,r (f) via the classical modulus of continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' He also studied the spectral properties of Ln,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' In the cases r = 0 and r = 1, the operators Ln,r reduce to the classical Bernstein operators Bn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=', Bn (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) = n � k=0 pn,k(x)f �k n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' What also has to be mentioned: Stancu himself in his 1983 paper observed that ”we can optimize the error bound of the approximation of the function f by means of Ln,rf if we take r = 0 or r = 1, when the operator Ln,r reduces to Bernstein’s.” So there is a shortcoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Since Bernstein polynomials are not appropriate for approximation of discontinuous functions (see [14, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='9]), by replacing the point evalu- ations f � k n � with the integral means over small intervals around the knots k n, Kantorovich [12] generalized the Bernstein operators as Kn (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) = n � k=0 pn,k (x) (n + 1) k+1 n+1 � k n+1 f (t) dt, x ∈ [0, 1], n ∈ N, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='5) for Lebesgue integrable functions f on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' On p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 239 of his mathematical memoirs [13] Kantorovich writes: ”While I was waiting for a student who was late, I was looking over vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' XIII of Fundamenta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' and saw in it a note from the Moscow Mathematician Khlodovskii related to Bernstein polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' In it I first caught sight of Bernstein polynomials, which he proposed in 1912 for an elementary proof of the well known Weierstrass theorem .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' I at once wondered if it is not possible in these polynomials to change the values of the function at certain points into the more stable average of the function in the corresponding interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' It turned out that this was possible, and the polynomials could be written in such a form not only for a continuous function but also for any Lebesgue-summable function.” Lorentz [14] proved that lim n→∞ ∥Kn(f) − f∥p = 0, f ∈ Lp[0, 1], 1 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' There are a lot of articles dealing with classical Kantorovich operators, and, in particular, their degree of approximation and the importance of BRASS-STANCU-KANTOROVICH OPERATORS ON A HYPERCUBE 3 second order moduli of different types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=', the work of Berens and DeVore [5], [6], Swetits and Wood [25] and Gonska and Zhou [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' It is beyond the scope of this note to further discuss this matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' As further work on the classical case here we only mention the 1976 work of M¨uller [16], Maier [15], and Altomare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' [1], see also the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Similarly to Kantorovich operators Bodur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' [7] constructed a Kan- torovich type modification of BSB operators as Kn,r (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) := n � k=0 wn,k,r(x) \uf8eb \uf8ec \uf8ec \uf8ed(n + 1) k+1 n+1 � k n+1 f (t) dt \uf8f6 \uf8f7 \uf8f7 \uf8f8 , x ∈ [0, 1], (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='6) for f ∈ L1 [0, 1], where r is a non-negative integer parameter, n is a natural number such that n > 2r and wn,k,r(x) are given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' And, it was shown that If f ∈ Lp[0, 1], 1 ≤ p < ∞, then lim n→∞ ∥Kn,r(f) − f∥p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' In addition, it was obtained that each Kn,r is variation detracting as well [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Throughout the paper, we shall call the operators Kn,r given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='6) ”Brass-Stancu-Kantorovich”, BSK operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Notice that from the definition of wn,k,r, Kn,r (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) can be expressed as Kn,r (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='7) = n−r � k=0 pn−r,k (x) (n + 1) \uf8ee \uf8ef\uf8ef\uf8f0(1 − x) k+1 n+1 � k n+1 f (t) dt + x k+r+1 n+1 � k+r n+1 f (t) dt \uf8f9 \uf8fa\uf8fa\uf8fb and in the cases r = 0 and r = 1 they reduce to the Kantorovich operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Kn,0 = Kn,1 = Kn given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Again they are defined for all n ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' MULTIVARIATE SITUATION Some work has been done in the multivariate setting for BSB and BSK operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' For the standard simplex this was done, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=', by Yang, Xiong and Cao [27] and Cao [9], For example, Cao proved that multivariate Stancu operators preserve the properties of multivariate moduli of continuity and obtained the rate of convergence with the help of Ditzian-Totik’s modulus of continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' In this work, motivated by the work Altomare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' [3], we deal with a multivariate extension of the BSK operators on a d-dimensional unit hy- percube and we study Lp -approximation by these operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' For the rate of convergence we provide an estimate in terms of the so called first order multivariate τ-modulus, a quantity coming from the Bulgarian school of Approximation Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Also, inspired by M¨uller’s approach in [17], we give estimates for differentiable functions and such in terms of the Lp-modulus of smoothness, using properties of the τ-modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Here the work of Quak [20], [21] was helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 4 G¨ULEN BAS¸CANBAZ-TUNCA AND HEINER GONSKA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Preliminaries Consider the space Rd, d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Let ∥x∥∞ denote the max-norm of a point x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , xd) ∈ Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' ∥x∥∞ := ∥x∥max = max i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=',d} |xi| and let 1 denote the constant function 1 : Rd → R such that 1 (x) = 1 for x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' And, for each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , d, let prj : Rd → R stand for the jth coordinate function defined for x ∈ Rd by prj (x) = xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' A multi-index is a d-tuple α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , αd) of non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Its norm (length) is the quantity |α| = d � i=1 αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' The differential operator Dα is defined by Dαf = Dα1 1 · · · Dαd d f, where Di, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , d, is the corresponding partial derivative operator (see [4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 335]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Throughout the paper Qd := [0, 1]d, d ∈ N, will denote the d-dimensional unit hypercube and we consider the space Lp (Qd) = {f : Qd → R | f p-integrable on Qd} , 1 ≤ p < ∞, with the standard norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Recall the following definition of the usual Lp-modulus of smoothness of first order: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Let f ∈ Lp (Qd) , 1 ≤ p < ∞, h ∈ Rd and δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' The modulus of smoothness of the first order for the function f and step δ in Lp-norm is given by ω1 (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' δ)p = sup 0<∥h∥∞≤δ \uf8eb \uf8ec \uf8ed � Qd |f (x + h) − f (x)|p dx \uf8f6 \uf8f7 \uf8f8 1/p if x, x + h ∈ Qd [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Let M (Qd) := {f | f bounded and measurable on Qd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Below, we present the concept of the first order averaged modulus of smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Let f ∈ M (Qd) , h ∈ Rd and δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' The multivariate averaged modulus of smoothness, or τ-modulus, of the first order for function f and step δ in Lp-norm is given by τ 1 (f, δ)p := ∥ω1 (f, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' δ)∥p , 1 ≤ p < ∞, where ω1 (f, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' δ) = sup � |f (t + h) − f (t)| : t, t + h ∈ Qd, ∥t − x∥∞ ≤ δ 2, ∥t + h − x∥∞ ≤ δ 2 � BRASS-STANCU-KANTOROVICH OPERATORS ON A HYPERCUBE 5 is the multivariate local modulus of smoothness of first order for the function f at the point x ∈ Qd and for step δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' For our future purposes, we need the following properties of first order multivariate averaged modulus of smoothness: For f ∈ M (Qd) , 1 ≤ p < ∞ and δ, λ, γ ∈ R+, there hold τ 1) τ 1 (f, δ)p ≤ τ 1 (f, λ)p for 0 < δ ≤ λ, τ 2) τ 1 (f, λδ)p ≤ (2 ⌊λ⌋ + 2)d+1 τ 1 (f, δ)p, where ⌊λ⌋ is the greatest inte- ger that does not exceed λ, τ 3) τ 1 (f, δ)p ≤ 2 � |α|≥1 δ|α| ∥Dαf∥p , αi = 0 or 1, if Dαf ∈ Lp (Qd) for all multi-indices α with |α| ≥ 1 and αi = 0 or 1 (see [19] or [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' For a detailed knowledge concerning averaged modulus of smoothness, we refer to the book of Sendov and Popov [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Now, consider the Sobolev space W p 1 (Qd) of functions f ∈ Lp (Qd) , 1 ≤ p < ∞, with (distributional) derivatives Dαf belong to Lp (Qd), where |α| ≤ 1, with the seminorm |f|W p 1 = � |α|=1 ∥Dαf∥p (see [4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 336]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Recall that for all f ∈ Lp (Qd) the K-functional, in Lp- norm, is defined as K1,p (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' t) := inf � ∥f − g∥p + t |g|W p 1 : g ∈ W p 1 (Qd) � (t > 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1) K1,p (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' t) is equivalent with the usual first order modulus of smoothness of f, ω1 (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' t)p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' namely, there are positive constants c1 and c2 such that c1K1,p (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' t) ≤ ω1 (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' t)p ≤ c2K1,p (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' t) (t > 0) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2) holds for all f ∈ Lp (Qd) (see [4, Formula 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='42 in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 341]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' The following result due to Quak [21] is an upper estimate for the Lp-norm of the approximation error by the multivariate positive linear operators in terms of the first order averaged modulus of smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Note that this idea was used first by Popov for the univariate case in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Let L : M (Qd) → M (Qd) be a positive linear operator that preserves the constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Then for every f ∈ M (Qd) and 1 ≤ p < ∞, the following estimate holds: ∥L(f) − f∥p ≤ Cτ1 � f, 2d√ A � p , where C is a positive constant and A := sup � L � (pri ◦ ψx)2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x � : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , d, x ∈ Qd � , in which ψx (y) := y − x for fixed x ∈ Qd and for every y ∈ Qd and A ≤ 1 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 6 G¨ULEN BAS¸CANBAZ-TUNCA AND HEINER GONSKA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Multivariate BSK-Operators In this section, motivated by the works of Altomare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' [1] and Al- tomare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' [3], we consider the multivariate extension of BSK-operators on Lp (Qd) and study approximation properties of these operators in Lp- norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' We investigate the rate of the convergence in terms of the first order τ-modulus and the usual Lp-modulus of smoothness of the first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Let r be a given non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' For any n ∈ N such that n > 2r, k = (k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , kd) ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , n}d and x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , xd) ∈ Qd, we set wn,k,r(x) := d � i=1 wn,ki,r(xi), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1) where, wn,ki,r(xi) is Stancu’s fundamental function given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2), written for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , d, 0 ≤ ki ≤ n and xi ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Thus, for x ∈ Qd, we have wn,k,r(x) ≥ 0 and � k∈{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=',n}d wn,k,r(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2) For f ∈ L1 (Qd) and x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , xd) ∈ Qd we consider the following multivariate extension of the BSK-operators Kn,r given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='6): Kd n,r (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) = n � k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=',kd=0 d � i=1 wn,ki,r(xi) � Qd f �k1 + u1 n + 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , kd + ud n + 1 � du1 · · · dud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Notice that from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1), and denoting, as usual, any f ∈ L1 (Qd) of x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , xd) ∈ Qd by f (x) = f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , xd), we can express these operators in compact form as Kd n,r (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) = � k∈{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=',n}d wn,k,r(x) � Qd f �k + u n + 1 � du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='3) It is clear that multivariate BSK-operators are positive and linear and the cases r = 0 and 1 give the multivariate Kantorovich operators on the hyper- cube Qd, which can be captured from [1] as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' For x ∈ Qd, we have Kd n,r (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) = 1, Kd n,r (pri;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) = n n + 1xi + 1 2 (n + 1), Kd n,r � pr2 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x � = n2 (n + 1)2 � x2 i + � 1 + r (r − 1) n � xi (1 − xi) n � + 3nxi + 1 3 (n + 1)2 , for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Taking this lemma into consideration, by the well-known theorem of Volkov [26], we immediately get that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Let r be a non-negative fixed integer and f ∈ C (Qd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Then lim n→∞ Kd n,r (f) = f uniformly on Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' BRASS-STANCU-KANTOROVICH OPERATORS ON A HYPERCUBE 7 Now, we need the following evaluations for the subsequent result: For 0 ≤ xi ≤ 1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , d, we have 1 � 0 (1 − xi) pn−r,ki (xi) dxi = �n − r ki � 1 � 0 xki i (1 − xi)n−r−ki+1 dxi = n − r − ki + 1 (n − r + 2) (n − r + 1) when 0 ≤ ki < r and 1 � 0 xipn−r,ki−r (xi) dxi = �n − r ki − r � 1 � 0 xki−r+1 i (1 − xi)n−ki dxi = ki − r + 1 (n − r + 2) (n − r + 1) when n − r < ki ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Thus, from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2), it follows that 1 � 0 wn,ki,r(xi)dxi = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 n−r−ki+1 (n−r+2)(n−r+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 0 ≤ ki < r n−2r+2 (n−r+2)(n−r+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' r ≤ ki ≤ n − r ki−r+1 (n−r+2)(n−r+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' n − r < ki ≤ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='4) Note that we can write the following estimates n − r − ki + 1 ≤ n − r + 1 when 0 ≤ ki < r, n − 2r + 2 ≤ n − r + 1 when r ≤ ki ≤ n − r, ki − r + 1 ≤ n − r + 1 when n − r < ki ≤ n (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='5) for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , d, where in the middle term, we have used the hypothesis n > 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Making use of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='5), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1), we obtain � Qd wn,k,r(x)dx = d � i=1 1 � 0 wn,ki,r(xi)dxi ≤ 1 (n − r + 2)d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='6) Lp-approximation by the sequence of the multivariate Stancu-Kantorovich operators is presented in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Let r be a non-negative fixed integer and f ∈ Lp (Qd) , 1 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Then lim n→∞ ��Kd n,r(f) − f �� p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Since the cases r = 0 and 1 correspond to the multivariate Kan- torovich operators (see [1] or [3]), we consider only the cases r > 1, which is taken as fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1, we obtain that lim n→∞ ��Kd n,r(f) − f �� p = 0 for any f ∈ C (Qd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Since C (Qd) is dense in Lp (Qd), denoting the norm of the operator Kd n,r acting on Lp (Qd) onto itself by ��Kd n,r ��, it remains to show that there exists an Mr, where Mr is a positive constant that maybe depends on r, such that ��Kd n,r �� ≤ Mr for all n > 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Now, as in [3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='604], we adopt the notation Qn,k := d � i=1 � ki n + 1, ki + 1 n + 1 � ⊂ Qd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' � k∈{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=',n}d Qn,k = Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 8 G¨ULEN BAS¸CANBAZ-TUNCA AND HEINER GONSKA Making use of the convexity of the function ϕ (t) := |t|p , t ∈ R, 1 ≤ p < ∞ (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=', [2]), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2), for every f ∈ Lp (Qd) , n > 2r, and x ∈ Qd, we obtain ���Kd n,r (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) ��� p ≤ � k∈{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=',n}d wn,k,r(x) � Qd ����f �k + u n + 1 ����� p du = � k∈{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=',n}d wn,k,r(x) (n + 1)d � Qn,k |f (v)|p dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Taking (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='6) into consideration, we reach to � Qd ���Kd n,r (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) ��� p dx ≤ � k∈{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=',n}d � n + 1 n − r + 2 �d � Qn,k |f (v)|p dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Since sup n>2r � n+1 n−r+2 �d = � 2r+2 r+3 �d := Mr for r > 1, where 1 < 2r+2 r+3 < 2, we get � Qd ���Kd n,r (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x) ��� p dx ≤ Mr � Qd |f (v)|p dv, which implies that ��Kd n,r (f) �� p ≤ M1/p r ∥f∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Note that for the cases r = 0 and 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' we have Mr = 1 (see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Therefore, the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Estimates for the rate of convergence In [17], M¨uller studied Lp-approximation by the sequence of the Cheney- Sharma-Kantorovich operators (CSK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' The author gave an estimate for this approximation in terms of the univariate τ-modulus and moreover, using some properties of the τ-modulus, he also obtained upper estimates for the Lp-norm of the error of approximation for first order differentiable functions as well as for continuous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' In this part, we show that similar estimates can also be obtained for ��Kd n,r (f) − f �� p in the multivariate setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Our first result is an application of Quak’s method in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Let r be a non-negative fixed integer, f ∈ M (Qd) and 1 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Then ���Kd n,r (f) − f ��� p ≤ Cτ 1 � f, 2d � 3n + 1 + 3r (r − 1) 12 (n + 1)2 � p (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1) for all n ∈ N such that n > 2r, where the positive constant C does not depend on f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' According to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' by taking ψx (y) = y − x for fixed x ∈ Qd and for every y ∈ Qd, and defining An,r := sup � Kd n,r � (pri ◦ ψx)2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x � : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , d, x ∈ Qd � , where (pri ◦ ψx)2 = pr2 i − 2xipri + x2 i 1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , d, we get the following estimate ���Kd n,r (f) − f ��� p ≤ Cτ1 � f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 2d� An,r � BRASS-STANCU-KANTOROVICH OPERATORS ON A HYPERCUBE 9 for any f ∈ M (Qd), under the condition that An,r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Now, applying the operators Kd n,r and making use of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1, for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' , d and x ∈ Qd, we obtain Kd n,r � (pri ◦ ψx)2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' x � = n − 1 + r (r − 1) (n + 1)2 xi (1 − xi) + 1 3 (n + 1)2 ≤ n − 1 + r (r − 1) 4 (n + 1)2 + 1 3 (n + 1)2 = 3n + 1 + 3r (r − 1) 12 (n + 1)2 for all n ∈ N such that n > 2r, where r ∈ N ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Therefore, since we have n ≥ 2r + 1, we take r ≤ n−1 2 and obtain that An,r ≤ 3n+1+3r(r−1) 12(n+1)2 ≤ 1 is satisfied, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' □ Now, making use of the properties τ1)-τ3) of the multivariate first order τ-modulus, we obtain Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Let r be a non-negative fixed integer, f ∈ Lp (Qd) , 1 ≤ p < ∞, and Dαf ∈ Lp (Qd) for all multi-indices α with |α| ≥ 1, αi = 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Then ���Kd n,r (f) − f ��� p ≤ 2Cr � |α|≥1 � 1 2d√n + 1 �|α| ∥Dαf∥p , for all n ∈ N such that n > 2r, where Cr is a positive constant depending on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Since n > 2r, we immediately have n + 1 ≥ 2 (r + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Thus, the term appearing inside the 2dth root in the formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1) can be estimated, respectively, for r > 1, and r = 0, 1, as 3n + 1 + 3r (r − 1) 12 (n + 1)2 = 3n + 3 + 3r (r − 1) − 2 12(n + 1)2 = 1 n + 1 �1 4 + 3r (r − 1) − 2 12(n + 1) � ≤ 1 n + 1 �1 4 + 3r (r − 1) − 2 24(r + 1) � = 1 n + 1 �3r2 + 3r + 4 24(r + 1) � and 3n + 1 12 (n + 1)2 = 1 n + 1 3n + 1 4 (3n + 3) < 1 4 (n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Now, defining Br := � 3r2+3r+4 24(r+1) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' r > 1, 1 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' r = 0, 1, 10 G¨ULEN BAS¸CANBAZ-TUNCA AND HEINER GONSKA and making use of the properties τ 1)-τ 3) of τ-modulus, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1), we arrive at ���Kd n,r (f) − f ��� p ≤ Cτ1 � f, 2d � 3n + 1 + 3r (r − 1) 12 (n + 1)2 � p ≤ Cτ1 � f, 2d� Br 1 2d√n + 1 � p ≤ C � 2 � 2d� Br � + 2 �d+1 τ 1 � f, 1 2d√n + 1 � p ≤ 2Cr � |α|≥1 � 1 2d√n + 1 �|α| ∥Dαf∥p , where the positive constant Cr is defined as Cr := C � 2 � 2d√Br � + 2 �d+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' □ For non-differentiable functions we have the following estimate in terms of the first order modulus of smoothness, in Lp-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Let r be a non-negative fixed integer and f ∈ Lp (Qd) , 1 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Then ���Kd n,r (f) − f ��� p ≤ c2Cr,pω1 � f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 1 2d√n + 1 � p , where ω1 is the first order multivariate modulus of smoothness of f and Cr,p is a constant depending on r and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2, since Kd n,r is bounded, with ��Kd n,r �� p ≤ M1/p r , for all n ∈ N such that n > 2r, we have ��Kd n,r (g) − g �� p ≤ � M1/p r + 1 � ∥g∥p for g ∈ Lp (Qd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Moreover, from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2, we can write ���Kd n,r (g) − g ��� p ≤ 2Cr � |α|≥1 � 1 2d√n + 1 �|α| ∥Dαg∥p for those g such that Dαg ∈ Lp (Qd), for all multi-indices α with |α| ≥ 1 and αi = 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Hence, for f ∈ Lp (Qd), it readily follows that ���Kd n,r (f) − f ��� p ≤ ���Kd n,r (f − g) − (f − g) ��� p + ���Kd n,r (g) − g ��� p ≤ � M1/p r + 1 � \uf8f1 \uf8f2 \uf8f3∥f − g∥p + 2Cr � |α|≥1 � 1 2d√n + 1 �|α| ∥Dαg∥p \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' BRASS-STANCU-KANTOROVICH OPERATORS ON A HYPERCUBE 11 Passing to the infimum for all g ∈ W p 1 (Qd) in the last formula, since the infimum of a superset does not exceed that of subset, we obtain ���Kd n,r (f) − f ��� p ≤ � M1/p r + 1 � inf \uf8f1 \uf8f2 \uf8f3∥f − g∥p + 2Cr 2d√n + 1 � |α|=1 ∥Dαg∥p : g ∈ W p 1 (Qd) \uf8fc \uf8fd \uf8fe = � M1/p r + 1 � inf � ∥f − g∥p + 2Cr 2d√n + 1 |g|W p 1 : g ∈ W p 1 (Qd) � = � M1/p r + 1 � K1,p � f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 2Cr 2d√n + 1 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2) where K1,p is the K-functional given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' The proof follows from the equivalence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2) of the K-functional and the first order modulus of smooth- ness in Lp-norm and the non-decreasingness property of the modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' In- deed, we get K1,p � f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 2Cr 2d√n + 1 � ≤ c2ω1 � f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 2Cr 2d√n + 1 � p ≤ c2 (2Cr + 1) ω1 � f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' 1 2d√n + 1 � p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='3) Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='3) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2) and defining Cr,p := � M1/p r + 1 � (2Cr + 1), where M1/p r and Cr are the same as in Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='2, respectively, we obtain the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' □ References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Altomare, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Cappelletti Montano, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' Leonessa, On a generalization of Kan- torovich operators on simplices and hypercubes, Adv.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content=' D¨ogol 06100, Bes¸evler, Ankara, Turkey Email address: tunca@science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='ankara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='tr University of Duisburg-Essen, Faculty of Mathematics, Forsthausweg 2, D-47057 Duisburg, Germany Email address: heiner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='gonska@uni-due.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='de and gonska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='sibiu@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'} diff --git a/FtE1T4oBgHgl3EQfXARM/content/tmp_files/2301.03121v1.pdf.txt b/FtE1T4oBgHgl3EQfXARM/content/tmp_files/2301.03121v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0714fc5dc928b607db3cd1016c25d9bdc6c0d172 --- /dev/null +++ b/FtE1T4oBgHgl3EQfXARM/content/tmp_files/2301.03121v1.pdf.txt @@ -0,0 +1,1863 @@ +AdS/BCFT correspondence and Horndeski gravity in the presence of gauge +fields: from holographic paramagnetism/ferromagnetism phase transition +Fabiano F. Santosa,∗ Mois´es Bravo-Gaeteb,† Oleksii Sokoliuk c,d,‡ and Alexander Baransky c§ +aInstituto de F´ısica, Universidade Federal do Rio de Janeiro, +Caixa Postal 68528, Rio de Janeiro-RJ, 21941-972 – Brazil. +bFacultad de Ciencias B´asicas, Universidad Cat´olica del Maule, Casilla 617, Talca, Chile. +c Astronomical Observatory, Taras Shevchenko National University of Kyiv, +3 Observatorna St., 04053 Kyiv, Ukraine, and +dMain Astronomical Observatory of the NAS of Ukraine (MAO NASU), Kyiv, 03143, Ukraine. +This paper presents a dual gravity model for a (2+1)-dimensional system with a limit +on finite charge density and temperature, which will be used to study the properties of the +holographic phase transition to paramagnetism-ferromagnetism in the presence of Horndeski +gravity terms. In our model, the non-zero charge density is supported by a magnetic field. +As a result, the radius ρ/B indicates a localized condensate, as we increase the Horndeski +gravity parameter, that is represented by γ. Furthermore, such condensate shows quantum +Hall-type behavior. This radius is also inversely related to the total action coefficients of +our model. It was observed that increasing the Horndeski parameter decreases the critical +temperature of the holographic model and leads to the harder formation of the magnetic +moment at the bottom of the black hole. +However, when removing the magnetic field, +the ferromagnetic material presents a disorder of its magnetic moments, which is observed +through the entropy of the system. We also found that at low temperatures, spontaneous +magnetization and ferromagnetic phase transition. +I. +INTRODUCTION +For almost thirty years, the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence +has been a bridge that allows us to relate gravity and strongly coupled conformal field theories +[1, 2]. Following this spirit, a new holographic dual of a CFT arises, which is defined on a manifold +M with a boundary ∂M, denoted as Boundary Conformal Field Theory (BCFT), proposed by +Takayanagi [3] and Takayanagi et al. [4], extending the AdS/CFT duality. This new holographic +∗Electronic address: fabiano.ffs23-at-gmail.com +†Electronic address: mbravo-at-ucm.cl +‡Electronic address: oleksii.sokoliuk-at-mao.kiev.ua +§Electronic address: abaransky-at-ukr.net +arXiv:2301.03121v1 [hep-th] 8 Jan 2023 + +2 +dual denoted as AdS/BCFT correspondence, is defined on a manifold boundary in a D-dimensional +manifold M to a (D+1)-dimensional asymptotically AdS space N in order to ∂N = M∪Q. Here, +Q corresponds to a D-dimensional manifold that satisfies ∂Q = ∂M (see Figure 1). +FIG. 1: Schematic representation of the AdS/BCFT correspondence. Here, M represents the manifold with +boundary ∂M where the CFT is present. On the other hand, the gravity side is represented by N, which is +asymptotically AdS is M. Together with the above, ∂M is extended into the bulk AdS, which constitutes +the boundary of the D−dimensional manifold Q. +At the moment to explore the AdS/CFT correspondence, we impose the Dirichlet boundary +condition at the boundary of AdS, and therefore we require the Dirichlet boundary condition on M. +Nevertheless, according to [3, 4], for AdS/BCFT duality a Neumann boundary condition (NBC) +on Q is required, given that this boundary should be dynamical, from the viewpoint of holography, +and there is no natural definite metric on Q specified from the CFT side [5]. +On the other hand, the AdS/BCFT conjecture appears in many scenarios of the transport +coefficients, where black holes take a providential role, such for example Hawking-Page phase +transition, the Hall conductivity and the fluid/gravity correspondence [4, 6–11]. Together with the +above, this duality finds its natural roots in the holographic derivation of entanglement entropy +[12] as well as in the Randall-Sundrum model [13]. In fact, this extension of the CFT’s boundary +inside the bulk of the AdS-space is considered a modification of a thin Randall-Sundrum brane, +which intersects the AdS boundary. For this brane to be a dynamical object, we need to impose, as +was shown before, NBC where the discontinuity in the bulk extrinsic curvature across the defect, +is compensated by the tension from the brane. Furthermore, these boundaries are known as the +Randall-Sundrum (RS) branes in the literature. +Following the above, Fujita et al. [14] propose a model with gauge fields in the AdS4 background +with boundary RS branes. In this setup, the authors show that the additional boundary conditions +impose relevant constraints on the gauge field parameters, deriving the Hall conductivity behavior + +M +Q +N3 +in the dual field theory. Nevertheless, this approach does not consider the back reaction of the gauge +fields on the geometry, constraining the geometry of the empty AdS space. A natural extension +and generalization from the above work was constructed in [6]. +In the present paper, we are interested in constructing configurations describing a physical sys- +tem at finite temperature and charge density. For this, we consider the most common playground, +provided by the charged AdS4 black holes. This background has already been shown to encode +many interesting condensed-matter-like phenomena such as superconductivity/superfluidity [15, 16] +and strange metallic behaviors [17], via an action characterized by the well-known Einstein-Hilbert +structure together with a cosmological constant and Abelian gauge fields. It is interesting to note +that the above toy model can be extended in the presence of boundaries within a special case of +the Horndeski gravity [18], (see for example [19–26]). Here, the gravity theory is given through the +Lagrangian +LH = κ +� +(R − 2Λ) − 1 +2(αgµν − γ Gµν)∇µφ∇νφ +� +, +(1) +where R, Gµν and Λ are the scalar curvature, the Einstein tensor, and the cosmological constant +respectively, φ = φ(r) is a scalar field, α and γ are coupling constants, while that κ = 1/(16πGN), +where GN is the Newton Gravitational constant. The Lagrangian (1) has been exhaustively ex- +plored from the perspective of hairy black hole configurations [27–31], boson and neutron stars +[33–35], Hairy Taub-NUT/Bolt-AdS solutions [36], as well as holographic applications such that +quantum complexity and shear viscosity [37–41]. +On the other hand, through this work the physical system analyzed is based on the model +proposed by [6, 14]. Here, as we will see in the following lines, we start from the same Lagrangian +for a Horndeski-Maxwell system, this is (1), together with the Maxwell Lagrangian +LM = − κ +4e2 F µνFµν, +(2) +where e is a coupling constant and Fµν = ∂µAν − ∂νAµ is the Maxwell stress tensor, describing +the gravity dual of a field theory on a half-plane. In the simple plane-symmetric black hole ansatz, +we have that only tensionless RS branes are allowed, and that the background solution must be +not allowed to model the situation with external electric fields, as in [14]. Even more, as a result +of the NBC for the gauge fields, and showing in [6], the charge density ρ in the dual field theory +must be supported by an external magnetic field B, where the ratio ρ/B, which is equal to the +Hall conductivity, is a constant inversely proportional to the coefficients. In our prescription, this +represents the topological terms present in the gravity action: namely, a m2 in the bulk action, that + +4 +is, an antisymmetric tensor field Mµν which is the effective polarization tensor of the term in the +boundary action on the RS branes [42–44]. Such behaviors are expected for a quantum Hall system +tuned to a quantized value of the conductivity. Furthermore, we provided similar results in the +AdS/BCFT holographic model, where, for example, we will see how accurately it can account for +the physical behaviors expected in a quantum Hall system where, as was showed before, through +AdS/BCFT construction the Hall conductivity is inversely proportional to the coefficients of the +terms that appear in the gravity Lagrangian. Additionally, the ratio ρ/B will indicate a localized +condensate [45, 46]. +Just for completeness, as discussed in [6], for the classical Hall effect, the charge density and the +external magnetic field are independent quantities, that is, the ρ/B ratio depends on the density +of conductance electrons. On the other hand, in the quantum Hall Effect (QHE) the transverse +conductivity given by σH, has plateaus that are independent of either ρ or B. These plateaus are +generally attributed to disorder [47–49], being responsible for the existence of localized electron +states [6]. Here, the localized states fill the gaps between the Landau levels. Nevertheless, there is +no active participation in the Hall conductivity. +Finally, we study the properties of holographic paramagnetism-ferromagnetism phase transition +in the presence of Horndeski gravity (1). Here, from the matter field part, we consider the effects +of the Maxwell field (2) on the phase transition of this system, following [50, 51], introducing a +massive 2-form coupled field, and neglect the effects of this 2-form field and gauge fields on the +background geometry. In our analysis, we observe that increasing the strength of parameter γ, +given in (1), decreases the temperature of the holographic model and leads to a harder formation +of the magnetic moment in the black hole background. On the other hand, at low temperatures, +spontaneous magnetization, and ferromagnetic phase transition happen, but when removes the +external magnetic field, this magnetization disappears. As we know, ferromagnetic materials have +coercivity, which is the ability to keep their elementary magnets stuck in a certain position. This +position can be modified by placing the magnetized material in the presence of an external magnetic +field. In this way, a material with high coercivity its elementary magnets resists the change of +position. In the material science, experimental framework [52], there is a close relationship between +the magnetic related to viscosity and coercivity, this relationship was predicted theoretically and +observed experimentally. Thus, we have a fundamental role in both cases, that is, between viscosity +and coercivity, where they play the so-called activation volume, which is the relevant volume where +thermally activated and field-induced magnetization processes occur, respectively. In our work, we +will study this way for the paramagnetic material to resist the external magnetic field, through the + +5 +viscosity/entropy ratio. In our model, this relationship depends on the external magnetic field, the +Horndeski parameters, and the boundary size ∆ yQ of the RS brane in a non-trivial way. +This work is organized as follows: In Section II we consider the gravitational setup, which con- +tains all the information with respect to the AdS4/BCFT3 duality, showing the solution. Together +with the above, in Section III the charge density is obtained for then, in Section IV to present +the boundary Q profile. In Section V, we perform a holographic renormalization, computing the +Euclidean on-shell action, which is related to the free energy of the corresponding thermodynamic +system, where in particular we will focus on the black hole entropy, present in Section VI, and +the holographic paramagnetism/ferromagnetism phase transition, given in Section VII. Finally, +Section VIII is devoted to our conclusions and discussions. +II. +BLACK HOLE AS A PROBE OF ADS/BCFT +As was shown in the introduction, we will present our setup starting with the total action, +which contains all information related to AdS4/BCFT3 correspondence with probe approximation, +so that: +S = SN +H + SN +M + SN +2−FF + SN +mat + SQ +bdry + SQ +mat + SQ +ct, +(3) +where +SN +H = +� +N +d4x√−g LH, +SN +M = +� +N +d4x√−g LM, +(4) +with LH and LM given previously in (1)-(2) respectively, while that SN +mat is the action associated +to matter sources and: +SQ +bdry = 2κ +� +Q +d3x +√ +−hLbdry +SQ +mat = 2 +� +Q +d3x +√ +−hLmat, +SQ +ct = 2κ +� +ct +d3x +√ +−hLct , +(5) +with +Lbdry = (K − Σ) − γ +4(∇µφ∇νφnµnν − (∇φ)2)K − γ +4∇µφ∇νφKµν , +(6) +Lct = c0 + c1R + c2RijRij + c3R2 + b1(∂iφ∂iφ)2 + · · · , +(7) +where in our notations (∇φ)2 = ∇µφ∇µφ. In Eq.(6), Lbdry corresponds to the Gibbons-Hawking γ- +dependent terms associated with the Horndeski gravity (1), where Kµν = h β +µ ∇βnν is the extrinsic + +6 +curvature, K = hµνKµν is the trace of the extrinsic curvature, hµν is the induced metric, nµ is an +outward pointing unit normal vector to the boundary of the hypersurface Q, Σ is the boundary +tension on Q. +Lmat is the matter Lagrangian on Q, while that in Eq. (7) Lct represents the +boundary counterterms, which do not affect the bulk dynamics and will be neglected. +Following the procedures presented by [3, 4, 6, 10, 11] we have imposed the NBC: +Kαβ − hαβ(K − Σ) − γ +4Hαβ = κSQ +αβ , +(8) +where +Hαβ ≡ (∇σφ∇ρφ nσnρ − (∇φ)2)(Kαβ − hαβK) − (∇αφ∇βφ)K , +(9) +SQ +αβ = − +2 +√ +−h +δSQ +mat +δhαβ . +(10) +Considering the matter stress-energy tensor on Q as a constant (this is SQ +αβ = 0), we can write +Kαβ − hαβ(K − Σ) − γ +4Hαβ = 0 . +(11) +On the other hand, from the gravitational part, given by the Einstein-Horndeski theory and as- +suming that SN +mat is constant, varying SN +H and SQ +bdry with respect to the dynamical fields, we have: +Eαβ = − +2 +√−g +δSN +δgαβ , +Eφ = − +2 +√−g +δSN +δφ , +Fφ = − +2 +√ +−h +δSQ +bdry +δφ +, +(12) +where +Eµν = Gµν + Λgµν − α +2 +� +∇µφ∇νφ − 1 +2gµν∇λφ∇λφ +� ++ γ +2 +�1 +2∇µφ∇νφR − 2∇λφ∇(µφRλ +ν) − ∇λφ∇ρφRµλνρ +� ++ γ +2 +� +−(∇µ∇λφ)(∇ν∇λφ) + (∇µ∇νφ)2φ + 1 +2Gµν(∇φ)2 +� +− γgµν +2 +� +−1 +2(∇λ∇ρφ)(∇λ∇ρφ) + 1 +2(2φ)2 − (∇λφ∇ρφ)Rλρ +� +, +(13) +Eφ = ∇µ [(αgµν − γGµν) ∇νφ] , +(14) +Fφ = −γ +4(∇µ∇νφnµnν − (∇2φ))K − γ +4(∇µ∇νφ)Kµν , +(15) +and note that, Eφ = Fφ, from the Euler-Lagrange equation. +Together with the above, and according to [27–31] , we have a condition that deals to static +black hole configurations, avoiding no-hair theorems [32]. Here, we need to require that the square + +7 +of the radial component of the conserved current must vanish identically without restricting the +radial dependence of the scalar field. Such discussion implies that in Eq. (14): +αgrr − γGrr = 0 , +(16) +and defining φ′(r) ≡ ψ(r), where (′) denotes the derivative with respect to r, we can show that the +equations Eφ = 0 = Err are satisfied. In our setup, the four dimensional metric is defined via the +following line element +ds2 = L2 +r2 +� +−f(r) dt2 + dx2 + dy2 + dr2 +f(r) +� +, +(17) +where x1 ≤ x ≤ x2 and y1 ≤ y ≤ y2, while that from Refs.[10, 20, 30], f(r) is the metric function +which takes the form +f(r) = αL2 +3γ +� +1 − +� r +rh +�3� +, +(18) +while that ψ(r) reads +ψ2(r) = (φ′(r))2 = −2L2(α + γΛ) +αγr2f(r) +, +(19) +where +φ(r) = ±2 +� +−6(α + Λγ) +3α +tanh−1 +�� +1 − r3 +r3 +h +� ++ φ0. +(20) +Here, φ0 and rh are integration constants, where the last one is related to the location of the event +horizon. Following the steps of [10, 20], performing the transformations +f(r) → αL2 +3γ f(r), +t → 3γ +αL2 t, +x → +� +3γ +αL2 x, +y → +� +3γ +αL2 y, +L → +� α +3γ L2, +(21) +we have that the line element (17) is invariant, but now the metric function f(r) takes the form +f(r) = 1 − +� r +rh +�3 +(22) +while the square of the derivative of the scalar field ψ2(r) takes the form given previously in (19). +Here is important to note that from Eqs. (19)-(20) we can see that to have a real scalar field, +α + Λγ ≤ 0, +where it vanishes when α = −Λγ. +It is important to note that, from the action (3), we can see that there is another contribution, +denoted as SN +2−FF, which is responsible to construct the ferromagnetic/paramagnetic model. The +above will be explained in the following section. + +8 +III. +THE FINITE CHARGE DENSITY +As was shown in the previous section, in the action (3) appears the additional contribution +SN +2−FF = λ2 +� +N +d4x√−g L2−FF, +where +L2−FF = − 1 +12(dM)2 − m2 +4 MµνMµν − 1 +2MµνFµν − J +8 V (M). +(23) +Here, the above action defined from the seminal works [42, 43], is coupled through the constant λ +and constructed via the 2-form Mµν, dM is the exterior differential of the 2-form field Mµν, this is +(dM)τµν = 3∇[τMµν] and (dM)2 = 9∇[τMµν]∇[τMµν], m is a constant related to the mass, while +that V (M) describes the self-interaction of polarization tensor, with J a constant, which reads +V (M) = (∗MµνMµν)2 = [∗(M ∧ M)]2, +(24) +where (∗) is the Hodge star operator, this is ∗Mµν = +1 +2!εαβ +µνMαβ and εαβ +µν is the Levi-Civita +Tensor. In the following lines, will restrict our analysis to the probe approximation, that is, from +the action Eq. (3), one can derive the corresponding equations of motions for matter fields in the +probe approximation, that is, e2 → ∞ and λ → 0, so that: +∇µ +� +Fµν + λ2 +4 Mµν +� += 0, +(25) +∇τ(dM)τµν − m2Mµν − J(∗MτσMτσ)(∗Mµν) − Fµν = 0 . +(26) +Given that we are focusing on the probe limit approximation, we are going to disregard any +back reaction coming from the two-form field Mµν. In order to analyze the holographic paramag- +netism/ferromagnetism and paraelectric/ferroelectric phase transition, we consider the gauge fields +Mµν and Aµ we consider the following ansatz: +Mµν = −p(r) dt ∧ dr + ρ(r) dx ∧ dy, +(27) +Aµ = At(r) dt + Bx dy, +F = dA, +(28) +where B is the external magnetic field. Using (17), (27)-(28) in the background (22), the field +equations (25) and (26) are given by +A′ +t + +� +m2 − 4 J r4 ρ2 +L4 +� +p = 0, +(29) +ρ′′ +L2 + +�f′ +f + 2 +r +� ρ′ +L2 − +�4 J r2 p2 +fL4 ++ m2 +r2 f +� +ρ − B +r2 f = 0, +(30) +A′′ +t + λ2 +4 p′ = 0 , +(31) + +9 +As we work with probe approximation, the back reaction can be neglected. Together with the +above, given that the behaviors are asymptotically AdS4, we can solve the field equations (29)-(31) +near to the boundary (this is r → 0). Here, asymptotic solutions are given by +At(r) ∼ µ − σr, +(32) +p(r) ∼ σ +m2 , +(33) +ρ(r) ∼ ρ+r∆+ + ρ−r∆− − B +m2 , +(34) +∆± = −1 ± +√ +1 + 4L2m2 +2 +. +(35) +Here, ρ+ and ρ− correspond to the source and vacuum expectation value of the dual operator in +the boundary field theory (up to a normalization factor), respectively. It is worth pointing out +that one should take ρ+ = 0, in order to obtain condensation spontaneously [43]. From Eq. (34), +we can define ρ+ and ρ− as +ρ+ ≡ r−∆+ +h +, +ρ− ≡ r−∆− +h +, +(36) +yielding to the asymptotic solution ρ(r) the following structure +ρ(r) ∼ +� r +rh +�∆+ ++ +� r +rh +�∆− +− B +m2 . +(37) +Additionally, and according to [8], we can to analyze the electromagnetic field, extracted from the +four dimensional electromagnetic duality, in a sense that the theory is invariant under +Fµν →∗ Fµν = 1 +2εµναβF αβ, +(38) +where, as before, εαβµν is the Levi-Civita Tensor, transforming the electric field into a magnetic field +and vice versa. Such duality gives that, from the action (2), FµνF µν = (∗Fµν)(∗F µν), showing that +is invariant under (38). Besides, the transformation (38) shows that Frt → (∗Frt) = Fxy = σ = B, +where σ (B) is the constant related to the electric (magnetic) field. +IV. +Q-BOUNDARY PROFILE +In this section, we present the boundary Q profile, we assume that Q is parameterized by the +equation y = yQ(r), analyzing the influence of the Horndeski action (1), (4). For this, to find the +extrinsic curvature, one has to consider the induced metric on this surface, which reads +ds2 +ind = L2 +r2 +� +−f(r)dt2 + dx2 + g2(r)dr2 +f(r) +� +, +(39) + +10 +where g2(r) = 1 + y′2(r)f(r) and (′) denotes the derivative with respect to the coordinate r. Here, +the normal vectors on Q are represented by +nµ = +r +Lg(r) +� +0, 0, 1, −f(r)y′(r) +� +. +(40) +Considering the field equation Fφ = 0 (15), one can solve the Eq. (11), yielding +y′(r) = +(ΣL) +� +� +� +�4 + γψ2(r) +4 +− (ΣL)2 +� +1 − +� r +rh +�3� , +(41) +and, with ψ2(r) given previously in Eq. (19), we have +y′(r) = +(ΣL) +� +� +� +� +� +� +4 − +ξL2 +2r2 +� +1 − +� r +rh +�3� − (ΣL)2 +� +1 − +� r +rh +�3� , +(42) +where we define +ξ = α + γΛ +α +. +(43) +With all this information, we can plot the yQ profile from Eq. (42), representing the holographic +description of BCFT considering the theory (1). +FIG. 2: The figure shows the numerical solution for Q boundary profile from Eq. (42) for the black hole +within Horndeski gravity, considering the values for θ′ = 2π/3, θ = π − θ′, Λ = −1, α = 8/3 with γ = 0 +(pink curve), γ = 0.1 (blue dashed curve ), γ = 0.2 (red dot dashed curve), and γ = 0.3 (green thick curve). +The dashed parallel vertical lines represent the UV solution, Eq. (46), that is, Randall-Sundrum branes. +The region between the curves Q represents the bulk N. + +Yo +-yo +0 +- +Q +N +N +rh +r11 +On the other hand, following the steps of [6, 7], we have that the NBC on the gauge field +is nµFµν|Q = 0, and B = σ. The holographic model (AdS4/BCFT3) predicts that a constant +boundary current in the bulk induces a constant current on the boundary Q. +Such boundary +current can be measured in materials graphene-like. Furthermore, nµMµν|Q = 0 provide +ρ(r) +B += f(r)y′(r) +m2 +. +(44) +Here, the density ρ and the magnetic field B are no longer two independent parameters. As the +ratio is the Hall conductivity, this is very reminiscent of the quantum Hall effect (QHE), where this +ratio is independent of both ρ and B and is inversely proportional to the topological coefficients, +which in our case are the coupling constant γ presents in the Horndeski gravity, together with the +parameter from the antisymmetric tensor field Mµν, this is m2. In our case, the equation of y′ from +(42) and then the ρ/B ratio (44) can be analyzed by numerical calculations, being represented in +Fig. 3. Here, we show the ratio ρ/B with respect to external magnetic field B for different values +of the Horndeski gravity parameter γ, where we introduced ΣL = cos(θ′), where θ′ represents +the angle between the positive direction of the y axis and Q. At the boundary Q, the curves of +solutions in the (ρ, B) plane will be a localized condensate [45, 46]. +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +0.11 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +r +Ρ +B +FIG. 3: Graphic of the ratio ρ/B with respect to external magnetic field B versus r, for different values of +the Horndeski parameter γ. Here, we consider the values rh = 0.1, L = 1, θ′ = 2π/3, Λ = −1, α = 0.5, +m = 1, and γ = 1 (represented through the blue curve), γ = 4 (represented through the red curve), and +γ = 8 (represented through the green curve). + +12 +Together with the above, in addition to the above numerical solution, we can analyze some +particular cases regarding the study of the UV and IR regimes. Thus, for the first case, performing +an expansion at r → 0 with, as before, ΣL = cos(θ′), the equation (42) becomes +yUV (r) = y0 + +� +2 +−ξL2 r cos(θ′), +(45) +where y0 is an integration constant. In the above equation, considering ξ → −∞, we have +yUV (r) = y0 = constant. +(46) +This is equivalent to keeping ξ finite and a zero tension limit Σ → 0, considering the cases θ′ = π/2 +and θ′ = 3π/2. Now, for this regime, we have that the ρ/B ratio takes the form +ρ +B = +� +2 +−ξL2 +cos(θ′) +m2 +. +(47) +Here, it turns out a straightforward generalization of a known AdS4/CFT3 solution, given by +the plane-symmetric charged four-dimensional AdS black hole, where only allows for tensionless +RS branes in the AdS4/BCFT3 construction [6]. In this case, requires that the static uniform +charge density is supported by a magnetic field. Specifically, we found that ρ/B is a constant +proportional to a ratio of the coefficients appearing in the Horndeski gravity. +These analyses +indicate a generalization of the AdS4 black hole can describe a quantum Hall system at a plateau +of the transverse conductivity. Additionally, the AdS/BCFT setup yields that the Hall conductivity +is inversely proportional to a sum of the coefficients of the topological terms appearing in the gravity +Lagrangian. That is, we obtain that σH = ρ/B, which from the equation (47) +σH = +� +2 +−ξL2 +cos(θ′) +m2 +, +(48) +where, as was shown in the introduction, in QHE the conductivity is related to the number of filled +Landau levels (filling fraction), namely, by +h +e2 σH = +� +2 +−ξL2 +cos(θ′) +m2 +, +(49) +where e2/h is the magnetic flux quantum. +In this way, the holographic description seems to +provide results similar to the description of the QHE obtained in [48, 49]. In our case, we have an +extension of the covariant form of the Hall relation ρ = σHB. +For the IR case, we take r → +∞ so that from Eq. (42) implies that limr→+∞(φ′(r))2 = 0, and +then φ = constant, which ensures a genuine vacuum solution. Plugging this result in Eq. (42), in +the limit r → ∞, we have +y′ +IR(r) ∼ +�rh +r +�3/2 ++ O +� 1 +r2 +� +, +(50) + +13 +and y +′ +IR(r) → 0 when r → +∞, which implies from (47) that ρ/B → 0. Such value becomes the +on-shell action finite. +For the sake of completeness, an approximate analytical solution for y(r) can be obtained by +performing an expansion for ξ very small from Eq. (42), this is +y′ +Q = +cos(θ′) +� +4 − cos2(θ′)f(r) ++ +L2 cos(θ)ξ +4r2f(r)(4 − cos2(θ′)f(r))3/2 + O(ξ2), +with f given previously in (22), and considering this expansion up to the first order, we obtain +yQ(r) = y0 + +r cos(θ′) +� +(r3 − r3 +h) cos2(θ′) + 4r3 +h +� +4r3 +h − (r3 − r3 +h) cos(2θ′) +4 − cos2(θ′) +×2F1 +�1 +3, 1 +2; 4 +3; − +r3 cos2(θ′) +r3 +h(4 − cos2(θ′)) +� ++ ξ +� +L2 cos(θ) +4r2f(r)(4 − cos2(θ′)f(r))3/2 dr + O(ξ2), +(51) +where 2F1(a, b; c; x) is the hypergeometric function. +V. +HOLOGRAPHIC RENORMALIZATION +In our setup, we will compute the Euclidean on-shell action, which is related to the free energy +of the corresponding thermodynamic system. Thus, our holographic scheme takes into account +the contributions of AdS4/BCFT3 correspondence within Horndeski gravity. Let us start with the +Euclidean action given by IE = Ibulk + 2Ibdry, i.e., +Ibulk = − +1 +16πGN +� +N +√gd4x +� +R − 2Λ + γ +2Gµν∇µφ∇νφ +� +− +1 +8πGN +� +M +d3x√¯γLM, +(52) +LM = K(¯γ) − Σ(¯γ) − γ +4(∇µφ∇νφnµnν − (∇φ)2)K(¯γ) − γ +4∇µφ∇νφK(¯γ) +µν . +(53) +Together with the above, g is the determinant of the metric gµν on the bulk N, while that ¯γ is the +induced metric, the surface tension on M is represented with Σ(¯γ), and K(¯γ) corresponds to the +extrinsic curvature on M. On the other hand, for the boundary, we have the following expressions +Ibdry = − +1 +16πGN +� +N +√gd4x +� +R − 2Λ + γ +2Gµν∇µφ∇νφ +� +− +1 +8πGN +� +Q +d3x +√ +hLbdry, +(54) +Lbdry = (K − Σ) − γ +4(∇µφ∇νφnµnν − (∇φ)2)K − γ +4∇µφ∇νφKµν. +(55) +Thus, in order to compute the bulk action Ibulk, we consider the induced metric on the bulk, which +is obtained from (17) after the transformation τ = it, given by +ds2 +ind = ¯γµνdxµdxν = L2 +r2 +� +f(r)dτ 2 + dx2 + dy2 + dr2 +f(r) +� +. +(56) + +14 +Here, we have that 0 ≤ τ ≤ β, where from Eq. (22) +β = 1 +T = +�|f′(rh)| +4π +�−1 += 4πrh +3 +, +(57) +where T is the Hawking Temperature, the above allows us to obtain the following quantities: +R = − 12 +L2 , +Λ = − 3 +L2 , +K(¯γ) = 3 +L, +Σ(¯γ) = 2 +L. +Thus, we have all elements needed to construct the bulk action Ibulk. In the process of holographic +renormalization, we need to introduce a cutoff ϵ to remove the IR divergence on the bulk side and +we can provide that: +Ibulk = +1 +16πGN +� +d2x +� +4πrh +3 +0 +dτ +� rh +ϵ +dr√g +� +R − 2Λ + γ +2Grrψ2(r) +� ++ +1 +16πGN +� +d2x +� +4πrh +3 +0 +dτ L2� +f(ϵ) +ϵ3 +, +(58) +Ibulk = − L2V +8r2 +hG +� +1 − ξ +4 +� +, +(59) +with ξ given previously in (61) and, in our notations, V = +� +d2x = ∆x∆y = (x2 − x1)(y2 − y1). +Now, computing the Ibdry, we introduce a cutoff ϵ to remove the UV divergence on the boundary +side, and with this information, we have: +Ibdry = rhL2∆yQ +2GN +� +1 − ξ +4 +� � rh +ϵ +∆yQ(r) +r4 +dr − rhL2 sec(θ′)∆yQ +2GN +� rh +ϵ +∆yQ(r) +r3 +dr +(60) +Here, ∆yQ is a constant and ∆yQ(r) := yQ(r) − y0 is obtained from the equation (51). As we +know, from the point of view of AdS/CFT correspondence, IR divergences in AdS correspond to +UV divergences in CFT. This relationship is known as the IR-UV connection. Thus, based on +this duality, we can reduce the above equation (60) after some eliminations of terms that produce +divergences to the following form: +Ibdry = −L2∆ yQ +2GN +� +1 − ξ +4 +� � +ξ L2b(θ′) +5r4 +h ++ q(θ +′) +4r2 +h +� ++L2 sec(θ′)∆ yQ +2GN +� +ξ L2b(θ′) +4r3 +h ++ q(θ +′) +2rh +� +, +(61) +where +b(θ′) = +cos(θ′) +4(4 − cos2(θ′))3/2 , +q(θ′) = +cos(θ′) +� +4 − cos2(θ′) +. +(62) +With all the above information, from Eqs. (59) and (61)-(62), we can compute IE = Ibulk + 2Ibdry +as: + +15 +IE = − L2V +8r2 +hGN +� +1 − ξ +4 +� +− L2∆ yQ +GN +� +1 − ξ +4 +� � +ξ L2b(θ′) +5r4 +h ++ q(θ +′) +4r2 +h +� ++L2 sec(θ′)∆ yQ +GN +� +ξ L2b(θ′) +4r3 +h ++ q(θ +′) +2rh +� +. +(63) +Here, IE is the approximated analytical expression for the Euclidean action. +This equation is +essential to construct the free energy and extract all thermodynamic quantities in our setup, as we +show in the next section. +VI. +BLACK HOLE ENTROPY +Now, we will compute the entropy related to the black hole considering the contributions of the +AdS/BCFT correspondence in the Horndeski gravity. Free energy is defined as +Ω = TIE , +(64) +one can obtain the corresponding entropy as: +S = −∂ Ω +∂T +(65) +where T is the Hawking Temperature. By plugging the Euclidean on-shell action IE from Eq.(63), +and replacing T obtained previously in (57), we have +Stotal = Sbulk + Sbdry, +(66) +where +Sbulk = +L2V +4r2 +hGN +� +1 − ξ +4 +� +, +(67) +Sbdry = L2∆ yQ +GN +� +1 − ξ +4 +� � +ξ L2b(θ′) +5r4 +h ++ q(θ +′) +4r2 +h +� +− L2 sec(θ′)∆ yQ +GN +� +ξ L2b(θ′) +4r3 +h ++ q(θ +′) +2rh +� +. +(68) +The interpretation for this total entropy can be identified with the Bekenstein-Hawking formula +for the black hole: +SBH = +A +4GN +, +(69) + +16 +where, in this case +A = L2V +2r2 +h +� +1 − ξ +4 +� ++ 4L2∆ yQ +� +1 − ξ +4 +� � +ξ L2b(θ′) +5r4 +h ++ q(θ +′) +4r2 +h +� +−4L2 sec(θ′)∆ yQ +� +ξ L2b(θ′) +4r3 +h ++ q(θ +′) +2rh +� +. +(70) +Here, A is the total area of the AdS black hole in the Horndeski contribution terms for the bulk +and the boundary Q. We can see that the information is bounded by the black hole area. Then, +the equation (70) suggests that the information storage increases with increasing |ξ|, as long as +ξ < 0. +Together with the above, with respect to the boundary contribution of (68), we have that this +expression is the entropy of the BCFT corrected by the Horndeski terms parametrized by ξ, given +previously in (43). In this case, the results presented in Refs. [6, 11] are recovered in the limit +ξ → 0. Besides, still analyzing Eq. (68), due to the effects of the Horndeski gravity, there is a +non-zero boundary entropy even if we consider the zero temperature scenario, similar to an extreme +black hole. This can be seen if one takes the limit T → 0 (or rh → ∞) in Eq.(68), then we do not +get the denominated residual boundary entropy, as discussed in [10]. +On the other hand, through Eq. (47) we have +Smagnetic +bdry += L2∆ yQ +GN +� +1 − ξ +4 +� � +−2B2 cos2(θ +′) +m2ρ2 +b(θ′) +5r4 +h ++ q(θ +′) +4r2 +h +� +− L2 sec(θ′)∆ yQ +GN +� +−2B2 cos2(θ +′) +m2ρ2 +b(θ′) +4r3 +h ++ q(θ +′) +2rh +� +, +(71) +where m2 > −1/(4L2). For the entropy bound, the restriction on m2 comes from Eq. (35). A +well-defined probe limit demands that the charge density contributed by the polarization should +be finite. At low temperatures, below the critical, in the ferromagnetic region, we can observe that +our entropy is Sbdry +magnetic ∝ B2, that is, has a square dependence on the external magnetic field +and this is a characteristic of ferromagnetic systems. Furthermore, we can observe that Smagnetic +bdry +is the magnetic entropy of the boundary Q, and we can observe that for ferromagnetic materials, +the magnetic entropy is associated with the disorder of the magnetic moments. In addition, these +materials have spontaneous magnetization. So when we remove the applied magnetic field, they +still show magnetization. + +17 +VII. +HOLOGRAPHIC PARAMAGNETISM/FERROMAGNETISM PHASE +TRANSITION +In this section, we present the holographic paramagnetism/ferromagnetism phase transition +through the boundary contribution from the entropy (71). For this, we start considering the free +energy Ω from (63) -(64), where the first law of black holes thermodynamics, considering the +canonical ensemble, takes the form +dΩ = −PdV − SdT, +(72) +where, in addition to the entropy S as well as the Hawking temperature T, the pressure P and the +volume V appear, yielding +Ω = ϵ − TS, +where ϵ takes the role of the energy density. +As a first thermodynamic quantity to study, we will consider the entropy S, from Eq. (66), +calculated in the previous section, and represented graphically in Fig. +4, with respect to the +Hawking temperature T (57). +Here, in the right panel (left panel) there is (not) an external +magnetic field B. Concretely, we see that the right panel exhibit similar behavior as analyzed in +[53], as for example ferromagnetic materials with nearly zero coercivity and hysteresis. On the +other hand, in the left panel, when the external magnetic field is removed (this is B = 0), we still +have a disorder of magnetic moments, this is a characteristic of paramagnetism. +The second parameter that we analyze is the heat capacity CV , which allows us to analyze local +thermodynamic stability, defined in the following form +CV = T +�∂S +∂T +� +V += −T +�∂2Ω +∂T 2 +� +V +, +(73) +where the sub-index V from Eq. (73) represents at volume constant. From Fig. 5, we can see +that in the right panel, the black hole can switch between stable (CV > 0) and unstable (CV < 0) +phases, depending on the sign of heat capacity CV . +This phase transition occurs, due to the +spontaneous electric polarization, which was realized in our model from the application of the +magnetic external field. Moreover, in the region CV > 0, we have structures built like magnetic +domes on the boundary Q. Additionally, in Fig. 5, one can see the influence of Horndeski gravity +(represented via the constant γ) with respect to the temperature T, where the phase transition +occurs for some ranges of values for T when the external magnetic field is null, that is, B = 0. + +18 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.00 +0.05 +0.10 +0.15 +0.20 +T +S +0 +2 +4 +6 +8 +10 +0 +20 +40 +60 +80 +100 +120 +T +S�B�0� +FIG. 4: Right panel: The behavior of the entropy S with the temperature T with different values for +α = 8/3, m = 1/8, B = (4/5)T, ρ = 1/4, Λ = −1, V = 1, GN = 1, θ′ = 2π/3 with γ = 1 (pink curve), +γ = 4 (red dot dashed curve), γ = 8 (green thick curve). Left panel: The behavior of the entropy S with +respect the temperature T, with different values for B = 0. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +�6 +�4 +�2 +0 +T +CV +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.5 +1.0 +1.5 +T +CV �B�0� +FIG. 5: Right panel: The behavior of the heat capacity CV with the temperature T with different values +for α = 8/3, m = 1/8, B = (4/5)T, ρ = 1/4, Λ = −1, θ′ = 2π/3 with γ = 1 (pink curve), γ = 4 (red dot +dashed curve), γ = 8 (green thick curve). Left panel: The behavior of the heat capacity CV with respect +the temperature T, with different values for B = 0. +Additionally, we can obtain the heat capacity at constant pressure CP , which reads +CP = T +�∂S +∂T +� +P +, +(74) +and, from Fig. 6, we can see that in the right panel, the black hole can switch between stable +(CP > 0), describing a ferromagnetic material, and unstable (CP < 0), describing a paramagnetic +material, depending on the sign of heat capacity. This phase transition occurs, as in the previous +case, due to spontaneous electric polarization. Moreover, in the region CP > 0, we have structures + +19 +built like magnetic domes on the boundary Q, wherein the experimental specific frame, these heat +curves without magnetic field can represent a material like DyAl2 [53]. On the other hand, the +left panel represents the heat capacity CP where B = 0, where we can see, that is locally unstable +(CP < 0). +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +�4 +�2 +0 +2 +4 +T +CP +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +�4 +�2 +0 +2 +4 +T +CP �B�0� +FIG. 6: Right panel: The behavior of the CP with respect to the temperature T with different values for +α = 8/3, m = 1/8, B = (4/5)T, ρ = 1/4, Λ = −1, θ′ = 2π/3 with γ = 1 (pink curve), γ = 4 (red dot dashed +curve), γ = 8 (green thick curve). Left panel: The behavior of CP with respect T, with different values for +B = 0. +Additionally, we can derive other quantities, as for example the magnetization density m, and +magnetic susceptibility χ, following the steps of [46], given by +m = − +�∂ Ω +∂B +� += L2∆ yQT +GN +� +1 − ξ +4 +� � +4 cos2(θ +′) +m2ρ2 +b(θ′) +5r4 +h +� +− L2 sec(θ′)∆ yQT +GN +� +cos(θ +′) +m2ρ2 +b(θ′) +4r3 +h +� +, +(75) +χ = +� ∂2Ω +∂B2 +� += −L2∆ yQT +GN +� +1 − ξ +4 +� � +4B cos2(θ +′) +m2ρ2 +b(θ′) +5r4 +h +� ++ L2 sec(θ′)∆ yQT +GN +� +B cos(θ +′) +m2ρ2 +b(θ′) +4r3 +h +� +. (76) +As we can see from equations (75) and (76), the RS brane behaves like a paramagnetism material, +that is, when we remove the external magnetic field, the equation (76) disappears and the entropy +linked disorder increases, as shown in Fig. 4. On the other hand, from the equation (75), the +magnetization density is not null for zero magnetic fields (this is B = 0). Thus, we can conclude +that paramagnetic materials have a low coercivity, that is, their ability to remain magnetized is +very low. Thus, one way to analyze coercivity is through viscosity η in our model [52]. + +20 +In order to be as clear as possible, the details about the computation of the shear viscosity and +entropy density ratio are present in Appendix A. In particular, we will focus on the η/S ratio, where +from Eq. A11 and Fig. 7, we can analyze the dependence of the viscosity on the magnetic field, +characterizing a magnetic side effect, and describing the slow relaxation of the magnetization of +paramagnetic materials when they acquire magnetization in the presence of an external magnetic +field B (left panel of Fig. 7). In the right panel, we can observe that under an interval of the +temperature T, the η/S ratio is an increasing function when B = 0. +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +T +Η +S +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +20 +40 +60 +80 +100 +T +Η +S +�B�0� +FIG. 7: Right panel: The behavior of the η/S ratio as a function of the temperature T for different values +for α = 8/3, B = (4/5)T, ρ = 1/4, Λ = −1, γ = 1 (pink curve), γ = 2 (red dot dashed curve), γ = 2.5 +(green thick curve). Left panel: The behavior of η/s for B = 0. +0.0 +0.5 +1.0 +1.5 +2.0 +0 +1 +2 +3 +4 +B +Η +S +FIG. 8: The behavior of η/S with respect to the magnetic field B, for different values for α = 8/3, T = 4/5, +ρ = 1/4, Λ = −1, γ = 1 (pink curve), γ = 2 (red dot dashed curve), γ = 2.5 (green thick curve). +On the other hand, and as we can see from Fig. 8 at a temperature T fixed when we observe +as the paramagnetic material, represented by the RS brane, we can obtain a relation between η/S + +21 +with respect to the magnetic field B, which is a decreasing function. Here, when B becomes large, +we have that η/S → 0. +We finalize this section showing the magnetic moment N at a low temperature T, corresponding +to order parameter ρ in the absence of an external magnetic field, setting B = 0, and then compute +the value of N, defined as +N = λ2rh +2L +� 1 +0 +ρ(r)dr = −λ2rh +2L +� +− B +m2 + +1 +(∆+ + 1)r∆+ +h ++ +1 +(∆− + 1)r∆− +h +� +. +(77) +In Fig. 9, it can be found that as the temperature decreases, the magnetization increases and +the system is in the perfect order with the maximum magnetization at zero temperature. Thus, +increasing the Horndeski parameters lowers the magnetization value and the critical temperature. +Indeed, we have that the effect of a larger value of the parameters γ and m2 makes the magnetization +harder and the ferromagnetic phase transition happen, which is in good agreement with previous +works [50, 51]. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0 +2 +4 +6 +8 +10 +T +N +Λ2 +FIG. 9: The behavior of magnetic moment N with different values for B = 0, α = 8/3 with γ = 1; m2 = 2 +(blue curve), γ = 4; m2 = 4 (red curve), γ = 8; m2 = 6 (green curve). We consider in the Eq. 77 the +transformations Eq.∼(21). +Finally, we present the susceptibility density χ of the materials as a response to the magnetic +moment. Thus, this behavior is an essential property of ferromagnetic materials. In order to study +χ of the ferromagnetic materials in the Horndeski gravity and to consider the transformations Eq. +(21), we follow the definition +χ +λ2 = lim +B→0 +∂N +∂B = +� +3 +8πm2L2 +� 1 +T . +(78) + +22 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0.0 +0.5 +1.0 +1.5 +2.0 +T +Λ2 +Χ +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0 +2 +4 +6 +8 +10 +T +Χ +Λ2 +FIG. 10: The behavior of 1/χ in the function of the temperature T with different values for α = 8/3 with +γ = 1; m2 = 2 (blue curve), γ = 4; m2 = 4 (red curve), γ = 8; m2 = 6 (green curve). We consider in the Eq. +(78) the transformations given in Eq.(21). +In Fig.10, we have the behavior of 1/χ and χ as a function of the temperature T for different +choices of m2 and γ. In our case, in the right panel, we have that increasing each one of these pa- +rameters makes the susceptibility value decrease when the temperature increases. This fact agrees +with our expectation of paramagnetic materials because when we remove the external magnetic +field, the paramagnetic substance loses its magnetism. Its magnetic susceptibility is very small, +but positive, and decreases with increasing temperature. In fact, this magnetic susceptibility is +only part of the background black hole and the other part of the polarization field. For pure dionic +Reissner-Nordstr¨om-AdS black hole, we have a diamagnetic material. In this sense, in the chemical +reference, we have that a particle (atom, ion, or molecule) is paramagnetic or diamagnetic when +the electrons in the particle are paired due to the external magnetic field [50, 51]. +VIII. +CONCLUSIONS AND DISCUSSIONS +In four dimensions, we analyzed an AdS/BCFT model of a condensed matter system at finite +temperature and charge density living on a 2+1-dimensional space with a boundary, showing an +extension of the previous work presented in [10], where in addition to the contributions of the +theory together with the boundary terms, we include the components Aµ and Mµν, responsible to +construct the ferromagnetic/paramagnetic model. +Via the resolution of the field equations, and using the no-hair theorem, we extend to the +four-dimensional configuration obtained in [10, 30]. From the above solution, we present the Q +profile, found a numerical solution, and present it in Fig. 9, where the Horndeski parameter γ + +23 +takes an important role. Together with the above, we show that components of Mµν can be viewed +as dual fields of the order parameter in the paraelectric/ferroelectric phase transition in dielectric +materials. Through the NBC over nµM|Q, we found the ratio ρ/B, where for some particular +cases is a constant proportional to a ratio of the coefficients appearing in the gravity action. These +properties resemble a quantum Hall system, which suggests at the boundary Q in the (ρ, B) plane +will be a localized condensate. +Additionally, via the solution we performed a holographic renormalization, calculating the Eu- +clidean on-shell action, which is related to the free energy Ω, and allowing us to obtain the entropy +S and the heat capacities CV , CP , thanks to the contribution to the bulk as well as the boundary. +With respect to the entropy S, we show that when the magnetic field is present we see it exhibits +similar behavior as for example ferromagnetic materials with nearly zero coercivity and hysteresis. +Nevertheless, when B = 0 the disorder entropy of the magnetic moments increases, being a char- +acteristic of paramagnetism. Together with the above, with respect to CV and CP , we obtained +for both cases stable and unstable phases, due to the spontaneous electric polarization, which was +realized in our model from the application of the magnetic external field B, being influence via +the Horndeski gravity, represented through γ. We also show that the specific heat CP behaves +like a material of the type DyAl2, having a growth behavior similar to that expected from the +experimental point of view, as presented by [53]. +Currently, we can observe that the microscopic differences between real experimental systems, +in relation to theories with gravitational dual suggest that, in the near future, we will have measure- +ments of these values for experimental quantities obtained holographically. So many measurements +can realistically aspire to more than useful benchmarks. Furthermore, it is important to highlight +in this regard the need to take the big limit N in holographic calculations [1]. We now have a +clarity of the value of the ratio between shear viscosity and entropy density, η/S = 1/4π, which is +universal in classical gravity to usual classical gravity [54]. Furthermore, in the Horndeski gravity, +these relations are modified by the parameter γ. However, there are controlled corrections 1/N +for this result, which can be both positive and negative and which for realistic values of N show +significant changes in the numerical value of the ratio. As we show in our model, the violation of +this universal bound in the Horndeski gravity with gauge fields changes the η/S ratio (see Fig.7 +and Fig.8), where this behavior is similar to the results of [55]. Furthermore, as γ increases, we +can observe a translational symmetry breaking that survives the lower energy scales. According to +Fig. 8, we have η/S → 0 at low temperatures. +One of the strongest motivations for working with AdS/BCFT for condensed matter physics + +24 +rests on two pillars. The first is that, although theories with holographic duals may exhibit spe- +cific exotic features, they also have features that are expected to be generic to tightly coupled +theories, for example, the quantum critiques. In this sense, theories with gravitational duals are +computationally tractable examples of generic tightly coupled field theories, and we can use them +both to test our generic expectations and to guide us in refining those expectations. Thus, the +examples discussed here are special cases of the fact that real-time finite temperature transport is +much easier to calculate via AdS/BCFT than almost any other microscopic theory. +Acknowledgments +F.S. would like to thank the group of Instituto de F´ısica da UFRJ for fruitful discussions about +the paramagnetic systems. In special to the E. Capossoli, Diego M. Rodrigues and Henrique Boschi- +Filho. S.O. performed the work in the frame of the ”Mathematical modeling in interdisciplinary +research of processes and systems based on intelligent supercomputer, grid and cloud technologies” +program of the NAS of Ukraine. M.B. is supported by PROYECTO INTERNO UCM-IN-22204, +L´ıNEA REGULAR. +Appendix A: Shear viscosity and entropy density ratio with magnetic field +We will present the calculation of the ratio η/S following the procedures [20, 38, 39, 54, 55]. +For this, we consider a perturbation along the xy direction in the metric Eq.17 [20, 38], in this +sense, we have +ds2 = L2 +r2 +� +−f(r)dt2 + dx2 + dy2 + 2Ψ(r, t)dxdy + dr2 +f(r) +� +. +(A1) +From the overview point of the holographic dictionary, this procedure maps the fluctuation of the +diagonal in the bulk metric in the off-diagonal components of the dual energy-momentum tensor. +In this sense, we have a linear regime where fluctuations are associated with shear waves in the +boundary fluid. Thus, substituting this metric (A1) in the Horndeski equation (Eµν = 0) for µ = x +and ν = y, one obtains: +P1Ψ +′′(r, t) + P2Ψ +′(r, t) + P3 ¨Ψ(r, t) = 0 , +(A2) +where we defined +P1 = 9γ2(α − γΛ)f2(r), +P2 = −3γ(α − γΛ)f(r)(2αL2 − 6γr3/r3 +h), + +25 +P3 = −9γ2r(3α + γΛ). +(A3) +Using the ansatz: +Ψ(r, t) = e−iωtΦ(r), +(A4) +Φ(r) = exp +� +−iωK ln +�6γ2r3f(r) +G +�� +, +G = L2V +GN +� +1 − ξ +4 +� +, +(A5) +we obtain +K = +1 +4πT +� +3α + γΛ +α − γΛ , +(A6) +with T the Hawking temperature given previously in (57). At this point, we must evaluate the +Lagrangian (1), using the metric function from (22), and expand it up to quadratic terms in Ψ +and its derivatives [38]. In this way, we can study the boundary field theory using the AdS/CFT +correspondence where the quadratic terms in the Lagrangian, after removing the second derivative +contributions using the Gibbons-Hawking term, can be written as +Hshear = P1Ψ2(r, t) + P2 ˙Ψ(r, t) + P3Ψ +′2(r, t) + P4Ψ(r, t)Ψ +′(r, t), +(A7) +where +P1 = − 48L2 +9r7f(r), +P2 = 4γ L2 +r7 +, +P3 = +6γ2 +r3f(r), +P4 = (α + γΛ) 2γ2L4 +α r7f(r). +(A8) +Here, (˙) denotes the derivative with respect t. Finally, viscosity η is determined from the term +P3Ψ(r, t)Ψ +′(r, t) which reads +η = 1 +4π +G +4r2 +h +� +3α + γΛ +α − γΛ , +(A9) +where the entropy, from (66)-(68), can be written as +S = GF +4r2 +h +, +(A10) +with +F = 1 + +� +B2 cos2(θ′)b(θ′) +5m2ρ2 +�4πT +3 +�4 ++ q(θ +′) +4 +�4πT +3 +�2� +− +sec(θ′) +� +1 − ξ +4 +� +� +−B2 cos2(θ′)b(θ′) +2m2ρ2 +�4πT +3 +�3 ++ q(θ +′) +2 +�4πT +3 +�� +, + +26 +and T given in (57). 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Santos, Entropy production, viscosity bounds and bumpy black +holes, JHEP 03, 170 (2016) doi:10.1007/JHEP03(2016)170 [arXiv:1601.02757 [hep-th]]. + diff --git a/FtE1T4oBgHgl3EQfXARM/content/tmp_files/load_file.txt b/FtE1T4oBgHgl3EQfXARM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8bb2ce1916a1b209a88927bbd1deef01e9772b16 --- /dev/null +++ b/FtE1T4oBgHgl3EQfXARM/content/tmp_files/load_file.txt @@ -0,0 +1,993 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf,len=992 +page_content='AdS/BCFT correspondence and Horndeski gravity in the presence of gauge fields: from holographic paramagnetism/ferromagnetism phase transition Fabiano F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Santosa,∗ Mois´es Bravo-Gaeteb,† Oleksii Sokoliuk c,d,‡ and Alexander Baransky c§ aInstituto de F´ısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro-RJ, 21941-972 – Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' bFacultad de Ciencias B´asicas, Universidad Cat´olica del Maule, Casilla 617, Talca, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' c Astronomical Observatory, Taras Shevchenko National University of Kyiv, 3 Observatorna St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=', 04053 Kyiv, Ukraine, and dMain Astronomical Observatory of the NAS of Ukraine (MAO NASU), Kyiv, 03143, Ukraine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' This paper presents a dual gravity model for a (2+1)-dimensional system with a limit on finite charge density and temperature, which will be used to study the properties of the holographic phase transition to paramagnetism-ferromagnetism in the presence of Horndeski gravity terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In our model, the non-zero charge density is supported by a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' As a result, the radius ρ/B indicates a localized condensate, as we increase the Horndeski gravity parameter, that is represented by γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Furthermore, such condensate shows quantum Hall-type behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' This radius is also inversely related to the total action coefficients of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' It was observed that increasing the Horndeski parameter decreases the critical temperature of the holographic model and leads to the harder formation of the magnetic moment at the bottom of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' However, when removing the magnetic field, the ferromagnetic material presents a disorder of its magnetic moments, which is observed through the entropy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' We also found that at low temperatures, spontaneous magnetization and ferromagnetic phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' INTRODUCTION For almost thirty years, the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence has been a bridge that allows us to relate gravity and strongly coupled conformal field theories [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Following this spirit, a new holographic dual of a CFT arises, which is defined on a manifold M with a boundary ∂M, denoted as Boundary Conformal Field Theory (BCFT), proposed by Takayanagi [3] and Takayanagi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' [4], extending the AdS/CFT duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' This new holographic ∗Electronic address: fabiano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='ffs23-at-gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='com †Electronic address: mbravo-at-ucm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='cl ‡Electronic address: oleksii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='sokoliuk-at-mao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='kiev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='ua §Electronic address: abaransky-at-ukr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='net arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='03121v1 [hep-th] 8 Jan 2023 2 dual denoted as AdS/BCFT correspondence, is defined on a manifold boundary in a D-dimensional manifold M to a (D+1)-dimensional asymptotically AdS space N in order to ∂N = M∪Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Here, Q corresponds to a D-dimensional manifold that satisfies ∂Q = ∂M (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 1: Schematic representation of the AdS/BCFT correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Here, M represents the manifold with boundary ∂M where the CFT is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' On the other hand, the gravity side is represented by N, which is asymptotically AdS is M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Together with the above, ∂M is extended into the bulk AdS, which constitutes the boundary of the D−dimensional manifold Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' At the moment to explore the AdS/CFT correspondence, we impose the Dirichlet boundary condition at the boundary of AdS, and therefore we require the Dirichlet boundary condition on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Nevertheless, according to [3, 4], for AdS/BCFT duality a Neumann boundary condition (NBC) on Q is required, given that this boundary should be dynamical, from the viewpoint of holography, and there is no natural definite metric on Q specified from the CFT side [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' On the other hand, the AdS/BCFT conjecture appears in many scenarios of the transport coefficients, where black holes take a providential role, such for example Hawking-Page phase transition, the Hall conductivity and the fluid/gravity correspondence [4, 6–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Together with the above, this duality finds its natural roots in the holographic derivation of entanglement entropy [12] as well as in the Randall-Sundrum model [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In fact, this extension of the CFT’s boundary inside the bulk of the AdS-space is considered a modification of a thin Randall-Sundrum brane, which intersects the AdS boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' For this brane to be a dynamical object, we need to impose, as was shown before, NBC where the discontinuity in the bulk extrinsic curvature across the defect, is compensated by the tension from the brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Furthermore, these boundaries are known as the Randall-Sundrum (RS) branes in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Following the above, Fujita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' [14] propose a model with gauge fields in the AdS4 background with boundary RS branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In this setup, the authors show that the additional boundary conditions impose relevant constraints on the gauge field parameters, deriving the Hall conductivity behavior M Q N3 in the dual field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Nevertheless, this approach does not consider the back reaction of the gauge fields on the geometry, constraining the geometry of the empty AdS space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' A natural extension and generalization from the above work was constructed in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In the present paper, we are interested in constructing configurations describing a physical sys- tem at finite temperature and charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' For this, we consider the most common playground, provided by the charged AdS4 black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' This background has already been shown to encode many interesting condensed-matter-like phenomena such as superconductivity/superfluidity [15, 16] and strange metallic behaviors [17], via an action characterized by the well-known Einstein-Hilbert structure together with a cosmological constant and Abelian gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' It is interesting to note that the above toy model can be extended in the presence of boundaries within a special case of the Horndeski gravity [18], (see for example [19–26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Here, the gravity theory is given through the Lagrangian LH = κ � (R − 2Λ) − 1 2(αgµν − γ Gµν)∇µφ∇νφ � , (1) where R, Gµν and Λ are the scalar curvature, the Einstein tensor, and the cosmological constant respectively, φ = φ(r) is a scalar field, α and γ are coupling constants, while that κ = 1/(16πGN), where GN is the Newton Gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' The Lagrangian (1) has been exhaustively ex- plored from the perspective of hairy black hole configurations [27–31], boson and neutron stars [33–35], Hairy Taub-NUT/Bolt-AdS solutions [36], as well as holographic applications such that quantum complexity and shear viscosity [37–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' On the other hand, through this work the physical system analyzed is based on the model proposed by [6, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Here, as we will see in the following lines, we start from the same Lagrangian for a Horndeski-Maxwell system, this is (1), together with the Maxwell Lagrangian LM = − κ 4e2 F µνFµν, (2) where e is a coupling constant and Fµν = ∂µAν − ∂νAµ is the Maxwell stress tensor, describing the gravity dual of a field theory on a half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In the simple plane-symmetric black hole ansatz, we have that only tensionless RS branes are allowed, and that the background solution must be not allowed to model the situation with external electric fields, as in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Even more, as a result of the NBC for the gauge fields, and showing in [6], the charge density ρ in the dual field theory must be supported by an external magnetic field B, where the ratio ρ/B, which is equal to the Hall conductivity, is a constant inversely proportional to the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In our prescription, this represents the topological terms present in the gravity action: namely, a m2 in the bulk action, that 4 is, an antisymmetric tensor field Mµν which is the effective polarization tensor of the term in the boundary action on the RS branes [42–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Such behaviors are expected for a quantum Hall system tuned to a quantized value of the conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Furthermore, we provided similar results in the AdS/BCFT holographic model, where, for example, we will see how accurately it can account for the physical behaviors expected in a quantum Hall system where, as was showed before, through AdS/BCFT construction the Hall conductivity is inversely proportional to the coefficients of the terms that appear in the gravity Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Additionally, the ratio ρ/B will indicate a localized condensate [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Just for completeness, as discussed in [6], for the classical Hall effect, the charge density and the external magnetic field are independent quantities, that is, the ρ/B ratio depends on the density of conductance electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' On the other hand, in the quantum Hall Effect (QHE) the transverse conductivity given by σH, has plateaus that are independent of either ρ or B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' These plateaus are generally attributed to disorder [47–49], being responsible for the existence of localized electron states [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Here, the localized states fill the gaps between the Landau levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Nevertheless, there is no active participation in the Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Finally, we study the properties of holographic paramagnetism-ferromagnetism phase transition in the presence of Horndeski gravity (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Here, from the matter field part, we consider the effects of the Maxwell field (2) on the phase transition of this system, following [50, 51], introducing a massive 2-form coupled field, and neglect the effects of this 2-form field and gauge fields on the background geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In our analysis, we observe that increasing the strength of parameter γ, given in (1), decreases the temperature of the holographic model and leads to a harder formation of the magnetic moment in the black hole background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' On the other hand, at low temperatures, spontaneous magnetization, and ferromagnetic phase transition happen, but when removes the external magnetic field, this magnetization disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' As we know, ferromagnetic materials have coercivity, which is the ability to keep their elementary magnets stuck in a certain position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' This position can be modified by placing the magnetized material in the presence of an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In this way, a material with high coercivity its elementary magnets resists the change of position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In the material science, experimental framework [52], there is a close relationship between the magnetic related to viscosity and coercivity, this relationship was predicted theoretically and observed experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Thus, we have a fundamental role in both cases, that is, between viscosity and coercivity, where they play the so-called activation volume, which is the relevant volume where thermally activated and field-induced magnetization processes occur, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In our work, we will study this way for the paramagnetic material to resist the external magnetic field, through the 5 viscosity/entropy ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In our model, this relationship depends on the external magnetic field, the Horndeski parameters, and the boundary size ∆ yQ of the RS brane in a non-trivial way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' This work is organized as follows: In Section II we consider the gravitational setup, which con- tains all the information with respect to the AdS4/BCFT3 duality, showing the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Together with the above, in Section III the charge density is obtained for then, in Section IV to present the boundary Q profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In Section V, we perform a holographic renormalization, computing the Euclidean on-shell action, which is related to the free energy of the corresponding thermodynamic system, where in particular we will focus on the black hole entropy, present in Section VI, and the holographic paramagnetism/ferromagnetism phase transition, given in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Finally, Section VIII is devoted to our conclusions and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' BLACK HOLE AS A PROBE OF ADS/BCFT As was shown in the introduction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' we will present our setup starting with the total action,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' which contains all information related to AdS4/BCFT3 correspondence with probe approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' so that: S = SN H + SN M + SN 2−FF + SN mat + SQ bdry + SQ mat + SQ ct,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (3) where SN H = � N d4x√−g LH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' SN M = � N d4x√−g LM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (4) with LH and LM given previously in (1)-(2) respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' while that SN mat is the action associated to matter sources and: SQ bdry = 2κ � Q d3x √ −hLbdry SQ mat = 2 � Q d3x √ −hLmat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' SQ ct = 2κ � ct d3x √ −hLct ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (5) with Lbdry = (K − Σ) − γ 4(∇µφ∇νφnµnν − (∇φ)2)K − γ 4∇µφ∇νφKµν ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (6) Lct = c0 + c1R + c2RijRij + c3R2 + b1(∂iφ∂iφ)2 + · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (7) where in our notations (∇φ)2 = ∇µφ∇µφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (6), Lbdry corresponds to the Gibbons-Hawking γ- dependent terms associated with the Horndeski gravity (1), where Kµν = h β µ ∇βnν is the extrinsic 6 curvature, K = hµνKµν is the trace of the extrinsic curvature, hµν is the induced metric, nµ is an outward pointing unit normal vector to the boundary of the hypersurface Q, Σ is the boundary tension on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Lmat is the matter Lagrangian on Q, while that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (7) Lct represents the boundary counterterms, which do not affect the bulk dynamics and will be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Following the procedures presented by [3, 4, 6, 10, 11] we have imposed the NBC: Kαβ − hαβ(K − Σ) − γ 4Hαβ = κSQ αβ , (8) where Hαβ ≡ (∇σφ∇ρφ nσnρ − (∇φ)2)(Kαβ − hαβK) − (∇αφ∇βφ)K , (9) SQ αβ = − 2 √ −h δSQ mat δhαβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (10) Considering the matter stress-energy tensor on Q as a constant (this is SQ αβ = 0), we can write Kαβ − hαβ(K − Σ) − γ 4Hαβ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (11) On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' from the gravitational part,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' given by the Einstein-Horndeski theory and as- suming that SN mat is constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' varying SN H and SQ bdry with respect to the dynamical fields,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' we have: Eαβ = − 2 √−g δSN δgαβ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Eφ = − 2 √−g δSN δφ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Fφ = − 2 √ −h δSQ bdry δφ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (12) where Eµν = Gµν + Λgµν − α 2 � ∇µφ∇νφ − 1 2gµν∇λφ∇λφ � + γ 2 �1 2∇µφ∇νφR − 2∇λφ∇(µφRλ ν) − ∇λφ∇ρφRµλνρ � + γ 2 � −(∇µ∇λφ)(∇ν∇λφ) + (∇µ∇νφ)2φ + 1 2Gµν(∇φ)2 � − γgµν 2 � −1 2(∇λ∇ρφ)(∇λ∇ρφ) + 1 2(2φ)2 − (∇λφ∇ρφ)Rλρ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (13) Eφ = ∇µ [(αgµν − γGµν) ∇νφ] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (14) Fφ = −γ 4(∇µ∇νφnµnν − (∇2φ))K − γ 4(∇µ∇νφ)Kµν ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (15) and note that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Eφ = Fφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' from the Euler-Lagrange equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Together with the above, and according to [27–31] , we have a condition that deals to static black hole configurations, avoiding no-hair theorems [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Here, we need to require that the square 7 of the radial component of the conserved current must vanish identically without restricting the radial dependence of the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Such discussion implies that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (14): αgrr − γGrr = 0 , (16) and defining φ′(r) ≡ ψ(r), where (′) denotes the derivative with respect to r, we can show that the equations Eφ = 0 = Err are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In our setup, the four dimensional metric is defined via the following line element ds2 = L2 r2 � −f(r) dt2 + dx2 + dy2 + dr2 f(r) � , (17) where x1 ≤ x ≤ x2 and y1 ≤ y ≤ y2, while that from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' [10, 20, 30], f(r) is the metric function which takes the form f(r) = αL2 3γ � 1 − � r rh �3� , (18) while that ψ(r) reads ψ2(r) = (φ′(r))2 = −2L2(α + γΛ) αγr2f(r) , (19) where φ(r) = ±2 � −6(α + Λγ) 3α tanh−1 �� 1 − r3 r3 h � + φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (20) Here, φ0 and rh are integration constants, where the last one is related to the location of the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Following the steps of [10, 20], performing the transformations f(r) → αL2 3γ f(r), t → 3γ αL2 t, x → � 3γ αL2 x, y → � 3γ αL2 y, L → � α 3γ L2, (21) we have that the line element (17) is invariant, but now the metric function f(r) takes the form f(r) = 1 − � r rh �3 (22) while the square of the derivative of the scalar field ψ2(r) takes the form given previously in (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Here is important to note that from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (19)-(20) we can see that to have a real scalar field, α + Λγ ≤ 0, where it vanishes when α = −Λγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' It is important to note that, from the action (3), we can see that there is another contribution, denoted as SN 2−FF, which is responsible to construct the ferromagnetic/paramagnetic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' The above will be explained in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 8 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' THE FINITE CHARGE DENSITY As was shown in the previous section, in the action (3) appears the additional contribution SN 2−FF = λ2 � N d4x√−g L2−FF, where L2−FF = − 1 12(dM)2 − m2 4 MµνMµν − 1 2MµνFµν − J 8 V (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (23) Here, the above action defined from the seminal works [42, 43], is coupled through the constant λ and constructed via the 2-form Mµν, dM is the exterior differential of the 2-form field Mµν, this is (dM)τµν = 3∇[τMµν] and (dM)2 = 9∇[τMµν]∇[τMµν], m is a constant related to the mass, while that V (M) describes the self-interaction of polarization tensor, with J a constant, which reads V (M) = (∗MµνMµν)2 = [∗(M ∧ M)]2, (24) where (∗) is the Hodge star operator, this is ∗Mµν = 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='εαβ µνMαβ and εαβ µν is the Levi-Civita Tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In the following lines, will restrict our analysis to the probe approximation, that is, from the action Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (3), one can derive the corresponding equations of motions for matter fields in the probe approximation, that is, e2 → ∞ and λ → 0, so that: ∇µ � Fµν + λ2 4 Mµν � = 0, (25) ∇τ(dM)τµν − m2Mµν − J(∗MτσMτσ)(∗Mµν) − Fµν = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (26) Given that we are focusing on the probe limit approximation, we are going to disregard any back reaction coming from the two-form field Mµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In order to analyze the holographic paramag- netism/ferromagnetism and paraelectric/ferroelectric phase transition, we consider the gauge fields Mµν and Aµ we consider the following ansatz: Mµν = −p(r) dt ∧ dr + ρ(r) dx ∧ dy, (27) Aµ = At(r) dt + Bx dy, F = dA, (28) where B is the external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Using (17), (27)-(28) in the background (22), the field equations (25) and (26) are given by A′ t + � m2 − 4 J r4 ρ2 L4 � p = 0, (29) ρ′′ L2 + �f′ f + 2 r � ρ′ L2 − �4 J r2 p2 fL4 + m2 r2 f � ρ − B r2 f = 0, (30) A′′ t + λ2 4 p′ = 0 , (31) 9 As we work with probe approximation, the back reaction can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Together with the above, given that the behaviors are asymptotically AdS4, we can solve the field equations (29)-(31) near to the boundary (this is r → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Here, asymptotic solutions are given by At(r) ∼ µ − σr, (32) p(r) ∼ σ m2 , (33) ρ(r) ∼ ρ+r∆+ + ρ−r∆− − B m2 , (34) ∆± = −1 ± √ 1 + 4L2m2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (35) Here, ρ+ and ρ− correspond to the source and vacuum expectation value of the dual operator in the boundary field theory (up to a normalization factor), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' It is worth pointing out that one should take ρ+ = 0, in order to obtain condensation spontaneously [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (34), we can define ρ+ and ρ− as ρ+ ≡ r−∆+ h , ρ− ≡ r−∆− h , (36) yielding to the asymptotic solution ρ(r) the following structure ρ(r) ∼ � r rh �∆+ + � r rh �∆− − B m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (37) Additionally, and according to [8], we can to analyze the electromagnetic field, extracted from the four dimensional electromagnetic duality, in a sense that the theory is invariant under Fµν →∗ Fµν = 1 2εµναβF αβ, (38) where, as before, εαβµν is the Levi-Civita Tensor, transforming the electric field into a magnetic field and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Such duality gives that, from the action (2), FµνF µν = (∗Fµν)(∗F µν), showing that is invariant under (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Besides, the transformation (38) shows that Frt → (∗Frt) = Fxy = σ = B, where σ (B) is the constant related to the electric (magnetic) field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Q-BOUNDARY PROFILE In this section, we present the boundary Q profile, we assume that Q is parameterized by the equation y = yQ(r), analyzing the influence of the Horndeski action (1), (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' For this, to find the extrinsic curvature, one has to consider the induced metric on this surface, which reads ds2 ind = L2 r2 � −f(r)dt2 + dx2 + g2(r)dr2 f(r) � , (39) 10 where g2(r) = 1 + y′2(r)f(r) and (′) denotes the derivative with respect to the coordinate r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Here, the normal vectors on Q are represented by nµ = r Lg(r) � 0, 0, 1, −f(r)y′(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (40) Considering the field equation Fφ = 0 (15), one can solve the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (11), yielding y′(r) = (ΣL) � � � �4 + γψ2(r) 4 − (ΣL)2 � 1 − � r rh �3� , (41) and, with ψ2(r) given previously in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (19), we have y′(r) = (ΣL) � � � � � � 4 − ξL2 2r2 � 1 − � r rh �3� − (ΣL)2 � 1 − � r rh �3� , (42) where we define ξ = α + γΛ α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (43) With all this information, we can plot the yQ profile from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (42), representing the holographic description of BCFT considering the theory (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 2: The figure shows the numerical solution for Q boundary profile from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (42) for the black hole within Horndeski gravity, considering the values for θ′ = 2π/3, θ = π − θ′, Λ = −1, α = 8/3 with γ = 0 (pink curve), γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='1 (blue dashed curve ), γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='2 (red dot dashed curve), and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='3 (green thick curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' The dashed parallel vertical lines represent the UV solution, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (46), that is, Randall-Sundrum branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' The region between the curves Q represents the bulk N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Yo yo 0 Q N N rh r11 On the other hand, following the steps of [6, 7], we have that the NBC on the gauge field is nµFµν|Q = 0, and B = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' The holographic model (AdS4/BCFT3) predicts that a constant boundary current in the bulk induces a constant current on the boundary Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Such boundary current can be measured in materials graphene-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Furthermore, nµMµν|Q = 0 provide ρ(r) B = f(r)y′(r) m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (44) Here, the density ρ and the magnetic field B are no longer two independent parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' As the ratio is the Hall conductivity, this is very reminiscent of the quantum Hall effect (QHE), where this ratio is independent of both ρ and B and is inversely proportional to the topological coefficients, which in our case are the coupling constant γ presents in the Horndeski gravity, together with the parameter from the antisymmetric tensor field Mµν, this is m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In our case, the equation of y′ from (42) and then the ρ/B ratio (44) can be analyzed by numerical calculations, being represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Here, we show the ratio ρ/B with respect to external magnetic field B for different values of the Horndeski gravity parameter γ, where we introduced ΣL = cos(θ′), where θ′ represents the angle between the positive direction of the y axis and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' At the boundary Q, the curves of solutions in the (ρ, B) plane will be a localized condensate [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='030 r Ρ B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 3: Graphic of the ratio ρ/B with respect to external magnetic field B versus r, for different values of the Horndeski parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Here, we consider the values rh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='1, L = 1, θ′ = 2π/3, Λ = −1, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='5, m = 1, and γ = 1 (represented through the blue curve), γ = 4 (represented through the red curve), and γ = 8 (represented through the green curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 12 Together with the above, in addition to the above numerical solution, we can analyze some particular cases regarding the study of the UV and IR regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Thus, for the first case, performing an expansion at r → 0 with, as before, ΣL = cos(θ′), the equation (42) becomes yUV (r) = y0 + � 2 −ξL2 r cos(θ′), (45) where y0 is an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In the above equation, considering ξ → −∞, we have yUV (r) = y0 = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (46) This is equivalent to keeping ξ finite and a zero tension limit Σ → 0, considering the cases θ′ = π/2 and θ′ = 3π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Now, for this regime, we have that the ρ/B ratio takes the form ρ B = � 2 −ξL2 cos(θ′) m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (47) Here, it turns out a straightforward generalization of a known AdS4/CFT3 solution, given by the plane-symmetric charged four-dimensional AdS black hole, where only allows for tensionless RS branes in the AdS4/BCFT3 construction [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In this case, requires that the static uniform charge density is supported by a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Specifically, we found that ρ/B is a constant proportional to a ratio of the coefficients appearing in the Horndeski gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' These analyses indicate a generalization of the AdS4 black hole can describe a quantum Hall system at a plateau of the transverse conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Additionally, the AdS/BCFT setup yields that the Hall conductivity is inversely proportional to a sum of the coefficients of the topological terms appearing in the gravity Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' That is, we obtain that σH = ρ/B, which from the equation (47) σH = � 2 −ξL2 cos(θ′) m2 , (48) where, as was shown in the introduction, in QHE the conductivity is related to the number of filled Landau levels (filling fraction), namely, by h e2 σH = � 2 −ξL2 cos(θ′) m2 , (49) where e2/h is the magnetic flux quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In this way, the holographic description seems to provide results similar to the description of the QHE obtained in [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In our case, we have an extension of the covariant form of the Hall relation ρ = σHB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' For the IR case, we take r → +∞ so that from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (42) implies that limr→+∞(φ′(r))2 = 0, and then φ = constant, which ensures a genuine vacuum solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Plugging this result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (42), in the limit r → ∞, we have y′ IR(r) ∼ �rh r �3/2 + O � 1 r2 � , (50) 13 and y ′ IR(r) → 0 when r → +∞, which implies from (47) that ρ/B → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Such value becomes the on-shell action finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' For the sake of completeness, an approximate analytical solution for y(r) can be obtained by performing an expansion for ξ very small from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (42), this is y′ Q = cos(θ′) � 4 − cos2(θ′)f(r) + L2 cos(θ)ξ 4r2f(r)(4 − cos2(θ′)f(r))3/2 + O(ξ2), with f given previously in (22), and considering this expansion up to the first order, we obtain yQ(r) = y0 + r cos(θ′) � (r3 − r3 h) cos2(θ′) + 4r3 h � 4r3 h − (r3 − r3 h) cos(2θ′) 4 − cos2(θ′) ×2F1 �1 3, 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 4 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' − r3 cos2(θ′) r3 h(4 − cos2(θ′)) � + ξ � L2 cos(θ) 4r2f(r)(4 − cos2(θ′)f(r))3/2 dr + O(ξ2), (51) where 2F1(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' x) is the hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' HOLOGRAPHIC RENORMALIZATION In our setup, we will compute the Euclidean on-shell action, which is related to the free energy of the corresponding thermodynamic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Thus, our holographic scheme takes into account the contributions of AdS4/BCFT3 correspondence within Horndeski gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Let us start with the Euclidean action given by IE = Ibulk + 2Ibdry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=', Ibulk = − 1 16πGN � N √gd4x � R − 2Λ + γ 2Gµν∇µφ∇νφ � − 1 8πGN � M d3x√¯γLM, (52) LM = K(¯γ) − Σ(¯γ) − γ 4(∇µφ∇νφnµnν − (∇φ)2)K(¯γ) − γ 4∇µφ∇νφK(¯γ) µν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (53) Together with the above, g is the determinant of the metric gµν on the bulk N, while that ¯γ is the induced metric, the surface tension on M is represented with Σ(¯γ), and K(¯γ) corresponds to the extrinsic curvature on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' On the other hand, for the boundary, we have the following expressions Ibdry = − 1 16πGN � N √gd4x � R − 2Λ + γ 2Gµν∇µφ∇νφ � − 1 8πGN � Q d3x √ hLbdry, (54) Lbdry = (K − Σ) − γ 4(∇µφ∇νφnµnν − (∇φ)2)K − γ 4∇µφ∇νφKµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (55) Thus, in order to compute the bulk action Ibulk, we consider the induced metric on the bulk, which is obtained from (17) after the transformation τ = it, given by ds2 ind = ¯γµνdxµdxν = L2 r2 � f(r)dτ 2 + dx2 + dy2 + dr2 f(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (56) 14 Here, we have that 0 ≤ τ ≤ β, where from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (22) β = 1 T = �|f′(rh)| 4π �−1 = 4πrh 3 , (57) where T is the Hawking Temperature, the above allows us to obtain the following quantities: R = − 12 L2 , Λ = − 3 L2 , K(¯γ) = 3 L, Σ(¯γ) = 2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Thus, we have all elements needed to construct the bulk action Ibulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In the process of holographic renormalization, we need to introduce a cutoff ϵ to remove the IR divergence on the bulk side and we can provide that: Ibulk = 1 16πGN � d2x � 4πrh 3 0 dτ � rh ϵ dr√g � R − 2Λ + γ 2Grrψ2(r) � + 1 16πGN � d2x � 4πrh 3 0 dτ L2� f(ϵ) ϵ3 , (58) Ibulk = − L2V 8r2 hG � 1 − ξ 4 � , (59) with ξ given previously in (61) and, in our notations, V = � d2x = ∆x∆y = (x2 − x1)(y2 − y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Now, computing the Ibdry, we introduce a cutoff ϵ to remove the UV divergence on the boundary side, and with this information, we have: Ibdry = rhL2∆yQ 2GN � 1 − ξ 4 � � rh ϵ ∆yQ(r) r4 dr − rhL2 sec(θ′)∆yQ 2GN � rh ϵ ∆yQ(r) r3 dr (60) Here, ∆yQ is a constant and ∆yQ(r) := yQ(r) − y0 is obtained from the equation (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' As we know, from the point of view of AdS/CFT correspondence, IR divergences in AdS correspond to UV divergences in CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' This relationship is known as the IR-UV connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Thus, based on this duality, we can reduce the above equation (60) after some eliminations of terms that produce divergences to the following form: Ibdry = −L2∆ yQ 2GN � 1 − ξ 4 � � ξ L2b(θ′) 5r4 h + q(θ ′) 4r2 h � +L2 sec(θ′)∆ yQ 2GN � ξ L2b(θ′) 4r3 h + q(θ ′) 2rh � , (61) where b(θ′) = cos(θ′) 4(4 − cos2(θ′))3/2 , q(θ′) = cos(θ′) � 4 − cos2(θ′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (62) With all the above information, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (59) and (61)-(62), we can compute IE = Ibulk + 2Ibdry as: 15 IE = − L2V 8r2 hGN � 1 − ξ 4 � − L2∆ yQ GN � 1 − ξ 4 � � ξ L2b(θ′) 5r4 h + q(θ ′) 4r2 h � +L2 sec(θ′)∆ yQ GN � ξ L2b(θ′) 4r3 h + q(θ ′) 2rh � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (63) Here, IE is the approximated analytical expression for the Euclidean action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' This equation is essential to construct the free energy and extract all thermodynamic quantities in our setup, as we show in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' BLACK HOLE ENTROPY Now, we will compute the entropy related to the black hole considering the contributions of the AdS/BCFT correspondence in the Horndeski gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Free energy is defined as Ω = TIE , (64) one can obtain the corresponding entropy as: S = −∂ Ω ∂T (65) where T is the Hawking Temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' By plugging the Euclidean on-shell action IE from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (63), and replacing T obtained previously in (57), we have Stotal = Sbulk + Sbdry, (66) where Sbulk = L2V 4r2 hGN � 1 − ξ 4 � , (67) Sbdry = L2∆ yQ GN � 1 − ξ 4 � � ξ L2b(θ′) 5r4 h + q(θ ′) 4r2 h � − L2 sec(θ′)∆ yQ GN � ξ L2b(θ′) 4r3 h + q(θ ′) 2rh � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (68) The interpretation for this total entropy can be identified with the Bekenstein-Hawking formula for the black hole: SBH = A 4GN , (69) 16 where, in this case A = L2V 2r2 h � 1 − ξ 4 � + 4L2∆ yQ � 1 − ξ 4 � � ξ L2b(θ′) 5r4 h + q(θ ′) 4r2 h � −4L2 sec(θ′)∆ yQ � ξ L2b(θ′) 4r3 h + q(θ ′) 2rh � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (70) Here, A is the total area of the AdS black hole in the Horndeski contribution terms for the bulk and the boundary Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' We can see that the information is bounded by the black hole area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Then, the equation (70) suggests that the information storage increases with increasing |ξ|, as long as ξ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Together with the above, with respect to the boundary contribution of (68), we have that this expression is the entropy of the BCFT corrected by the Horndeski terms parametrized by ξ, given previously in (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In this case, the results presented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' [6, 11] are recovered in the limit ξ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Besides, still analyzing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (68), due to the effects of the Horndeski gravity, there is a non-zero boundary entropy even if we consider the zero temperature scenario, similar to an extreme black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' This can be seen if one takes the limit T → 0 (or rh → ∞) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (68), then we do not get the denominated residual boundary entropy, as discussed in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' On the other hand, through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (47) we have Smagnetic bdry = L2∆ yQ GN � 1 − ξ 4 � � −2B2 cos2(θ ′) m2ρ2 b(θ′) 5r4 h + q(θ ′) 4r2 h � − L2 sec(θ′)∆ yQ GN � −2B2 cos2(θ ′) m2ρ2 b(θ′) 4r3 h + q(θ ′) 2rh � , (71) where m2 > −1/(4L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' For the entropy bound, the restriction on m2 comes from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' A well-defined probe limit demands that the charge density contributed by the polarization should be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' At low temperatures, below the critical, in the ferromagnetic region, we can observe that our entropy is Sbdry magnetic ∝ B2, that is, has a square dependence on the external magnetic field and this is a characteristic of ferromagnetic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Furthermore, we can observe that Smagnetic bdry is the magnetic entropy of the boundary Q, and we can observe that for ferromagnetic materials, the magnetic entropy is associated with the disorder of the magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In addition, these materials have spontaneous magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' So when we remove the applied magnetic field, they still show magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 17 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' HOLOGRAPHIC PARAMAGNETISM/FERROMAGNETISM PHASE TRANSITION In this section, we present the holographic paramagnetism/ferromagnetism phase transition through the boundary contribution from the entropy (71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' For this, we start considering the free energy Ω from (63) -(64), where the first law of black holes thermodynamics, considering the canonical ensemble, takes the form dΩ = −PdV − SdT, (72) where, in addition to the entropy S as well as the Hawking temperature T, the pressure P and the volume V appear, yielding Ω = ϵ − TS, where ϵ takes the role of the energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' As a first thermodynamic quantity to study, we will consider the entropy S, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (66), calculated in the previous section, and represented graphically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 4, with respect to the Hawking temperature T (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Here, in the right panel (left panel) there is (not) an external magnetic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Concretely, we see that the right panel exhibit similar behavior as analyzed in [53], as for example ferromagnetic materials with nearly zero coercivity and hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' On the other hand, in the left panel, when the external magnetic field is removed (this is B = 0), we still have a disorder of magnetic moments, this is a characteristic of paramagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' The second parameter that we analyze is the heat capacity CV , which allows us to analyze local thermodynamic stability, defined in the following form CV = T �∂S ∂T � V = −T �∂2Ω ∂T 2 � V , (73) where the sub-index V from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (73) represents at volume constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 5, we can see that in the right panel, the black hole can switch between stable (CV > 0) and unstable (CV < 0) phases, depending on the sign of heat capacity CV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' This phase transition occurs, due to the spontaneous electric polarization, which was realized in our model from the application of the magnetic external field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Moreover, in the region CV > 0, we have structures built like magnetic domes on the boundary Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Additionally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 5, one can see the influence of Horndeski gravity (represented via the constant γ) with respect to the temperature T, where the phase transition occurs for some ranges of values for T when the external magnetic field is null, that is, B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='20 T S 0 2 4 6 8 10 0 20 40 60 80 100 120 T S�B�0� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 4: Right panel: The behavior of the entropy S with the temperature T with different values for α = 8/3, m = 1/8, B = (4/5)T, ρ = 1/4, Λ = −1, V = 1, GN = 1, θ′ = 2π/3 with γ = 1 (pink curve), γ = 4 (red dot dashed curve), γ = 8 (green thick curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Left panel: The behavior of the entropy S with respect the temperature T, with different values for B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 �6 �4 �2 0 T CV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='5 T CV �B�0� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 5: Right panel: The behavior of the heat capacity CV with the temperature T with different values for α = 8/3, m = 1/8, B = (4/5)T, ρ = 1/4, Λ = −1, θ′ = 2π/3 with γ = 1 (pink curve), γ = 4 (red dot dashed curve), γ = 8 (green thick curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Left panel: The behavior of the heat capacity CV with respect the temperature T, with different values for B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Additionally, we can obtain the heat capacity at constant pressure CP , which reads CP = T �∂S ∂T � P , (74) and, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 6, we can see that in the right panel, the black hole can switch between stable (CP > 0), describing a ferromagnetic material, and unstable (CP < 0), describing a paramagnetic material, depending on the sign of heat capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' This phase transition occurs, as in the previous case, due to spontaneous electric polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Moreover, in the region CP > 0, we have structures 19 built like magnetic domes on the boundary Q, wherein the experimental specific frame, these heat curves without magnetic field can represent a material like DyAl2 [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' On the other hand, the left panel represents the heat capacity CP where B = 0, where we can see, that is locally unstable (CP < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 �4 �2 0 2 4 T CP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 �4 �2 0 2 4 T CP �B�0� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 6: Right panel: The behavior of the CP with respect to the temperature T with different values for α = 8/3, m = 1/8, B = (4/5)T, ρ = 1/4, Λ = −1, θ′ = 2π/3 with γ = 1 (pink curve), γ = 4 (red dot dashed curve), γ = 8 (green thick curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Left panel: The behavior of CP with respect T, with different values for B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Additionally, we can derive other quantities, as for example the magnetization density m, and magnetic susceptibility χ, following the steps of [46], given by m = − �∂ Ω ∂B � = L2∆ yQT GN � 1 − ξ 4 � � 4 cos2(θ ′) m2ρ2 b(θ′) 5r4 h � − L2 sec(θ′)∆ yQT GN � cos(θ ′) m2ρ2 b(θ′) 4r3 h � , (75) χ = � ∂2Ω ∂B2 � = −L2∆ yQT GN � 1 − ξ 4 � � 4B cos2(θ ′) m2ρ2 b(θ′) 5r4 h � + L2 sec(θ′)∆ yQT GN � B cos(θ ′) m2ρ2 b(θ′) 4r3 h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (76) As we can see from equations (75) and (76), the RS brane behaves like a paramagnetism material, that is, when we remove the external magnetic field, the equation (76) disappears and the entropy linked disorder increases, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' On the other hand, from the equation (75), the magnetization density is not null for zero magnetic fields (this is B = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Thus, we can conclude that paramagnetic materials have a low coercivity, that is, their ability to remain magnetized is very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Thus, one way to analyze coercivity is through viscosity η in our model [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 20 In order to be as clear as possible, the details about the computation of the shear viscosity and entropy density ratio are present in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In particular, we will focus on the η/S ratio, where from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' A11 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 7, we can analyze the dependence of the viscosity on the magnetic field, characterizing a magnetic side effect, and describing the slow relaxation of the magnetization of paramagnetic materials when they acquire magnetization in the presence of an external magnetic field B (left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In the right panel, we can observe that under an interval of the temperature T, the η/S ratio is an increasing function when B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='2 T Η S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0 20 40 60 80 100 T Η S �B�0� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 7: Right panel: The behavior of the η/S ratio as a function of the temperature T for different values for α = 8/3, B = (4/5)T, ρ = 1/4, Λ = −1, γ = 1 (pink curve), γ = 2 (red dot dashed curve), γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='5 (green thick curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Left panel: The behavior of η/s for B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0 1 2 3 4 B Η S FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 8: The behavior of η/S with respect to the magnetic field B, for different values for α = 8/3, T = 4/5, ρ = 1/4, Λ = −1, γ = 1 (pink curve), γ = 2 (red dot dashed curve), γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='5 (green thick curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' On the other hand, and as we can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 8 at a temperature T fixed when we observe as the paramagnetic material, represented by the RS brane, we can obtain a relation between η/S 21 with respect to the magnetic field B, which is a decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Here, when B becomes large, we have that η/S → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' We finalize this section showing the magnetic moment N at a low temperature T, corresponding to order parameter ρ in the absence of an external magnetic field, setting B = 0, and then compute the value of N, defined as N = λ2rh 2L � 1 0 ρ(r)dr = −λ2rh 2L � − B m2 + 1 (∆+ + 1)r∆+ h + 1 (∆− + 1)r∆− h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (77) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 9, it can be found that as the temperature decreases, the magnetization increases and the system is in the perfect order with the maximum magnetization at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Thus, increasing the Horndeski parameters lowers the magnetization value and the critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Indeed, we have that the effect of a larger value of the parameters γ and m2 makes the magnetization harder and the ferromagnetic phase transition happen, which is in good agreement with previous works [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='2 0 2 4 6 8 10 T N Λ2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 9: The behavior of magnetic moment N with different values for B = 0, α = 8/3 with γ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' m2 = 2 (blue curve), γ = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' m2 = 4 (red curve), γ = 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' m2 = 6 (green curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' We consider in the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 77 the transformations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='∼(21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Finally, we present the susceptibility density χ of the materials as a response to the magnetic moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Thus, this behavior is an essential property of ferromagnetic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In order to study χ of the ferromagnetic materials in the Horndeski gravity and to consider the transformations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (21), we follow the definition χ λ2 = lim B→0 ∂N ∂B = � 3 8πm2L2 � 1 T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (78) 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 T Λ2 Χ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='2 0 2 4 6 8 10 T Χ Λ2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 10: The behavior of 1/χ in the function of the temperature T with different values for α = 8/3 with γ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' m2 = 2 (blue curve), γ = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' m2 = 4 (red curve), γ = 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' m2 = 6 (green curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' We consider in the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (78) the transformations given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='10, we have the behavior of 1/χ and χ as a function of the temperature T for different choices of m2 and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In our case, in the right panel, we have that increasing each one of these pa- rameters makes the susceptibility value decrease when the temperature increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' This fact agrees with our expectation of paramagnetic materials because when we remove the external magnetic field, the paramagnetic substance loses its magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Its magnetic susceptibility is very small, but positive, and decreases with increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In fact, this magnetic susceptibility is only part of the background black hole and the other part of the polarization field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' For pure dionic Reissner-Nordstr¨om-AdS black hole, we have a diamagnetic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In this sense, in the chemical reference, we have that a particle (atom, ion, or molecule) is paramagnetic or diamagnetic when the electrons in the particle are paired due to the external magnetic field [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' CONCLUSIONS AND DISCUSSIONS In four dimensions, we analyzed an AdS/BCFT model of a condensed matter system at finite temperature and charge density living on a 2+1-dimensional space with a boundary, showing an extension of the previous work presented in [10], where in addition to the contributions of the theory together with the boundary terms, we include the components Aµ and Mµν, responsible to construct the ferromagnetic/paramagnetic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Via the resolution of the field equations, and using the no-hair theorem, we extend to the four-dimensional configuration obtained in [10, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' From the above solution, we present the Q profile, found a numerical solution, and present it in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 9, where the Horndeski parameter γ 23 takes an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Together with the above, we show that components of Mµν can be viewed as dual fields of the order parameter in the paraelectric/ferroelectric phase transition in dielectric materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Through the NBC over nµM|Q, we found the ratio ρ/B, where for some particular cases is a constant proportional to a ratio of the coefficients appearing in the gravity action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' These properties resemble a quantum Hall system, which suggests at the boundary Q in the (ρ, B) plane will be a localized condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Additionally, via the solution we performed a holographic renormalization, calculating the Eu- clidean on-shell action, which is related to the free energy Ω, and allowing us to obtain the entropy S and the heat capacities CV , CP , thanks to the contribution to the bulk as well as the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' With respect to the entropy S, we show that when the magnetic field is present we see it exhibits similar behavior as for example ferromagnetic materials with nearly zero coercivity and hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Nevertheless, when B = 0 the disorder entropy of the magnetic moments increases, being a char- acteristic of paramagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Together with the above, with respect to CV and CP , we obtained for both cases stable and unstable phases, due to the spontaneous electric polarization, which was realized in our model from the application of the magnetic external field B, being influence via the Horndeski gravity, represented through γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' We also show that the specific heat CP behaves like a material of the type DyAl2, having a growth behavior similar to that expected from the experimental point of view, as presented by [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Currently, we can observe that the microscopic differences between real experimental systems, in relation to theories with gravitational dual suggest that, in the near future, we will have measure- ments of these values for experimental quantities obtained holographically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' So many measurements can realistically aspire to more than useful benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Furthermore, it is important to highlight in this regard the need to take the big limit N in holographic calculations [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' We now have a clarity of the value of the ratio between shear viscosity and entropy density, η/S = 1/4π, which is universal in classical gravity to usual classical gravity [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Furthermore, in the Horndeski gravity, these relations are modified by the parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' However, there are controlled corrections 1/N for this result, which can be both positive and negative and which for realistic values of N show significant changes in the numerical value of the ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' As we show in our model, the violation of this universal bound in the Horndeski gravity with gauge fields changes the η/S ratio (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='7 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='8), where this behavior is similar to the results of [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Furthermore, as γ increases, we can observe a translational symmetry breaking that survives the lower energy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 8, we have η/S → 0 at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' One of the strongest motivations for working with AdS/BCFT for condensed matter physics 24 rests on two pillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' The first is that, although theories with holographic duals may exhibit spe- cific exotic features, they also have features that are expected to be generic to tightly coupled theories, for example, the quantum critiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In this sense, theories with gravitational duals are computationally tractable examples of generic tightly coupled field theories, and we can use them both to test our generic expectations and to guide us in refining those expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Thus, the examples discussed here are special cases of the fact that real-time finite temperature transport is much easier to calculate via AdS/BCFT than almost any other microscopic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Acknowledgments F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' would like to thank the group of Instituto de F´ısica da UFRJ for fruitful discussions about the paramagnetic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In special to the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Capossoli, Diego M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Rodrigues and Henrique Boschi- Filho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' performed the work in the frame of the ”Mathematical modeling in interdisciplinary research of processes and systems based on intelligent supercomputer, grid and cloud technologies” program of the NAS of Ukraine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' is supported by PROYECTO INTERNO UCM-IN-22204, L´ıNEA REGULAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Appendix A: Shear viscosity and entropy density ratio with magnetic field We will present the calculation of the ratio η/S following the procedures [20, 38, 39, 54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' For this, we consider a perturbation along the xy direction in the metric Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content='17 [20, 38], in this sense, we have ds2 = L2 r2 � −f(r)dt2 + dx2 + dy2 + 2Ψ(r, t)dxdy + dr2 f(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (A1) From the overview point of the holographic dictionary, this procedure maps the fluctuation of the diagonal in the bulk metric in the off-diagonal components of the dual energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In this sense, we have a linear regime where fluctuations are associated with shear waves in the boundary fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Thus, substituting this metric (A1) in the Horndeski equation (Eµν = 0) for µ = x and ν = y, one obtains: P1Ψ ′′(r, t) + P2Ψ ′(r, t) + P3 ¨Ψ(r, t) = 0 , (A2) where we defined P1 = 9γ2(α − γΛ)f2(r), P2 = −3γ(α − γΛ)f(r)(2αL2 − 6γr3/r3 h), 25 P3 = −9γ2r(3α + γΛ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (A3) Using the ansatz: Ψ(r, t) = e−iωtΦ(r), (A4) Φ(r) = exp � −iωK ln �6γ2r3f(r) G �� , G = L2V GN � 1 − ξ 4 � , (A5) we obtain K = 1 4πT � 3α + γΛ α − γΛ , (A6) with T the Hawking temperature given previously in (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' At this point, we must evaluate the Lagrangian (1), using the metric function from (22), and expand it up to quadratic terms in Ψ and its derivatives [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' In this way, we can study the boundary field theory using the AdS/CFT correspondence where the quadratic terms in the Lagrangian, after removing the second derivative contributions using the Gibbons-Hawking term, can be written as Hshear = P1Ψ2(r, t) + P2 ˙Ψ(r, t) + P3Ψ ′2(r, t) + P4Ψ(r, t)Ψ ′(r, t), (A7) where P1 = − 48L2 9r7f(r), P2 = 4γ L2 r7 , P3 = 6γ2 r3f(r), P4 = (α + γΛ) 2γ2L4 α r7f(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' (A8) Here, (˙) denotes the derivative with respect t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Finally, viscosity η is determined from the term P3Ψ(r, t)Ψ ′(r, t) which reads η = 1 4π G 4r2 h � 3α + γΛ α − γΛ , (A9) where the entropy, from (66)-(68), can be written as S = GF 4r2 h , (A10) with F = 1 + � B2 cos2(θ′)b(θ′) 5m2ρ2 �4πT 3 �4 + q(θ ′) 4 �4πT 3 �2� − sec(θ′) � 1 − ξ 4 � � −B2 cos2(θ′)b(θ′) 2m2ρ2 �4πT 3 �3 + q(θ ′) 2 �4πT 3 �� , 26 and T given in (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Thus, after algebraic manipulation and imposing V = 1, we have: η S = 1 4πF � 3α + γΛ α − γΛ , (A11) where B = 0 and θ′ = π/2, we recover the result of [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' Maldacena, The Large N limit of superconformal field theories and supergravity, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfXARM/content/2301.03121v1.pdf'} +page_content=' 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Massachusetts Institute of Technology, Cambridge, MA 20139 +Abstract +In last-mile delivery, drivers frequently deviate from planned delivery routes because of their tacit knowledge +of the road and curbside infrastructure, customer availability, and other characteristics of the respective service +areas. Hence, the actual stop sequences chosen by an experienced human driver may be potentially preferable +to the theoretical shortest-distance routing under real-life operational conditions. Thus, being able to predict +the actual stop sequence that a human driver would follow can help to improve route planning in last-mile +delivery. This paper proposes a pair-wise attention-based pointer neural network for this prediction task using +drivers’ historical delivery trajectory data. In addition to the commonly used encoder-decoder architecture +for sequence-to-sequence prediction, we propose a new attention mechanism based on an alternative specific +neural network to capture the local pair-wise information for each pair of stops. To further capture the global +efficiency of the route, we propose a new iterative sequence generation algorithm that is used after model +training to identify the first stop of a route that yields the lowest operational cost. Results from an extensive +case study on real operational data from Amazon’s last-mile delivery operations in the US show that our +proposed method can significantly outperform traditional optimization-based approaches and other machine +learning methods (such as the Long Short-Term Memory encoder-decoder and the original pointer network) +in finding stop sequences that are closer to high-quality routes executed by experienced drivers in the field. +Compared to benchmark models, the proposed model can increase the average prediction accuracy of the +first four stops from around 0.2 to 0.312, and reduce the disparity between the predicted route and the actual +route by around 15%. +Keywords: Route planning, Trajectory prediction, Sequence-to-sequence model, Last-mile delivery, +Pointer network, Attention +1. Introduction +The optimal planning and efficient execution of last-mile delivery routes is becoming increasingly +important for the business operations of many logistics service providers around the globe for a variety of +reasons. E-commerce volumes are growing rapidly and make up a constantly growing share of overall retail +sales. For instance, in the US, the share of e-commerce sales in total retail sales has grown from around 4% in +2010 to around 13% in 2021. Even by the end of 2019, i.e., before the outbreak of the COVID-19 pandemic, +∗Corresponding author +Preprint submitted to Elsevier +January 11, 2023 +arXiv:2301.03802v1 [cs.LG] 10 Jan 2023 + +it had reached 11% (US Census Bureau, 2021). Undoubtedly, the pandemic further accelerated the growth +of e-commerce (postnord, 2021; McKinsey & Company, 2021). In the medium to long run, its growth will +continue to be fueled by an ongoing trend towards further urbanization, which is particularly pronounced in +developing and emerging economies (United Nations Department of Economic and Social Affairs, 2019). +The share of the global population living in urban areas is currently projected to rise from around 55% in +2018 to around 68% by 2050. The associated increase in population density in most urban areas will likely +lead to growing operational uncertainties for logistics service providers, as increasing congestion levels, less +predictable travel times, and scarce curb space make efficient and reliable transport of goods into and out of +urban markets increasingly challenging (Rose et al., 2016). +As a result of the continued boom of e-commerce and constantly growing cities, global parcel delivery +volumes have been increasing rapidly in recent years and are expected to continue to do so. Across the +13 largest global markets, including the US, Brazil, and China, the volume of parcels delivered more than +tripled from 43 billion in 2014 to 131 billion in 2020 (Pitney Bowes, 2020). At the same time, customer +expectations towards last-mile logistics services are rising. For instance, there is a growing demand for +shorter delivery lead times, including instant delivery services and same-day delivery, as well as customer- +defined delivery preferences when it comes to the time and place of delivery (Lim and Winkenbach, 2019; +Cortes and Suzuki, 2021; Snoeck and Winkenbach, 2021). The rapid growth and increasing operational +complexity of urban parcel delivery operations also amplifies their negative externalities, including their +contribution to greenhouse gas and other pollutant emissions, public health safety risks, as well as overall +urban congestion and a corresponding decline in overall mobility and accessibility of cities (Jaller et al., +2013; World Economic Forum, 2020). +When applied to realistically sized instances of a last-mile delivery problem, solving the underlying +traveling salesman problem (TSP) or vehicle routing problem (VRP) to (near) optimality becomes chal- +lenging, as both problem classes are known to be NP-hard. Traditional TSP and VRP formulations aim to +minimize the total distance or duration of the route(s) required to serve a given set of delivery stops. The +operations research literature has covered the TSP, VRP, and their many variants extensively, and in recent +years important advances have been made with regards to solution quality and computational cost. However, +in practice, many drivers, with their own tacit knowledge of delivery routes and service areas, divert from +seemingly optimal routes for reasons that are difficult to encode in an optimization model directly. For exam- +ple, experienced drivers may have a better understanding of which roads are hard to navigate, at which times +traffic is likely to be bad, when and where they can easily find parking, and which stops can be conveniently +served together. Therefore, compared to the theoretically optimal (i.e., distance or time minimizing) route, +the deviated actual route sequence chosen by an experienced human driver is potentially preferable under +real-life operational conditions. +An important challenge in today’s last-mile delivery route planning is therefore to leverage historical +route execution data to propose planned route sequences that are close to the actual trajectories that would +be executed by drivers, given the delivery requests and their characteristics. Note that, while distance and +time-based route efficiency is still an important factor for planning route sequences, it is not the sole objective, +as tacit driver knowledge is also incorporated in the proposed route sequences. Unlike a typical VRP in +which the number of vehicles and their respective route sequences need to be determined simultaneously, in +this study, we focus on solving a problem that is similar to a TSP at the individual vehicle level. That is, +we aim to solve a stop sequence to serve a given set of delivery requests, and expect that the proposed stop +sequence is as close to the actual trajectories that would be executed by drivers as possible. +2 + +To this end, we propose a pair-wise attention-based pointer neural network to predict the actual route +sequence taken by delivery drivers using drivers’ historical delivery trajectory data. The proposed model +follows a typical encoder-decoder architecture for the sequence-to-sequence prediction. However, unlike +previous studies, we propose a new attention mechanism based on an alternative specific neural network +(ASNN) to capture the local pair-wise information for each stop pair. To further capture the global efficiency +of the route (i.e., its operational cost in terms of total distance or duration), after model training, we propose a +new sequence generation algorithm that iterates over different first stops and selects the route with the lowest +operational cost. +The main contribution of this paper is three-fold: First, we propose a new ASNN-based attention +mechanism to capture the local information between pairs of stops (e.g., travel time, geographical relation), +which can be well adapted to the original pointer network framework for sequence prediction. Second, we +propose a new sequence generation algorithm that iterates over different first stops in the predicted route +sequences and selects the lowest operational cost route. The intuition is that the stop-to-stop relationship +(referred to as the local view) is easier to learn from data than the stop sequence of the route as a whole +(referred to as the global view). Lastly, we apply our proposed method to a large set of routes executed by +Amazon delivery drivers in the US. The results show that our proposed model can outperform traditional +optimization-based approaches and other machine learning methods in finding stop sequences that are closer +to high-quality routes executed by experienced drivers in the field. +The remainder of this paper is structured as follows. In Section 2 we define the problem setting under +investigation in a more formal way. Section 3 then reviews previous studies in the literature related to this +paper. Section 4 presents our methodology and elaborates on the detailed architecture of the proposed +pair-wise attention-based pointer neural network. Section 5 presents the experimental setup and numerical +results of our case study, applying our proposed method to real-world data made available by the Amazon +Last-Mile Routing Research Challenge (Merchán et al., 2022; Winkenbach et al., 2021). Section 6 concludes +this paper and discusses future research directions. +2. Problem Setting +In the last-mile delivery routing problem considered here, a set of stops S = {s1, ..., sn} to be served +by a given delivery vehicle is given to the route planner. The planner’s objective is to find the optimal +stop sequence that has the minimal operational cost. In this case, we consider total cost as total travel +time. +The planner is given the expected operational cost (i.e., travel times) between all pairs of stops +(si, sj). The theoretically optimal stop sequence, denoted by (sT +(1), ..., sT +(n)), can be found by solving a TSP +formulation. This stop sequence is referred to as the planned stop sequence. However, as discussed in +Section 1, minimizing the theoretical operational cost (i.e., total travel time) of the route may not capture +drivers’ tacit knowledge about the road network, infrastructure, and recipients. Therefore, the actual driver +executed stop sequence (s(1), ..., s(n)) can be different from the planned route sequence. Note that here, +s(i) ∈ S denotes the i-th stop that is actually visited by the driver. +The objective of the model presented in this study is to predict the actual driver executed sequence +(s(1), ..., s(n)) given a set of stops S and the corresponding delivery requests and characteristics XS (such +as the number of packages, estimated service time for each package, geographical information for each stop, +travel time between each stop pairs, etc.). All drivers are assumed to start their routes from a known depot +DS and return back to DS. Therefore, the complete trajectory should be a tour (DS, s(1), ..., s(n), DS). For +the convenience of model description, we ignore the depot station in the sequence. +3 + +Figure 1 provides a simple example for illustration. In this example, we are given four stops S = +{s1, s2, s3, s4} and a depot DS. The planned stop sequence for the driver is (s4, s1, s2, s3), while the actual +stop sequence executed by the driver is (s4, s2, s1, s3). The proposed model aims to predict the actual +sequence (s4, s2, s1, s3) given the depot location DS, the set of stops to be visited S, and characteristics of +the stops XS. This problem setup is inspired by the Amazon Last-Mile Routing Research Challenge (cf., +Winkenbach et al., 2021). Note that this study only focuses on the stop sequence prediction. The routing +between stops is not considered. It is assumed that the drivers always take the optimal route between stops, +which is reflected by the travel time matrix between stops in our problem setup. +Figure 1: Illustrative example of the problem setting. +3. Literature Review +The problem setting defined in Section 2 involves both solving a cost-minimizing routing problem (i.e., +the TSP) and capturing tacit driver knowledge to learn systematic deviation of drivers from the planned and +theoretically optimal stop sequences. Therefore, we will first review the extant literature on the TSPs and +its most relevant variants. We will then go through various machine learning approaches that have been +proposed by the extant literature to generate sequences, with a section on methods specifically for solving the +TSP. Note that although these machine learning approaches are used to solve the TSP instead of the actual +routes taken by drivers, their architectures may be helpful to learn the actual route as well. +3.1. Travelling salesman problems +First, given the travel times between stops, a solution to the TSP, which finds the route with the minimum +cost or distance (i.e., the planned route), can be a close approximation of the actual route. Since the drivers +are paid for the number of packages delivered, all drivers’ goal is to deliver the packages in the minimum +amount of time. Most of the drivers do follow large parts of the planned routes. +The TSP is a well-known NP-hard problem that has been studied extensively over the last century, with a +lot of books and review papers published on its history, formulations, solution approaches, and applications +(Applegate et al., 2006; Matai et al., 2010; Davendra and Bialic-Davendra, 2020). An overview of the +relevant TSP variants and solution approaches are presented below. +The basic setup of TSP has one traveler and requires the traveler to return to the starting point after +visiting each node exactly once, and that the traveling cost matrix (represented by distance and/or time) is +4 + +Depot Ds +Input +Output +S,Xs,Ds +Model +Actual route +S3 +S1 +Actual route +S2 +- Planned routesymmetric (cost between i and j is the same with that between j and i). In most real-world applications, +the basic setup needs to be modified. For example, the cost matrix, if represented by travel times, is likely +asymmetric. This variant of TSP is thus named asymmetric TSP (ATSP) (Jonker and Volgenant, 1983). +In some applications, the vehicle does not need to return to the original depot (Traub et al., 2021), or +it can charge/refuel and potentially load additional delivery items at intermediate stops (Küçükoğlu et al., +2019). In many last-mile delivery applications, some packages are time-sensitive, and therefore time window +constraints to their delivery need to be considered in a so-called TSP with time windows (TSPTW) (da Silva +and Urrutia, 2010; Mladenović et al., 2012). In large systems, there might be more than one salesman serving +a set of stops, resulting in multiple traveling salesmen problems (MTSPs) (Cheikhrouhou and Khoufi, 2021). +Different variants of TSP further impose different constraints on the solution. While some problems +can be reduced to the basic setup in the formulation stage, others require more versatile solution algorithms. +In general, the solution approaches to the TSP can be divided into exact approaches and approximate +approaches. Exact approaches include branch-and-cut (Yuan et al., 2020) and branch-and-bound (Salman +et al., 2020). Since the TSP is a well-known NP-hard problem, exact approaches can only be applied on +problems of smaller scale, or aid in heuristics to cut the solution space. Among approximate approaches, +there are heuristics designed for the TSP specifically, as well as meta-heuristics that are generic and treat the +problem like a blackbox. The most commonly used heuristics and meta-heuristics include nearest neighbor +searches, local searches, simulated annealing, and genetic algorithms. A more comprehensive review of +existing solution approaches can be found in Halim and Ismail (2017); Purkayastha et al. (2020). Despite +the TSP being NP-hard, modern mixed-integer optimization solvers (e.g., Gurobi, CPLEX, or GLPK) can +solve it efficiently for real-world instances by combining exact approaches with heuristics. +3.2. Sequence-to-sequence prediction using deep learning +The TSP and its variants are a viable option for sequence generation only when the objective is clearly de- +fined. They fall short when the sequence generation problem does not have a well-defined cost-minimization +objective. In a lot of applications, the rule of sequence generation cannot be simply defined and optimized. +A standard example for a sequence learning problem is machine translation, where a sequence of words +in one language needs to be translated to another language. Another type of sequence learning is time series +modeling, where a sequence of historical observations is given to predict future states of the system. In both +cases, the primary modeling task is to learn the sequence generation rules. In recent years, deep learning +has successfully achieved great performance in various settings of sequence generation. These models are +often referred to as sequence-to-sequence (seq2seq) models. +seq2seq models often consist of an encoder and a decoder, where the encoder encodes the input sequence +into a fixed-length vector representation, and the decoder generates a sequence based on the generated vector +representation. Most encoder-decoder architectures adopt recurrent neural network (RNN) layers and its +variants such as Long Short-Term Memory (LSTM) (Hochreiter and Schmidhuber, 1997) and gated recurrent +layers (GRU) (Cho et al., 2014) to learn long-range dependencies. Early works using LSTM alone were able +to generate plausible texts (Graves, 2013) and translate between English and French (Sutskever et al., 2014) +with long-range dependencies. Chung et al. (2014) demonstrate the superiority of GRU compared to LSTMs +in music and speech signal modeling. +Attention-based mechanisms, first introduced by Bahdanau et al. (2015), have been shown to be a great +addition since it allows the decoder to selectively attend to parts of the input sequence and relieves the encoder +of the task of encoding all the information into a fixed-length vector representation. Most sequence generation +5 + +problems benefit from keeping track of long-range dependencies and global context while decoding. To +address that, multi-level attention was proposed to capture the local and global dependency, and has shown +to be effective in speech recognition (Chorowski et al., 2015), text generation (Liu et al., 2018), and machine +translation tasks (Luong et al., 2015). +The encoder-decoder architecture combined with attention is very versatile, and it can be combined +with other deep learning architectures to perform sequence learning in addition to language tasks. The +LSTM and attention architecture is applied to semantic trajectory prediction (Karatzoglou et al., 2018), +text summarization (Liang et al., 2020), demand modelling (Ren et al., 2020), and wind power forecasting +(Zhang et al., 2020). +When the goal is set to recover the original sequence, unsupervised learning of +molecule embedding can be obtained for downstream classification tasks (Xu et al., 2017). When the spatial +dimension is added, a convolutional neural network (CNN) layer can be added, and the dimension of the +sequence generated can be expanded. For example, Wang et al. (2020a) predict a city’s crowd flow patterns, +and Wu et al. (2020) generate 3D shapes via sequentially assembling different parts of the shapes. +While RNN-based architectures are still a widely adopted choice for seq2seq modeling, attention can also +be used as a standalone mechanism for seq2seq translations independent of RNNs. The idea was proposed +by Vaswani et al. (2017) in an architecture named transformer. Without recurrence, the network allows for +significantly more parallelization, and is shown to achieve superior performance in experiments, and powered +the popularity of transformer-based architectures in various sequence generation tasks (Huang et al., 2018; +Lu et al., 2021). A separate line of work by Zhang et al. (2019) also demonstrated that a hierarchical CNN +model with attention outperforms the traditional RNN-based models. +3.3. Using deep learning to generate TSP solutions +The above seq2seq translation mechanisms work well when the input data is naturally organized as +a sequence, and the output sequence corresponds to the input sequence, such as in music and language. +However, in our paper, the input is an unordered sequence, and the output has the same but re-ordered +elements of the same input sequence. +In this case, the concept of attention is helpful and has been +successfully used to produce solutions to the TSP. The pointer network, proposed by Vinyals et al. (2015) +and further developed in Vinyals et al. (2016), uses attention to select a member of the input sequence at +each decoder step. While it is not required that the input sequence is ordered, an informative ordering could +improve the performance (Vinyals et al., 2016). +While the original pointer network was solved as a classification problem and cross-entropy loss was +used, it is not necessarily the most efficient choice. The cross-entropy loss only distinguishes between +a correct prediction and an incorrect prediction. But in instances like routing, the distances between the +predicted position and the correct position, as well as the ordering of subsequences, could incur different +costs in practice. Further developments in solving TSP with machine learning methods involve reinforcement +learning (RL), which enables the optimization of custom evaluation metrics (Bello et al., 2019; Kool et al., +2019; Ma et al., 2019; Liu et al., 2020). Joshi et al. (2019) compared the performance of RL and supervised +learning (SL) on TSP solutions and found that SL and RL models achieve similar performance when the +graphs are of similar sizes in training and testing, whereas RL models have better generalizability over variable +graph sizes. However, RL models require significantly more data points and computational resources, which +is not always feasible. +Although this seq2seq and attention framework has only been used to reproduce TSP solutions, it provides +an opportunity to learn and incorporate additional information beyond the given travel times and potentially +6 + +learn individual differences when more information is given to the neural network. In this paper, we combine +the ideas of seq2seq modeling and attention to predict the actual route executed by a driver. +4. Methodology +This section details the methodology proposed to address the problem. First, the high-level seq2seq +modelling framework is introduced, followed by the explanation of the novel pair-wise attention and sequence +generation and selection mechanism used within the modelling framework. +4.1. Sequence-to-sequence modeling framework +Let the input sequence be an arbitrarily-ordered sequence (s1, ..., sn). Denote the output sequence as +(ˆs(1), ..., ˆs(n)). Let ci indicate the “position index” of stop ˆs(i) with respect to the input sequence (where +ci ∈ {1, ..., n}). For example, for input sequence (B, A, C) and output sequence (A, B, C), we have c1 = 2, +c2 = 1, c3 = 3, which means the first output stop A is in the second position of the input sequence (B, A, C) +and so on. +The seq2seq model computes the conditional probability P(c1, ..., cn | S; θ) using a parametric neural +network (e.g., recurrent neural network) with parameter θ, i.e., +P(c1, ..., cn | S, XS; θ) = P(c1 | S, XS; θ) · +n +� +i=2 +P(ci | c1, ..., ci−1, S, XS; θ) +(1) +The parameters of the model are learnt by empirical risk minimization (maximizing the conditional +probabilities on the training set), i.e., +θ∗ = arg max +θ +� +S +P(c1, ..., cn | S, XS; θ) +(2) +where the summation of S is over all training routes. In the following section, we will elaborate how +P(ci | c1, ..., ci−1, S, XS; θ) is calculated using the pair-wise attention-based pointer neural network. +4.2. Pair-wise attention-based pointer neural network +Figure 2 uses a four-stop example to illustrate the architecture of the proposed model. The whole model +is based on the LSTM encoder and decoder structure. In particular, we use one LSTM (i.e., encoder) to +read the input sequence, one time step at a time, to obtain a large fixed dimensional vector representation, +and then to use another LSTM (i.e., decoder) to extract the output sequence. However, different from the +typical seq2seq model, we borrow the idea of the pointer network (Vinyals et al., 2015) to add a pair-wise +attention mechanism to predict the output sequence based on the attention mask over the input sequence. The +pair-wise attention is calculated based on an ASNN which was previously used for travel mode prediction +(Wang et al., 2020b). Model details will be shown in the following sections. +Intuitively, the LSTM encoder and decoder aim to capture the global view of the input information +(i.e., overall sequence pattern) by embedding the input sequence to hidden vector representation. While the +ASNN-based pair-wise attention aims to capture the local view (i.e., the relationship between two stops). +Our experiments in Section 5 demonstrate the importance of both global and local views in the sequence +prediction. +7 + +Figure 2: Overall architecture of the pair-wise attention-based pointer neural network (adapted from Vinyals et al. (2015)) +4.2.1. LSTM encoder. +Given an arbitrary stop sequence (s1, ..., sn) as the input, let xi ∈ RK be the features of stop si, where +xi may include the package information, the customer information, and the geographical information of the +stop si. K is the number of features. The encoder computes a sequence of encoder output vectors (e1, ..., en) +by iterating the following: +hE +i , ei = LSTM(xi, hE +i−1; θE) +∀i = 1, ..., n +(3) +where hE +i ∈ RKE +h is the encoder hidden vector with hE +0 := 0. ei ∈ RKe is the encoder output vector. KE +h +and Ke are corresponding vector dimensions. θE is the learnable parameters in an encoder LSTM cell. +The calculation details of an LSTM cell can be found in Appendix A. The encoding process transforms a +sequence of features (x1, ..., xn) into a sequence of embedded representation (e1, ..., en). And the hidden +vector of the last time step (hE +n) includes the global information of the whole sequence, which will be used +for the LSTM decoder. +Figure 3: Illustration of LSTM ecnoder +4.2.2. LSTM decoder. +The role of a decoder in the traditional seq2seq model (Figure 4) is to predict a new sequence one time step +at a time. However, in the pointer network structure with attention, the role of the decoder becomes producing +8 + +AsNN Attention Component +Predict next is S3 +Predict next is S1 +Predict next is S2 +Predict next is S4 +S4 +S4 +Encoder +Decoderen +e1 +e2 +he +h2 +he +LSTM +LSTM +LSTM +Decoder +X2 +x1 +Xna vector to modulate the pair-wise attention over inputs. Denote the output sequence as (ˆs(1), ..., ˆs(n)). Let +x(i) be the feature of stop ˆs(i). +At decoder step i, we have +hD +(i+1), d(i) = LSTM +�� +x(i) +w(i) +� +, hD +(i); θD +� +∀i = 0, 1, ..., n +(4) +where hD +(i) ∈ RKD +h is the decoder hidden vector with hD +(0) = hE +n, d(i) ∈ RKd is the decoder output vector, KD +h +and Kd are corresponding vector dimensions, and θD are learnable parameters of the decoder LSTM cell. +Note that we set x(0) = xD and d(0) = dD, representing the features and the decoder output of the depot, +respectively. w(i) is the context vector calculated from the attention component, which will be explained in +the next section. +Figure 4: Illustration of LSTM decoder +4.2.3. ASNN-based pair-wise attention. +The pair-wise attention aims to aggregate the global and local information to predict the next stop. +Specifically, at each decoder time step i ∈ {0, ..., n}, we know that the last predicted stop is ˆs(i). To predict +ˆs(i+1), we consider all candidate stops sj ∈ S, which is the set of all stops not yet visited. We want to +evaluate how possible that sj will be the next stop of ˆs(i). The information of the stop pair ˆs(i) and sj can be +represented by the following concatenated vector: +vj +(i) = concat(zj +(i), φ(x(i), xj), d(i), ej) +(5) +where zj +(i) is a vector of features associated with the stop pair (such as travel time from ˆs(i) to sj), and +φ(x(i), xj) represents a feature processing function to extract the pair-wise information from x(i) and xj. For +example, φ(·) may return geographical relationship between stops ˆs(i) and sj, and it may also drop features +not useful for the attention calculation. Intuitively, zj +(i) and φ(x(i), xj) contains only local information of the +stop pair, while d(i) and ej contain the global information of the whole stop set and previously visited stops. +9 + +Predict next is S(n) +Output W(n) +. +Predict next is S(3) +ASNN +Output W(3) +Attention +Component +Predict next is S(2) +Output W(2) +Predict next is S(1) +Output W(1) +dp +d(1) +d(2) +d(n) +d(n-1) +4 +Encoder +LSTM +LSTM +LSTM +LSTM +LSTM +[] +[x(1) +x(2) +x(n-1) +x(n) +W(1) +W(2)) +W(n-1)) +W(n)]Figure 5: Illustration of ASNN-based pair-wise attention +Given the pair-wise information vector vj +(i), we can calculate the attention of stop ˆs(i) to stop sj as: +uj +(i) = ASNN(vj +(i); θA) +∀i, j = 1, ..., n +(6) +aj +(i) = +exp(uj +(i)) +�n +j′=1 exp(uj′ +(i)) +∀i, j = 1, ..., n +(7) +where aj +(i) ∈ R is attention of stop ˆs(i) to stop sj. ASNN(·; θA)) is a multilayer perception (MLP) with +the output dimension of one (i.e., uj +(i) ∈ R). θA are the learnable parameters of the ASNN. The name +“alternative specific” is because the same parametric network will be applied on all alternative stops sj ∈ S +separately (Wang et al., 2020b). Finally, we calculate the conditional probability to make the prediction: +P(ci+1 = j | c1, ..., ci, S, XS; θ) = aj +(i) +∀i = 0, 1, ..., n, j = 1, ..., n +(8) +ˆs(i+1) = arg max +sj∈S\SV +(i) +aj +(i) +∀i = 0, 1, ..., n +(9) +where SV +(i) = {ˆs(1), ..., ˆs(i)} is the set of stops that have been predicted (i.e., previously visited) until decoder +step i. Eqs. 8 and 9 indicate that the predicted next stop at step i is the one with highest attention among all +stops that have not been visited. +The pair-wise attention framework also leverages the attention information as the input for the next step. +This was achieved by introducing the context vector (Bahdanau et al., 2015): +w(i) = +n +� +j=1 +aj +(i) · ej +(10) +The context vector is a weighted sum of all the encoder output vectors with attention as the weights. As +the attention provides the emphasis for stop prediction, w(i) helps to incorporate the encoded representation +of the last predicted stop for the next stop prediction. The inputs for the next LSTM cell thus will be the +10 + +Predict next is S(i+1) +Output W(i+1) +Softmax +u +ASNN +ASNN +ASNN +e1 +e2 +en +he +D +LSTM +LSTM +LSTM +LSTM +X1 +X2 +Xn +x(i) +W(i)concatenation of the stop features and w(i), i.e., +� +x(i) +w(i) +� +. +It is worth noting that, the specific architecture of ASNN(·; θA)) can be flexible depending on the input +pair-wise information. For example, if the information includes images or networks, convolutional neural +network or graph convolutional networks can be used for better extract features. In this study, we use the +MLP for simplification as it already outperforms benchmark models. The key idea is of the ASNN is to +share the same trainable parameter θA for all stop pairs so as to better capture various pair-wise information +in the training process. +4.3. Sequence generation and selection +During inference, given a stop set S, the trained model with learned parameters θ∗ are used to generate +the sequence. Typically, in the seq2seq modeling framework, the final output sequence is selected as the one +with the highest probability, i.e., +(sj∗ +1, ..., sj∗n), where j∗ +1, ..., j∗ +n = arg max +j1,...,jn∈CS P(c1 = j1, ..., cn = jn | S, XS; θ∗) +(11) +where CS = {All permutations of {1, ..., n}} +Finding this optimal sequence is computationally impractical because of the combinatorial number of +possible output sequences. And so it is usually done with the greedy algorithm (i.e., always select the +most possible next stop) or the beam search procedure (i.e., find the best possible sequence among a set of +generated sequences given a beam size). However, in this study, we observe that the first predicted stop ˆs(1) +is critical for the quality of the generated sequence. The reason may be that the local relationship between a +stop pair (i.e., given the last stop to predict the next one) is easier to learn than the global relationship (i.e., +predict the whole sequence). Hence, in this study, we first generate sequences using the greedy algorithm +with different initial stops, and select the one with the lowest operational cost. The intuition behind this +process is that, once the first stop is given, the model can follow the learned pair-wise relationship to generate +the sequence with relatively high accuracy. For all the generated sequences with different first stops, the +one with the lowest operation cost captures the global view of the sequence’s quality. Therefore, the final +sequence generation and selection algorithm is as follows: +Algorithm 1 Sequence generation +Input: Trained model, S +Output: Predicted stop sequence +1: for s in S do +2: +Let the first predicted stop be ˆs(1) = s +3: +Predict the following stop sequence (ˆs(2), ..., ˆs(n)) using the greedy algorithm. Denote the predicted sequence +as Ps. +4: +Calculate the total operation cost of the whole sequence (including depot), denoted as OCs. +return Ps∗ where s∗ = arg mins∈S OCs +5. Case Study +5.1. Dataset +The data used in our case study was made available as part of the Amazon Last Mile Routing Research +Challenge (Merchán et al., 2022). The dataset contains a total of 6,112 actual Amazon driver trajectories +11 + +for the last-mile delivery from 5 major cities in the US: Austin, Boston, Chicago, Los Angeles, and Seattle. +Each route consists of a sequence of stops. Each stop represents the actual parking location of the driver, and +the package information (package numbers, package size, and planned service time) associated with each +stop is given. The stops are characterized by their latitudes and longitudes, and expected travel time between +stops are known. +Figure 6 shows the distribution of the number of stops per route and an example route. Most routes have +around 120 to 180 stops, and the maximum observed number of stops is around 250. Figure 6b shows an +example of an actual driver trajectory in Boston. Since the depot is far from the delivery stops, we attach the +complete route (with the depot indicated by a red dot) at the bottom left of the figure, while the main plot +only shows the delivery stops. +In this data set, each stop is associated with a zone ID (indicated by different colors in Figure 6b). When +Amazon generates planned routes for drivers, they usually expect drivers to finish the delivery for one zone +first, then go to another zone. And the actual driver trajectories also follow this pattern as shown in Figure +6b (but the actual zone sequence may be different from the planned one). Therefore, in this study, we focus +on the problem of zone sequence prediction. That is, si in the case study section now represents a specific +zone, S represents the set of zones, and XS represents zone features. This transformation does not affect the +model structure proposed in Section 4. The only difference is that the new problem has a relatively smaller +scale compared to the stop sequence prediction because the number of zones in a route is smaller than that +of stops. The zone-to-zone travel time is calculated as the average travel time of all stop pairs between the +two zones. Figure 7 presents an illustrative example of the relationship between zone and stop sequences. +As the dataset does not contain the original planned sequence, we assume the planned zone sequence is the +one with the lowest total travel time (generated by a TSP solver, (sT +1, ..., sT +n)). After generating the zone +sequence, we can restore the whole stop sequence by assuming that drivers within a specific zone follow an +optimal TSP tour. Details of the zone sequence to stop sequence generation can be found in Appendix B. +Figure 7: Relationship between stop sequence and zone sequence. +5.2. Experimental setup +We randomly select 4,889 routes for model training and cross-validation, and the remaining 1,223 routes +are used to evaluate/test model performance. +We consider a one-layer LSTM for both the encoder and decoder with the hidden unit sizes of 32 (i.e., +KD +h = Ke = KE +h = Kd = 32). And the ASNN is set with 2 hidden layers with 128 hidden units in each +layer. We train the model using Adam optimizer with a default learning rate of 0.001 and 30 training epochs. +To utilize the planned route information, the input zone sequence for the LSTM encoder is set as the TSP +12 + +Zone sequence +Zone 1 +Zone 2 +Zone 3 +B +C +D +E +G +H +A +Depot +Depot +Stop sequence(a) Number of stops distribution +(b) Actual route example +Figure 6: Description of dataset +13 + +400 +350 +300 +250 +Counts +200 +150 +100 +50 +0 +50 +100 +150 +200 +Number of stops per routeASTBOSTON +Air +LOPREST +ZOVESTE-ET +Complete route +CHELSEA +CHARLESTOWN +BOSTONresult (i.e., lowest travel time). That is, the input sequence (s1, ..., sn) = (sT +1, ..., sT +n). +In the case study, xi represents zone features, including the latitude and longitude of the zone center, +number of stops in the zone, number of intersections in the zone, number of packages in the zone, total +service time in the zone, total package size in the zone, and the travel time from this zone to all other zones. +The zone pair features zj +(i) includes the travel time from ˆs(i) to sj and zone ID relationship characteristics. +For example, the zone IDs “B-6.2C” and “B-6.3A” signal that they belong to the higher-level cluster “B-6”. +As we assume all pair-wise features are captured by zj +(i), φ(x(i), xj) is not specified in this case study. +Consistent with the Amazon Last Mile Routing Research Challenge, we evaluate the quality of the +predicted stop sequences using a “disparity score” defined as follows: +R(A, B) = SD(A, B) · ERPnorm(A, B) +ERPe(A, B) +(12) +where R(A, B) is the disparity score for the actual sequence A and predicted sequence B, and SD(A, B) is +the sequence deviation defined as +SD(A, B) = +2 +n(n − 1) +n +� +i=2 +� +|c[Bi] − c[Bi−1]| − 1 +� +(13) +where n is the total number of stops, Bi is the i-th stop of sequence B, c[Bi] is the index of stop Bi in the +actual sequence A (i.e., its position in sequence A). In the case of A = B (i.e., perfectly predicted), we have +c[Bi] − c[Bi−1] = 1 for all i = 2, ..., n, and SD(A, B) = 0. +ERPnorm(A, B) is the Edited Distance with Real Penalty (ERP) defined by the following recursive +formula: +ERPnorm(A, B) = ERPnorm(A2:|A|, B2:|B|) + Timenorm(A1, B1) +(14) +where Timenorm(si, sj) = +Time(si,sj) +� +j′∈{1,...,n} Time(si,sj′) is the normalized travel time from stop si to stop +sj. +ERPe(A, B) is the number of edit operations (insertions, substitutions, or deletions) required to +transform sequence A to sequence B as when executing the recursive ERPnorm formulation. Hence, the +ratio ERPnorm(A,B) +ERPe(A,B) represents the average normalized travel time between the two stops involved in each ERP +edit operation. In the case of A = B, we have ERPnorm(A,B) +ERPe(A,B) += 0. +The disparity score R(A, B) describes how well the model-estimated sequence matches the known actual +sequence. Lower score indicates better model performance. A score of zero means perfect prediction. The +final model performance is evaluated by the mean score over all routes in the test set. +In addition to the disparity score, we also evaluate the prediction accuracy of the first four zones in each +route. We choose the first four because the minimum number of zones in a route is four. +5.3. Benchmark models +The following optimization and machine learning models are used as benchmarks to compare with the +proposed approach. +Conventional TSP. The first benchmark model is the zone sequence generated by conventional TSP, +which we treat as the planned route with the lowest travel time. +ASNN model. The ASNN component can be trained to predict the next zone given the current zone, and +the prediction sequence can be constructed in a greedy way starting from the given depot. The training zone +14 + +pairs (including from depot to the first zone) are extracted from all sequences in the training routes. And the +input features are the same as the ASNN component in the proposed model except for (d(i), ej) (i.e., output +vectors from LSTM decoder and encoder, respectively). All hyper-parameters of the ASNN model are the +same as the attention component. +Inspired by the importance of the first zone, we also implement another sequence generation method +similar to Section 4.3. That is, we go through all zones in a route and assume it is the first zone, then use the +trained ASNN to predict the remaining sequence. The final sequence is selected as the one with the lowest +travel time. +LSTM-encoder-decoder. The LSTM-encoder-decoder (LSTM-E-D) architecture is a typical seq2seq +model proposed by Sutskever et al. (2014). The model structure is shown in Figure 8. In the decoder stage, +the model outputs the predicted zone based on last predicted zone’s information. The model formulation can +be written as +hE +i , ei = LSTM(xi, hE +i−1; θE) +∀i = 1, ..., n +(15) +hD +(i+1), d(i) = LSTM(x(i), hD +(i); θD) +∀i = 0, 1, ..., n +(16) +The decoder output vector d(i) are, then feed into a fully-connected (FC) layer to calculate probability of the +next stop: +g(i) = FC(d(i); θFC) +∀i = 1, ..., n +(17) +P(ci+1 | c1, ..., ci, S, XS; θ) = Softmax(g(i)) +∀i = 1, ..., n +(18) +where g(i) ∈ RKz, Kz is the maximum number of zones in the dataset. And the next predicted stop is +selected by maximizing P(ci+1 = j | c1, ..., ci, S, XS; θ) for all sj ∈ S \ SV +(i) (i.e., the zones that are not in +the route and that have been visited are excluded). +Figure 8: Model architecture of the LSTM-E-D seq2seq prediction model. +Original Pointer Network. Another benchmark model is the original pointer network (Pnt Net) proposed +by (Vinyals et al., 2015). The overall architecture of the pointer network is similar to the proposed model +15 + +S4 +S2 +S1 +S3 +End +FC + Softmax +S1 +S2 +S3 +S4 +Ds +S4 +S2 +S3 +S +Encoder +Decoderexcept for the attention component. Specifically, the pointer network calculates attention as: +uj +(i) = W T +1 tanh(W2ej + W3d(i)) +∀i, j = 1, ..., n +(19) +aj +(i) = +exp(uj +(i)) +�n +j′=1 exp(uj′ +(i)) +∀i, j = 1, ..., n +(20) +The original pointer network does not include the pair-wise local information (zj +(i), φ(x(i), xj)), and the +attention calculation is only quantified from three learnable parameters W1, W2, and W3, which may limit +its capacity in prediction. We observe that the original pointer network without local information performs +extremely badly. For a fair comparison, we add the local information with the similar format in Eq. 19 as: +uj +(i) = W T +1 tanh(W2ej + W3d(i)) + W4 +� +zj +(i) +φ(x(i), xj) +� +∀i, j = 1, ..., n +(21) +After training the model, we generate the final sequence with the greedy algorithm and Algorithm 1, +respectively. +5.4. Results +5.4.1. Model comparison. +Table 1 presents the performance of different models. Note that for all approaches except for the TSP, we +generate sequences based on two different methods (greedy and Algorithm 1) for comparison. The standard +deviation of disparity scores is taken over all testing routes. Results show that sequence generation with +Algorithm 1 (i.e., iterating different first zones) can consistently reduce the disparity score for all machine +learning methods. +It implies that the first zone prediction and the global view (i.e., shortest path) are +important for estimating the driver’s trajectory. +The proposed method outperforms all other models, both in disparity scores and prediction accuracy. +This means the proposed pair-wise ASNN-based attention (Eq. 6) has better performance than the original +content-based attention (Eq. 21). The comparison between LSTM-E-D and Pnt Net models demonstrates +the effectiveness of the attention mechanism. +All machine learning models except for LSTM-E-D can +outperform the baseline TSP sequence with Algorithm 1 sequence generation method, suggesting that the +hidden trajectory patterns can be learned from the data. +Another observation is that, the prediction accuracy and disparity score do not always move in the same +direction. For example, the LSTM-E-D model with Algorithm 1 sequence generation, though has lower +accuracy, shows a better disparity score. This is because the accuracy metric does not differentiate “how +wrong an erroneous prediction is”. By the definition of disparity score, if a stop is si but the prediction is +sj, and sj and si are geographically close to each other, the score does not worsen too much. This suggests +a future research direction in using disparity score as the loss function (e.g., training by RL) instead of +cross-entropy loss. +Figure 9 shows the distribution of disparity scores for our proposed method with Algorithm 1 sequence +generation (i.e., the best model). We observe that the prediction performance varies a lot across different +routes. There is a huge proportion of routes with very small disparity scores (less than 0.01). The mean +score is impacted by outlier routes. The median score is 0.0340, which is smaller than the mean value. +16 + +Table 1: Model performance +Sequence generation +Model +Disparity score +Prediction accuracy +Mean +Std. Dev +1st zone +2nd zone +3rd zone +4th zone +- +TSP +0.0443 +0.0289 +0.207 +0.185 +0.163 +0.168 +Greedy +ASNN +0.0470 +0.0289 +0.150 +0.141 +0.119 +0.123 +LSTM-E-D +0.0503 +0.0313 +0.207 +0.183 +0.161 +0.166 +Pnt Net +0.0460 +0.0309 +0.224 +0.204 +0.186 +0.165 +Ours +0.0417 +0.0306 +0.241 +0.231 +0.224 +0.221 +Algorithm 1 +ASNN +0.0429 +0.0299 +0.221 +0.213 +0.203 +0.195 +LSTM-E-D +0.0501 +0.0305 +0.182 +0.156 +0.142 +0.149 +Pnt Net +0.0382 +0.0301 +0.286 +0.273 +0.262 +0.274 +Ours +0.0369 +0.0301 +0.320 +0.310 +0.303 +0.314 +Figure 9: Disparity score distribution of the best model +5.4.2. Factors on trajectory predictability. +As our proposed model exhibits various levels of predictability across different routes, we aim to +investigate which attributes of a route cause high (or low) predictability. This can be done by running a +regression model with the disparity score as the dependent variable and route attributes (e.g., locations, +departure time, package numbers) as independent variables. The variables used are defined as follows: +• Total planned service time: The estimated time to deliver all packages in the route (service time only, +excluding travel time). +• Earliest time window constraint: The earliest due time to deliver packages with time window constraint +minus the vehicle departure time. The smaller the value, the tighter the time limit. +• Avg. # traffic signals: Average number of traffic signals in each zone of the route (obtained from +OpenStreetMap data). +17 + +250 +Mean = 0.0369 +Median = 0.034 +200 +150 +Counts +100 +50 +0.00 +0.05 +0.10 +0.15 +0.20 +Disparity scores• If high-quality route: A dummy variable indicating whether the route is labeled as “high quality” by +Amazon or not (Yes = 1). High quality means the actual travel time of the route is similar to or better +than Amazon’s expectation. +• If in Location: A dummy variable indicating whether the route is in a specific city or not (Yes = 1). +• If departure Time: A dummy variable indicating the (local) departure time (e.g., before 7AM, after +10AM). +Table 2 shows the results of the regression. Since the dependent variable is disparity scores, a negative +sign indicates a positive impact on the predictability. We observe that routes with tighter time window +constraints and more stops are easier to predict. This may be due to the fact that these routes are usually +harder to deliver. Hence, to avoid the risk of violating time constraints or delay, drivers tend to follow the +planned routes and thus the route sequences are easier predict. We also find that routes associated with larger +vans (i.e., larger vehicle capacity) are more predictable. The reason may be that larger vans are less flexible +in choosing different routes, thus drivers are more likely to follow the navigation. Another important factor +for better predictability is high-quality routes. This may be because high-quality routes are closer to the TSP +sequence which we use as inputs. Finally, routes in LA are more predictable than in other areas such as +Chicago and Boston. +Table 2: Factors on trajectory predictability +Variables +Coefficients (×10−3) +Variables +Coefficients (×10−3) +Intercept +91.07 ** +If high quality route +-1.66×10−14 ** +Total # of packages +0.059 +If in LA +-4.998 * +Total planned service time +-0.476 +If in Chicago +0.783 +Earliest time window constraint +-3.047 ** +If in Boston +-3.354 +Avg. # traffic signals +-3.255 +If on weekends +1.775 +Total # of stops +-0.142 ** +If departure before 7AM +0.582 +Vehicle capacity (m3) +-6.041 * +If departure after 10AM +-2.704 +Number of routes: 1,002. +R2: 0.065; +∗∗: p-value < 0.01; ∗: p-value < 0.05. +5.4.3. Impact of input sequence. +All machine learning models in Table 1 (except for ASNN) have the LSTM encoder component, which +requires the specification of input zone sequence. As mentioned in Section 5.2, we currently use the TSP +sequence as input. It is worth exploring the model performance if we use a random zone sequence instead, +which corresponds to the scenario without planned route information. Table 3 shows the model performance +without the TSP sequence information. Since the ASNN result does not rely on TSP information, it is not +listed in the table. Results show that the LSTM-E-D model becomes much worse with a random sequence as +inputs, while the performance of Pnt Net and our method is only slightly affected. Even without the planned +route information, the proposed model can still provide a reasonable estimation of driver trajectories. +18 + +Table 3: Model performance without TSP information +Sequence generation +Model +Disparity score +Prediction accuracy +Mean +Std. Dev +1st zone +2nd zone +3rd zone +4th zone +Greedy +LSTM-E-D +0.1176 +0.0498 +0.045 +0.047 +0.041 +0.050 +Pnt Net +0.0512 +0.0323 +0.090 +0.096 +0.097 +0.096 +Ours +0.0426 +0.0311 +0.204 +0.192 +0.195 +0.196 +Algorithm 1 +LSTM-E-D +0.1054 +0.0463 +0.103 +0.061 +0.049 +0.052 +Pnt Net +0.0398 +0.0311 +0.298 +0.284 +0.273 +0.273 +Ours +0.0376 +0.0307 +0.316 +0.298 +0.302 +0.298 +5.5. Summary +Our numerical results show that our proposed model outperforms its benchmarks in terms of disparity +scores and prediction accuracy, meaning that it can better predict the actual route trajectories taken by drivers. +The comparison with benchmark models shows that our proposed ASNN-based pair-wise attention mecha- +nism and our sequence generation algorithm (Algorithm 1) are both helpful for the prediction. Moreover, +we can observe that the predictive performance varies across different routes. Factors such as route quality, +delivery time windows, and the total number of stops of a route affect predictability. Finally, the proposed +model is insensitive to the input sequence. The prediction performance only slightly decreases when the +input sequence is changed from the TSP solution to a random stop sequence. This property implies that we +only need the set of stops to implement the model and obtain high-quality solution, while information on the +planned route sequence is not strictly required. +6. Conclusion and Future Research +In this paper, we propose a pair-wise attention-based pointer neural network that predicts actual driver +trajectories on last-mile delivery routes for given sets of delivery stops. Compared to previously proposed +pointer networks, this study leverages a new alternative specific neural network-based attention mechanism +to incorporate pair-wise local information (such as relative distances and locations of stops) for the attention +calculation. To better capture the global efficiency of a route in terms of operational cost (i.e., total travel +time), we further propose a new sequence generation algorithm that finds the lowest-cost route sequence by +iterating through different first stops. +We apply our proposed method to a large set of real operational route data provided by the Amazon +Last-Mile Routing Research Challenge in 2021. The results show that our proposed method can outperform +a wide range of benchmark models in terms of both the disparity score and prediction accuracy, meaning +that the predicted route sequence is closer to the actual sequence executed by drivers. Compared to the best +benchmark model (original pointer network), our method reduces the disparity score from 0.0382 to 0.0369, +and increases the average prediction accuracy of the first four zones from 0.229 to 0.312. Moreover, our +proposed sequence generation method can consistently improve the prediction performance for all models. +The disparity scores are reduced by 10-20% across different models. Lastly, we show that the proposed +methodology is robust against changes in the input sequence pattern. Compared to an optimal TSP solution +as the input sequence, a random input sequence only slightly increases the disparity score from 0.0369 to +0.0376. +19 + +The data-driven route planning method proposed in this paper has several highly relevant practical +implications. First, our proposed model performs well at predicting stop sequences that would be preferable +to delivery drivers in a real operational environment, even if it is not provided with a theoretically optimal +(i.e., minimal route duration) planned TSP sequence as an input. Therefore, the model can be used to +generate a predicted actual stop sequence that a driver would likely be taking for a given set of delivery +stops. The prediction can serve as a new ‘empirical’ planned route that is informed by historical driver +behavior and thus more consistent with the driver’s experience and preferences. Second, by comparing +the stop sequence predicted by our model with the traditional, TSP-based planned stop sequence, a route +planner may infer potential reasons for the drivers’ deviations and adjust the company’s planning procedures +and/or driver incentives if necessary. Third, as stop sequence generation using machine learning models is +computationally more efficient than traditional optimization-based approaches, a trained machine learning +model can be applied in real-time to quickly re-optimize routes when drivers are unexpectedly forced to +deviate from their original stop sequence (e.g., due to road closures) and need updated routing strategies. +Based on the work presented in this paper, a number of fruitful future research avenues arise. First, instead +of focusing on stop sequence prediction, future work may improve the interpretability of such prediction +models and develop machine learning approaches that better explain which factors cause drivers to deviate +from a planned stop sequence and how they affect their actual route trajectories. Second, future work should +attempt to combine the strengths of optimization-based route planning approaches and machine learning by +incorporating tacit driver knowledge learned via machine learning models into route optimization algorithms. +Appendices +Appendix A. Mathematical Formulation of a LSTM Cell +The details of an LSTM cell, ht, et = LSTM(xt, ht−1; θ), is shown below: +ft = σg(Wfxt + Ufht−1 + bf) +(A.1) +it = σg(Wixt + Uiht−1 + bi) +(A.2) +ot = σg(Woxt + Uoht−1 + bo) +(A.3) +˜ct = σc(Wcxt + Ucht−1 + bc) +(A.4) +ct = ft ◦ ct−1 + it ◦ ˜ct +(A.5) +ht = ot ◦ σh(ct) +(A.6) +et = ht (if this is a single layer one-directional LSTM) +(A.7) +where [Wf, Wi, Wo, Wc, Uf, Ui, Uo, Uc, bf, bi, bo, bc] = θ is the vector of learnable parameters. xt is the +input vector to the LSTM unit. ft is the forget gate’s activation vector. it is the input/update gate’s activation +vector. ot is the output gate’s activation vector. ht is the hidden state vector. et is the output vector of +the LSTM. Note that for a multi-layer or bidirectional LSTM, et may not equal to ht. In this study, we +use a single layer one-directional LSTM and thus have et = ht. More details on the output vector can be +found in Pytorch (2021). ˜ct is the cell input activation vector. ct is the cell state vector. “◦” indicates the +component-wise multiplication. +20 + +Appendix B. From Zone Sequence to Stop Sequence +The complete stop sequence is generated based on the given zone sequence. The detailed generation +process is shown in Algorithm 2. +Algorithm 2 Complete sequence generation. Input: zone sequence (ˆz(1), .., ˆz(n)), depot DS, set of stops in +each zone S(i), i = 1, ..., n. PathTSP(S, sfirst, slast) and TourTSP(S) are two oracle functions for solving +path and tour TSP problems given the set of stops S, first stop sfirst and last stop slast to be visited. +1: function CompleteSeqGeneration((ˆz(1), .., ˆz(n)), {S(i), i = 1, , , n}) +2: +sprev ← DS +3: +s∗ +complete ← (sprev) +▷ Initialize the complete stop sequence with depot +4: +for i ∈ {1, ..., n − 1} do +5: +Sfirst ← Set of three stops in S(i) that are closest to sprev +6: +Slast ← Set of three stops in S(i) that are closest to all stops in Si+1 on average +7: +P(i) ← ∅ +▷ Initialize the set of optimal paths in zone ˆz(i) +8: +for sfirst ∈ Sfirst do +9: +for slast ∈ Slast do +10: +if sfirst = slast then +11: +ˆptemp, ttemp = TourTSP(S(i)) +▷ Solve the optimal tour and travel time for zone ˆz(i) +12: +Delete the last edge back to sfirst in the tour ˆptemp. Let the new path and travel time be ˆp′ +temp +and t′ +temp +13: +Add ˆp′ +temp and t′ +temp to P(i) +14: +else +15: +ˆptemp, ttemp = PathTSP(S(i), sfirst, slast) ▷ Solve the optimal path and travel time for zone i +16: +Add ˆptemp and ttemp to P(i) +17: +ˆp(i) ← Path in P(i) with the minimum travel time +18: +s∗ +complete ← (s∗ +complete, ˆp(i)) +▷ Concatenate two sequence +19: +sprev ← Last stop of path ˆp(i) +20: +s∗ +complete ← (s∗ +complete, DS) +▷ Concatenate the last stop as the depot +21: +return s∗ +complete +Consider an optimal zone sequence, (ˆz(1), .., ˆz(n)), generated from the proposed machine learning +method. We can always add the depot before the first and after last zone (i.e., (DS, ˆz(1), .., ˆz(n), DS)) and +make the whole zone sequence a loop. For each zone ˆz(i), we aim to generate a within-zone path ˆp(i), and +the final stop sequence will be (DS, ˆp(i), ..., ˆp(n), DS). +When generating ˆp(i) for zone ˆz(i), we assume ˆp(i−1) is known (generated from the last step and +ˆp(0) = (DS)). Let the set of all stops in zone ˆz(i) be S(i). We identify three potential first stops and last +stops of path ˆp(i) based on following rules: +• Three potential first stops of ˆp(i) are the three most closest stops (in travel time) to ˆp(i−1)’s last stop. +• Three potential last stops of ˆp(i) are the three most closest stops (in travel time) to all stops in S(i+1) +on average. Note that S(n+1) = {DS} +With three potential first stops and last stops, we then solve path TSP problems between any first and last +stop pair to generate the potential optimal inner zone path with the shortest travel time. In this step, at most +21 + +nine small-scale path TSP problems will be solved since there might be overlapping between the first and +the last stops. If the first and the last stops are identical, we solve a tour TSP problem and output the path by +deleting the last edge which traverses back to the first stop in the tour. +After having all potential inner zone paths and total path travel time between any first and last stop pair, +we keep the path with the minimum travel time as the inner zone sequence, ˆp(i). The key assumption we +make here about drivers is that they will deliver packages within a zone following a path that minimizes their +total travel time. With the optimal inner zone stop sequence of the current zone, we then move to the next +visited zone in the optimal zone sequence and repeat the same procedure until we generate the complete stop +sequence. +References +Applegate, D.L., Bixby, R.E., Chvatal, V., Cook, W.J., 2006. The Travelling Salesman Problem. 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Xiaotong Guoa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Matthias Winkenbachb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Jinhua Zhaoc aDepartment of Civil and Environmental Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' MA 02139 bCenter for Transportation and Logistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' MA 20139 cDepartment of Urban Studies and Planning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' MA 20139 Abstract In last-mile delivery,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' drivers frequently deviate from planned delivery routes because of their tacit knowledge of the road and curbside infrastructure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' customer availability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' and other characteristics of the respective service areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Hence, the actual stop sequences chosen by an experienced human driver may be potentially preferable to the theoretical shortest-distance routing under real-life operational conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Thus, being able to predict the actual stop sequence that a human driver would follow can help to improve route planning in last-mile delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' This paper proposes a pair-wise attention-based pointer neural network for this prediction task using drivers’ historical delivery trajectory data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In addition to the commonly used encoder-decoder architecture for sequence-to-sequence prediction, we propose a new attention mechanism based on an alternative specific neural network to capture the local pair-wise information for each pair of stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' To further capture the global efficiency of the route, we propose a new iterative sequence generation algorithm that is used after model training to identify the first stop of a route that yields the lowest operational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Results from an extensive case study on real operational data from Amazon’s last-mile delivery operations in the US show that our proposed method can significantly outperform traditional optimization-based approaches and other machine learning methods (such as the Long Short-Term Memory encoder-decoder and the original pointer network) in finding stop sequences that are closer to high-quality routes executed by experienced drivers in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Compared to benchmark models, the proposed model can increase the average prediction accuracy of the first four stops from around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='312, and reduce the disparity between the predicted route and the actual route by around 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Keywords: Route planning, Trajectory prediction, Sequence-to-sequence model, Last-mile delivery, Pointer network, Attention 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Introduction The optimal planning and efficient execution of last-mile delivery routes is becoming increasingly important for the business operations of many logistics service providers around the globe for a variety of reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' E-commerce volumes are growing rapidly and make up a constantly growing share of overall retail sales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' For instance, in the US, the share of e-commerce sales in total retail sales has grown from around 4% in 2010 to around 13% in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Even by the end of 2019, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', before the outbreak of the COVID-19 pandemic, ∗Corresponding author Preprint submitted to Elsevier January 11, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='03802v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='LG] 10 Jan 2023 it had reached 11% (US Census Bureau, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Undoubtedly, the pandemic further accelerated the growth of e-commerce (postnord, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' McKinsey & Company, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In the medium to long run, its growth will continue to be fueled by an ongoing trend towards further urbanization, which is particularly pronounced in developing and emerging economies (United Nations Department of Economic and Social Affairs, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The share of the global population living in urban areas is currently projected to rise from around 55% in 2018 to around 68% by 2050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The associated increase in population density in most urban areas will likely lead to growing operational uncertainties for logistics service providers, as increasing congestion levels, less predictable travel times, and scarce curb space make efficient and reliable transport of goods into and out of urban markets increasingly challenging (Rose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' As a result of the continued boom of e-commerce and constantly growing cities, global parcel delivery volumes have been increasing rapidly in recent years and are expected to continue to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Across the 13 largest global markets, including the US, Brazil, and China, the volume of parcels delivered more than tripled from 43 billion in 2014 to 131 billion in 2020 (Pitney Bowes, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' At the same time, customer expectations towards last-mile logistics services are rising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' For instance, there is a growing demand for shorter delivery lead times, including instant delivery services and same-day delivery, as well as customer- defined delivery preferences when it comes to the time and place of delivery (Lim and Winkenbach, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Cortes and Suzuki, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Snoeck and Winkenbach, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The rapid growth and increasing operational complexity of urban parcel delivery operations also amplifies their negative externalities, including their contribution to greenhouse gas and other pollutant emissions, public health safety risks, as well as overall urban congestion and a corresponding decline in overall mobility and accessibility of cities (Jaller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' World Economic Forum, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' When applied to realistically sized instances of a last-mile delivery problem, solving the underlying traveling salesman problem (TSP) or vehicle routing problem (VRP) to (near) optimality becomes chal- lenging, as both problem classes are known to be NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Traditional TSP and VRP formulations aim to minimize the total distance or duration of the route(s) required to serve a given set of delivery stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The operations research literature has covered the TSP, VRP, and their many variants extensively, and in recent years important advances have been made with regards to solution quality and computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' However, in practice, many drivers, with their own tacit knowledge of delivery routes and service areas, divert from seemingly optimal routes for reasons that are difficult to encode in an optimization model directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' For exam- ple, experienced drivers may have a better understanding of which roads are hard to navigate, at which times traffic is likely to be bad, when and where they can easily find parking, and which stops can be conveniently served together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Therefore, compared to the theoretically optimal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', distance or time minimizing) route, the deviated actual route sequence chosen by an experienced human driver is potentially preferable under real-life operational conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' An important challenge in today’s last-mile delivery route planning is therefore to leverage historical route execution data to propose planned route sequences that are close to the actual trajectories that would be executed by drivers, given the delivery requests and their characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Note that, while distance and time-based route efficiency is still an important factor for planning route sequences, it is not the sole objective, as tacit driver knowledge is also incorporated in the proposed route sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Unlike a typical VRP in which the number of vehicles and their respective route sequences need to be determined simultaneously, in this study, we focus on solving a problem that is similar to a TSP at the individual vehicle level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' That is, we aim to solve a stop sequence to serve a given set of delivery requests, and expect that the proposed stop sequence is as close to the actual trajectories that would be executed by drivers as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 2 To this end, we propose a pair-wise attention-based pointer neural network to predict the actual route sequence taken by delivery drivers using drivers’ historical delivery trajectory data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The proposed model follows a typical encoder-decoder architecture for the sequence-to-sequence prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' However, unlike previous studies, we propose a new attention mechanism based on an alternative specific neural network (ASNN) to capture the local pair-wise information for each stop pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' To further capture the global efficiency of the route (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', its operational cost in terms of total distance or duration), after model training, we propose a new sequence generation algorithm that iterates over different first stops and selects the route with the lowest operational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The main contribution of this paper is three-fold: First, we propose a new ASNN-based attention mechanism to capture the local information between pairs of stops (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', travel time, geographical relation), which can be well adapted to the original pointer network framework for sequence prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Second, we propose a new sequence generation algorithm that iterates over different first stops in the predicted route sequences and selects the lowest operational cost route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The intuition is that the stop-to-stop relationship (referred to as the local view) is easier to learn from data than the stop sequence of the route as a whole (referred to as the global view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Lastly, we apply our proposed method to a large set of routes executed by Amazon delivery drivers in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The results show that our proposed model can outperform traditional optimization-based approaches and other machine learning methods in finding stop sequences that are closer to high-quality routes executed by experienced drivers in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The remainder of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In Section 2 we define the problem setting under investigation in a more formal way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Section 3 then reviews previous studies in the literature related to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Section 4 presents our methodology and elaborates on the detailed architecture of the proposed pair-wise attention-based pointer neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Section 5 presents the experimental setup and numerical results of our case study, applying our proposed method to real-world data made available by the Amazon Last-Mile Routing Research Challenge (Merchán et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Winkenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Section 6 concludes this paper and discusses future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Problem Setting In the last-mile delivery routing problem considered here, a set of stops S = {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', sn} to be served by a given delivery vehicle is given to the route planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The planner’s objective is to find the optimal stop sequence that has the minimal operational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In this case, we consider total cost as total travel time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The planner is given the expected operational cost (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', travel times) between all pairs of stops (si, sj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The theoretically optimal stop sequence, denoted by (sT (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', sT (n)), can be found by solving a TSP formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' This stop sequence is referred to as the planned stop sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' However, as discussed in Section 1, minimizing the theoretical operational cost (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', total travel time) of the route may not capture drivers’ tacit knowledge about the road network, infrastructure, and recipients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Therefore, the actual driver executed stop sequence (s(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', s(n)) can be different from the planned route sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Note that here, s(i) ∈ S denotes the i-th stop that is actually visited by the driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The objective of the model presented in this study is to predict the actual driver executed sequence (s(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', s(n)) given a set of stops S and the corresponding delivery requests and characteristics XS (such as the number of packages, estimated service time for each package, geographical information for each stop, travel time between each stop pairs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' All drivers are assumed to start their routes from a known depot DS and return back to DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Therefore, the complete trajectory should be a tour (DS, s(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', s(n), DS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' For the convenience of model description, we ignore the depot station in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 3 Figure 1 provides a simple example for illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In this example, we are given four stops S = {s1, s2, s3, s4} and a depot DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The planned stop sequence for the driver is (s4, s1, s2, s3), while the actual stop sequence executed by the driver is (s4, s2, s1, s3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The proposed model aims to predict the actual sequence (s4, s2, s1, s3) given the depot location DS, the set of stops to be visited S, and characteristics of the stops XS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' This problem setup is inspired by the Amazon Last-Mile Routing Research Challenge (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', Winkenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Note that this study only focuses on the stop sequence prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The routing between stops is not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' It is assumed that the drivers always take the optimal route between stops, which is reflected by the travel time matrix between stops in our problem setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Figure 1: Illustrative example of the problem setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Literature Review The problem setting defined in Section 2 involves both solving a cost-minimizing routing problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', the TSP) and capturing tacit driver knowledge to learn systematic deviation of drivers from the planned and theoretically optimal stop sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Therefore, we will first review the extant literature on the TSPs and its most relevant variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' We will then go through various machine learning approaches that have been proposed by the extant literature to generate sequences, with a section on methods specifically for solving the TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Note that although these machine learning approaches are used to solve the TSP instead of the actual routes taken by drivers, their architectures may be helpful to learn the actual route as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Travelling salesman problems First, given the travel times between stops, a solution to the TSP, which finds the route with the minimum cost or distance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', the planned route), can be a close approximation of the actual route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Since the drivers are paid for the number of packages delivered, all drivers’ goal is to deliver the packages in the minimum amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Most of the drivers do follow large parts of the planned routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The TSP is a well-known NP-hard problem that has been studied extensively over the last century, with a lot of books and review papers published on its history, formulations, solution approaches, and applications (Applegate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Matai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Davendra and Bialic-Davendra, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' An overview of the relevant TSP variants and solution approaches are presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The basic setup of TSP has one traveler and requires the traveler to return to the starting point after visiting each node exactly once, and that the traveling cost matrix (represented by distance and/or time) is 4 Depot Ds Input Output S,Xs,Ds Model Actual route S3 S1 Actual route S2 Planned routesymmetric (cost between i and j is the same with that between j and i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In most real-world applications, the basic setup needs to be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' For example, the cost matrix, if represented by travel times, is likely asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' This variant of TSP is thus named asymmetric TSP (ATSP) (Jonker and Volgenant, 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In some applications, the vehicle does not need to return to the original depot (Traub et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2021), or it can charge/refuel and potentially load additional delivery items at intermediate stops (Küçükoğlu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In many last-mile delivery applications, some packages are time-sensitive, and therefore time window constraints to their delivery need to be considered in a so-called TSP with time windows (TSPTW) (da Silva and Urrutia, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Mladenović et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In large systems, there might be more than one salesman serving a set of stops, resulting in multiple traveling salesmen problems (MTSPs) (Cheikhrouhou and Khoufi, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Different variants of TSP further impose different constraints on the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' While some problems can be reduced to the basic setup in the formulation stage, others require more versatile solution algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In general, the solution approaches to the TSP can be divided into exact approaches and approximate approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Exact approaches include branch-and-cut (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2020) and branch-and-bound (Salman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Since the TSP is a well-known NP-hard problem, exact approaches can only be applied on problems of smaller scale, or aid in heuristics to cut the solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Among approximate approaches, there are heuristics designed for the TSP specifically, as well as meta-heuristics that are generic and treat the problem like a blackbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The most commonly used heuristics and meta-heuristics include nearest neighbor searches, local searches, simulated annealing, and genetic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' A more comprehensive review of existing solution approaches can be found in Halim and Ismail (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Purkayastha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Despite the TSP being NP-hard, modern mixed-integer optimization solvers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', Gurobi, CPLEX, or GLPK) can solve it efficiently for real-world instances by combining exact approaches with heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Sequence-to-sequence prediction using deep learning The TSP and its variants are a viable option for sequence generation only when the objective is clearly de- fined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' They fall short when the sequence generation problem does not have a well-defined cost-minimization objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In a lot of applications, the rule of sequence generation cannot be simply defined and optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' A standard example for a sequence learning problem is machine translation, where a sequence of words in one language needs to be translated to another language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Another type of sequence learning is time series modeling, where a sequence of historical observations is given to predict future states of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In both cases, the primary modeling task is to learn the sequence generation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In recent years, deep learning has successfully achieved great performance in various settings of sequence generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' These models are often referred to as sequence-to-sequence (seq2seq) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' seq2seq models often consist of an encoder and a decoder, where the encoder encodes the input sequence into a fixed-length vector representation, and the decoder generates a sequence based on the generated vector representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Most encoder-decoder architectures adopt recurrent neural network (RNN) layers and its variants such as Long Short-Term Memory (LSTM) (Hochreiter and Schmidhuber, 1997) and gated recurrent layers (GRU) (Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2014) to learn long-range dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Early works using LSTM alone were able to generate plausible texts (Graves, 2013) and translate between English and French (Sutskever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2014) with long-range dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' (2014) demonstrate the superiority of GRU compared to LSTMs in music and speech signal modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Attention-based mechanisms, first introduced by Bahdanau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' (2015), have been shown to be a great addition since it allows the decoder to selectively attend to parts of the input sequence and relieves the encoder of the task of encoding all the information into a fixed-length vector representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Most sequence generation 5 problems benefit from keeping track of long-range dependencies and global context while decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' To address that, multi-level attention was proposed to capture the local and global dependency, and has shown to be effective in speech recognition (Chorowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2015), text generation (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2018), and machine translation tasks (Luong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The encoder-decoder architecture combined with attention is very versatile, and it can be combined with other deep learning architectures to perform sequence learning in addition to language tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The LSTM and attention architecture is applied to semantic trajectory prediction (Karatzoglou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2018), text summarization (Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2020), demand modelling (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2020), and wind power forecasting (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' When the goal is set to recover the original sequence, unsupervised learning of molecule embedding can be obtained for downstream classification tasks (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' When the spatial dimension is added, a convolutional neural network (CNN) layer can be added, and the dimension of the sequence generated can be expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' For example, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' (2020a) predict a city’s crowd flow patterns, and Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' (2020) generate 3D shapes via sequentially assembling different parts of the shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' While RNN-based architectures are still a widely adopted choice for seq2seq modeling, attention can also be used as a standalone mechanism for seq2seq translations independent of RNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The idea was proposed by Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' (2017) in an architecture named transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Without recurrence, the network allows for significantly more parallelization, and is shown to achieve superior performance in experiments, and powered the popularity of transformer-based architectures in various sequence generation tasks (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' A separate line of work by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' (2019) also demonstrated that a hierarchical CNN model with attention outperforms the traditional RNN-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Using deep learning to generate TSP solutions The above seq2seq translation mechanisms work well when the input data is naturally organized as a sequence, and the output sequence corresponds to the input sequence, such as in music and language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' However, in our paper, the input is an unordered sequence, and the output has the same but re-ordered elements of the same input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In this case, the concept of attention is helpful and has been successfully used to produce solutions to the TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The pointer network, proposed by Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' (2015) and further developed in Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' (2016), uses attention to select a member of the input sequence at each decoder step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' While it is not required that the input sequence is ordered, an informative ordering could improve the performance (Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' While the original pointer network was solved as a classification problem and cross-entropy loss was used, it is not necessarily the most efficient choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The cross-entropy loss only distinguishes between a correct prediction and an incorrect prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' But in instances like routing, the distances between the predicted position and the correct position, as well as the ordering of subsequences, could incur different costs in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Further developments in solving TSP with machine learning methods involve reinforcement learning (RL), which enables the optimization of custom evaluation metrics (Bello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Kool et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' (2019) compared the performance of RL and supervised learning (SL) on TSP solutions and found that SL and RL models achieve similar performance when the graphs are of similar sizes in training and testing, whereas RL models have better generalizability over variable graph sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' However, RL models require significantly more data points and computational resources, which is not always feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Although this seq2seq and attention framework has only been used to reproduce TSP solutions, it provides an opportunity to learn and incorporate additional information beyond the given travel times and potentially 6 learn individual differences when more information is given to the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In this paper, we combine the ideas of seq2seq modeling and attention to predict the actual route executed by a driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Methodology This section details the methodology proposed to address the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' First, the high-level seq2seq modelling framework is introduced, followed by the explanation of the novel pair-wise attention and sequence generation and selection mechanism used within the modelling framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Sequence-to-sequence modeling framework Let the input sequence be an arbitrarily-ordered sequence (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Denote the output sequence as (ˆs(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', ˆs(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Let ci indicate the “position index” of stop ˆs(i) with respect to the input sequence (where ci ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' For example, for input sequence (B, A, C) and output sequence (A, B, C), we have c1 = 2, c2 = 1, c3 = 3, which means the first output stop A is in the second position of the input sequence (B, A, C) and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The seq2seq model computes the conditional probability P(c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', cn | S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θ) using a parametric neural network (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', recurrent neural network) with parameter θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', P(c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', cn | S, XS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θ) = P(c1 | S, XS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θ) · n � i=2 P(ci | c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', ci−1, S, XS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θ) (1) The parameters of the model are learnt by empirical risk minimization (maximizing the conditional probabilities on the training set), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', θ∗ = arg max θ � S P(c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', cn | S, XS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θ) (2) where the summation of S is over all training routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In the following section, we will elaborate how P(ci | c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', ci−1, S, XS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θ) is calculated using the pair-wise attention-based pointer neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Pair-wise attention-based pointer neural network Figure 2 uses a four-stop example to illustrate the architecture of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The whole model is based on the LSTM encoder and decoder structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In particular, we use one LSTM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', encoder) to read the input sequence, one time step at a time, to obtain a large fixed dimensional vector representation, and then to use another LSTM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', decoder) to extract the output sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' However, different from the typical seq2seq model, we borrow the idea of the pointer network (Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2015) to add a pair-wise attention mechanism to predict the output sequence based on the attention mask over the input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The pair-wise attention is calculated based on an ASNN which was previously used for travel mode prediction (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Model details will be shown in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Intuitively, the LSTM encoder and decoder aim to capture the global view of the input information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', overall sequence pattern) by embedding the input sequence to hidden vector representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' While the ASNN-based pair-wise attention aims to capture the local view (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', the relationship between two stops).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Our experiments in Section 5 demonstrate the importance of both global and local views in the sequence prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 7 Figure 2: Overall architecture of the pair-wise attention-based pointer neural network (adapted from Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' (2015)) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' LSTM encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Given an arbitrary stop sequence (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', sn) as the input, let xi ∈ RK be the features of stop si, where xi may include the package information, the customer information, and the geographical information of the stop si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' K is the number of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The encoder computes a sequence of encoder output vectors (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', en) by iterating the following: hE i , ei = LSTM(xi, hE i−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θE) ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n (3) where hE i ∈ RKE h is the encoder hidden vector with hE 0 := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' ei ∈ RKe is the encoder output vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' KE h and Ke are corresponding vector dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θE is the learnable parameters in an encoder LSTM cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The calculation details of an LSTM cell can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The encoding process transforms a sequence of features (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', xn) into a sequence of embedded representation (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', en).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' And the hidden vector of the last time step (hE n) includes the global information of the whole sequence, which will be used for the LSTM decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Figure 3: Illustration of LSTM ecnoder 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' LSTM decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The role of a decoder in the traditional seq2seq model (Figure 4) is to predict a new sequence one time step at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' However, in the pointer network structure with attention, the role of the decoder becomes producing 8 AsNN Attention Component Predict next is S3 Predict next is S1 Predict next is S2 Predict next is S4 S4 S4 Encoder Decoderen e1 e2 he h2 he LSTM LSTM LSTM Decoder X2 x1 Xna vector to modulate the pair-wise attention over inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Denote the output sequence as (ˆs(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', ˆs(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Let x(i) be the feature of stop ˆs(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' At decoder step i, we have hD (i+1), d(i) = LSTM �� x(i) w(i) � , hD (i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θD � ∀i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n (4) where hD (i) ∈ RKD h is the decoder hidden vector with hD (0) = hE n, d(i) ∈ RKd is the decoder output vector, KD h and Kd are corresponding vector dimensions, and θD are learnable parameters of the decoder LSTM cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Note that we set x(0) = xD and d(0) = dD, representing the features and the decoder output of the depot, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' w(i) is the context vector calculated from the attention component, which will be explained in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Figure 4: Illustration of LSTM decoder 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' ASNN-based pair-wise attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The pair-wise attention aims to aggregate the global and local information to predict the next stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Specifically, at each decoder time step i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n}, we know that the last predicted stop is ˆs(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' To predict ˆs(i+1), we consider all candidate stops sj ∈ S, which is the set of all stops not yet visited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' We want to evaluate how possible that sj will be the next stop of ˆs(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The information of the stop pair ˆs(i) and sj can be represented by the following concatenated vector: vj (i) = concat(zj (i), φ(x(i), xj), d(i), ej) (5) where zj (i) is a vector of features associated with the stop pair (such as travel time from ˆs(i) to sj), and φ(x(i), xj) represents a feature processing function to extract the pair-wise information from x(i) and xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' For example, φ(·) may return geographical relationship between stops ˆs(i) and sj, and it may also drop features not useful for the attention calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Intuitively, zj (i) and φ(x(i), xj) contains only local information of the stop pair, while d(i) and ej contain the global information of the whole stop set and previously visited stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 9 Predict next is S(n) Output W(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Predict next is S(3) ASNN Output W(3) Attention Component Predict next is S(2) Output W(2) Predict next is S(1) Output W(1) dp d(1) d(2) d(n) d(n-1) 4 Encoder LSTM LSTM LSTM LSTM LSTM [] [x(1) x(2) x(n-1) x(n) W(1) W(2)) W(n-1)) W(n)]Figure 5: Illustration of ASNN-based pair-wise attention Given the pair-wise information vector vj (i), we can calculate the attention of stop ˆs(i) to stop sj as: uj (i) = ASNN(vj (i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θA) ∀i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n (6) aj (i) = exp(uj (i)) �n j′=1 exp(uj′ (i)) ∀i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n (7) where aj (i) ∈ R is attention of stop ˆs(i) to stop sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' ASNN(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θA)) is a multilayer perception (MLP) with the output dimension of one (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', uj (i) ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θA are the learnable parameters of the ASNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The name “alternative specific” is because the same parametric network will be applied on all alternative stops sj ∈ S separately (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Finally, we calculate the conditional probability to make the prediction: P(ci+1 = j | c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', ci, S, XS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θ) = aj (i) ∀i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n (8) ˆs(i+1) = arg max sj∈S\\SV (i) aj (i) ∀i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n (9) where SV (i) = {ˆs(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', ˆs(i)} is the set of stops that have been predicted (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', previously visited) until decoder step i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 8 and 9 indicate that the predicted next stop at step i is the one with highest attention among all stops that have not been visited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The pair-wise attention framework also leverages the attention information as the input for the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' This was achieved by introducing the context vector (Bahdanau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2015): w(i) = n � j=1 aj (i) · ej (10) The context vector is a weighted sum of all the encoder output vectors with attention as the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' As the attention provides the emphasis for stop prediction, w(i) helps to incorporate the encoded representation of the last predicted stop for the next stop prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The inputs for the next LSTM cell thus will be the 10 Predict next is S(i+1) Output W(i+1) Softmax u ASNN ASNN ASNN e1 e2 en he D LSTM LSTM LSTM LSTM X1 X2 Xn x(i) W(i)concatenation of the stop features and w(i), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', � x(i) w(i) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' It is worth noting that, the specific architecture of ASNN(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θA)) can be flexible depending on the input pair-wise information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' For example, if the information includes images or networks, convolutional neural network or graph convolutional networks can be used for better extract features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In this study, we use the MLP for simplification as it already outperforms benchmark models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The key idea is of the ASNN is to share the same trainable parameter θA for all stop pairs so as to better capture various pair-wise information in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Sequence generation and selection During inference, given a stop set S, the trained model with learned parameters θ∗ are used to generate the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Typically, in the seq2seq modeling framework, the final output sequence is selected as the one with the highest probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', (sj∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', sj∗n), where j∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', j∗ n = arg max j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=',jn∈CS P(c1 = j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', cn = jn | S, XS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θ∗) (11) where CS = {All permutations of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n}} Finding this optimal sequence is computationally impractical because of the combinatorial number of possible output sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' And so it is usually done with the greedy algorithm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', always select the most possible next stop) or the beam search procedure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', find the best possible sequence among a set of generated sequences given a beam size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' However, in this study, we observe that the first predicted stop ˆs(1) is critical for the quality of the generated sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The reason may be that the local relationship between a stop pair (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', given the last stop to predict the next one) is easier to learn than the global relationship (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', predict the whole sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Hence, in this study, we first generate sequences using the greedy algorithm with different initial stops, and select the one with the lowest operational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The intuition behind this process is that, once the first stop is given, the model can follow the learned pair-wise relationship to generate the sequence with relatively high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' For all the generated sequences with different first stops, the one with the lowest operation cost captures the global view of the sequence’s quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Therefore, the final sequence generation and selection algorithm is as follows: Algorithm 1 Sequence generation Input: Trained model, S Output: Predicted stop sequence 1: for s in S do 2: Let the first predicted stop be ˆs(1) = s 3: Predict the following stop sequence (ˆs(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', ˆs(n)) using the greedy algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Denote the predicted sequence as Ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 4: Calculate the total operation cost of the whole sequence (including depot), denoted as OCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' return Ps∗ where s∗ = arg mins∈S OCs 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Case Study 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Dataset The data used in our case study was made available as part of the Amazon Last Mile Routing Research Challenge (Merchán et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The dataset contains a total of 6,112 actual Amazon driver trajectories 11 for the last-mile delivery from 5 major cities in the US: Austin, Boston, Chicago, Los Angeles, and Seattle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Each route consists of a sequence of stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Each stop represents the actual parking location of the driver, and the package information (package numbers, package size, and planned service time) associated with each stop is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The stops are characterized by their latitudes and longitudes, and expected travel time between stops are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Figure 6 shows the distribution of the number of stops per route and an example route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Most routes have around 120 to 180 stops, and the maximum observed number of stops is around 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Figure 6b shows an example of an actual driver trajectory in Boston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Since the depot is far from the delivery stops, we attach the complete route (with the depot indicated by a red dot) at the bottom left of the figure, while the main plot only shows the delivery stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In this data set, each stop is associated with a zone ID (indicated by different colors in Figure 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' When Amazon generates planned routes for drivers, they usually expect drivers to finish the delivery for one zone first, then go to another zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' And the actual driver trajectories also follow this pattern as shown in Figure 6b (but the actual zone sequence may be different from the planned one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Therefore, in this study, we focus on the problem of zone sequence prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' That is, si in the case study section now represents a specific zone, S represents the set of zones, and XS represents zone features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' This transformation does not affect the model structure proposed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The only difference is that the new problem has a relatively smaller scale compared to the stop sequence prediction because the number of zones in a route is smaller than that of stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The zone-to-zone travel time is calculated as the average travel time of all stop pairs between the two zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Figure 7 presents an illustrative example of the relationship between zone and stop sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' As the dataset does not contain the original planned sequence, we assume the planned zone sequence is the one with the lowest total travel time (generated by a TSP solver, (sT 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', sT n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' After generating the zone sequence, we can restore the whole stop sequence by assuming that drivers within a specific zone follow an optimal TSP tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Details of the zone sequence to stop sequence generation can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Figure 7: Relationship between stop sequence and zone sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Experimental setup We randomly select 4,889 routes for model training and cross-validation, and the remaining 1,223 routes are used to evaluate/test model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' We consider a one-layer LSTM for both the encoder and decoder with the hidden unit sizes of 32 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', KD h = Ke = KE h = Kd = 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' And the ASNN is set with 2 hidden layers with 128 hidden units in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' We train the model using Adam optimizer with a default learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='001 and 30 training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' To utilize the planned route information, the input zone sequence for the LSTM encoder is set as the TSP 12 Zone sequence Zone 1 Zone 2 Zone 3 B C D E G H A Depot Depot Stop sequence(a) Number of stops distribution (b) Actual route example Figure 6: Description of dataset 13 400 350 300 250 Counts 200 150 100 50 0 50 100 150 200 Number of stops per routeASTBOSTON Air LOPREST ZOVESTE-ET Complete route CHELSEA CHARLESTOWN BOSTONresult (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', lowest travel time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' That is, the input sequence (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', sn) = (sT 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', sT n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In the case study, xi represents zone features, including the latitude and longitude of the zone center, number of stops in the zone, number of intersections in the zone, number of packages in the zone, total service time in the zone, total package size in the zone, and the travel time from this zone to all other zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The zone pair features zj (i) includes the travel time from ˆs(i) to sj and zone ID relationship characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' For example, the zone IDs “B-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='2C” and “B-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='3A” signal that they belong to the higher-level cluster “B-6”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' As we assume all pair-wise features are captured by zj (i), φ(x(i), xj) is not specified in this case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Consistent with the Amazon Last Mile Routing Research Challenge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' we evaluate the quality of the predicted stop sequences using a “disparity score” defined as follows: R(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' B) = SD(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' B) · ERPnorm(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' B) ERPe(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' B) (12) where R(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' B) is the disparity score for the actual sequence A and predicted sequence B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' and SD(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' B) is the sequence deviation defined as SD(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' B) = 2 n(n − 1) n � i=2 � |c[Bi] − c[Bi−1]| − 1 � (13) where n is the total number of stops,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Bi is the i-th stop of sequence B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' c[Bi] is the index of stop Bi in the actual sequence A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', its position in sequence A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In the case of A = B (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', perfectly predicted), we have c[Bi] − c[Bi−1] = 1 for all i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n, and SD(A, B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' ERPnorm(A, B) is the Edited Distance with Real Penalty (ERP) defined by the following recursive formula: ERPnorm(A, B) = ERPnorm(A2:|A|, B2:|B|) + Timenorm(A1, B1) (14) where Timenorm(si, sj) = Time(si,sj) � j′∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=',n} Time(si,sj′) is the normalized travel time from stop si to stop sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' ERPe(A, B) is the number of edit operations (insertions, substitutions, or deletions) required to transform sequence A to sequence B as when executing the recursive ERPnorm formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Hence, the ratio ERPnorm(A,B) ERPe(A,B) represents the average normalized travel time between the two stops involved in each ERP edit operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In the case of A = B, we have ERPnorm(A,B) ERPe(A,B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The disparity score R(A, B) describes how well the model-estimated sequence matches the known actual sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Lower score indicates better model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' A score of zero means perfect prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The final model performance is evaluated by the mean score over all routes in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In addition to the disparity score, we also evaluate the prediction accuracy of the first four zones in each route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' We choose the first four because the minimum number of zones in a route is four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Benchmark models The following optimization and machine learning models are used as benchmarks to compare with the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Conventional TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The first benchmark model is the zone sequence generated by conventional TSP, which we treat as the planned route with the lowest travel time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' ASNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The ASNN component can be trained to predict the next zone given the current zone, and the prediction sequence can be constructed in a greedy way starting from the given depot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The training zone 14 pairs (including from depot to the first zone) are extracted from all sequences in the training routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' And the input features are the same as the ASNN component in the proposed model except for (d(i), ej) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', output vectors from LSTM decoder and encoder, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' All hyper-parameters of the ASNN model are the same as the attention component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Inspired by the importance of the first zone, we also implement another sequence generation method similar to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' That is, we go through all zones in a route and assume it is the first zone, then use the trained ASNN to predict the remaining sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The final sequence is selected as the one with the lowest travel time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' LSTM-encoder-decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The LSTM-encoder-decoder (LSTM-E-D) architecture is a typical seq2seq model proposed by Sutskever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The model structure is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In the decoder stage, the model outputs the predicted zone based on last predicted zone’s information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The model formulation can be written as hE i , ei = LSTM(xi, hE i−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θE) ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n (15) hD (i+1), d(i) = LSTM(x(i), hD (i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θD) ∀i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n (16) The decoder output vector d(i) are, then feed into a fully-connected (FC) layer to calculate probability of the next stop: g(i) = FC(d(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θFC) ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n (17) P(ci+1 | c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', ci, S, XS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θ) = Softmax(g(i)) ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n (18) where g(i) ∈ RKz, Kz is the maximum number of zones in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' And the next predicted stop is selected by maximizing P(ci+1 = j | c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', ci, S, XS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θ) for all sj ∈ S \\ SV (i) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', the zones that are not in the route and that have been visited are excluded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Figure 8: Model architecture of the LSTM-E-D seq2seq prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Original Pointer Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Another benchmark model is the original pointer network (Pnt Net) proposed by (Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The overall architecture of the pointer network is similar to the proposed model 15 S4 S2 S1 S3 End FC + Softmax S1 S2 S3 S4 Ds S4 S2 S3 S Encoder Decoderexcept for the attention component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Specifically, the pointer network calculates attention as: uj (i) = W T 1 tanh(W2ej + W3d(i)) ∀i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n (19) aj (i) = exp(uj (i)) �n j′=1 exp(uj′ (i)) ∀i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n (20) The original pointer network does not include the pair-wise local information (zj (i), φ(x(i), xj)), and the attention calculation is only quantified from three learnable parameters W1, W2, and W3, which may limit its capacity in prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' We observe that the original pointer network without local information performs extremely badly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' For a fair comparison, we add the local information with the similar format in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 19 as: uj (i) = W T 1 tanh(W2ej + W3d(i)) + W4 � zj (i) φ(x(i), xj) � ∀i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n (21) After training the model, we generate the final sequence with the greedy algorithm and Algorithm 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Model comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Table 1 presents the performance of different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Note that for all approaches except for the TSP, we generate sequences based on two different methods (greedy and Algorithm 1) for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The standard deviation of disparity scores is taken over all testing routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Results show that sequence generation with Algorithm 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', iterating different first zones) can consistently reduce the disparity score for all machine learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' It implies that the first zone prediction and the global view (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', shortest path) are important for estimating the driver’s trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The proposed method outperforms all other models, both in disparity scores and prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' This means the proposed pair-wise ASNN-based attention (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 6) has better performance than the original content-based attention (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The comparison between LSTM-E-D and Pnt Net models demonstrates the effectiveness of the attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' All machine learning models except for LSTM-E-D can outperform the baseline TSP sequence with Algorithm 1 sequence generation method, suggesting that the hidden trajectory patterns can be learned from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Another observation is that, the prediction accuracy and disparity score do not always move in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' For example, the LSTM-E-D model with Algorithm 1 sequence generation, though has lower accuracy, shows a better disparity score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' This is because the accuracy metric does not differentiate “how wrong an erroneous prediction is”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' By the definition of disparity score, if a stop is si but the prediction is sj, and sj and si are geographically close to each other, the score does not worsen too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' This suggests a future research direction in using disparity score as the loss function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', training by RL) instead of cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Figure 9 shows the distribution of disparity scores for our proposed method with Algorithm 1 sequence generation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', the best model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' We observe that the prediction performance varies a lot across different routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' There is a huge proportion of routes with very small disparity scores (less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The mean score is impacted by outlier routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The median score is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0340, which is smaller than the mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 16 Table 1: Model performance Sequence generation Model Disparity score Prediction accuracy Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Dev 1st zone 2nd zone 3rd zone 4th zone TSP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0443 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0289 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='168 Greedy ASNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0470 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0289 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='123 LSTM-E-D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0503 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0313 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='161 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='166 Pnt Net 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0460 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='165 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0417 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='241 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='221 Algorithm 1 ASNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0429 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0299 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='195 LSTM-E-D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0501 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='149 Pnt Net 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0382 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='262 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='274 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0369 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='320 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='303 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='314 Figure 9: Disparity score distribution of the best model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Factors on trajectory predictability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' As our proposed model exhibits various levels of predictability across different routes, we aim to investigate which attributes of a route cause high (or low) predictability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' This can be done by running a regression model with the disparity score as the dependent variable and route attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', locations, departure time, package numbers) as independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The variables used are defined as follows: Total planned service time: The estimated time to deliver all packages in the route (service time only, excluding travel time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Earliest time window constraint: The earliest due time to deliver packages with time window constraint minus the vehicle departure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The smaller the value, the tighter the time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' # traffic signals: Average number of traffic signals in each zone of the route (obtained from OpenStreetMap data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 17 250 Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0369 Median = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='034 200 150 Counts 100 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='20 Disparity scores• If high-quality route: A dummy variable indicating whether the route is labeled as “high quality” by Amazon or not (Yes = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' High quality means the actual travel time of the route is similar to or better than Amazon’s expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' If in Location: A dummy variable indicating whether the route is in a specific city or not (Yes = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' If departure Time: A dummy variable indicating the (local) departure time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', before 7AM, after 10AM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Table 2 shows the results of the regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Since the dependent variable is disparity scores, a negative sign indicates a positive impact on the predictability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' We observe that routes with tighter time window constraints and more stops are easier to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' This may be due to the fact that these routes are usually harder to deliver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Hence, to avoid the risk of violating time constraints or delay, drivers tend to follow the planned routes and thus the route sequences are easier predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' We also find that routes associated with larger vans (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', larger vehicle capacity) are more predictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The reason may be that larger vans are less flexible in choosing different routes, thus drivers are more likely to follow the navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Another important factor for better predictability is high-quality routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' This may be because high-quality routes are closer to the TSP sequence which we use as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Finally, routes in LA are more predictable than in other areas such as Chicago and Boston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Table 2: Factors on trajectory predictability Variables Coefficients (×10−3) Variables Coefficients (×10−3) Intercept 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='07 ** If high quality route 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='66×10−14 ** Total # of packages 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='059 If in LA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='998 * Total planned service time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='476 If in Chicago 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='783 Earliest time window constraint 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='047 ** If in Boston 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='354 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' # traffic signals 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='255 If on weekends 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='775 Total # of stops 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='142 ** If departure before 7AM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='582 Vehicle capacity (m3) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='041 * If departure after 10AM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='704 Number of routes: 1,002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='065;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' ∗∗: p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' ∗: p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Impact of input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' All machine learning models in Table 1 (except for ASNN) have the LSTM encoder component, which requires the specification of input zone sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' As mentioned in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='2, we currently use the TSP sequence as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' It is worth exploring the model performance if we use a random zone sequence instead, which corresponds to the scenario without planned route information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Table 3 shows the model performance without the TSP sequence information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Since the ASNN result does not rely on TSP information, it is not listed in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Results show that the LSTM-E-D model becomes much worse with a random sequence as inputs, while the performance of Pnt Net and our method is only slightly affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Even without the planned route information, the proposed model can still provide a reasonable estimation of driver trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 18 Table 3: Model performance without TSP information Sequence generation Model Disparity score Prediction accuracy Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Dev 1st zone 2nd zone 3rd zone 4th zone Greedy LSTM-E-D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='1176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0498 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='050 Pnt Net 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0323 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='096 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='196 Algorithm 1 LSTM-E-D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='1054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0463 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='052 Pnt Net 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0398 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='298 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='273 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0376 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='298 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='298 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Summary Our numerical results show that our proposed model outperforms its benchmarks in terms of disparity scores and prediction accuracy, meaning that it can better predict the actual route trajectories taken by drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The comparison with benchmark models shows that our proposed ASNN-based pair-wise attention mecha- nism and our sequence generation algorithm (Algorithm 1) are both helpful for the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Moreover, we can observe that the predictive performance varies across different routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Factors such as route quality, delivery time windows, and the total number of stops of a route affect predictability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Finally, the proposed model is insensitive to the input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The prediction performance only slightly decreases when the input sequence is changed from the TSP solution to a random stop sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' This property implies that we only need the set of stops to implement the model and obtain high-quality solution, while information on the planned route sequence is not strictly required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Conclusion and Future Research In this paper, we propose a pair-wise attention-based pointer neural network that predicts actual driver trajectories on last-mile delivery routes for given sets of delivery stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Compared to previously proposed pointer networks, this study leverages a new alternative specific neural network-based attention mechanism to incorporate pair-wise local information (such as relative distances and locations of stops) for the attention calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' To better capture the global efficiency of a route in terms of operational cost (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', total travel time), we further propose a new sequence generation algorithm that finds the lowest-cost route sequence by iterating through different first stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' We apply our proposed method to a large set of real operational route data provided by the Amazon Last-Mile Routing Research Challenge in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The results show that our proposed method can outperform a wide range of benchmark models in terms of both the disparity score and prediction accuracy, meaning that the predicted route sequence is closer to the actual sequence executed by drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Compared to the best benchmark model (original pointer network), our method reduces the disparity score from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0382 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0369, and increases the average prediction accuracy of the first four zones from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='229 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Moreover, our proposed sequence generation method can consistently improve the prediction performance for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The disparity scores are reduced by 10-20% across different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Lastly, we show that the proposed methodology is robust against changes in the input sequence pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Compared to an optimal TSP solution as the input sequence, a random input sequence only slightly increases the disparity score from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0369 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='0376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 19 The data-driven route planning method proposed in this paper has several highly relevant practical implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' First, our proposed model performs well at predicting stop sequences that would be preferable to delivery drivers in a real operational environment, even if it is not provided with a theoretically optimal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', minimal route duration) planned TSP sequence as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Therefore, the model can be used to generate a predicted actual stop sequence that a driver would likely be taking for a given set of delivery stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The prediction can serve as a new ‘empirical’ planned route that is informed by historical driver behavior and thus more consistent with the driver’s experience and preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Second, by comparing the stop sequence predicted by our model with the traditional, TSP-based planned stop sequence, a route planner may infer potential reasons for the drivers’ deviations and adjust the company’s planning procedures and/or driver incentives if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Third, as stop sequence generation using machine learning models is computationally more efficient than traditional optimization-based approaches, a trained machine learning model can be applied in real-time to quickly re-optimize routes when drivers are unexpectedly forced to deviate from their original stop sequence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', due to road closures) and need updated routing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Based on the work presented in this paper, a number of fruitful future research avenues arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' First, instead of focusing on stop sequence prediction, future work may improve the interpretability of such prediction models and develop machine learning approaches that better explain which factors cause drivers to deviate from a planned stop sequence and how they affect their actual route trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Second, future work should attempt to combine the strengths of optimization-based route planning approaches and machine learning by incorporating tacit driver knowledge learned via machine learning models into route optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Appendices Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Mathematical Formulation of a LSTM Cell The details of an LSTM cell, ht, et = LSTM(xt, ht−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' θ), is shown below: ft = σg(Wfxt + Ufht−1 + bf) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='1) it = σg(Wixt + Uiht−1 + bi) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='2) ot = σg(Woxt + Uoht−1 + bo) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='3) ˜ct = σc(Wcxt + Ucht−1 + bc) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='4) ct = ft ◦ ct−1 + it ◦ ˜ct (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='5) ht = ot ◦ σh(ct) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='6) et = ht (if this is a single layer one-directional LSTM) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='7) where [Wf, Wi, Wo, Wc, Uf, Ui, Uo, Uc, bf, bi, bo, bc] = θ is the vector of learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' xt is the input vector to the LSTM unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' ft is the forget gate’s activation vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' it is the input/update gate’s activation vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' ot is the output gate’s activation vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' ht is the hidden state vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' et is the output vector of the LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Note that for a multi-layer or bidirectional LSTM, et may not equal to ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In this study, we use a single layer one-directional LSTM and thus have et = ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' More details on the output vector can be found in Pytorch (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' ˜ct is the cell input activation vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' ct is the cell state vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' “◦” indicates the component-wise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 20 Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' From Zone Sequence to Stop Sequence The complete stop sequence is generated based on the given zone sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The detailed generation process is shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Algorithm 2 Complete sequence generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Input: zone sequence (ˆz(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='., ˆz(n)), depot DS, set of stops in each zone S(i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' PathTSP(S, sfirst, slast) and TourTSP(S) are two oracle functions for solving path and tour TSP problems given the set of stops S, first stop sfirst and last stop slast to be visited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' 1: function CompleteSeqGeneration((ˆz(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='., ˆz(n)), {S(i), i = 1, , , n}) 2: sprev ← DS 3: s∗ complete ← (sprev) ▷ Initialize the complete stop sequence with depot 4: for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', n − 1} do 5: Sfirst ← Set of three stops in S(i) that are closest to sprev 6: Slast ← Set of three stops in S(i) that are closest to all stops in Si+1 on average 7: P(i) ← ∅ ▷ Initialize the set of optimal paths in zone ˆz(i) 8: for sfirst ∈ Sfirst do 9: for slast ∈ Slast do 10: if sfirst = slast then 11: ˆptemp, ttemp = TourTSP(S(i)) ▷ Solve the optimal tour and travel time for zone ˆz(i) 12: Delete the last edge back to sfirst in the tour ˆptemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Let the new path and travel time be ˆp′ temp and t′ temp 13: Add ˆp′ temp and t′ temp to P(i) 14: else 15: ˆptemp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' ttemp = PathTSP(S(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' sfirst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' slast) ▷ Solve the optimal path and travel time for zone i 16: Add ˆptemp and ttemp to P(i) 17: ˆp(i) ← Path in P(i) with the minimum travel time 18: s∗ complete ← (s∗ complete,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' ˆp(i)) ▷ Concatenate two sequence 19: sprev ← Last stop of path ˆp(i) 20: s∗ complete ← (s∗ complete,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' DS) ▷ Concatenate the last stop as the depot 21: return s∗ complete Consider an optimal zone sequence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' (ˆz(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='., ˆz(n)), generated from the proposed machine learning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' We can always add the depot before the first and after last zone (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', (DS, ˆz(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='., ˆz(n), DS)) and make the whole zone sequence a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' For each zone ˆz(i), we aim to generate a within-zone path ˆp(i), and the final stop sequence will be (DS, ˆp(i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=', ˆp(n), DS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' When generating ˆp(i) for zone ˆz(i), we assume ˆp(i−1) is known (generated from the last step and ˆp(0) = (DS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Let the set of all stops in zone ˆz(i) be S(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' We identify three potential first stops and last stops of path ˆp(i) based on following rules: Three potential first stops of ˆp(i) are the three most closest stops (in travel time) to ˆp(i−1)’s last stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Three potential last stops of ˆp(i) are the three most closest stops (in travel time) to all stops in S(i+1) on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' Note that S(n+1) = {DS} With three potential first stops and last stops, we then solve path TSP problems between any first and last stop pair to generate the potential optimal inner zone path with the shortest travel time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' In this step, at most 21 nine small-scale path TSP problems will be solved since there might be overlapping between the first and the last stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' If the first and the last stops are identical, we solve a tour TSP problem and output the path by deleting the last edge which traverses back to the first stop in the tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' After having all potential inner zone paths and total path travel time between any first and last stop pair, we keep the path with the minimum travel time as the inner zone sequence, ˆp(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' The key assumption we make here about drivers is that they will deliver packages within a zone following a path that minimizes their total travel time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' With the optimal inner zone stop sequence of the current zone, we then move to the next visited zone in the optimal zone sequence and repeat the same procedure until we generate the complete stop sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content=' References Applegate, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE2T4oBgHgl3EQfTQfv/content/2301.03802v1.pdf'} +page_content='L.' metadata={'source': 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Various datasets have been proposed for simultaneous local- +ization and mapping (SLAM) and related problems. Existing datasets +often include small environments, have incomplete ground truth, or lack +important sensor data, such as depth and infrared images. We propose +an easy-to-use framework for acquiring building-scale 3D reconstruction +using a consumer depth camera. Unlike complex and expensive acquisi- +tion setups, our system enables crowd-sourcing, which can greatly bene- +fit data-hungry algorithms. Compared to similar systems, we utilize raw +depth maps for odometry computation and loop closure refinement which +results in better reconstructions. We acquire a building-scale 3D dataset +(BS3D) and demonstrate its value by training an improved monocular +depth estimation model. As a unique experiment, we benchmark visual- +inertial odometry methods using both color and active infrared images. +Keywords: Depth camera · SLAM · Large-scale. +1 +Introduction +Simultaneous localization and mapping (SLAM) is an essential component in +robot navigation, virtual reality (VR), and augmented reality (AR) systems. Var- +ious datasets and benchmarks have been proposed for SLAM [11,35,39] and re- +lated problems, including visual-intertial odometry [30,6], camera re-localization +[29,32,15], and depth estimation [21,33]. Currently, there exists only a few building- +scale SLAM datasets [28] that include ground truth camera poses and dense 3D +geometry. Such datasets enable, for example, evaluation of algorithms needed in +large-scale AR applications. +The lack of building-scale SLAM datasets is explained by the difficulty of +acquiring ground truth data. Some have utilized a high-end LiDAR for obtaining +3D geometry of the environment [26,2,4,28]. Ground truth camera poses may +be acquired using a motion capture (MoCap) system when the environment is +small enough [35,40]. The high cost of equipment, complex sensor setup, and +slow capturing process make these approaches less attractive and inconvenient +for crowd-sourced data collection. +arXiv:2301.01057v1 [cs.CV] 3 Jan 2023 + +2 +Mustaniemi et al. +An alternative is to perform 3D reconstruction using a monocular, stereo, or +depth camera. Consumer RGB-D cameras, in particular, are interesting because +of their relatively good accuracy, fast acquisition speed, low-cost, and effective- +ness in textureless environments. RGB-D cameras have been used to collect +datasets for depth estimation [21,33], scene understanding [8], and camera re- +localization [32,38], among other tasks. The problem is that existing RGB-D +reconstruction systems (e.g. [22,9,5]) are limited to room-scale and apartment- +scale environments. +Synthetic SLAM datasets have also been proposed [20,39,27] that include per- +fect ground truth. The challenge is that data such as time-of-flight (ToF) depth +maps and infrared images are difficult to synthesize realistically. Consequently, +training and evaluation done using synthetic data may not reflect algorithm’s +real-world performance. To address the domain gap problem, algorithms are +often fine-tuned using real data. +We propose a framework to create building-scale 3D reconstructions using a +consumer depth camera (Azure Kinect). Unlike existing approaches, we register +color images and depth maps using color-to-depth (C2D) strategy. This allows us +to directly utilize the raw depth maps captured by the wide field-of-view (FoV) +infrared camera. Coupled with an open-source SLAM library [19], we acquire a +building-scale 3D vision dataset (BS3D) that is considerably larger than similar +datasets as shown in Fig. 1. The BS3D dataset includes 392k synchronized color +images, depth maps and infrared images, inertial measurements, camera poses, +enhanced depth maps, surface reconstructions, and laser scans. Our framework +will be released for the public to enable fast, easy and affordable indoor 3D +reconstruction. +240 m +80 m +8 m +Zoomed +Fig. 1. Building-scale 3D reconstruction (4300 m2) obtained using an RGB-D camera +and the proposed framework. The magnified area (90 m2) is larger than any recon- +struction in the ScanNet dataset [8]. +2 +Related work +This section introduces commonly used RGB-D SLAM datasets and correspond- +ing data acquisition processes. A summary of the datasets is provided in Table + +BS3D: Building-scale 3D Reconstruction from RGB-D Images +3 +1. As there exist countless SLAM datasets, the scope is restricted to real-world +indoor scenarios. We leave out datasets focusing on aerial scenarios (e.g. Eu- +RoC MAV [2]) and autonomous driving (e.g. KITTI [11]). We also omit RGB-D +datasets captured with a stationary scanner (e.g. Matterport3D [4]) as they can- +not be used for SLAM evaluation. Synthetic datasets, such as SceneNet RGB-D +[20], TartanAir [39], and ICL [27] are also omitted. +ADVIO [6] dataset is a realistic visual-inertial odometry benchmark that in- +cludes building-scale environments. Ground truth trajectory is computed using +an inertial navigation system (INS) together with manual location fixes. The +main limitation of the dataset is that it does not come with ground truth 3D +geometry. LaMAR [28] is a large-scale SLAM benchmark that utilizes high-end +mapping platforms (NavVis M6 trolley and VLX backpack) for ground truth +generation. Although the capturing setup includes a variety of devices (e.g. +HoloLens2 and iPad Pro), it does not include a dedicated RGB-D camera. +OpenLORIS-Scene [31] focuses on the lifelong SLAM scenario where environ- +ments are dynamic and changing, similar to LaMAR [28]. The data is collected +over an extended period of time using wheeled robots equipped with various +sensors, including RGB-D, stereo, IMU, wheel odometry, and LiDAR. Ground +truth poses are acquired using an external motion capture (MoCap) system, or +with a 2D laser SLAM method. The dataset is not ideal for handheld SLAM +evaluation because of the limited motion patterns of a ground robot. +TUM RGB-D SLAM [35] is one of the most popular SLAM datasets. The +RGB-D images are acquired using a consumer depth camera Microsoft Kinect +v1. Ground truth trajectory is incomplete because the MoCap system can only +cover a small part of the environment. CoRBS [40] consists of four room-scale +environments. It also utilizes MoCap for acquiring ground truth trajectories for +Microsoft Kinect v2. Unlike [35], CoRBS provides ground truth 3D geometry +acquired using a laser scanner. The data also includes infrared images, but not +inertial measurements, unlike our dataset. +7-Scenes [32] and 12-Scenes [38] are commonly used for evaluating camera lo- +calization. 7-Scenes includes seven scenes captured using Kinect v1. KinectFusion +[22] is used to obtain ground truth poses and dense 3D models from the RGB- +D images. 12-Scenes consists of multiple rooms captured using the Structure.io +depth sensor and iPad color camera. The reconstructions are larger compared +to 7-Scenes, about 37 m3 on average. They are acquired using the VoxelHashing +framework [23], an alternative to KinectFusion with better scalability. +ScanNet [8] is an RGB-D dataset containing 2.5M views acquired in 707 +distinct spaces. It includes estimated calibration parameters, camera poses, 3D +surface reconstructions, textured meshes, and object-level semantic segmenta- +tions. The hardware consists of a Structure.io depth sensor attached to a tablet +computer. Pose estimation is done using BundleFusion [9], after which volumet- +ric integration is performed through VoxelHashing [23]. +Sun3D [43] is a large RGB-D database with camera poses, point clouds, +object labels, and refined depth maps. The reconstruction process is based on +structure from motion (SfM) where manual object annotations are utilized to + +4 +Mustaniemi et al. +reduce drift and loop-closure failures. Refined depth maps are obtained via vol- +umetric fusion similar to KinectFusion [22]. We emphasize that ScanNet [8] and +Sun3D [43] reconstructions are considerably smaller and have lower quality than +those provided in our dataset. Unlike [28,31,35], our system also does not require +a complex and expensive capturing setup, or manual annotation [6,43]. +Table 1. List of indoor RGB-D SLAM datasets. The BS3D acquisition setup does +not require high-end LiDARs [40,31,28], MoCap systems [40,31,36], or manual annota- +tion [43,6]. BS3D is building-scale, unlike [32,36,8,40,38,43]. Note that ADVIO [6] and +LaMAR [28] do not have a dedicated depth camera. +Dataset +Scale +Depth +IMU +IR +Ground truth +7-Scenes [32] +room +Kinect v1 +- +- +RGBD-recons. +TUM RGBD [36] +room +Kinect v1 +✓ +- +MoCap +ScanNet [8] +room +Structure.io +✓ +- +RGBD-recons. +CoRBS [40] +room +Kinect v2 +- +✓ +MoCap+LiDAR +12-Scenes [38] +apartment +Structure.io +- +- +RGBD-recons. +Sun3D [43] +apartment +Xtion Pro Live +- +- +RGBD+manual +OpenLORIS [31] +building +RS-D435i +✓ +- +MoCap+LiDAR +ADVIO [6] +building +Tango +✓ +- +INS+manual +LaMAR [28] +building +HoloLens2 +✓ +✓ +LiDAR+VIO+SfM +BS3D (ours) +building +Azure Kinect +✓ +✓ +RGBD-recons. +3 +Reconstruction framework +In this section, we introduce the RGB-D reconstruction framework shown in +Fig. 2. The framework produces accurate 3D reconstructions of building-scale +environments using low-cost hardware. The system is fully automatic and robust +against poor lighting conditions and fast motions. Color images are only used for +loop closure detection as they are susceptible to motion blur and rolling shutter +distortion. Raw depth maps enable accurate odometry and the refinement of +loop closure transformations. +3.1 +Hardware +Data is captured using the Azure Kinect depth camera, which is well-suited for +crowd-sourcing due to its popularity and affordability. We capture synchronized +depth, color, and infrared images at 30 Hz using the official recorder application +running on a laptop computer. We use the wide FoV mode of the infrared camera +with 2x2 binning to extend the Z-range. The resolution of raw depth maps and +IR images is 512 x 512 pixels. Auto-exposure is enabled when capturing color +images at the resolution of 720 x 1280 pixels. We also record accelerometer and +gyroscope readings at 1.6 kHz. + +BS3D: Building-scale 3D Reconstruction from RGB-D Images +5 +Fig. 2. Overview of the RGB-D reconstruction system. +. +3.2 +Color-to-depth alignment +Most RGB-D reconstruction systems expect that color images and depth maps +have been spatially and temporally aligned. Modern depth cameras typically +produce temporally synchronized images so the main concern is the spatial align- +ment. Conventionally, raw depth maps are transformed to the coordinate system +of the color camera, which we refer to as the depth-to-color (D2C) alignment. +In the case of Azure Kinect, the color camera’s FoV is much narrower (90 +x 59 degrees) compared to the infrared camera (120 x 120 degrees). Thus, the +D2C alignment would not take advantage of the infrared camera’s wide FoV +because depth maps would be heavily cropped. Moreover, the D2C alignment +might introduce artefacts to the raw depth maps. +We propose an alternative called color-to-depth (C2D) alignment where color +images are transformed instead. In the experiments, we show that this drastically +improves the quality of the reconstructions. The main challenge of C2D is that +it requires a fully dense depth map. Fortunately, a reasonably good alignment +can be achieved even with a low quality depth map. This is because the baseline +between the cameras is narrow and missing depths often appear in areas that +are far away from the camera. +For the C2D alignment, we first perform depth inpainting using linear in- +terpolation. Then, the color image is transformed to the raw depth frame. To +keep as much of the color information as possible, the output resolution will be +higher (1024 x 1024 pixels) compared to the raw depth maps . After that, holes +in the color image due to occlusions are inpainted using the OpenCV library’s +implementation of [37]. We note that minor artefacts in the aligned color images +will have little impact on the SIFT-based loop closure detection. +3.3 +RGB-D Mapping +We process the RGB-D sequences using an open-source SLAM library called +RTAB-Map [19]. Odometry is computed from the raw depth maps using the +point-to-plane variant of the iterative closest point (ICP) algorithm [25]. We use +the scan-to-map odometry strategy [19] where incoming frames are registered +against a point cloud map created from past keyframes. The wide FoV ensures +that ICP-odometry rarely fails, but in case it does, a new map is initialized. + +RGBD +RGB +Depth +RGBD +Color-to-depth +Loop closures +Volumetric +(C2D) +(PnP + ICP) +fusion +Poses +Normals +Depth (raw) +Mesh +(optimized) +Poses +Odometry +(odometry) +Graph +Render +(ICP) +optimization6 +Mustaniemi et al. +Loop closure detection is needed for drift correction and merging of individual +maps. For this purpose, SIFT features are extracted from the valid area of the +aligned color images. Loop closures are detected using the bag-of-words approach +[18], and transformations are estimated using the Perspective-n-Point RANSAC +algorithm and refined using ICP [25]. Graph optimization is done using the +GTSAM library [10] and Gauss-Newton algorithm. +RTAB-Map supports multi-session mapping which is a necessary feature +when reconstructing building-scale environments. It is not practical to collect +possibly hours of data at once. Furthermore, having the ability to later update +and expand the map is a useful feature. In practise, individual sequences are +first processed separately, followed by multi-session mapping. The sessions are +merged by finding loop closures and by performing graph optimization. The in- +put is a sequence of keyframes along with odometry poses and SIFT features +computed during single-session mapping. The sessions are processed in such or- +der that there is at least some overlap between the current session and the global +map build so far. +3.4 +Surface reconstruction +It is often useful to have a 3D surface representation of the environment. There +exists many classical [14,22] and learning-based [41,1] surface reconstruction ap- +proaches. Methods that utilize deep neural networks, such as NeuralFusion [41], +have produced impressive results on the task of depth map fusion. Neural ra- +diance fields (NeRFs) have also been adapted to RGB-D imagery [1] showing +good performance. We did not use learning-based approaches in this work be- +cause they are limited to small scenes, at least for the time being. Moreover, +scene-specific learning [1] takes several hours even with powerful hardware. +Surface reconstruction is done in segments due to the large scale of the en- +vironment and the vast number of frames. To that end, we first create a point +cloud from downsampled raw depth maps. Every point includes a view index +along with 3D coordinates. The point cloud is partitioned into manageable seg- +ments using the K-means algorithm. A mesh is created for each segment using +the scalable TSDF fusion implementation [46] that is based on [7,22]. It uses a +hierarchical hashing structure to support large scenes. +4 +BS3D dataset +The BS3D dataset was collected at the university campus using Azure Kinect +(Section 3.1). Figure 3 shows example frames from the dataset. The collection +was done in multiple sessions due to large scale of the environment. The record- +ings were processed using the framework described in Section 3. +4.1 +Dataset features +The reconstruction shown in Fig. 1 consists of 47 overlapping recording sessions. +Additional 14 sessions, including 3D laser scans, were recorded for evaluation + +BS3D: Building-scale 3D Reconstruction from RGB-D Images +7 +Cafeteria +Stairs +Study +Corridor +Lobby +Fig. 3. Example frames from the dataset. Environments are diverse and challenging, +including cafeterias, stairs, study areas, corridors, and lobbies. +purposes. Most sessions begin and end at the same location to encourage loop +closure detection. The total duration of the sessions is 3 hours and 38 minutes +and the combined trajectory length is 6.4 kilometers. The reconstructed floor +area is approximately 4300 m2. +The dataset consists of 392k frames, including color images, raw depth maps, +and infrared images. Color images and depth maps are provided in both coordi- +nate frames (color and infrared camera). The images have been undistorted for +convenience, but the original recordings are also included. We provide camera +poses in a global reference frame for every image. Data also includes inertial mea- +surements, enhanced depth maps and surface normals that have been rendered +from the mesh as visualized in Fig. 4. +Color +Infrared +Normals (render) +Mesh +Depth +Depth (raw) +Depth (render) +Fig. 4. The BS3D dataset includes color and infrared images, depth maps, IMU data, +camera parameters, and surface reconstructions. Enhanced depth maps and surface +normals are rendered from the mesh. +4.2 +Laser scan +We utilize the FARO 3D X 130 laser scanner for acquiring ground truth 3D +geometry. The scanned area includes a lobby and corridors of different sizes (800 +m2). The 28 individual scans were registered using the SCENE software that +comes with the laser scanner. Noticeable artefacts, e.g. those caused by mirrors, + +8 +Mustaniemi et al. +were manually removed. The laser scan is used to evaluate the reconstruction +framework in Section 5. However, this data also enables, for example, training +and evaluation of RGB-D surface reconstruction algorithms. +5 +Experiments +We compare our framework with the state-of-the-art RGB-D reconstruction +methods [5,9,3]. The value of the BS3D dataset is demonstrated by training +a recent monocular depth estimation model [44]. We also benchmark visual- +inertial odometry approaches [12,34,3] using either color or infrared images to +further highlight the unique aspects of the BS3D dataset. +5.1 +Reconstruction framework +In this experiment, we compare the framework against Redwood [5], Bundle- +Fusion [9], and ORB-SLAM3 [3]. RGBD images are provided for [5,9,3] in the +coordinate frame of the color camera. Given the estimated camera poses, we cre- +ate a point cloud and compare it to the laser scan (Section 4.2). The evaluation +metrics include overlap of the point clouds and RMSE of inlier correspondences. +Before comparison, we create uniformly sampled point clouds using voxel down- +sampling (1 cm3 voxel) that computes the centroid of the points in each voxel. +The overlap is defined as the ratio of inlier correspondences and the number of +ground truth points. A 3D point is considered to be an inlier if the distance to +the closest ground truth point is below threshold γ. +Table 2 shows the results for environments of different sizes. All methods +are able to reconstruct the small environment (35 m2) consisting of 2.8k frames. +The differences between the methods become more evident when reconstructing +the medium-size environment (160 m2) consisting of 7.3k frames. BundleFusion +[9] only produces a partial reconstruction because of odometry failures. The +proposed approach gives the most accurate reconstructions as visualized in Fig. +5. Note that it is not possible to achieve 100 % overlap because the depth camera +does not observe all parts of the ground truth. +The largest environment (350 m2) consists of 24k frames acquired in four +sessions. Redwood [5] does not scale to input sequences of this long. ORB-SLAM3 +[3] frequently loses the odometry in open spaces which leads to incomplete and +less accurate reconstructions. Our method suffers the same problem when C2D +is disabled. Unreliable odometry is likely due to the color camera’s limited FoV, +rolling shutter distortion, and motion blur. The C2D alignment improves the +accuracy and robustness of ICP-based odometry and loop closures. Without +C2D, the frequent odometry failures result in disconnected maps and noticeable +drift. We note that the reconstruction in Fig. 1 was computed from ∼300k frames +which is far more than [5,9,3] can handle. + +BS3D: Building-scale 3D Reconstruction from RGB-D Images +9 +Table 2. Comparison of RGB-D reconstruction methods in small, medium and large- +scale environments (from top to bottom). Overlap of the point clouds and inlier RMSE +computed for distance thresholds γ (mm). Some methods only work in small and/or +medium scale environments. +γ = 10 (mm) +γ = 20 (mm) +γ = 50 (mm) +Method +Overlap ↑ +RMSE ↓ +Overlap ↑ +RMSE ↓ +Overlap ↑ +RMSE ↓ +Redwood [5] +66.5 +5.6 +77.9 +7.6 +87.1 +12.6 +BundleFusion [9] +72.1 +5.5 +80.8 +6.9 +88.3 +11.7 +ORB-SLAM3 [3] +78.2 +5.3 +85.2 +6.5 +91.3 +10.6 +Prop. (w/o C2D) +66.8 +5.7 +77.8 +7.5 +87.0 +12.7 +Proposed +78.4 +5.2 +85.7 +6.5 +91.6 +10.6 +Redwood [5] +30.4 +6.2 +44.5 +9.8 +63.9 +19.9 +BundleFusion [9] +8.1 +6.2 +11.1 +9.2 +14.8 +18.8 +ORB-SLAM3 [3] +44.3 +6.0 +57.7 +8.7 +71.0 +16.2 +Prop. (w/o C2D) +36.5 +6.1 +49.2 +9.0 +64.3 +18.3 +Proposed +54.1 +5.7 +64.8 +7.7 +73.2 +13.4 +ORB-SLAM3 [3] +9.5 +6.3 +14.4 +9.9 +20.8 +20.7 +Prop. (w/o C2D) +23.1 +6.7 +40.6 +10.9 +64.7 +22.4 +Proposed +34.7 +6.4 +52.7 +10.0 +75.0 +19.8 +ORB-SLAM3 [3] +Proposed +Redwood [5] +Proposed (w/o C2D) +ϵ < 20 mm +20 ≤ ϵ < 50 +50 ≤ ϵ < 100 +100 ≤ ϵ < 200 +ϵ ≥ 200 mm +Fig. 5. Reconstructions obtained using Redwood [5], ORB-SLAM3 [3], and the pro- +posed method. Colors depict errors (distance to the closest ground truth point). + +10 +Mustaniemi et al. +5.2 +Depth estimation +We investigate whether the BS3D dataset can be used to train better models +for monocular depth estimation. For this experiment, we use the state-of-the- +art LeReS model [44] based on ResNet50. The original model has been trained +using 354k samples taken from various datasets [45,24,16,13,42]. We finetune +the model using 16.5k samples from BS3D. We set the learning rate to 2e-5 and +train only 4 epochs to avoid overfitting. Other training details, including loss +functions are the same as in [44]. +For testing, we use NYUD-v2 [21] and iBims-1 [17] that are not seen during +training. We also evaluate using BS3D by sampling 535 images from an unseen +part of the building. Table 3 shows that finetuning improves the performance on +iBims-1 and BS3D. The finetuned model performs marginally worse on NYUD- +v2 which is not surprising considering that NYUD-v2 mainly contains room-scale +scenes that are not present in BS3D. The qualitative comparison in Fig. 6 also +shows a clear improvement over the pretrained model on iBims-1 that contains +both small and large scenes. The model trained only using BS3D cannot compete +with other models, except on BS3D on which the performance is surprisingly +good. The poor performance on other datasets is not surprising because of the +small training set. +Table 3. Monocular depth estimation using LeReS [44] trained from scratch using +BS3D, pretrained model, and finetuned model. NUYD-v2 [21], iBims-1 [17], and BS3D +are used for testing. +NYUD-v2 [21] +iBims-1 [17] +BS3D +Training data +AbsRel ↓ +δ1 ↑ +AbsRel ↓ +δ1 ↑ +AbsRel ↓ +δ1 ↑ +BS3D +0.181 +0.764 +0.188 +0.763 +0.144 +0.828 +Pretrained +0.096 +0.913 +0.115 +0.890 +0.157 +0.785 +Pre. + BS3D +0.100 +0.907 +0.098 +0.901 +0.115 +0.881 +Color +Pretrained +Finetuned +Ground truth +Fig. 6. Comparison of pretrained and finetuned (BS3D) monocular depth estimation +model LeReS [44] on an independent iBims-1 [17] dataset unseen during training. + +BS3D: Building-scale 3D Reconstruction from RGB-D Images +11 +5.3 +Visual-inertial odometry +The BS3D dataset includes active infrared images along with color and IMU +data. This opens interesting possibilities, for example, the comparison of color +and infrared as inputs for visual-inertial odometry. Infrared-inertial odometry +is an attractive approach in the sense that it does not require external light, +meaning it would work in completely dark environments. +We evaluate OpenVINS [12], ORB-SLAM3 [3], and DM-VIO [34] using color- +inertial and infrared-inertial inputs. Note that ORB-SLAM3 has an unfair ad- +vantage because it has a loop closure detector that cannot be disabled. In the +case of infrared images, we apply a power law transformation (I = 0.04 · I0.6) +to increase brightness. As supported by [34], we provide a mask of valid pix- +els to ignore black areas near the edges of the infrared images. We adjust the +parameters related to feature detection when using infrared images with [12,3]. +We use the standard error metrics, namely absolute trajectory error (ATE) and +relative pose error (RPE) which measures the drift per second. The methods are +evaluated 5 times on each of the 10 sequences (Table 4). +From the results in Table 5, we can see that ORB-SLAM3 has the lowest +ATE when evaluating color-inertial odometry, mainly because of loop closure +detection. In most cases, ORB-SLAM3 and OpenVINS fail to initialize when +using infrared images. We conclude that off-the-shelve feature detectors (FAST +and ORB) are quite poor at detecting good features from infrared images. Inter- +estingly, DM-VIO performs better when using infrared images instead of color +which is likely due to the infrared camera’s global shutter and wider FoV. This +result reveals the great potential of using active infrared images for visual-inertial +odometry and the need for new research. +Table 4. Evaluation sequences used in the visual-inertial odometry experiment. Last +column shows if the camera returns to the starting point (chance for a loop closure). +Sequence +Duration (s) +Length (m) +Dimensions (m) +Loop +cafeteria +200 +90.0 +12.4 x 15.7 x 0.8 +✓ +central +242 +155.0 +25.5 x 42.1 x 5.3 +✓ +dining +192 +109.2 +33.8 x 25.0 x 5.5 +✓ +corridor +174 +77.6 +31.1 x 4.7 x 2.4 +✓ +foobar +75 +37.1 +5.4 x 14.4 x 0.6 +✓ +hub +124 +52.3 +11.4 x 5.9 x 0.7 +- +juice +103 +42.7 +6.3 x 8.6 x 0.5 +- +lounge +222 +94.2 +14.4 x 10.3 x 1.1 +✓ +study +87 +40.0 +5.6 x 9.8 x 0.6 +- +waiting +139 +60.1 +9.8 x 6.7 x 0.9 +✓ + +12 +Mustaniemi et al. +Table 5. Comparison of visual-inertial odometry methods using color-inertial and +infrared-inertial inputs. Average absolute trajectory error (ATE) and relative pose error +(RPE). Last column shows the percentage of successful runs. +Color-inertial odometry +Infrared-inertial odometry +Method +ATE ↓ +(m) +RPE ↓ +(deg/s) +RPE ↓ +(m/s) +Succ. ↑ +(%) +ATE ↓ +(m) +RPE ↓ +(deg/s) +RPE ↓ +(m/s) +Succ. ↑ +(%) +OpenVINS [12] +0.347 +0.37 +0.031 +76.0 +0.597 +0.42 +0.057 +36.0 +ORB-SLAM3 [3] +0.298 +0.29 +0.026 +100.0 +0.193 +0.29 +0.025 +24.0 +DM-VIO [34] +0.491 +0.29 +0.033 +100.0 +0.433 +0.29 +0.025 +100.0 +6 +Conclusion +We presented a framework for acquiring high-quality 3D reconstructions using +a consumer depth camera. The ability to produce building-scale reconstructions +is a significant improvement over existing methods that are limited to smaller +environments such as rooms or apartments. The proposed C2D alignment en- +ables the use of raw depth maps, resulting in more accurate 3D reconstructions. +Our approach is fast, easy to use, and requires no expensive hardware, making +it ideal for crowd-sourced data collection. 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Zhou, Q.Y., Park, J., Koltun, V.: Open3D: A modern library for 3D data process- +ing. arXiv preprint arXiv:1801.09847 (2018) + diff --git a/JtAzT4oBgHgl3EQfIPso/content/tmp_files/load_file.txt b/JtAzT4oBgHgl3EQfIPso/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8882806c51026e938f352afb8f50e9c8a82362c5 --- /dev/null +++ b/JtAzT4oBgHgl3EQfIPso/content/tmp_files/load_file.txt @@ -0,0 +1,888 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf,len=887 +page_content='BS3D: Building-scale 3D Reconstruction from RGB-D Images Janne Mustaniemi1, Juho Kannala2, Esa Rahtu3, Li Liu1, and Janne Heikkilä1 1 Center for Machine Vision and Signal Analysis, University of Oulu, Finland 2 Department of Computer Science, Aalto University, Finland 3 Tampere University, Finland janne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='mustaniemi@oulu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='fi Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Various datasets have been proposed for simultaneous local- ization and mapping (SLAM) and related problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Existing datasets often include small environments, have incomplete ground truth, or lack important sensor data, such as depth and infrared images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We propose an easy-to-use framework for acquiring building-scale 3D reconstruction using a consumer depth camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Unlike complex and expensive acquisi- tion setups, our system enables crowd-sourcing, which can greatly bene- fit data-hungry algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Compared to similar systems, we utilize raw depth maps for odometry computation and loop closure refinement which results in better reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We acquire a building-scale 3D dataset (BS3D) and demonstrate its value by training an improved monocular depth estimation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' As a unique experiment, we benchmark visual- inertial odometry methods using both color and active infrared images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Keywords: Depth camera · SLAM · Large-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 1 Introduction Simultaneous localization and mapping (SLAM) is an essential component in robot navigation, virtual reality (VR), and augmented reality (AR) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Var- ious datasets and benchmarks have been proposed for SLAM [11,35,39] and re- lated problems, including visual-intertial odometry [30,6], camera re-localization [29,32,15], and depth estimation [21,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Currently, there exists only a few building- scale SLAM datasets [28] that include ground truth camera poses and dense 3D geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Such datasets enable, for example, evaluation of algorithms needed in large-scale AR applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The lack of building-scale SLAM datasets is explained by the difficulty of acquiring ground truth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Some have utilized a high-end LiDAR for obtaining 3D geometry of the environment [26,2,4,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Ground truth camera poses may be acquired using a motion capture (MoCap) system when the environment is small enough [35,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The high cost of equipment, complex sensor setup, and slow capturing process make these approaches less attractive and inconvenient for crowd-sourced data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='01057v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='CV] 3 Jan 2023 2 Mustaniemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' An alternative is to perform 3D reconstruction using a monocular, stereo, or depth camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Consumer RGB-D cameras, in particular, are interesting because of their relatively good accuracy, fast acquisition speed, low-cost, and effective- ness in textureless environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' RGB-D cameras have been used to collect datasets for depth estimation [21,33], scene understanding [8], and camera re- localization [32,38], among other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The problem is that existing RGB-D reconstruction systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' [22,9,5]) are limited to room-scale and apartment- scale environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Synthetic SLAM datasets have also been proposed [20,39,27] that include per- fect ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The challenge is that data such as time-of-flight (ToF) depth maps and infrared images are difficult to synthesize realistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Consequently, training and evaluation done using synthetic data may not reflect algorithm’s real-world performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' To address the domain gap problem, algorithms are often fine-tuned using real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We propose a framework to create building-scale 3D reconstructions using a consumer depth camera (Azure Kinect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Unlike existing approaches, we register color images and depth maps using color-to-depth (C2D) strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' This allows us to directly utilize the raw depth maps captured by the wide field-of-view (FoV) infrared camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Coupled with an open-source SLAM library [19], we acquire a building-scale 3D vision dataset (BS3D) that is considerably larger than similar datasets as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The BS3D dataset includes 392k synchronized color images, depth maps and infrared images, inertial measurements, camera poses, enhanced depth maps, surface reconstructions, and laser scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Our framework will be released for the public to enable fast, easy and affordable indoor 3D reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 240 m 80 m 8 m Zoomed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Building-scale 3D reconstruction (4300 m2) obtained using an RGB-D camera and the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The magnified area (90 m2) is larger than any recon- struction in the ScanNet dataset [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 2 Related work This section introduces commonly used RGB-D SLAM datasets and correspond- ing data acquisition processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' A summary of the datasets is provided in Table BS3D: Building-scale 3D Reconstruction from RGB-D Images 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' As there exist countless SLAM datasets, the scope is restricted to real-world indoor scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We leave out datasets focusing on aerial scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Eu- RoC MAV [2]) and autonomous driving (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' KITTI [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We also omit RGB-D datasets captured with a stationary scanner (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Matterport3D [4]) as they can- not be used for SLAM evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Synthetic datasets, such as SceneNet RGB-D [20], TartanAir [39], and ICL [27] are also omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' ADVIO [6] dataset is a realistic visual-inertial odometry benchmark that in- cludes building-scale environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Ground truth trajectory is computed using an inertial navigation system (INS) together with manual location fixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The main limitation of the dataset is that it does not come with ground truth 3D geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' LaMAR [28] is a large-scale SLAM benchmark that utilizes high-end mapping platforms (NavVis M6 trolley and VLX backpack) for ground truth generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Although the capturing setup includes a variety of devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' HoloLens2 and iPad Pro), it does not include a dedicated RGB-D camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' OpenLORIS-Scene [31] focuses on the lifelong SLAM scenario where environ- ments are dynamic and changing, similar to LaMAR [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The data is collected over an extended period of time using wheeled robots equipped with various sensors, including RGB-D, stereo, IMU, wheel odometry, and LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Ground truth poses are acquired using an external motion capture (MoCap) system, or with a 2D laser SLAM method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The dataset is not ideal for handheld SLAM evaluation because of the limited motion patterns of a ground robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' TUM RGB-D SLAM [35] is one of the most popular SLAM datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The RGB-D images are acquired using a consumer depth camera Microsoft Kinect v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Ground truth trajectory is incomplete because the MoCap system can only cover a small part of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' CoRBS [40] consists of four room-scale environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' It also utilizes MoCap for acquiring ground truth trajectories for Microsoft Kinect v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Unlike [35], CoRBS provides ground truth 3D geometry acquired using a laser scanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The data also includes infrared images, but not inertial measurements, unlike our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 7-Scenes [32] and 12-Scenes [38] are commonly used for evaluating camera lo- calization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 7-Scenes includes seven scenes captured using Kinect v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' KinectFusion [22] is used to obtain ground truth poses and dense 3D models from the RGB- D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 12-Scenes consists of multiple rooms captured using the Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='io depth sensor and iPad color camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The reconstructions are larger compared to 7-Scenes, about 37 m3 on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' They are acquired using the VoxelHashing framework [23], an alternative to KinectFusion with better scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' ScanNet [8] is an RGB-D dataset containing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='5M views acquired in 707 distinct spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' It includes estimated calibration parameters, camera poses, 3D surface reconstructions, textured meshes, and object-level semantic segmenta- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The hardware consists of a Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='io depth sensor attached to a tablet computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Pose estimation is done using BundleFusion [9], after which volumet- ric integration is performed through VoxelHashing [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Sun3D [43] is a large RGB-D database with camera poses, point clouds, object labels, and refined depth maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The reconstruction process is based on structure from motion (SfM) where manual object annotations are utilized to 4 Mustaniemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' reduce drift and loop-closure failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Refined depth maps are obtained via vol- umetric fusion similar to KinectFusion [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We emphasize that ScanNet [8] and Sun3D [43] reconstructions are considerably smaller and have lower quality than those provided in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Unlike [28,31,35], our system also does not require a complex and expensive capturing setup, or manual annotation [6,43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' List of indoor RGB-D SLAM datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The BS3D acquisition setup does not require high-end LiDARs [40,31,28], MoCap systems [40,31,36], or manual annota- tion [43,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' BS3D is building-scale, unlike [32,36,8,40,38,43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Note that ADVIO [6] and LaMAR [28] do not have a dedicated depth camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Dataset Scale Depth IMU IR Ground truth 7-Scenes [32] room Kinect v1 RGBD-recons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' TUM RGBD [36] room Kinect v1 ✓ MoCap ScanNet [8] room Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='io ✓ RGBD-recons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' CoRBS [40] room Kinect v2 ✓ MoCap+LiDAR 12-Scenes [38] apartment Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='io RGBD-recons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Sun3D [43] apartment Xtion Pro Live RGBD+manual OpenLORIS [31] building RS-D435i ✓ MoCap+LiDAR ADVIO [6] building Tango ✓ INS+manual LaMAR [28] building HoloLens2 ✓ ✓ LiDAR+VIO+SfM BS3D (ours) building Azure Kinect ✓ ✓ RGBD-recons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 3 Reconstruction framework In this section, we introduce the RGB-D reconstruction framework shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The framework produces accurate 3D reconstructions of building-scale environments using low-cost hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The system is fully automatic and robust against poor lighting conditions and fast motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Color images are only used for loop closure detection as they are susceptible to motion blur and rolling shutter distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Raw depth maps enable accurate odometry and the refinement of loop closure transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='1 Hardware Data is captured using the Azure Kinect depth camera, which is well-suited for crowd-sourcing due to its popularity and affordability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We capture synchronized depth, color, and infrared images at 30 Hz using the official recorder application running on a laptop computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We use the wide FoV mode of the infrared camera with 2x2 binning to extend the Z-range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The resolution of raw depth maps and IR images is 512 x 512 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Auto-exposure is enabled when capturing color images at the resolution of 720 x 1280 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We also record accelerometer and gyroscope readings at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='6 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' BS3D: Building-scale 3D Reconstruction from RGB-D Images 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Overview of the RGB-D reconstruction system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='2 Color-to-depth alignment Most RGB-D reconstruction systems expect that color images and depth maps have been spatially and temporally aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Modern depth cameras typically produce temporally synchronized images so the main concern is the spatial align- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Conventionally, raw depth maps are transformed to the coordinate system of the color camera, which we refer to as the depth-to-color (D2C) alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' In the case of Azure Kinect, the color camera’s FoV is much narrower (90 x 59 degrees) compared to the infrared camera (120 x 120 degrees).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Thus, the D2C alignment would not take advantage of the infrared camera’s wide FoV because depth maps would be heavily cropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Moreover, the D2C alignment might introduce artefacts to the raw depth maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We propose an alternative called color-to-depth (C2D) alignment where color images are transformed instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' In the experiments, we show that this drastically improves the quality of the reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The main challenge of C2D is that it requires a fully dense depth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Fortunately, a reasonably good alignment can be achieved even with a low quality depth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' This is because the baseline between the cameras is narrow and missing depths often appear in areas that are far away from the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' For the C2D alignment, we first perform depth inpainting using linear in- terpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Then, the color image is transformed to the raw depth frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' To keep as much of the color information as possible, the output resolution will be higher (1024 x 1024 pixels) compared to the raw depth maps .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' After that, holes in the color image due to occlusions are inpainted using the OpenCV library’s implementation of [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We note that minor artefacts in the aligned color images will have little impact on the SIFT-based loop closure detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='3 RGB-D Mapping We process the RGB-D sequences using an open-source SLAM library called RTAB-Map [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Odometry is computed from the raw depth maps using the point-to-plane variant of the iterative closest point (ICP) algorithm [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We use the scan-to-map odometry strategy [19] where incoming frames are registered against a point cloud map created from past keyframes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The wide FoV ensures that ICP-odometry rarely fails, but in case it does, a new map is initialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' RGBD RGB Depth RGBD Color-to-depth Loop closures Volumetric (C2D) (PnP + ICP) fusion Poses Normals Depth (raw) Mesh (optimized) Poses Odometry (odometry) Graph Render (ICP) optimization6 Mustaniemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Loop closure detection is needed for drift correction and merging of individual maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' For this purpose, SIFT features are extracted from the valid area of the aligned color images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Loop closures are detected using the bag-of-words approach [18], and transformations are estimated using the Perspective-n-Point RANSAC algorithm and refined using ICP [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Graph optimization is done using the GTSAM library [10] and Gauss-Newton algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' RTAB-Map supports multi-session mapping which is a necessary feature when reconstructing building-scale environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' It is not practical to collect possibly hours of data at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Furthermore, having the ability to later update and expand the map is a useful feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' In practise, individual sequences are first processed separately, followed by multi-session mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The sessions are merged by finding loop closures and by performing graph optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The in- put is a sequence of keyframes along with odometry poses and SIFT features computed during single-session mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The sessions are processed in such or- der that there is at least some overlap between the current session and the global map build so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='4 Surface reconstruction It is often useful to have a 3D surface representation of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' There exists many classical [14,22] and learning-based [41,1] surface reconstruction ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Methods that utilize deep neural networks, such as NeuralFusion [41], have produced impressive results on the task of depth map fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Neural ra- diance fields (NeRFs) have also been adapted to RGB-D imagery [1] showing good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We did not use learning-based approaches in this work be- cause they are limited to small scenes, at least for the time being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Moreover, scene-specific learning [1] takes several hours even with powerful hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Surface reconstruction is done in segments due to the large scale of the en- vironment and the vast number of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' To that end, we first create a point cloud from downsampled raw depth maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Every point includes a view index along with 3D coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The point cloud is partitioned into manageable seg- ments using the K-means algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' A mesh is created for each segment using the scalable TSDF fusion implementation [46] that is based on [7,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' It uses a hierarchical hashing structure to support large scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 4 BS3D dataset The BS3D dataset was collected at the university campus using Azure Kinect (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Figure 3 shows example frames from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The collection was done in multiple sessions due to large scale of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The record- ings were processed using the framework described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='1 Dataset features The reconstruction shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 1 consists of 47 overlapping recording sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Additional 14 sessions, including 3D laser scans, were recorded for evaluation BS3D: Building-scale 3D Reconstruction from RGB-D Images 7 Cafeteria Stairs Study Corridor Lobby Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Example frames from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Environments are diverse and challenging, including cafeterias, stairs, study areas, corridors, and lobbies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Most sessions begin and end at the same location to encourage loop closure detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The total duration of the sessions is 3 hours and 38 minutes and the combined trajectory length is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='4 kilometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The reconstructed floor area is approximately 4300 m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The dataset consists of 392k frames, including color images, raw depth maps, and infrared images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Color images and depth maps are provided in both coordi- nate frames (color and infrared camera).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The images have been undistorted for convenience, but the original recordings are also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We provide camera poses in a global reference frame for every image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Data also includes inertial mea- surements, enhanced depth maps and surface normals that have been rendered from the mesh as visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Color Infrared Normals (render) Mesh Depth Depth (raw) Depth (render) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The BS3D dataset includes color and infrared images, depth maps, IMU data, camera parameters, and surface reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Enhanced depth maps and surface normals are rendered from the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='2 Laser scan We utilize the FARO 3D X 130 laser scanner for acquiring ground truth 3D geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The scanned area includes a lobby and corridors of different sizes (800 m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The 28 individual scans were registered using the SCENE software that comes with the laser scanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Noticeable artefacts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' those caused by mirrors, 8 Mustaniemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' were manually removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The laser scan is used to evaluate the reconstruction framework in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' However, this data also enables, for example, training and evaluation of RGB-D surface reconstruction algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 5 Experiments We compare our framework with the state-of-the-art RGB-D reconstruction methods [5,9,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The value of the BS3D dataset is demonstrated by training a recent monocular depth estimation model [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We also benchmark visual- inertial odometry approaches [12,34,3] using either color or infrared images to further highlight the unique aspects of the BS3D dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='1 Reconstruction framework In this experiment, we compare the framework against Redwood [5], Bundle- Fusion [9], and ORB-SLAM3 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' RGBD images are provided for [5,9,3] in the coordinate frame of the color camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Given the estimated camera poses, we cre- ate a point cloud and compare it to the laser scan (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The evaluation metrics include overlap of the point clouds and RMSE of inlier correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Before comparison, we create uniformly sampled point clouds using voxel down- sampling (1 cm3 voxel) that computes the centroid of the points in each voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The overlap is defined as the ratio of inlier correspondences and the number of ground truth points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' A 3D point is considered to be an inlier if the distance to the closest ground truth point is below threshold γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Table 2 shows the results for environments of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' All methods are able to reconstruct the small environment (35 m2) consisting of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='8k frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The differences between the methods become more evident when reconstructing the medium-size environment (160 m2) consisting of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='3k frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' BundleFusion [9] only produces a partial reconstruction because of odometry failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The proposed approach gives the most accurate reconstructions as visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Note that it is not possible to achieve 100 % overlap because the depth camera does not observe all parts of the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The largest environment (350 m2) consists of 24k frames acquired in four sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Redwood [5] does not scale to input sequences of this long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' ORB-SLAM3 [3] frequently loses the odometry in open spaces which leads to incomplete and less accurate reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Our method suffers the same problem when C2D is disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Unreliable odometry is likely due to the color camera’s limited FoV, rolling shutter distortion, and motion blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The C2D alignment improves the accuracy and robustness of ICP-based odometry and loop closures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Without C2D, the frequent odometry failures result in disconnected maps and noticeable drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We note that the reconstruction in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 1 was computed from ∼300k frames which is far more than [5,9,3] can handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' BS3D: Building-scale 3D Reconstruction from RGB-D Images 9 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Comparison of RGB-D reconstruction methods in small, medium and large- scale environments (from top to bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Overlap of the point clouds and inlier RMSE computed for distance thresholds γ (mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Some methods only work in small and/or medium scale environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' γ = 10 (mm) γ = 20 (mm) γ = 50 (mm) Method Overlap ↑ RMSE ↓ Overlap ↑ RMSE ↓ Overlap ↑ RMSE ↓ Redwood [5] 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='6 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='6 BundleFusion [9] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='9 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 ORB-SLAM3 [3] 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='5 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='6 Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' (w/o C2D) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 Proposed 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='5 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='6 Redwood [5] 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='2 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='9 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='9 BundleFusion [9] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='8 ORB-SLAM3 [3] 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='0 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='2 Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' (w/o C2D) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='1 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='3 Proposed 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='4 ORB-SLAM3 [3] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' (w/o C2D) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='9 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='4 Proposed 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='4 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='8 ORB-SLAM3 [3] Proposed Redwood [5] Proposed (w/o C2D) ϵ < 20 mm 20 ≤ ϵ < 50 50 ≤ ϵ < 100 100 ≤ ϵ < 200 ϵ ≥ 200 mm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Reconstructions obtained using Redwood [5], ORB-SLAM3 [3], and the pro- posed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Colors depict errors (distance to the closest ground truth point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 10 Mustaniemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='2 Depth estimation We investigate whether the BS3D dataset can be used to train better models for monocular depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' For this experiment, we use the state-of-the- art LeReS model [44] based on ResNet50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The original model has been trained using 354k samples taken from various datasets [45,24,16,13,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We finetune the model using 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='5k samples from BS3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We set the learning rate to 2e-5 and train only 4 epochs to avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Other training details, including loss functions are the same as in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' For testing, we use NYUD-v2 [21] and iBims-1 [17] that are not seen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We also evaluate using BS3D by sampling 535 images from an unseen part of the building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Table 3 shows that finetuning improves the performance on iBims-1 and BS3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The finetuned model performs marginally worse on NYUD- v2 which is not surprising considering that NYUD-v2 mainly contains room-scale scenes that are not present in BS3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The qualitative comparison in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 6 also shows a clear improvement over the pretrained model on iBims-1 that contains both small and large scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The model trained only using BS3D cannot compete with other models, except on BS3D on which the performance is surprisingly good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The poor performance on other datasets is not surprising because of the small training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Monocular depth estimation using LeReS [44] trained from scratch using BS3D, pretrained model, and finetuned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' NUYD-v2 [21], iBims-1 [17], and BS3D are used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' NYUD-v2 [21] iBims-1 [17] BS3D Training data AbsRel ↓ δ1 ↑ AbsRel ↓ δ1 ↑ AbsRel ↓ δ1 ↑ BS3D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='828 Pretrained 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='890 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='157 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='785 Pre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' + BS3D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='881 Color Pretrained Finetuned Ground truth Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Comparison of pretrained and finetuned (BS3D) monocular depth estimation model LeReS [44] on an independent iBims-1 [17] dataset unseen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' BS3D: Building-scale 3D Reconstruction from RGB-D Images 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='3 Visual-inertial odometry The BS3D dataset includes active infrared images along with color and IMU data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' This opens interesting possibilities, for example, the comparison of color and infrared as inputs for visual-inertial odometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Infrared-inertial odometry is an attractive approach in the sense that it does not require external light, meaning it would work in completely dark environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We evaluate OpenVINS [12], ORB-SLAM3 [3], and DM-VIO [34] using color- inertial and infrared-inertial inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Note that ORB-SLAM3 has an unfair ad- vantage because it has a loop closure detector that cannot be disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' In the case of infrared images, we apply a power law transformation (I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='04 · I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='6) to increase brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' As supported by [34], we provide a mask of valid pix- els to ignore black areas near the edges of the infrared images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We adjust the parameters related to feature detection when using infrared images with [12,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We use the standard error metrics, namely absolute trajectory error (ATE) and relative pose error (RPE) which measures the drift per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The methods are evaluated 5 times on each of the 10 sequences (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' From the results in Table 5, we can see that ORB-SLAM3 has the lowest ATE when evaluating color-inertial odometry, mainly because of loop closure detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' In most cases, ORB-SLAM3 and OpenVINS fail to initialize when using infrared images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We conclude that off-the-shelve feature detectors (FAST and ORB) are quite poor at detecting good features from infrared images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Inter- estingly, DM-VIO performs better when using infrared images instead of color which is likely due to the infrared camera’s global shutter and wider FoV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' This result reveals the great potential of using active infrared images for visual-inertial odometry and the need for new research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Evaluation sequences used in the visual-inertial odometry experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Last column shows if the camera returns to the starting point (chance for a loop closure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Sequence Duration (s) Length (m) Dimensions (m) Loop cafeteria 200 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='4 x 15.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='8 x 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='0 x 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='5 ✓ corridor 174 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='6 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='1 x 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='4 ✓ foobar 75 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 juice 103 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='3 x 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='6 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='5 lounge 222 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='4 x 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='3 x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='1 ✓ study 87 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='6 x 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='8 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='6 waiting 139 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='8 x 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='7 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='9 ✓ 12 Mustaniemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Comparison of visual-inertial odometry methods using color-inertial and infrared-inertial inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Average absolute trajectory error (ATE) and relative pose error (RPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Last column shows the percentage of successful runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Color-inertial odometry Infrared-inertial odometry Method ATE ↓ (m) RPE ↓ (deg/s) RPE ↓ (m/s) Succ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' ↑ (%) ATE ↓ (m) RPE ↓ (deg/s) RPE ↓ (m/s) Succ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' ↑ (%) OpenVINS [12] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='347 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='031 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='597 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='057 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='0 ORB-SLAM3 [3] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='298 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='026 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='193 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='025 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='0 DM-VIO [34] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='033 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='433 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='025 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content='0 6 Conclusion We presented a framework for acquiring high-quality 3D reconstructions using a consumer depth camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The ability to produce building-scale reconstructions is a significant improvement over existing methods that are limited to smaller environments such as rooms or apartments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' The proposed C2D alignment en- ables the use of raw depth maps, resulting in more accurate 3D reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' Our approach is fast, easy to use, and requires no expensive hardware, making it ideal for crowd-sourced data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We acquire building-scale 3D dataset (BS3D) and demonstrate its value for monocular depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' BS3D is unique also because it includes active infrared images, which are often miss- ing in other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAzT4oBgHgl3EQfIPso/content/2301.01057v1.pdf'} +page_content=' We employ infrared images for visual-inertial odometry, discovering a promising new research direction.' metadata={'source': 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sha256:a6cd59058df224e2b65d542185f30adca19ddaf73ed2248b926ecea35978203e +size 101343 diff --git a/OtAyT4oBgHgl3EQftfkF/content/tmp_files/2301.00594v1.pdf.txt b/OtAyT4oBgHgl3EQftfkF/content/tmp_files/2301.00594v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1ab525cb958da9e55b1f1a7712b8a6a5ca108ca0 --- /dev/null +++ b/OtAyT4oBgHgl3EQftfkF/content/tmp_files/2301.00594v1.pdf.txt @@ -0,0 +1,1043 @@ +Rate Region of MIMO RIS-assisted Broadcast +Channels with Rate Splitting and Improper +Signaling +Mohammad Soleymani∗, Ignacio Santamaria†, and Eduard Jorswieck‡ +*Signal & System Theory Group, Universit¨at Paderborn, Germany +†Dept. Communications Engineering, Universidad de Cantabria, Spain +‡ Institute for Communications Technology, Technische Universit¨at Braunschweig, Germany +Email: mohammad.soleymani@sst.upb.dei.santamaria@unican.es, jorswieck@ifn.ing.tu-bs.de +Abstract—In this paper, we study the achievable rate region of +1-layer rate splitting (RS) in the presence of hardware impair- +ment (HWI) and improper Gaussian signaling (IGS) for a single- +cell reconfigurable intelligent surface (RIS) assisted broadcast +channel (BC). We assume that the transceivers may suffer from +an imbalance in in-band and quadrature signals, which is known +as I/Q imbalance (IQI). The received signal and noise can be +improper when there exists IQI. Therefore, we employ IGS to +compensate for IQI as well as to manage interference. Our results +show that RS and RIS can significantly enlarge the rate region, +where the role of RS is to manage interference while RIS mainly +improves the coverage. +Index Terms—Achievable rate region, hardware impairment, +improper Gaussian signaling, MIMO broadcast channels, rate +splitting. +I. INTRODUCTION +The sixth generation (6G) of wireless communication sys- +tems should be much more spectral and energy efficient than +the existing communication systems [1]. This goal may not be +achieved without employing some emerging technologies such +as reconfigurable intelligent surfaces (RISs) and rate splitting +(RS), which have been shown to be able to improve the spec- +tral and energy efficiency of various wireless communication +systems [2]–[5]. +Interference has been always among the main restrictions +of modern wireless communication systems, and interference- +management techniques are thus expected to continue playing +a key role in such systems [6]. Even though there are many +studies on the performance of RIS, its role should be further +investigated in overloaded interference-limited systems, in +which the number of users is larger than the number of spatial, +temporal, or frequency resources. RIS can modulate channels, +canceling interference links or improving the strength of de- +sired links. In other words, RIS can be potentially employed to +manage and/or neutralize interference in some scenarios such +as multiple-user interference channels. Thus, the following +question may arise: Are other advanced interference manage- +ment techniques, such as rate-splitting or improper Gaussian +signaling (IGS), still necessary in RIS-assisted systems? We +answer this question in positive in this paper. +Rate splitting (RS) is a powerful technology to highly +improve +the +spectral +and +energy +efficiency +of +various +interference-limited systems [4], [5]. There are different RS +schemes such as 1-layer RS, hybrid RS and generalized +RS. The generalized RS scheme is the most complete RS +scheme and includes many other technologies/techniques such +as spatial division multiple access (SDMA), non-orthogonal +multiple access (NOMA), orthogonal multiple access (OMA) +and treating interference as noise (TIN) [7]. Implementing +generalized RS has high complexities when the number of +users grows. An alternative to the generalized RS is 1-layer +RS, which is a very practical scheme with much lower com- +plexities. 1-layer RS is very efficient and is able to improve +the performance of different interference-limited systems [8]– +[10]. +Another powerful interference-management technique is +improper Gaussian signaling (IGS), which can improve the +system performance when the receivers apply TIN or partial +successive cancellation (SIC) [11]–[16]. Moreover, IGS has +been shown to increase the degrees-of-freedom of the 3-user +single-input, single-output (SISO) interference channel (IC) +through interference alignment [17]. +Interference is not the only performance limiting factor +in wireless communication systems. Another limitation arises +from non-idealities in transceivers. A source of imperfections +in transceivers is I/Q imbalance (IQI), which is modeled +as a widely linear transformation of the input signal [12], +[18], [19]. Hardware impairments (HWI) can highly affect the +system performance especially when such imperfections are +not taken into account in the system design. +In this paper, we investigate the role of RIS, RS and IGS +in multiple-input, multiple-output (MIMO) broadcast channels +(BCs) with IQI. We show that although RIS can enlarge the +rate region, RS and IGS are still needed to manage interference +and compensate for IQI. Indeed, the role of RIS in this scenario +is mainly to improve the coverage, while RS is responsible for +handling interference. It is known that 1-layer RS with proper +Gaussian signaling (PGS) is the optimal scheme in the two- +user single-cell BC with perfect devices. However, it is not +the case in the presence of IQI. Our results show that IQI +shrinks the achievable rate region, and IGS with TIN may +outperform RS with PGS in some operational points/regimes. +In this case, the 1-layer RS with IGS outperforms the other +arXiv:2301.00594v1 [cs.IT] 2 Jan 2023 + +BS +RIS +2 +U +1 +U +1 +UK− +UK +3 +U +Fig. 1: A broadcast channel with RIS. +considered schemes. +This paper is organized as follows. Section II presents the +system model. Section III proposes a suboptimal scheme to +obtain the achievable rate region. Section IV presents some +numerical results, and Section V summarizes the main findings +of the paper. +II. SYSTEM MODEL +A. Network Scenario +We consider a single-cell RIS-assisted system with IQI at +transceivers, as shown in Fig. 1. We assume that there is a +BS with NBS transmit antennas, serving K users with Nu +receive antennas each. Additionally, there is a RIS with NRIS +components to assist the BS. The users and the BS may suffer +from IQI according to the model in [12], [18]. For the sake +of notational simplicity, we consider a symmetric scenario in +which all the users have the same number of antennas as well +as the same IQI parameters, although the model can be easily +extended to asymmetric scenarios. +B. Channel model +We employ the channel model in [20] for MIMO RIS- +assisted systems. In this paper, we briefly present the channel +model and refer the readers to [20], [21] for more detailed +discussions on the fading models of MIMO RIS-assisted +systems. The channel matrix between the BS and user k is +[22, Eq. (14)] +Hk (Θ) = +GkΘG +� �� � +Link through RIS ++ +Fk +���� +Direct link +∈ CNu×NBS, +(1) +where Fk is the channel matrix between the BS and user k, +Gk is the channel matrix between the RIS and user k, G is the +channel matrix between the BS and the RIS, and the matrix +Θ is +Θ = diag (θ1, θ2, · · · , θNRIS) , +(2) +where θis for all i are RIS components. In this paper, am- +plitudes of the RIS components are assumed to be fixed to +1, while the phases can take any value between 0 and 2π. In +other words, the constraint set for the RIS components is +T = {θi : |θi| = 1 ∀i} . +(3) +We refer the reader to [21] for a description of other common +constraints on the amplitudes and phases of the RIS elements. +C. 1-layer rate splitting scheme +We consider 1-layer RS scheme in which the BS broadcasts +a common message for all users and K private messages (one +for each user). Thus, the BS is intended to transmit +x = xc + +K +� +k=1 +xk, +(4) +where xc is the common message, and xk is the private +message of the BS intended for user k. Since the BS may +suffer from IQI, the actual transmit signal of the BS is a widely +linear transformation of x as [12], [18] +xt = V1x + V2x∗, +(5) +where the constant matrices V1 and V2 are, respectively, +defined in [12, Eq. (7)] and [12, Eq. (8)]. The receive signal +at the receiver of user k is +yk = Hkxt + nk, +(6) +where nk is a zero-mean proper white additive Gaussian noise +with variance σ2 at receiver k, and Hk is the effective channel +between the BS and user k, given by (1). Note the effective +channel is a function of Θ; however, we drop the dependency +to simplify the representation of equations. The receiver of +user k may suffer from IQI, which means the final output of +the received signal is a widely linear transformation of yk as +yk = Γ1 [Hk (V1xk + V2x∗ +k) + nk] ++ Γ2 [Hk (V1xk + V2x∗ +k) + nk]∗ , +(7) +where the constant matrices Γ1 and Γ2 are given by [12, Eq. +(12)] and [12, Eq. (13)], respectively. Note that the effective +noise at user k can be improper due to IQI, meaning that the +real and imaginary parts of the effective noise can be correlated +and/or have unequal powers [23]. To compensate for IQI, we +assume that the common and private messages, xc and xk, +can be improper Gaussian signals. Additionally, the signals +xc and xk are zero-mean and independent. Note that IGS can +also help to reduce or manage interference, as indicated before, +so its role is not only to compensate for IQI. +A way to model IGS is through the real-decomposition +method. +We +can +rewrite +(7) +by +employing +the +real- +decomposition method as +yk = Hkxi + nlk += +Hkxc +� �� � +Common M. ++ Hkxk +� �� � +Private M. ++ Hk +K +� +j=1,j̸=k +xj +� +�� +� +Interference ++ nk +���� +Noise +, +(8) +where Hk is the equivalent channel given by [22, Eq. (11)], +y = +� +R{y}T +I{y}T �T , and x = +� +R{x}T +I{x}T �T +are, respectively, the real decomposition of y and x. Addition- +ally, nk represents the effective improper noise and is given +by +nk = +� +R{Γ1n + Γ2n∗}T +I{Γ1n + Γ2n∗}T �T . +(9) + +We represent the covariance matrix of the noise by E{n nT } = +Cn. Moreover, the transmit covariance matrix of x, xc, and +xk are denoted as P, Pc, and Pk, respectively, where P = +Pc + � +k Pk. The constraint set of the transmit covariance +matrices is given by [21, Eq. (3)] for IGS (denoted by PI) +and by [21, Eq. (4)] for PGS schemes (denoted by PP ). Since +RS can be applied to both PGS and IGS cases, we represent +the constraint set of the transmit covariance matrices as P. +Note that the set P is convex. Moreover, note that PGS is +a special case of IGS, thus, an optimal IGS scheme never +performs worse than a PGS scheme. We refer the reader to +[22, Sec. II.A], [21, Appendix A] and [23] for further details +on modeling IQI and/or impropriety. +D. Rate expressions +Users firstly decode the common message and cancel it from +the received signal. Thus, the maximum decoding rate for the +common message at user k is [24, Eqs. (2)-(3)] +¯rck = 1 +2 log2 +������ +I + +� +Cn + +� +∀i +HkPiHT +k +�−1 +HkPcHT +k +������ += 1 +2 log2 +���Cn + HkPHT +k +��� +� +�� +� +¯rck,1 +− 1 +2 log2 +�����Cn + +� +∀i +HkPiHT +k +����� +� +�� +� +¯rck,2 +. +The common message must be transmitted at a rate that +is decodable for all users. Hence, the maximum rate for +transmitting the common message is +rc({P}, Θ) = min +k {¯rck({P}, Θ)} . +(10) +After decoding and canceling the common message, each user +decodes its own private message. Therefore, the maximum +decoding rate for the private message at user k is +rpk = 1 +2 log2 +������� +I + +� +�Cn + +� +∀i̸=k +HkPiHT +k +� +� +−1 +HkPkHT +k +������� +(11) += 1 +2log2 +�����Cn+ +� +∀i +HkPiHT +k +����� +� +�� +� +rpk,1 +−1 +2log2 +������ +Cn+ +� +∀i̸=k +HkPiHT +k +������ +� +�� +� +rpk,2 +. +(12) +Finally, the rate of user k is the summation of the decoding rate +of its private message and its dedicated rate from the common +message, i.e., [21] +rk({P}, Θ) = rpk({P}, Θ) + rck, +(13) +where rck ≥ 0 and � +k rck ≤ rc. Note that the rates rk, +rpk, ¯rck and rc are functions of {P} and Θ, while rc = +{rc1, rc2, · · · , rcK} is a design parameter. Due to notational +simplicity, we, hereafter, drop the dependency of the rates to +{P} and Θ in representing rk, rpk, ¯rck and rc. +E. Problem Statement +Employing the rate profile technique, the achievable rate +region can be obtained by solving [25] +max +Θ∈T ,{P}∈P,rc r +(14a) +s.t. rk = rpk + rck ≥ αkr +∀k, +(14b) +� +∀k +rck ≤ rc, +rck ≥ 0, +∀k, +(14c) +and varying the weights such that � +∀k αk = 1 with αk ≥ 0 +for all k. +III. PROPOSED ALGORITHM +We employ an approach based on the optimization frame- +work proposed in [21] to solve (14). That is, we first employ +an alternating optimization (AO) approach to sequentially +optimize over the transmit covariance matrices and RIS com- +ponents. Indeed, we first fix the RIS components to Θ(t−1) +and update the transmit covariance matrices as {P(t)}. Then, +we alternate and optimize over the reflecting coefficients +for fixed transmit covariance matrices. Unfortunately, even +after fixing either the RIS or the covariance matrices, the +resulting optimization problems are still non-convex. In the +following subsections, we proposed iterative algorithms to find +a suboptimal solution. +A. Optimizing transmit covariance matrices +In this subsection, we update the transmit covariance ma- +trices for a fixed Θ(t−1) by solving +max +{P}∈P,rc r +s.t. +(14b), (14c), +(15) +which is a non-convex problem. Note that the constraints in +(15) are linear in rc, however, the rates are not concave in {P}, +which makes the problem non-convex. Indeed, the rate rpk (or +rck) can be written as a difference of two concave functions +rpk,1 and rpk,2 (or rck,1 and rck,2). Thus, to solve (15), we can +employ difference of convex programming (DCP), which falls +into majorization minimization (MM). That is, we approximate +the rates by a suitable concave lower bound. To this end, we +keep rpk,1 (or rck,1) unchanged and employ the first-order +Taylor expansion to approximate rpk,2 (or rck,2) by an affine +(linear) function as +rpk ≥ ˜rpk = rpk,1 − rpk,2 +� +{P(t−1)} +� +− +� +∀j̸=k +Tr +� +HT +k (Cn+� +∀i̸=kHkP(t−1) +i +HT +k )−1Hk +2 ln 2 +� +Pj−P(t−1) +j +�� +. +(16) +Similarly, a concave lower bound for ¯rck is +rck ≥ ˜rck = ¯rck,1 − ¯rck,2 +� +{P(t−1)} +� +− +� +∀j +Tr +� +HT +k (Cn+� +∀iHkP(t−1) +i +HT +k )−1Hk +2 ln 2 +� +Pj−P(t−1) +j +�� +. +(17) + +We refer the reader to [21, Corollary 1] for the proof. Substi- +tuting the concave lower bounds in (15) results in the following +convex optimization problem +max +{P}∈P,rc r +(18a) +s.t. ˜rk = ˜rpk + rkc ≥ αkr +∀k, (18b) +� +∀k +rck ≤ ˜rc = min +k {˜rck} , +rck ≥ 0, +∀k. (18c) +This problem can be solved by existing numerical tools, which +yields {P(t)}. +B. Optimizing RIS components +Now we update the RIS components Θ for fixed transmit +covariance matrices {P(t)} by solving +max +Θ∈T ,rc r +s.t. +(14b), (14c), +(19) +This problem is non-convex since the constraint set for the RIS +components is not a convex set, and additionally, the rates are +not concave in Θ. Thus, to solve (19), we first find suitable +concave lower bounds for the rates and then, convexify the +constraint set T . To this end, we employ the concave lower +bound given by [21, Lemma 4], which results in the following +concave lower bound for rpk +rpk ≥ ˆrpk = rpk +� +Θ(t−1)� +− +1 +2 ln 2Tr +� ¯Vk ¯VT +k ¯Y−1 +k +� +− +1 +2 ln 2Tr +� +[ ¯Y−1 +k − ( ¯Vk ¯VT +k + ¯Yk)−1]T [VkVT +k +Yk] +� ++ +1 +ln 2Tr +� ¯VT +k ¯Y−1 +k Vk +� +, +(20) +where Vk = Hk (Θ) P(t)1/2 +k +, ¯Vk = Hk +� +Θ(t−1)� +P(t)1/2 +k +and +Yk = Cn + +� +∀i̸=k +Hk (Θ) P(t) +i HT +k (Θ) +(21) +¯Yk = Cn + +� +∀i̸=k +Hk +� +Θ(t−1)� +P(t) +i HT +k +� +Θ(t−1)� +. +(22) +Similarly, a concave lower bound for ¯rck can be found as +¯rck ≥ ˆrck = ¯rck +� +Θ(t−1)� +− +1 +2 ln 2Tr +� ¯Vck ¯VT +ck ¯Y−1 +ck +� +− +1 +2 ln 2Tr +� +[ ¯Y−1 +ck − ( ¯Vck ¯VT +ck+ ¯Yck)−1]T [VckVT +ck+Yck] +� ++ +1 +ln 2Tr +� ¯VT +ck ¯Y−1 +ck Vck +� +, +(23) +where Vck = Hk (Θ) P(t)1/2 +c +, ¯Vck = Hk +� +Θ(t−1)� +P(t)1/2 +c +and +Yck = Cn + +� +∀i +Hk (Θ) P(t) +i HT +k (Θ) +(24) +¯Yck = Cn + +� +∀i +Hk +� +Θ(t−1)� +P(t) +i HT +k +� +Θ(t−1)� +. +(25) +Now we propose a suboptimal approach to convexify the +constraint |θn| = 1 for all n. This constraint can be rewritten +as |θn|2 ≤ 1, and |θn|2 ≥ 1, where the former is convex, +while the latter is not. To convexify |θn|2 ≥ 1, we employ the +first-order Taylor expansion since |θn|2 is a convex function, +which results in +|θn|2 ≥ |θ(t−1) +n +|2 − 2R{θ(t−1)∗ +n +(θn − θ(t−1) +n +)} ≥ 1. +(26) +To converge faster, we relax the constraint in (26) as +|θn|2 ≥|θ(t−1) +n +|2− 2R{θ(t−1)∗ +n +(θn − θ(t−1) +n +)}≥ 1 − ϵ, +(27) +where ϵ > 0. The constraint (27) is linear. Hence, the following +surrogate optimization problem is convex +max +Θ,rc r +(28a) +s.t. ˆrk = ˆrpk + rkc ≥ αkr +∀k, +(28b) +� +∀k +rck ≤ ˆrc = min +k {ˆrck} , +rck ≥ 0, +∀k, +(28c) +|θn|2 ≤ 1, +and +(27), +∀n. +(28d) +The solution of (28), ˆΘ, may not necessarily be in T since +we relaxed the constraint in (26). To get a feasible point, we +normalize ˆΘ as θnew +n += ˆθn/|ˆθn|, for all n. Finally, we update +Θ based on the following rule +Θ(t) = +� +� +� +� +� +� +� +� +� +� +� +ˆΘnew +if +mink +� +rk({P(t)}, ˆΘnew) +αk +� +≥ +mink +� +rk({P(t)}, ˆΘ(t−1)) +αk +� +{Θ(t−1)} +Otherwise. +(29) +This updating rule guarantees the convergence since the al- +gorithm generates a sequence of non-decreasing minimum +weighted rates. +IV. NUMERICAL RESULTS +In this section, we provide some numerical examples to +clarify the role of RS, RIS and IGS in single-cell BCs. We +consider a line-of-sight (LoS) connection for the links reaching +to or departing from the RIS, and a non-LoS (NLoS) link for +the direct links between the users and the BS. It means that +the small-scale fading of the links related to RIS is Rician, +while that of direct links is Rayleigh. The large-scale path loss +component of RIS links is αRIS = 3.2. The other simulation +parameters are chosen based on [22]. The considered schemes +in the simulations are as follows: PT (or IT) denotes the +PGS (or IGS) scheme with TIN but without RIS. PR (or IR) +denotes the PGS-based (or IGS-based) RS scheme without +RIS. PRIR (or IRIR) denotes the PGS-based (or IGS-based) +RS scheme with RIS. Finally, TS denotes the time-division- +multiplexing-access (TDMA) with time sharing. +A. SISO systems +Fig. 2 shows the achievable rate region of a two-user SISO +BC with P = 10 dB and the channel realization +C1 : f1 = −1.3992 + 0.0292i, +f2 = 0.2353 − 0.1238i. + +0 +1 +2 +3 +4 +0 +0.2 +0.4 +0.6 +0.8 +r1 (b/s/Hz) +r2 (b/s/Hz) +PR +IT +PT +TS +(a) With perfect devices. +0 +1 +2 +3 +4 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +r1 (b/s/Hz) +r2 (b/s/Hz) +IR +PR +IT +PT +(b) With IQI. +Fig. 2: Achievable rate region of a two-user SISO BC with P = 10 +dB and the channel realization C1. +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +r1 (b/s/Hz) +r2 (b/s/Hz) +PR +IT +PT +TS +Fig. 3: Achievable rate region of a two-user SISO BC with P = 10 +dB and the channel realization C2. +As can be observed in Fig. 2a, PGS with TIN is very subop- +timal, and all other schemes can highly outperform PGS with +TIN. IGS with TIN can enlarge the rate region over the PGS +scheme with TIN as well as the TS scheme. Moreover, RS with +PGS is the optimal scheme, and RS with IGS performs the +same as RS with PGS when the devices are perfect. However, +it is not the case when there exists IQI, as can be observed in +Fig. 2b. In the presence of IQI, the noise is improper, and to +compensate for it, we should employ improper signaling. As +shown in Fig. 2b, IGS with TIN can outperform RS with PGS +in some operational points. Additionally, RS with IGS highly +outperforms RS with PGS. Furthermore, it can be observed in +Figs. 2a and 2b that the achievable rate region shrinks when +the devices are imperfect. +Fig. 3 shows the achievable rate region of a two-user SISO +BC with P = 10 dB and the channel realization +C2 : f1 = 0.3672 + 0.8681i, +f2 = 0.2798 + 0.9214i. +For this channel realization, the absolute values of the channels +are almost equal, and non-orthogonal multiple access schemes +cannot provide a significant gain over TDMA and TS. As can +be observed, RS with PGS is the optimal strategy and attains +all the points on the achievable rate region without employing +TS. However, PGS and IGS schemes with TIN are highly +suboptimal. Indeed, this example shows that the 1-layer RS +scheme includes OMA schemes. +0 +1 +2 +3 +4 +5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +r1 (b/s/Hz) +r2 (b/s/Hz) +PR +IT +PT +IRIR +Fig. 4: Achievable rate region of a two-user SISO BC with P = 10 +dB and the channel realization C3. +0 +1 +2 +3 +4 +5 +6 +7 +0 +2 +4 +6 +8 +r1 (b/s/Hz) +r2 (b/s/Hz) +IRIR +PR +IT +PT +TS +(a) C4. +0 +2 +4 +6 +8 +9 +0 +2 +4 +6 +8 +9 +r1 (b/s/Hz) +r2 (b/s/Hz) +IRIR +PR +IT +PT +TS +(b) C5. +Fig. 5: Achievable rate region of a two-user 2 × 2 MIMO BC with +P = 10 dB and different channel realizations. +Fig. 4 shows the achievable rate region of a two-user SISO +BC with P = 10 dB and the channel realization +C3 : f1 = 0.5909 − 1.0615i, +f2 = 0.2540 − 0.0052i. +As can be observed, RIS can highly enlarge the achievable +rate region. However, we should employ RIS with RS to get +the best performance out of RIS. Moreover, we can observe +that RIS provides more benefits for the rate of the user with +a weaker channel gain, which shown the ability of RIS to +significantly improve the coverage. +B. MIMO systems +Fig. 5 shows the achievable rate region of a two-user 2 × 2 +MIMO BC with P = 10 dB, αRIS = 3 and the following +channel realizations: +C4 : F1 = +� +−1.6952 + 1.7244i +−0.5196 − 0.1194i +0.0665 + 0.3475i +0.1105 + 0.3237i +� +, +F2 = +� −0.0233 + 0.6539i +0.2841 + 0.8593i +−0.2500 − 1.2059i +0.8494 + 0.5047i +� +, +C5 : F1 = +� +0.2949 − 0.7399i +−2.1314 + 0.5059i +−1.5491 + 0.3702i +−0.1943 + 0.9528i +� +, +F2 = +� −0.7849 + 2.4803i +0.0522 − 0.0681i +−1.5022 + 0.1034i +0.4433 − 1.0066i +� +, + +A two-user 2 × 2 MIMO BC can be considered as an +underloaded system since the sum of the number of transmit +and receive antennas is higher than the number of users. As +can be observed in Fig. 5, IGS with TIN can enlarge the rate +region over the PGS with TIN scheme. Furthermore, RS with +TIN outperforms the other schemes as it is the optimal scheme +in the considered system. We can also observe that RIS can +enlarge the rate region by improving the coverage. Since this +system is underloaded, the benefits of IGS and RS are less than +in the two-user SISO BC, which is a highly overloaded system. +However, there are still some benefits in the employment of +RS and/or IGS. +V. SUMMARY AND CONCLUSION +In this paper, we have characterized the achievable rate +region of RIS-assisted BCs with RS and IQI. Our main +findings can be summarized as follows: +• The role of RS is to manage interference. The 1-layer +PGS-based RS scheme is optimal in a two-user BC with +perfect devices. This scheme includes OMA, TIN and +NOMA. However, when the transceivers suffer from IQI, +PGS is unable to compensate for it, and we should +employ IGS. Interestingly, IGS with TIN may outperform +the 1-layer RS with PGS in some regimes. Thus, in the +presence of IQI, the 1-layer IGS-based RS scheme is +optimal in a two-user BC with and/or without RIS. +• The role of IGS is twofold: to manage interference and +to compensate for IQI. +• The role of RIS in this system is mainly to improve the +coverage, as it cannot completely manage interference in +a BC, which is in line with our previous studies [21], +[26]. Indeed, we have to employ advanced interference- +management techniques such as RS in highly overloaded +systems to use RIS more efficiently. +• RS and IGS as interference-management techniques can +provide considerable benefits in overloaded systems. +However, these benefits decrease (or may even vanish) +in underloaded systems. +ACKNOWLEDGMENT +The work of I. Santamaria has been partly supported +by the project ADELE PID2019-104958RB-C43, funded by +MCIN/AEI/10.13039/501100011033. 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Rezvani, “NOMA- +based improper signaling for multicell MISO RIS-assisted broadcast +channels,” arXiv preprint arXiv:2206.03795, 2022. + diff --git a/OtAyT4oBgHgl3EQftfkF/content/tmp_files/load_file.txt b/OtAyT4oBgHgl3EQftfkF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3eef246c57e62a988adf307a530db4d1814775b8 --- /dev/null +++ b/OtAyT4oBgHgl3EQftfkF/content/tmp_files/load_file.txt @@ -0,0 +1,564 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf,len=563 +page_content='Rate Region of MIMO RIS-assisted Broadcast Channels with Rate Splitting and Improper Signaling Mohammad Soleymani∗, Ignacio Santamaria†, and Eduard Jorswieck‡ Signal & System Theory Group, Universit¨at Paderborn, Germany †Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Communications Engineering, Universidad de Cantabria, Spain ‡ Institute for Communications Technology, Technische Universit¨at Braunschweig, Germany Email: mohammad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='soleymani@sst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='upb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='dei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='santamaria@unican.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='es, jorswieck@ifn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='tu-bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='de Abstract—In this paper, we study the achievable rate region of 1-layer rate splitting (RS) in the presence of hardware impair- ment (HWI) and improper Gaussian signaling (IGS) for a single- cell reconfigurable intelligent surface (RIS) assisted broadcast channel (BC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' We assume that the transceivers may suffer from an imbalance in in-band and quadrature signals, which is known as I/Q imbalance (IQI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' The received signal and noise can be improper when there exists IQI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Therefore, we employ IGS to compensate for IQI as well as to manage interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Our results show that RS and RIS can significantly enlarge the rate region, where the role of RS is to manage interference while RIS mainly improves the coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Index Terms—Achievable rate region, hardware impairment, improper Gaussian signaling, MIMO broadcast channels, rate splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' INTRODUCTION The sixth generation (6G) of wireless communication sys- tems should be much more spectral and energy efficient than the existing communication systems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' This goal may not be achieved without employing some emerging technologies such as reconfigurable intelligent surfaces (RISs) and rate splitting (RS), which have been shown to be able to improve the spec- tral and energy efficiency of various wireless communication systems [2]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Interference has been always among the main restrictions of modern wireless communication systems, and interference- management techniques are thus expected to continue playing a key role in such systems [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Even though there are many studies on the performance of RIS, its role should be further investigated in overloaded interference-limited systems, in which the number of users is larger than the number of spatial, temporal, or frequency resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' RIS can modulate channels, canceling interference links or improving the strength of de- sired links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' In other words, RIS can be potentially employed to manage and/or neutralize interference in some scenarios such as multiple-user interference channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Thus, the following question may arise: Are other advanced interference manage- ment techniques, such as rate-splitting or improper Gaussian signaling (IGS), still necessary in RIS-assisted systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' We answer this question in positive in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Rate splitting (RS) is a powerful technology to highly improve the spectral and energy efficiency of various interference-limited systems [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' There are different RS schemes such as 1-layer RS, hybrid RS and generalized RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' The generalized RS scheme is the most complete RS scheme and includes many other technologies/techniques such as spatial division multiple access (SDMA), non-orthogonal multiple access (NOMA), orthogonal multiple access (OMA) and treating interference as noise (TIN) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Implementing generalized RS has high complexities when the number of users grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' An alternative to the generalized RS is 1-layer RS, which is a very practical scheme with much lower com- plexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 1-layer RS is very efficient and is able to improve the performance of different interference-limited systems [8]– [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Another powerful interference-management technique is improper Gaussian signaling (IGS), which can improve the system performance when the receivers apply TIN or partial successive cancellation (SIC) [11]–[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Moreover, IGS has been shown to increase the degrees-of-freedom of the 3-user single-input, single-output (SISO) interference channel (IC) through interference alignment [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Interference is not the only performance limiting factor in wireless communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Another limitation arises from non-idealities in transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' A source of imperfections in transceivers is I/Q imbalance (IQI), which is modeled as a widely linear transformation of the input signal [12], [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Hardware impairments (HWI) can highly affect the system performance especially when such imperfections are not taken into account in the system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' In this paper, we investigate the role of RIS, RS and IGS in multiple-input, multiple-output (MIMO) broadcast channels (BCs) with IQI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' We show that although RIS can enlarge the rate region, RS and IGS are still needed to manage interference and compensate for IQI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Indeed, the role of RIS in this scenario is mainly to improve the coverage, while RS is responsible for handling interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' It is known that 1-layer RS with proper Gaussian signaling (PGS) is the optimal scheme in the two- user single-cell BC with perfect devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' However, it is not the case in the presence of IQI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Our results show that IQI shrinks the achievable rate region, and IGS with TIN may outperform RS with PGS in some operational points/regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' In this case, the 1-layer RS with IGS outperforms the other arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='00594v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='IT] 2 Jan 2023 BS RIS 2 U 1 U 1 UK− UK 3 U Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 1: A broadcast channel with RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' considered schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Section II presents the system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Section III proposes a suboptimal scheme to obtain the achievable rate region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Section IV presents some numerical results, and Section V summarizes the main findings of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' SYSTEM MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Network Scenario We consider a single-cell RIS-assisted system with IQI at transceivers, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' We assume that there is a BS with NBS transmit antennas, serving K users with Nu receive antennas each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Additionally, there is a RIS with NRIS components to assist the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' The users and the BS may suffer from IQI according to the model in [12], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' For the sake of notational simplicity, we consider a symmetric scenario in which all the users have the same number of antennas as well as the same IQI parameters, although the model can be easily extended to asymmetric scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Channel model We employ the channel model in [20] for MIMO RIS- assisted systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' In this paper, we briefly present the channel model and refer the readers to [20], [21] for more detailed discussions on the fading models of MIMO RIS-assisted systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' The channel matrix between the BS and user k is [22, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (14)] Hk (Θ) = GkΘG � �� � Link through RIS + Fk ���� Direct link ∈ CNu×NBS, (1) where Fk is the channel matrix between the BS and user k, Gk is the channel matrix between the RIS and user k, G is the channel matrix between the BS and the RIS, and the matrix Θ is Θ = diag (θ1, θ2, · · · , θNRIS) , (2) where θis for all i are RIS components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' In this paper, am- plitudes of the RIS components are assumed to be fixed to 1, while the phases can take any value between 0 and 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' In other words, the constraint set for the RIS components is T = {θi : |θi| = 1 ∀i} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (3) We refer the reader to [21] for a description of other common constraints on the amplitudes and phases of the RIS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 1-layer rate splitting scheme We consider 1-layer RS scheme in which the BS broadcasts a common message for all users and K private messages (one for each user).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Thus, the BS is intended to transmit x = xc + K � k=1 xk, (4) where xc is the common message, and xk is the private message of the BS intended for user k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Since the BS may suffer from IQI, the actual transmit signal of the BS is a widely linear transformation of x as [12], [18] xt = V1x + V2x∗, (5) where the constant matrices V1 and V2 are, respectively, defined in [12, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (7)] and [12, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (8)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' The receive signal at the receiver of user k is yk = Hkxt + nk, (6) where nk is a zero-mean proper white additive Gaussian noise with variance σ2 at receiver k, and Hk is the effective channel between the BS and user k, given by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Note the effective channel is a function of Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' however, we drop the dependency to simplify the representation of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' The receiver of user k may suffer from IQI, which means the final output of the received signal is a widely linear transformation of yk as yk = Γ1 [Hk (V1xk + V2x∗ k) + nk] + Γ2 [Hk (V1xk + V2x∗ k) + nk]∗ , (7) where the constant matrices Γ1 and Γ2 are given by [12, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (12)] and [12, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (13)], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Note that the effective noise at user k can be improper due to IQI, meaning that the real and imaginary parts of the effective noise can be correlated and/or have unequal powers [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' To compensate for IQI, we assume that the common and private messages, xc and xk, can be improper Gaussian signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Additionally, the signals xc and xk are zero-mean and independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Note that IGS can also help to reduce or manage interference, as indicated before, so its role is not only to compensate for IQI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' A way to model IGS is through the real-decomposition method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' We can rewrite (7) by employing the real- decomposition method as yk = Hkxi + nlk = Hkxc � �� � Common M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' + Hkxk � �� � Private M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' + Hk K � j=1,j̸=k xj � �� � Interference + nk ���� Noise , (8) where Hk is the equivalent channel given by [22, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (11)], y = � R{y}T I{y}T �T , and x = � R{x}T I{x}T �T are, respectively, the real decomposition of y and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Addition- ally, nk represents the effective improper noise and is given by nk = � R{Γ1n + Γ2n∗}T I{Γ1n + Γ2n∗}T �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (9) We represent the covariance matrix of the noise by E{n nT } = Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Moreover, the transmit covariance matrix of x, xc, and xk are denoted as P, Pc, and Pk, respectively, where P = Pc + � k Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' The constraint set of the transmit covariance matrices is given by [21, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (3)] for IGS (denoted by PI) and by [21, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (4)] for PGS schemes (denoted by PP ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Since RS can be applied to both PGS and IGS cases, we represent the constraint set of the transmit covariance matrices as P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Note that the set P is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Moreover, note that PGS is a special case of IGS, thus, an optimal IGS scheme never performs worse than a PGS scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' We refer the reader to [22, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='A], [21, Appendix A] and [23] for further details on modeling IQI and/or impropriety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Rate expressions Users firstly decode the common message and cancel it from the received signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Thus, the maximum decoding rate for the common message at user k is [24, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (2)-(3)] ¯rck = 1 2 log2 ������ I + � Cn + � ∀i HkPiHT k �−1 HkPcHT k ������ = 1 2 log2 ���Cn + HkPHT k ��� � �� � ¯rck,1 − 1 2 log2 �����Cn + � ∀i HkPiHT k ����� � �� � ¯rck,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' The common message must be transmitted at a rate that is decodable for all users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Hence, the maximum rate for transmitting the common message is rc({P}, Θ) = min k {¯rck({P}, Θ)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (10) After decoding and canceling the common message, each user decodes its own private message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Therefore, the maximum decoding rate for the private message at user k is rpk = 1 2 log2 ������� I + � �Cn + � ∀i̸=k HkPiHT k � � −1 HkPkHT k ������� (11) = 1 2log2 �����Cn+ � ∀i HkPiHT k ����� � �� � rpk,1 −1 2log2 ������ Cn+ � ∀i̸=k HkPiHT k ������ � �� � rpk,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (12) Finally, the rate of user k is the summation of the decoding rate of its private message and its dedicated rate from the common message, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=', [21] rk({P}, Θ) = rpk({P}, Θ) + rck, (13) where rck ≥ 0 and � k rck ≤ rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Note that the rates rk, rpk, ¯rck and rc are functions of {P} and Θ, while rc = {rc1, rc2, · · · , rcK} is a design parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Due to notational simplicity, we, hereafter, drop the dependency of the rates to {P} and Θ in representing rk, rpk, ¯rck and rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Problem Statement Employing the rate profile technique, the achievable rate region can be obtained by solving [25] max Θ∈T ,{P}∈P,rc r (14a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' rk = rpk + rck ≥ αkr ∀k, (14b) � ∀k rck ≤ rc, rck ≥ 0, ∀k, (14c) and varying the weights such that � ∀k αk = 1 with αk ≥ 0 for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' PROPOSED ALGORITHM We employ an approach based on the optimization frame- work proposed in [21] to solve (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' That is, we first employ an alternating optimization (AO) approach to sequentially optimize over the transmit covariance matrices and RIS com- ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Indeed, we first fix the RIS components to Θ(t−1) and update the transmit covariance matrices as {P(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Then, we alternate and optimize over the reflecting coefficients for fixed transmit covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Unfortunately, even after fixing either the RIS or the covariance matrices, the resulting optimization problems are still non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' In the following subsections, we proposed iterative algorithms to find a suboptimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Optimizing transmit covariance matrices In this subsection, we update the transmit covariance ma- trices for a fixed Θ(t−1) by solving max {P}∈P,rc r s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (14b), (14c), (15) which is a non-convex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Note that the constraints in (15) are linear in rc, however, the rates are not concave in {P}, which makes the problem non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Indeed, the rate rpk (or rck) can be written as a difference of two concave functions rpk,1 and rpk,2 (or rck,1 and rck,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Thus, to solve (15), we can employ difference of convex programming (DCP), which falls into majorization minimization (MM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' That is, we approximate the rates by a suitable concave lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' To this end, we keep rpk,1 (or rck,1) unchanged and employ the first-order Taylor expansion to approximate rpk,2 (or rck,2) by an affine (linear) function as rpk ≥ ˜rpk = rpk,1 − rpk,2 � {P(t−1)} � − � ∀j̸=k Tr � HT k (Cn+� ∀i̸=kHkP(t−1) i HT k )−1Hk 2 ln 2 � Pj−P(t−1) j �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (16) Similarly, a concave lower bound for ¯rck is rck ≥ ˜rck = ¯rck,1 − ¯rck,2 � {P(t−1)} � − � ∀j Tr � HT k (Cn+� ∀iHkP(t−1) i HT k )−1Hk 2 ln 2 � Pj−P(t−1) j �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (17) We refer the reader to [21, Corollary 1] for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Substi- tuting the concave lower bounds in (15) results in the following convex optimization problem max {P}∈P,rc r (18a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' ˜rk = ˜rpk + rkc ≥ αkr ∀k, (18b) � ∀k rck ≤ ˜rc = min k {˜rck} , rck ≥ 0, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (18c) This problem can be solved by existing numerical tools, which yields {P(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Optimizing RIS components Now we update the RIS components Θ for fixed transmit covariance matrices {P(t)} by solving max Θ∈T ,rc r s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (14b), (14c), (19) This problem is non-convex since the constraint set for the RIS components is not a convex set, and additionally, the rates are not concave in Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Thus, to solve (19), we first find suitable concave lower bounds for the rates and then, convexify the constraint set T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' To this end,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' we employ the concave lower bound given by [21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Lemma 4],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' which results in the following concave lower bound for rpk rpk ≥ ˆrpk = rpk � Θ(t−1)� − 1 2 ln 2Tr � ¯Vk ¯VT k ¯Y−1 k � − 1 2 ln 2Tr � [ ¯Y−1 k − ( ¯Vk ¯VT k + ¯Yk)−1]T [VkVT k +Yk] � + 1 ln 2Tr � ¯VT k ¯Y−1 k Vk � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (20) where Vk = Hk (Θ) P(t)1/2 k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' ¯Vk = Hk � Θ(t−1)� P(t)1/2 k and Yk = Cn + � ∀i̸=k Hk (Θ) P(t) i HT k (Θ) (21) ¯Yk = Cn + � ∀i̸=k Hk � Θ(t−1)� P(t) i HT k � Θ(t−1)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (22) Similarly, a concave lower bound for ¯rck can be found as ¯rck ≥ ˆrck = ¯rck � Θ(t−1)� − 1 2 ln 2Tr � ¯Vck ¯VT ck ¯Y−1 ck � − 1 2 ln 2Tr � [ ¯Y−1 ck − ( ¯Vck ¯VT ck+ ¯Yck)−1]T [VckVT ck+Yck] � + 1 ln 2Tr � ¯VT ck ¯Y−1 ck Vck � , (23) where Vck = Hk (Θ) P(t)1/2 c , ¯Vck = Hk � Θ(t−1)� P(t)1/2 c and Yck = Cn + � ∀i Hk (Θ) P(t) i HT k (Θ) (24) ¯Yck = Cn + � ∀i Hk � Θ(t−1)� P(t) i HT k � Θ(t−1)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (25) Now we propose a suboptimal approach to convexify the constraint |θn| = 1 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' This constraint can be rewritten as |θn|2 ≤ 1, and |θn|2 ≥ 1, where the former is convex, while the latter is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' To convexify |θn|2 ≥ 1, we employ the first-order Taylor expansion since |θn|2 is a convex function, which results in |θn|2 ≥ |θ(t−1) n |2 − 2R{θ(t−1)∗ n (θn − θ(t−1) n )} ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (26) To converge faster, we relax the constraint in (26) as |θn|2 ≥|θ(t−1) n |2− 2R{θ(t−1)∗ n (θn − θ(t−1) n )}≥ 1 − ϵ, (27) where ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' The constraint (27) is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Hence, the following surrogate optimization problem is convex max Θ,rc r (28a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' ˆrk = ˆrpk + rkc ≥ αkr ∀k, (28b) � ∀k rck ≤ ˆrc = min k {ˆrck} , rck ≥ 0, ∀k, (28c) |θn|2 ≤ 1, and (27), ∀n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (28d) The solution of (28), ˆΘ, may not necessarily be in T since we relaxed the constraint in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' To get a feasible point, we normalize ˆΘ as θnew n = ˆθn/|ˆθn|, for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Finally, we update Θ based on the following rule Θ(t) = � � � � � � � � � � � ˆΘnew if mink � rk({P(t)}, ˆΘnew) αk � ≥ mink � rk({P(t)}, ˆΘ(t−1)) αk � {Θ(t−1)} Otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' (29) This updating rule guarantees the convergence since the al- gorithm generates a sequence of non-decreasing minimum weighted rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, we provide some numerical examples to clarify the role of RS, RIS and IGS in single-cell BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' We consider a line-of-sight (LoS) connection for the links reaching to or departing from the RIS, and a non-LoS (NLoS) link for the direct links between the users and the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' It means that the small-scale fading of the links related to RIS is Rician, while that of direct links is Rayleigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' The large-scale path loss component of RIS links is αRIS = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' The other simulation parameters are chosen based on [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' The considered schemes in the simulations are as follows: PT (or IT) denotes the PGS (or IGS) scheme with TIN but without RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' PR (or IR) denotes the PGS-based (or IGS-based) RS scheme without RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' PRIR (or IRIR) denotes the PGS-based (or IGS-based) RS scheme with RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Finally, TS denotes the time-division- multiplexing-access (TDMA) with time sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' SISO systems Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 2 shows the achievable rate region of a two-user SISO BC with P = 10 dB and the channel realization C1 : f1 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='3992 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='0292i, f2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='2353 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='1238i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 0 1 2 3 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='8 r1 (b/s/Hz) r2 (b/s/Hz) PR IT PT TS (a) With perfect devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 0 1 2 3 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='7 r1 (b/s/Hz) r2 (b/s/Hz) IR PR IT PT (b) With IQI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 2: Achievable rate region of a two-user SISO BC with P = 10 dB and the channel realization C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='5 r1 (b/s/Hz) r2 (b/s/Hz) PR IT PT TS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 3: Achievable rate region of a two-user SISO BC with P = 10 dB and the channel realization C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' As can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 2a, PGS with TIN is very subop- timal, and all other schemes can highly outperform PGS with TIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' IGS with TIN can enlarge the rate region over the PGS scheme with TIN as well as the TS scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Moreover, RS with PGS is the optimal scheme, and RS with IGS performs the same as RS with PGS when the devices are perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' However, it is not the case when there exists IQI, as can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' In the presence of IQI, the noise is improper, and to compensate for it, we should employ improper signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 2b, IGS with TIN can outperform RS with PGS in some operational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Additionally, RS with IGS highly outperforms RS with PGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Furthermore, it can be observed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 2a and 2b that the achievable rate region shrinks when the devices are imperfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 3 shows the achievable rate region of a two-user SISO BC with P = 10 dB and the channel realization C2 : f1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='3672 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='8681i, f2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='2798 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='9214i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' For this channel realization, the absolute values of the channels are almost equal, and non-orthogonal multiple access schemes cannot provide a significant gain over TDMA and TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' As can be observed, RS with PGS is the optimal strategy and attains all the points on the achievable rate region without employing TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' However, PGS and IGS schemes with TIN are highly suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Indeed, this example shows that the 1-layer RS scheme includes OMA schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 0 1 2 3 4 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='2 r1 (b/s/Hz) r2 (b/s/Hz) PR IT PT IRIR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 4: Achievable rate region of a two-user SISO BC with P = 10 dB and the channel realization C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 0 2 4 6 8 r1 (b/s/Hz) r2 (b/s/Hz) IRIR PR IT PT TS (a) C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 0 2 4 6 8 9 0 2 4 6 8 9 r1 (b/s/Hz) r2 (b/s/Hz) IRIR PR IT PT TS (b) C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 5: Achievable rate region of a two-user 2 × 2 MIMO BC with P = 10 dB and different channel realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 4 shows the achievable rate region of a two-user SISO BC with P = 10 dB and the channel realization C3 : f1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='5909 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='0615i, f2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='2540 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='0052i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' As can be observed, RIS can highly enlarge the achievable rate region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' However, we should employ RIS with RS to get the best performance out of RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Moreover, we can observe that RIS provides more benefits for the rate of the user with a weaker channel gain, which shown the ability of RIS to significantly improve the coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' MIMO systems Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 5 shows the achievable rate region of a two-user 2 × 2 MIMO BC with P = 10 dB, αRIS = 3 and the following channel realizations: C4 : F1 = � −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='6952 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='7244i −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='5196 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='1194i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='0665 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='3475i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='1105 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='3237i � , F2 = � −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='0233 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='6539i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='2841 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='8593i −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='2500 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='2059i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='8494 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='5047i � , C5 : F1 = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='2949 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='7399i −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='1314 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='5059i −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='5491 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='3702i −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='1943 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='9528i � , F2 = � −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='7849 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='4803i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='0522 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='0681i −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='5022 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='1034i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='4433 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='0066i � , A two-user 2 × 2 MIMO BC can be considered as an underloaded system since the sum of the number of transmit and receive antennas is higher than the number of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' As can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' 5, IGS with TIN can enlarge the rate region over the PGS with TIN scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Furthermore, RS with TIN outperforms the other schemes as it is the optimal scheme in the considered system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' We can also observe that RIS can enlarge the rate region by improving the coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Since this system is underloaded, the benefits of IGS and RS are less than in the two-user SISO BC, which is a highly overloaded system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' However, there are still some benefits in the employment of RS and/or IGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' SUMMARY AND CONCLUSION In this paper, we have characterized the achievable rate region of RIS-assisted BCs with RS and IQI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Our main findings can be summarized as follows: The role of RS is to manage interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' The 1-layer PGS-based RS scheme is optimal in a two-user BC with perfect devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' This scheme includes OMA, TIN and NOMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' However, when the transceivers suffer from IQI, PGS is unable to compensate for it, and we should employ IGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Interestingly, IGS with TIN may outperform the 1-layer RS with PGS in some regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Thus, in the presence of IQI, the 1-layer IGS-based RS scheme is optimal in a two-user BC with and/or without RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' The role of IGS is twofold: to manage interference and to compensate for IQI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' The role of RIS in this system is mainly to improve the coverage, as it cannot completely manage interference in a BC, which is in line with our previous studies [21], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Indeed, we have to employ advanced interference- management techniques such as RS in highly overloaded systems to use RIS more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' RS and IGS as interference-management techniques can provide considerable benefits in overloaded systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' However, these benefits decrease (or may even vanish) in underloaded systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' ACKNOWLEDGMENT The work of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' Santamaria has been partly supported by the project ADELE PID2019-104958RB-C43, funded by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content='13039/501100011033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQftfkF/content/2301.00594v1.pdf'} +page_content=' The work of Eduard Jor- swieck was supported in part by the Federal Ministry of 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EFFICIENT INFER- +ENCE +Sayan Ghosh +Meta Platforms Inc. +sayanghosh@meta.com +Karthik Prasad +Meta Platforms Inc. +krp@meta.com +Xiaoliang Dai +Meta Platforms Inc. +xiaoliangdai@meta.com +Peizhao Zhang +Meta Platforms Inc. +stzpz@meta.com +Bichen Wu +Meta Platforms Inc. +wbc@meta.com +Graham Cormode +Meta Platforms Inc. +gcormode@meta.com +Peter Vajda +Meta Platforms Inc. +vajdap@meta.com +ABSTRACT +Neural network pruning is frequently used to compress over-parameterized net- +works by large amounts, while incurring only marginal drops in generalization +performance. However, the impact of pruning on networks that have been highly +optimized for efficient inference has not received the same level of attention. In +this paper, we analyze the effect of pruning for computer vision, and study state-of- +the-art ConvNets, such as the FBNetV3 family of models. We show that model +pruning approaches can be used to further optimize networks trained through NAS +(Neural Architecture Search). The resulting family of pruned models can consis- +tently obtain better performance than existing FBNetV3 models at the same level +of computation, and thus provide state-of-the-art results when trading off between +computational complexity and generalization performance on the ImageNet bench- +mark. In addition to better generalization performance, we also demonstrate that +when limited computation resources are available, pruning FBNetV3 models incur +only a fraction of GPU-hours involved in running a full-scale NAS. +1 +INTRODUCTION +Neural networks frequently suffer from the problem of over-parameterization, such that the model +can be compressed by a large factor to drastically reduce memory footprint, computation as well +as energy consumption while maintaining similar performance. This is especially pronounced for +models for computer vision (Simonyan & Zisserman, 2014), speech recognition (Pratap et al., 2020) +and large text understanding models such as BERT (Devlin et al., 2018). The improvements obtained +from intelligently reducing the number of model parameters has several benefits, such as reduction in +datacenter power consumption, faster inference and reduced memory footprint on edge devices such +as mobile phones which also enable decentralized techniques ex. federated learning (Kairouz et al., +2019). +There are several techniques to reduce model size while maintaining similar generalization perfor- +mance, such as model quantization (Polino et al., 2018), NAS (Neural Architecture Search) (Elsken +et al., 2019) and model distillation through teacher-student networks (Gou et al., 2021). For the scope +of this paper, we consider pruning as a technique to remove trainable weights in the network, and save +on computation costs for the FBNet family of models. The motivations for this are two-fold. Firstly, +state-of-the-art models such as FBNet (Wu et al., 2019) already adopt the best practices in the area +of efficient hardware-aware design of convolutional neural network based models, and are widely +used across different vision tasks. This makes them suitable baselines to understand whether pruning +can offer any performance gain over their already optimized behavior. While there has been limited +work on pruning for efficient convolution network models they investigate older architectures such as +1 +arXiv:2301.04502v1 [cs.CV] 11 Jan 2023 + +EfficientNet and MobileNet (Aflalo et al., 2020) or integrate pruning into expensive techniques such +as joint prune-and-architecture search (Wang et al., 2020). +For each of the constituent models of the FBNetV3 family (FBNetV3A, FBNetV3B,..., FBNetV3G) +we reduce the number of parameters using two pruning based approaches: (1) Global magnitude- +based pruning: Starting with the pre-trained model, we prune all weights whose magnitude is below +a threshold chosen in order to achieve a target number of FLOPs for the pruned model; (2) Uniform +magnitude-based pruning: Starting with the pre-trained model, we prune weights in each layer whose +magnitude is below a level-specific threshold in order to yield a pruned model achieving a target +number of FLOPs with the same sparsity in each layer. After either pruning method is applied, we +fine-tune the pruned model for a certain number of epochs until convergence is reached. Within the +scope of our study in this paper, we are mostly interested in the following research questions: +• RQ1: Pruning to improve computation vs. performance tradeoff. Can a model obtained by pruning +a larger FBNetV3 model M1 (optimized using NAS) achieve higher generalization performance +than a smaller FBNetV3 model M2 when the pruned model has the same number of FLOPs as M2? +• RQ2: Pruning as an efficient paradigm. When a larger FBNetV3 model M1 is available and +computational resources are limited, is pruning a faster and less computationally expensive approach +to obtain a model with higher accuracy at a desired computation level (FLOPs) than running a +full-fledged architecture search? +Pruning to improve computation vs. performance tradeoff (RQ1). There have been recent research +advances in the area of building hardware-aware efficient models (Deng et al., 2020). These can +provide good generalization performance while adhering to constraints on memory, inference latency +and battery power, which are often dictated by the hardware environment where inference happens. +Experiments described in existing work on efficient vision models such as ChamNet (Dai et al., 2019), +MobileNet (Howard et al., 2017), EfficientNet (Tan & Le, 2019) and FBNetV2 (Wan et al., 2020) +have shown that it is possible to achieve even higher performances on standard image recognition +tasks such as ImageNet (Deng et al., 2009) at a certain level of FLOPs. However the efficient design +of these models does not solve the over-parameterization problem completely, and none of these +approaches study how model pruning can be performed to obtain even better trade-offs between +computation and model accuracy. This paper is the first of its kind to understand how we can improve +on the state-of-the-art in this problem space. +Pruning as an efficient paradigm (RQ2). In addition to achieving state-of-the-art performance with +reduced FLOPs, we are also interested in understanding how such pruned models can be obtained +inexpensively with limited resources that are generally available to a machine learning practitioner +who has access to existing optimized models but limited computing resources. For example, the +FBNetV3 models are freely available through Facebook’s Mobile Model Zoo1, while EfficientNet +models can be obtained at GitHub2. While the techniques needed to obtain computation- and latency- +friendly models have been democratized through open-sourcing the source code as well as the models +themselves, fully applying these techniques necessitates costly operations such as finding an optimal +network topology through meta-learning approaches (You et al., 2020) and search algorithms such as +Genetic Algorithms (GAs) (Goldberg & Deb, 1991). +Given the high-degree of intractability of this problem, expensive computational resources are often +needed in this case, easily exceeding the budget available to a university research laboratory or an +angel-stage startup (Zoph & Le, 2016). When a starting model is already available, for example +through open-sourcing, the best option would be to perform a cheap modification of the model to fit +a certain target FLOPs/latency requirement. In this paper we have compared the NAS approaches +for training FBNetV3 models with our pruning techniques on a computational complexity metric +(GPU-hours) to effectively answer RQ2. +Benchmark results. In addition to experimental outcomes for answering RQ1 and RQ2, we also +benchmark pruned FBNetV3 models using available open-sourced quantized sparse kernels and +conduct ablation studies to obtain additional insights into pruning performance. These results augment +our main observations and demonstrate that with existing hardware support, it is possible to deploy +1FBNetV3 +models +available +here +http://https://github.com/facebookresearch/ +mobile_cv/model_zoo/models/model_info/fbnet_v2/model_info_fbnet_v3.json +2EfficientNet models available here https://github.com/mingxingtan/efficientnet +2 + +pruned cutting-edge computer vision models with practical latency reductions and improve further +beyond the performance vs. FLOPs trade-off. +We conduct our experiments on ImageNet, which is an object-recognition task on a large training +dataset of 1.2 million images. We show that computationally less intensive techniques such as uniform +and global magnitude-based pruning of larger FBNetV3 models can yield higher test accuracies than +small models while having the same number of FLOPs. Given a target computation budget for an +efficient model, we show that it is more practically advantageous (both in terms of performance and +running time) to simply prune the larger model than run a neural architecture search to find the target +model from scratch. +The technique we have employed for pruning (unstructured sparsity) is already tried and tested, +however our novelty lies in studying whether efficient image recognition models such as FBNetV3 +can be optimized further to improve on the FLOPs-accuracy curve, and the contributions are two +fold : (1) FBNets are themselves state-of-the-art in efficient vision models and we achieve better +accuracy-FLOPs tradeoff over these models and (2) from the standpoint of computational overhead, +we significantly reduce the amount of GPU hours required to obtain such models. Pruning a publicly +available NAS optimized model incurs ≈4x less GPU hours to achieve a target FLOPs level, compared +to training a full-fledged NAS to obtain a model which has less accuracy at the same FLOPs level. +Paper organization. The remainder of this paper is organized as follows. In Section 2, we describe +related work in the area of efficient vision model design and also provide an introduction to different +pruning techniques. In Section 3, we discuss our experimental setup, including a description of +the baseline models and the global and uniform pruning approaches we have employed. Section 4 +describes our main findings and we conclude the paper in Section 5. +2 +RELATED WORK +We discuss related literature in the areas of computationally efficient vision models and model +pruning. Within the scope of our work, we mainly focus on inference efficiency of models in contrast +to training efficiency. +Computationally efficient vision models: Neural networks for computer vision are generally char- +acterized by convolutional layers and fully-connected layers, along with blocks such as residual or +skip connections. This makes such networks resource intensive in terms of FLOPs, which affects +the memory storage and power consumed, and also leads to increased latency. It is of paramount +importance to design more efficient networks which can provide higher performance for the same +FLOPs or latency level, or even to optimize them appropriately to provide the same performance at +reduced FLOPs/latency. This can be performed either through the design of new simplified layers, for +example in deep residual learning (He et al., 2016) or though explicit model compression as in weight +quantization (Polino et al., 2018). Extremely deep networks for image recognition often suffer from +not only high complexity and inference latency, but also from the issue of vanishing gradients (Pas- +canu et al., 2013). This was addressed through deep residual networks which effectively simplified +network design through skip-connections. MobileNets (Howard et al., 2017) are one of the earlier +approaches to building small low-latency networks by using depthwise separable convolutions with +two parameters, width and resolution multipliers. They demonstrate the effectiveness of MobileNets +across different vision tasks, such as face embeddings and object detection. MobileNetV2 (Sandler +et al., 2018) extends MobileNets by utilizing inverted residual filter structures and linear bottlenecks, +obtaining improvements on state-of-the-art models both in terms of accuracy and computational com- +plexity. ShuffleNets (Zhang et al., 2018) propose dedicated residual units where 1×1 convolutions +are replaced with pointwise group convolutions and channel shuffling reducing FLOPs computations. +More recently, the focus on building efficient neural network models has shifted to techniques that +treat the design of efficient networks as a search problem, falling under the umbrella of Neural +Architecture Search (NAS). EfficientNets (Tan & Le, 2019) propose a novel scaling method which +adjusts the network’s length, width, and resolution to optimize performance subject to target memory +and FLOPs constraints. They also define a novel baseline that is optimized by a multi-objective neural +architecture search. The FBNet collections of models—FBNet (Wu et al., 2019), FBNetV2 (Wan et al., +2020) and FBNetV3 (Dai et al., 2021)—employ neural architecture search to obtain highly-optimized +models that improve on the state-of-the-art for different visual understanding tasks. FBNet frames the +3 + +architecture search as a differentiable meta-learning problem with gradient based techniques, namely +DNAS—Differentiable Neural Architecture Search—by Wu et al. (2019), and avoids selecting the +optimized model over a discrete set. The subsequent entry in this collection, FBNetV2, expands the +search space over conventional DNAS, and employs a masking scheme to maintain the same level of +computational complexity while searching over this expanded space. FBNetV3 further improves on +the state-of-the-art by employing Neural Architecture Recipe Search (NARS) and searching over the +space of not only architectures, but also corresponding recipes (which are generally hyper-parameters). +In this paper, we consider FBNetV3 models as our baselines as they are state-of-the-art. We are +interested in understanding if they are overparameterized and evaluate how much model pruning can +improve performance at a certain FLOPs level over the state-of-the-art in this family of models. +Model Pruning: Modern neural networks, particularly those processing complex sensory inputs (such +as speech, vision and language) for perception applications, are often over-parameterized. It is only +to be expected that we should be able to compress such networks significantly to maintain the same +level of performance at decreased level of computation (fewer weights and reduced FLOPs), memory +footprint and power consumption. Foundational efforts in this space include the Optimal Brain +Surgeon (Hassibi & Stork, 1993) and Optimal Brain Damage (LeCun et al., 1990). Recently the +idea of network pruning has been formalized through the lottery ticket hypothesis (Frankle & Carbin, +2018), which claims that randomly initialized, feed-forward networks have winning sub-networks that +perform just as well as the original network on an unseen test dataset. Model pruning is generally of +two types: unstructured and structured pruning. Unstructured pruning, as the name suggests, doesn’t +adhere to any structure and prunes neurons based on chosen criteria (such as magnitude). This has +the advantage of providing higher performance, but is difficult to implement in hardware, as it needs +dedicated support for efficient sparse matrix multiplications. Meanwhile, structured pruning is the +practice of removing entire groups of neurons (e.g., blocks within the weight matrix, or channels in +convolutional neural networks). This is easy to implement without dedicated hardware support, but +has the issue of lower generalization performance than unstructured pruning (Yao et al., 2019). In the +literature, there have also been several studies, for example investigating whether rewinding (training +from scratch with a fixed mask) can perform just as well as the fine-tuning on top of the original +unpruned network (Renda et al., 2020). Blalock et al. (2020) provide an overview survey of recent +advances and open problems in neural network pruning. +In the research area of designing efficient networks for computer vision, there has not been much +focus on understanding how pruning can be applied to the current generation of models. Most +literature on pruning is based on older networks such as VGGNet, ResNet (He et al., 2016), and +MobileNet (Sandler et al., 2018). Our work improves upon these existing studies by understanding +how pruning can improve the FLOPs-accuracy tradeoff over existing state-of-the-art networks. +3 +PRUNING TECHNIQUES AND SETUP +In this section, we describe the main components of our techniques and experimental setup, including +Baseline Models, Pruning Techniques, Latency Measurement and Metrics. We have mainly used +standard splits of the ImageNet dataset, further details are in Section A.1 of the appendix. +3.1 +BASELINE MODELS +Dai et al. (2020) address the previous limitations of NAS-based architecture search where these +approaches can only search over architectures given a training recipe (set of hyperparameters), and +thus cannot optimize over both. As described in Section 2, the most recent state-of-the-art models are +based on NARS (Neural Architecture-Recipe Search), which we select as baseline models. Table 3 +lists the accuracy of FBNetV3 models (Dai et al., 2021) on the ImageNet classification task, along +with the number of model parameters and computation complexity in terms of FLOPs. +Each baseline model consists of multiple IRF (Inverted Residual Filter) blocks, which contain +convolutional layers of different kernel sizes. For our experiments, we are mostly interested in 1×1 +convolutions as potentially prunable, since within each FBNetV3 model, the 1×1 convolution layers +constitute >80% of total model FLOPs for all models in the family, and the open-sourced sparsity +kernel support we use for latency benchmarking is available only for fully connected layers. A 1×1 +4 + +convolution can be transformed into an equivalent fully connected layer with a few tensor reshape +operations without any significant loss of performance or latency. +For each initial and target FBNetV3 model X and Y , where X is larger than Y , we prune X to a +sparsity level of S so that the FLOP count is the same as for Y . The number of FLOPs consumed by +a linear layer of sparsity S is proportional to the number of sparse matrix multiplications performed +and is given by S ∗F, where F is the corresponding dense FLOPs. Thus if F1×1(X) is the number of +FLOPs consumed by the 1×1 convolution layers and F(x) is the total number of FLOPs consumed +by model X, we have: +S = (F(X) − F(Y ))/F1×1(X) +(1) +Hence, sparsity measures the fraction of 1×1 convolution weights removed, and so higher sparsity +indicates a smaller model. For the uniform pruning scnario, Table 1 shows the amount of sparsity +required to prune each larger FBNetV3 model to a smaller one based on Eq. (1). For global pruning, +(1) does not hold, and we compute the target sparsities empirically from the layer shapes instead +with details provided in Section A.2. We prune each larger FBNetV3 model to a discrete FLOPs +target based on a defined set of smaller models in the family, and not to a continuous range of FLOPs +values, as it makes it easier to compare models directly based on a given computation budget. If we +can demonstrate that for the same computation level, the pruned larger FBNetV3 model has higher +performance than a smaller model with the same FLOPs, it is sufficient to demonstrate that we can +improve on the FLOPs-accuracy curve over the state-of-the-art. +3.2 +PRUNING TECHNIQUES +In this paper, we utilize a pre-trained FBNetV3 model with higher number of FLOPs without training +an image classification model from scratch with sparsity, which would be time consuming and +computationally intensive. There are several approaches in the literature such as prune-and-fine- +tune (Han et al., 2015) and iterative pruning with sparsity scheduling (Frankle & Carbin, 2018). +We have utilized the former for our experiments, as although studies have shown that iterative and +incremental pruning approaches lead to better generalization performance, they typically require +training for high number of epochs, need tuning and selection of optimal sparsity schedules and are +computationally resource intensive. We have therefore not considered them in our experiments. For +our prune and fine-tune experiments, we have used 8-GPU boxes, with each box having Nvidia V100 +(Volta) 32G GPUs. As described in Section 1, we perform both global and magnitude-based pruning +experiments. For the latency benchmarking, we also perform magnitude-based uniform pruning with +a sparse block size of 1 × 4 as explained in Section 3.3. +We have conducted a hyper-parameter tuning for the learning rate parameter, with LR values in the set +{4e-5, 8e-5, 1.6e-4}, as fine-tuning generally admits smaller learning rates than training from scratch. +We have found that using the same learning rate for all models, along with the same hyper-parameter +settings used for training the seed model is sufficient to obtain pruned networks which are superior +to the baseline FBNetV3 models. Hence minimal hyper-parameter tuning was required for our +experiments and we have used values of settings such as weight decay and momentum to be the +same as those used for training the baseline FBNetV3 models. During fine-tuning after pruning, we +have used a smoothed validation loss to stop the process early after a convergence tolerance (0.01%) +is reached between two consecutive epochs. Generally, we have observed fine-tuning to converge +around ∼250 epochs. +3.3 +LATENCY MEASUREMENTS AND METRICS +We are interested not only in the sparsity level of our pruned models and the image recognition +performance, but also in metrics which potentially improve due to model sparsity, such as number of +parameters, the FLOP count and the model latency. For reporting model performance under pruning, +we use standard image recognition metrics such as Top-1 and Top-5 test accuracies. We measure +overall model sparsity, which is different to the layer sparsity since we only prune 1×1 convolution +layers, as explained in Section 3.1. We report the model FLOPs, because this metric captures the +computational footprint of the model and its power consumption. +5 + +Last, we record the total latency (in ms.) under pruning. The sparse kernels used in our experiments +are already in open-source and released under the PyTorch sparse quantization library3. Prior to +using these kernels, we perform uniform layer-wise block-based pruning with block sizes of 1 × 4. +Magnitude based pruning is implemented at block level, and the model is quantized to 8-bit integers +(int8) before latency benchmarking, which is performed on Intel CPUs designed using the Skylake +micro-architecture. While we would expect sparsity to translate to tangible inference speedups, this +is highly dependent on the sparse kernel support provided by hardware. Current hardware is not well- +suited for unstructured randomly sparse matrix multiplications and tend to do better with structured +sparsity in models (Anwar et al., 2017). We have utilized block sparsity within the weight matrix for +latency experiments. However this often tends to come at a decreased level of model performance. +The design of highly performant sparse models under structured sparsity with reasonable inference +speedups remains an important research topic outside the scope of this paper. +4 +RESULTS +4.1 +PRUNED FBNETV3 MODEL PERFORMANCE +To answer RQ1, we consider the family of FBNetV3 models as baselines and seed models for further +pruning. For each pair of models X, Y in the family, we calculate the amount of sparsity required +to prune the larger model X to a model that consumes the same number of FLOPs as the target +smaller model Y , via Equation 1. There are 21 potential seed and target model pairs, however we +conduct pruning experiments only for a depth of 2 for tractability. For example, given FBNetV3E as +the seed, we only prune it to FLOPs targets corresponding to FBNetV3D and FBNetV3C. Table 1 +presents the accuracy and number of parameters of the pruned models at each target FLOPs level. +The improvement in performance is apparent even at lower FLOPs targets, where we might expect +baseline models such as FBNetV3A to not be over-parameterized. For example, pruning FBNetV3C +to a target of 356.6 MFLOPs obtains a network which is 1.43% better than FBNetV3A. Figure 1 plots +the Top-1 ImageNet testing accuracy vs. FLOPs for the best pruned models as seen from Table 1. +This clearly shows that pruning FBNetV3 models with minimal fine-tuning can significantly improve +on the state-of-the-art for FLOPs vs. Top-1 accuracy trade-off. This analysis is performed for both +uniform layer-wise and global magnitude-based prune with fine-tune settings. Global pruning ranks +the weights of the entire network in contrast to uniform layer-wise pruning, which ranks each layer’s +weights to determine the sparsity mask. It would be expected that global pruning performs better than +uniform pruning for the same target sparsity level or number of non-sparse parameters. However in +our experiments we determine the pruning threshold based on FLOPs targets, and find global pruning +to require higher sparsity levels, which results in uniform pruning outperforming global pruning in +Top-1 ImageNet accuracy in most cases. +4.2 +PRUNING COMPLEXITY +In addition to demonstrating the improvement over state-of-the-art obtained by pruning FBNetV3 +models, it is also important to quantify the reduction in computational complexity obtained in pruning +a larger FBNetV3 model compared to training an FBNetV3 model directly through NAS (Network +Architecture Search). RQ2 (pruning for efficient model search) asks if the pruning and subsequent +fine-tuning approach in Section 4.1 is faster than a full-fledged neural architecture search. During +pruning and subsequent fine-tuning, we train the pruned networks till the validation loss converges to +within a pre-specified tolerance, as described in Section 3.2. The time needed is generally less than +when training the original FBNetV3 models, which runs for 400 epochs. The number of GPU-hours +is computed as (number of training GPU nodes) * (number of GPUs per node) * (training time to +convergence) for each network. In Table 2, for each of the best performing uniformly-pruned models +in Section 4.1 we report the number of GPU-hours consumed by the prune and fine-tune strategy, +along with the GPU-hours consumed when obtaining a FBNetV3 model through architecture search +using the method described in Dai et al. (2020). The results are quite conclusive—we not only obtain +pruned models superior in performance to the original neural search optimized models, but also as +3https://github.com/pytorch/pytorch/blob/master/torch/ao/nn/sparse/quantized/linear.py +6 + +Table 1: Sparsity level (in percentage) and performance of pruned FBNetV3 networks on ImageNet +dataset for different target MFLOPs. The best accuracy obtained at each target FLOPs level is +highlighted in bold. +Seed +network +FBNetV3_ +Target +network +FBNetV3_ +Target +MFLOPs +Baseline +Accuracy +Uniform pruning +Global pruning +Sparsity +level(%) +Top-1 +Acc. +Gain(%) Sparsity +level(%) +Top-1 +Acc. +Gain(%) +B +A +356.6 +79.6 +26.59 +80.308 0.887 +39.5 +80.232 0.793 +C +A +356.6 +79.6 +40.7 +80.738 1.43 +57.9 +80.476 1.1 +C +B +461.6 +80.2 +19.4 +80.996 0.992 +28.9 +80.998 0.985 +D +B +461.6 +80.2 +31.47 +81.116 1.142 +43.7 +81.08 +1.097 +D +C +557.0 +80.8 +15.04 +81.278 0.591 +21.5 +81.208 1.256 +E +C +557.0 +80.8 +31.0 +81.282 0.596 +43.6 +81.184 0.475 +E +D +644.4 +81.0 +17.8 +81.118 0.145 +25.8 +81.388 0.479 +F +D +644.4 +81.0 +38.2 +82.00 +1.234 +67.8 +81.484 0.597 +F +E +762.0 +81.3 +29.8 +82.19 +1.094 +54.7 +81.97 +0.824 +G +E +762.0 +81.3 +71.67 +81.166 -0.16 +85.5 +79.934 -1.68 +G +F +1181.6 +82.0 +49.69 +82.528 0.643 +63.8 +82.454 0.553 +A +B +C +D +E +F +C->A +D->B +E->C +F->D +F->E +G->F +Figure 1: FLOPs vs. performance (ImageNet Top-1 acc.) for different pruned FBNetV3 networks. +For comparison, the existing FBNetV3 networks are also shown here. +described in Section 1, computational cost is significantly lower when starting from a pre-trained +model with higher FLOPs. Given the performance improvements obtained with lower computational +resources, this approach is beneficial for an experimental setting where researchers have access to +open-sourced pre-trained models and limited GPU resources, for example in a small startup or an +academic environment. We observe that the degree of speedup reduces as the network size gets bigger +(e.g., in FBNetV3A vs. FBNetV3C) due to higher training time to convergence. Nevertheless, we +still obtain a speedup of 3-5 times compared to a full NAS (Neural Architecture Search). +4.3 +LATENCY EXPERIMENTS +We also measure the latency-performance tradeoff for the pruned FBNetV3G models. FBNetV3G +is the largest model in the family and so is expected to have the best generalization performance +under high sparsity levels. As described in Section 3.3, we prune the network using block sparsity +(where the block size is 1 × 4) to sparsity levels in the set {40%, 50%, 60%}. We have not +utilized lower sparsity levels, as we have observed that for the selected kernels we need at least +7 + +.5 +LD +L +FBNetV3 +Pruned FBNetV3 (our models) +FLOPs (M]Table 2: Computation speedup in term of GPU-hours when comparing NAS (neural Architecture +Search) with pruning and fine-tuning approaches. The selected seed networks are drawn from those +in Table 1 with the best performance at target FLOPs. +Target FLOPs +(FBNetV3 Model) +GPU-hours +in NAS +GPU-hours +in pruning +and fine-tuning +Computational cost +speedup +356.6 (FBNetV3A) +10.7k +2.240k +4.77 +557.0 (FBNetV3C) +10.7k +2.496k +4.28 +762.0 (FBNetV3E) +10.7k +3.456k +3.09 +(a) Latency vs. Top-1 accuracy on ImageNet +(b) Layer-wise sparsity pattern for FBNetV3E +(c) Layer-wise FLOPs distribution for FBNetV3E +(d) Performance distribution for layer type +Figure 2: Latency benchmarking on FBNetV3G for different sparsity levels {40%, 50%, 60%} and +layer-wise sparsity/FLOPs/accuracy sensitivity for a pruned FBNetV3E network. +40% sparsity to yield any significant latency benefits. We have pruned all 1×1 convolution layers +uniformly here and subsequently converted them to fully-connected layers for compatibility with the +quantized sparse kernels. In Figure 2a, we present the Top-1 ImageNet accuracy vs. latency curve +after pruning the FBNetV3G network for different sparsity levels. The pruned FBNetV3G models +show marked performance reduction with lower latency as expected, with a sparsity level of 60% +translating to around 7% absolute accuracy reduction with a latency reduction of 18 ms (16% relative). +While the 1×1 convolution layers account for >80% of FLOPs, they only constitute 25% of overall +network latency. This is consistent with previous literature (Dudziak et al., 2020) which shows that +computational complexity (ex. FLOPs) and latency are not well-correlated, and indeed the latter is +more dependent on layer shapes. This result underscores the need to develop more latency-friendly +pruning techniques which can potentially improve on the state-of-the-art in this domain. +4.4 +INSIGHTS INTO PRUNING EXPERIMENTS +Our pruning experiments demonstrate that we can improve on the state-of-the-art FBNetV3 models +in generalization performance for a given FLOPs level. In this subsection, we obtain an insight into +(1) the sparsity pattern under global magnitude-based pruning and (2) the sensitivity of each layer +when pruned in isolation under uniform layer-wise magnitude pruning (sparsity level of 95%). For +(1) in Figure 2b, we plot the amount of sparsity obtained per 1×1 convolution layer. The model being +considered is an FBNetV3E network pruned to a sparsity level of 43.6%, to the same FLOPs level +as FBNetV3C and subsequently fine-tuned. We note that the sparsity level in lower layers is lower +which is potentially required for maintaining the performance . Higher sparsity can be admitted in +8 + +81 +80 +79 +Top-1 (ImageNet) +78 +77 +Accuracy. T +76 +75 +95.0 +97.5 +100.0 +102.5 +105.0 +107.5 +110.0 +112.5 +Latency fms.J1.0 +0.8 +Layer spars +0.6 +0.4 +0.2 +20 +40 +60 +80 +100 +1x1 conv. layer index25 +20 +Layer MFLOPs +15 +10 +5 +0 +20 +40 +60 +80 +100 +1x1 conv. layer index80 + Accuracy (ImageNet) +60 +40 +80 +20 +Top-1 +0 +PW +PWL +SE +Type of 1x1 conv layerupper layers of the network where it has learnt more redundant representations. SE (Squeeze and +Excitation) 1×1 convolution layers generally tend to get pruned more compared to other layers, with +the sparsity being >99% for two such SE layers in stage xif5_0. This indicates that we can also +consider revisiting SE layer role in FBNetV3 networks, and even remove entire layers in future work +to yield additional latency and FLOPs benefits. +For analysis (2) we prune each 1×1 convolution layer in isolation at a sparsity target of 95% and +record the Top-1 test accuracy obtained on ImageNet dataset. For each type of layer, PW:expansion, +PWL: bottleneck, SE: Squeeze-Excitation we plot the distribution of accuracies in Figure 2d. We +observe that the PW and PWL layers are most sensitive to high sparsity, while SE layers are able to +retain performance adequately. We could also avoid pruning the most sensitive layers (appearing as +outliers in the figure) to maintain generalization performance. This observation corroborates findings +from analysis (1), and motivates us to revisit the role of squeeze-excitation layers in future work. +5 +CONCLUSIONS +In this paper, we have investigated the problem of improving on the current state-of-the-art FLOPs +vs. performance trade-off for FBNets which have been pre-optimized by NAS (Neural Architecture +Search). We have employed network pruning techniques, and our results demonstrate that we can +further improve on performance over FBNetV3 at a given FLOPs target through global as well as +uniform magnitude-based pruning. This happens not only for relatively over-parameterized networks +such as FBNetV3G, but also smaller networks such as FBNetV3A which have lower computational +complexity. On average, the GPU-hours incurred during pruning is about ∼ 4× less than that +consumed by a full-scale NAS. 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Neural architecture search with reinforcement learning. arXiv preprint +arXiv:1611.01578, 2016. +11 + +Table 3: Baseline FBNetV3 models chosen for our experiments +Baseline +No. of +parameters +(in millions) +MFLOPs +Top-1 Accuracy +(ImageNet) +Top-5 Accuracy +(ImageNet) +FBNetV3A +8.5 +356.6 +79.6 +94.7 +FBNetV3B +8.5 +461.6 +80.2 +94.9 +FBNetV3C +9.9 +557.0 +80.8 +95.3 +FBNetV3D +10.2 +644.4 +81.0 +95.4 +FBNetV3E +10.7 +762.0 +81.3 +95.5 +FBNetV3F +13.8 +1181.6 +82.5 +95.9 +FBNetV3G +16.5 +2129.7 +83.2 +96.3 +A +APPENDIX +A.1 +DATASET +Our pruning experiments are conducted on the ImageNet dataset, which is commonly used in the +literature to evaluate performance of image classification models. It is a collection of millions +of images, where there is a defined taxonomy based on the WordNet hierarchy. The taxonomy +comprises of approximately 22,000 visual subcategories, making this a large-scale classification +problem. ImageNet was first introduced by Deng et al. (2009) and has been adopted by the machine +learning and computer vision communities to benchmark image classification models. We use the +entire dataset consisting of 14 million images for our experiments, and we utilize both the training +set and the validation set. We split the training set to also create a smaller validation set (of 50,000 +images evenly distributed across all image categories) for parameter tuning and setting the training +convergence criterion. The ImageNet validation set is used in our experiments as an testing set for +reporting model generalization performance. +A.2 +OBTAINING SPARSITY LEVELS FOR GLOBAL MAGNITUDE-BASED PRUNING +While the sparsity level required to prune the 1×1 convolution layers can be obtained from Equation 1 +for the uniform pruning case, for global magnitude-based pruning all weights in such layers are ranked +globally and then thresholded to determine the sparsity mask. There is no closed form expression +we can use to determine the sparsity level given the FLOPs target for this scenario. To obtain the +sparsity level, we have used the pre-trained seed model and pruned it to a sparsity level s. We can +calculate the overall FLOPs consumed due to sparsity s by plugging in the layer-wise shapes and +sparsity levels. Extrapolating backwards from target FLOPs, we can easily find out which sparsity +level corresponds to this. It is important to note that since we prune and fine-tune with a fixed sparsity +mask, the number of FLOPs estimated at sparsity level s does not change even after the network is +fine-tuned. +12 + diff --git a/TNE3T4oBgHgl3EQfaApj/content/tmp_files/load_file.txt b/TNE3T4oBgHgl3EQfaApj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bd5459013f228c603540233277357633499378a4 --- /dev/null +++ b/TNE3T4oBgHgl3EQfaApj/content/tmp_files/load_file.txt @@ -0,0 +1,591 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf,len=590 +page_content='PRUNING COMPACT CONVNETS FOR EFFICIENT INFER- ENCE Sayan Ghosh Meta Platforms Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' sayanghosh@meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='com Karthik Prasad Meta Platforms Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' krp@meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='com Xiaoliang Dai Meta Platforms Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' xiaoliangdai@meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='com Peizhao Zhang Meta Platforms Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' stzpz@meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='com Bichen Wu Meta Platforms Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' wbc@meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='com Graham Cormode Meta Platforms Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' gcormode@meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='com Peter Vajda Meta Platforms Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' vajdap@meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='com ABSTRACT Neural network pruning is frequently used to compress over-parameterized net- works by large amounts, while incurring only marginal drops in generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' However, the impact of pruning on networks that have been highly optimized for efficient inference has not received the same level of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' In this paper, we analyze the effect of pruning for computer vision, and study state-of- the-art ConvNets, such as the FBNetV3 family of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We show that model pruning approaches can be used to further optimize networks trained through NAS (Neural Architecture Search).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The resulting family of pruned models can consis- tently obtain better performance than existing FBNetV3 models at the same level of computation, and thus provide state-of-the-art results when trading off between computational complexity and generalization performance on the ImageNet bench- mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' In addition to better generalization performance, we also demonstrate that when limited computation resources are available, pruning FBNetV3 models incur only a fraction of GPU-hours involved in running a full-scale NAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' 1 INTRODUCTION Neural networks frequently suffer from the problem of over-parameterization, such that the model can be compressed by a large factor to drastically reduce memory footprint, computation as well as energy consumption while maintaining similar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' This is especially pronounced for models for computer vision (Simonyan & Zisserman, 2014), speech recognition (Pratap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2020) and large text understanding models such as BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The improvements obtained from intelligently reducing the number of model parameters has several benefits, such as reduction in datacenter power consumption, faster inference and reduced memory footprint on edge devices such as mobile phones which also enable decentralized techniques ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' federated learning (Kairouz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' There are several techniques to reduce model size while maintaining similar generalization perfor- mance, such as model quantization (Polino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2018), NAS (Neural Architecture Search) (Elsken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2019) and model distillation through teacher-student networks (Gou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For the scope of this paper, we consider pruning as a technique to remove trainable weights in the network, and save on computation costs for the FBNet family of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The motivations for this are two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Firstly, state-of-the-art models such as FBNet (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2019) already adopt the best practices in the area of efficient hardware-aware design of convolutional neural network based models, and are widely used across different vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' This makes them suitable baselines to understand whether pruning can offer any performance gain over their already optimized behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' While there has been limited work on pruning for efficient convolution network models they investigate older architectures such as 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='04502v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='CV] 11 Jan 2023 EfficientNet and MobileNet (Aflalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2020) or integrate pruning into expensive techniques such as joint prune-and-architecture search (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For each of the constituent models of the FBNetV3 family (FBNetV3A, FBNetV3B,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', FBNetV3G) we reduce the number of parameters using two pruning based approaches: (1) Global magnitude- based pruning: Starting with the pre-trained model, we prune all weights whose magnitude is below a threshold chosen in order to achieve a target number of FLOPs for the pruned model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' (2) Uniform magnitude-based pruning: Starting with the pre-trained model, we prune weights in each layer whose magnitude is below a level-specific threshold in order to yield a pruned model achieving a target number of FLOPs with the same sparsity in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' After either pruning method is applied, we fine-tune the pruned model for a certain number of epochs until convergence is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Within the scope of our study in this paper, we are mostly interested in the following research questions: RQ1: Pruning to improve computation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' performance tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Can a model obtained by pruning a larger FBNetV3 model M1 (optimized using NAS) achieve higher generalization performance than a smaller FBNetV3 model M2 when the pruned model has the same number of FLOPs as M2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' RQ2: Pruning as an efficient paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' When a larger FBNetV3 model M1 is available and computational resources are limited, is pruning a faster and less computationally expensive approach to obtain a model with higher accuracy at a desired computation level (FLOPs) than running a full-fledged architecture search?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Pruning to improve computation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' performance tradeoff (RQ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' There have been recent research advances in the area of building hardware-aware efficient models (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' These can provide good generalization performance while adhering to constraints on memory, inference latency and battery power, which are often dictated by the hardware environment where inference happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Experiments described in existing work on efficient vision models such as ChamNet (Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2019), MobileNet (Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2017), EfficientNet (Tan & Le, 2019) and FBNetV2 (Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2020) have shown that it is possible to achieve even higher performances on standard image recognition tasks such as ImageNet (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2009) at a certain level of FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' However the efficient design of these models does not solve the over-parameterization problem completely, and none of these approaches study how model pruning can be performed to obtain even better trade-offs between computation and model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' This paper is the first of its kind to understand how we can improve on the state-of-the-art in this problem space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Pruning as an efficient paradigm (RQ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' In addition to achieving state-of-the-art performance with reduced FLOPs, we are also interested in understanding how such pruned models can be obtained inexpensively with limited resources that are generally available to a machine learning practitioner who has access to existing optimized models but limited computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For example, the FBNetV3 models are freely available through Facebook’s Mobile Model Zoo1, while EfficientNet models can be obtained at GitHub2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' While the techniques needed to obtain computation- and latency- friendly models have been democratized through open-sourcing the source code as well as the models themselves, fully applying these techniques necessitates costly operations such as finding an optimal network topology through meta-learning approaches (You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2020) and search algorithms such as Genetic Algorithms (GAs) (Goldberg & Deb, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Given the high-degree of intractability of this problem, expensive computational resources are often needed in this case, easily exceeding the budget available to a university research laboratory or an angel-stage startup (Zoph & Le, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' When a starting model is already available, for example through open-sourcing, the best option would be to perform a cheap modification of the model to fit a certain target FLOPs/latency requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' In this paper we have compared the NAS approaches for training FBNetV3 models with our pruning techniques on a computational complexity metric (GPU-hours) to effectively answer RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Benchmark results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' In addition to experimental outcomes for answering RQ1 and RQ2, we also benchmark pruned FBNetV3 models using available open-sourced quantized sparse kernels and conduct ablation studies to obtain additional insights into pruning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' These results augment our main observations and demonstrate that with existing hardware support, it is possible to deploy 1FBNetV3 models available here http://https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='com/facebookresearch/ mobile_cv/model_zoo/models/model_info/fbnet_v2/model_info_fbnet_v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='json 2EfficientNet models available here https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='com/mingxingtan/efficientnet 2 pruned cutting-edge computer vision models with practical latency reductions and improve further beyond the performance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' FLOPs trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We conduct our experiments on ImageNet, which is an object-recognition task on a large training dataset of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='2 million images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We show that computationally less intensive techniques such as uniform and global magnitude-based pruning of larger FBNetV3 models can yield higher test accuracies than small models while having the same number of FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Given a target computation budget for an efficient model, we show that it is more practically advantageous (both in terms of performance and running time) to simply prune the larger model than run a neural architecture search to find the target model from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The technique we have employed for pruning (unstructured sparsity) is already tried and tested,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' however our novelty lies in studying whether efficient image recognition models such as FBNetV3 can be optimized further to improve on the FLOPs-accuracy curve,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' and the contributions are two fold : (1) FBNets are themselves state-of-the-art in efficient vision models and we achieve better accuracy-FLOPs tradeoff over these models and (2) from the standpoint of computational overhead,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' we significantly reduce the amount of GPU hours required to obtain such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Pruning a publicly available NAS optimized model incurs ≈4x less GPU hours to achieve a target FLOPs level, compared to training a full-fledged NAS to obtain a model which has less accuracy at the same FLOPs level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Paper organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' In Section 2, we describe related work in the area of efficient vision model design and also provide an introduction to different pruning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' In Section 3, we discuss our experimental setup, including a description of the baseline models and the global and uniform pruning approaches we have employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Section 4 describes our main findings and we conclude the paper in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' 2 RELATED WORK We discuss related literature in the areas of computationally efficient vision models and model pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Within the scope of our work, we mainly focus on inference efficiency of models in contrast to training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Computationally efficient vision models: Neural networks for computer vision are generally char- acterized by convolutional layers and fully-connected layers, along with blocks such as residual or skip connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' This makes such networks resource intensive in terms of FLOPs, which affects the memory storage and power consumed, and also leads to increased latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' It is of paramount importance to design more efficient networks which can provide higher performance for the same FLOPs or latency level, or even to optimize them appropriately to provide the same performance at reduced FLOPs/latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' This can be performed either through the design of new simplified layers, for example in deep residual learning (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2016) or though explicit model compression as in weight quantization (Polino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Extremely deep networks for image recognition often suffer from not only high complexity and inference latency, but also from the issue of vanishing gradients (Pas- canu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' This was addressed through deep residual networks which effectively simplified network design through skip-connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' MobileNets (Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2017) are one of the earlier approaches to building small low-latency networks by using depthwise separable convolutions with two parameters, width and resolution multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' They demonstrate the effectiveness of MobileNets across different vision tasks, such as face embeddings and object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' MobileNetV2 (Sandler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2018) extends MobileNets by utilizing inverted residual filter structures and linear bottlenecks, obtaining improvements on state-of-the-art models both in terms of accuracy and computational com- plexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' ShuffleNets (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2018) propose dedicated residual units where 1×1 convolutions are replaced with pointwise group convolutions and channel shuffling reducing FLOPs computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' More recently, the focus on building efficient neural network models has shifted to techniques that treat the design of efficient networks as a search problem, falling under the umbrella of Neural Architecture Search (NAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' EfficientNets (Tan & Le, 2019) propose a novel scaling method which adjusts the network’s length, width, and resolution to optimize performance subject to target memory and FLOPs constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' They also define a novel baseline that is optimized by a multi-objective neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The FBNet collections of models—FBNet (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2019), FBNetV2 (Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2020) and FBNetV3 (Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2021)—employ neural architecture search to obtain highly-optimized models that improve on the state-of-the-art for different visual understanding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' FBNet frames the 3 architecture search as a differentiable meta-learning problem with gradient based techniques, namely DNAS—Differentiable Neural Architecture Search—by Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' (2019), and avoids selecting the optimized model over a discrete set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The subsequent entry in this collection, FBNetV2, expands the search space over conventional DNAS, and employs a masking scheme to maintain the same level of computational complexity while searching over this expanded space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' FBNetV3 further improves on the state-of-the-art by employing Neural Architecture Recipe Search (NARS) and searching over the space of not only architectures, but also corresponding recipes (which are generally hyper-parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' In this paper, we consider FBNetV3 models as our baselines as they are state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We are interested in understanding if they are overparameterized and evaluate how much model pruning can improve performance at a certain FLOPs level over the state-of-the-art in this family of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Model Pruning: Modern neural networks, particularly those processing complex sensory inputs (such as speech, vision and language) for perception applications, are often over-parameterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' It is only to be expected that we should be able to compress such networks significantly to maintain the same level of performance at decreased level of computation (fewer weights and reduced FLOPs), memory footprint and power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Foundational efforts in this space include the Optimal Brain Surgeon (Hassibi & Stork, 1993) and Optimal Brain Damage (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Recently the idea of network pruning has been formalized through the lottery ticket hypothesis (Frankle & Carbin, 2018), which claims that randomly initialized, feed-forward networks have winning sub-networks that perform just as well as the original network on an unseen test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Model pruning is generally of two types: unstructured and structured pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Unstructured pruning, as the name suggests, doesn’t adhere to any structure and prunes neurons based on chosen criteria (such as magnitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' This has the advantage of providing higher performance, but is difficult to implement in hardware, as it needs dedicated support for efficient sparse matrix multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Meanwhile, structured pruning is the practice of removing entire groups of neurons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', blocks within the weight matrix, or channels in convolutional neural networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' This is easy to implement without dedicated hardware support, but has the issue of lower generalization performance than unstructured pruning (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' In the literature, there have also been several studies, for example investigating whether rewinding (training from scratch with a fixed mask) can perform just as well as the fine-tuning on top of the original unpruned network (Renda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Blalock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' (2020) provide an overview survey of recent advances and open problems in neural network pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' In the research area of designing efficient networks for computer vision, there has not been much focus on understanding how pruning can be applied to the current generation of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Most literature on pruning is based on older networks such as VGGNet, ResNet (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2016), and MobileNet (Sandler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Our work improves upon these existing studies by understanding how pruning can improve the FLOPs-accuracy tradeoff over existing state-of-the-art networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' 3 PRUNING TECHNIQUES AND SETUP In this section, we describe the main components of our techniques and experimental setup, including Baseline Models, Pruning Techniques, Latency Measurement and Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We have mainly used standard splits of the ImageNet dataset, further details are in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='1 of the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='1 BASELINE MODELS Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' (2020) address the previous limitations of NAS-based architecture search where these approaches can only search over architectures given a training recipe (set of hyperparameters), and thus cannot optimize over both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' As described in Section 2, the most recent state-of-the-art models are based on NARS (Neural Architecture-Recipe Search), which we select as baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Table 3 lists the accuracy of FBNetV3 models (Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2021) on the ImageNet classification task, along with the number of model parameters and computation complexity in terms of FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Each baseline model consists of multiple IRF (Inverted Residual Filter) blocks, which contain convolutional layers of different kernel sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For our experiments, we are mostly interested in 1×1 convolutions as potentially prunable, since within each FBNetV3 model, the 1×1 convolution layers constitute >80% of total model FLOPs for all models in the family, and the open-sourced sparsity kernel support we use for latency benchmarking is available only for fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' A 1×1 4 convolution can be transformed into an equivalent fully connected layer with a few tensor reshape operations without any significant loss of performance or latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For each initial and target FBNetV3 model X and Y , where X is larger than Y , we prune X to a sparsity level of S so that the FLOP count is the same as for Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The number of FLOPs consumed by a linear layer of sparsity S is proportional to the number of sparse matrix multiplications performed and is given by S ∗F, where F is the corresponding dense FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Thus if F1×1(X) is the number of FLOPs consumed by the 1×1 convolution layers and F(x) is the total number of FLOPs consumed by model X, we have: S = (F(X) − F(Y ))/F1×1(X) (1) Hence, sparsity measures the fraction of 1×1 convolution weights removed, and so higher sparsity indicates a smaller model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For the uniform pruning scnario, Table 1 shows the amount of sparsity required to prune each larger FBNetV3 model to a smaller one based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For global pruning, (1) does not hold, and we compute the target sparsities empirically from the layer shapes instead with details provided in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We prune each larger FBNetV3 model to a discrete FLOPs target based on a defined set of smaller models in the family, and not to a continuous range of FLOPs values, as it makes it easier to compare models directly based on a given computation budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' If we can demonstrate that for the same computation level, the pruned larger FBNetV3 model has higher performance than a smaller model with the same FLOPs, it is sufficient to demonstrate that we can improve on the FLOPs-accuracy curve over the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='2 PRUNING TECHNIQUES In this paper, we utilize a pre-trained FBNetV3 model with higher number of FLOPs without training an image classification model from scratch with sparsity, which would be time consuming and computationally intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' There are several approaches in the literature such as prune-and-fine- tune (Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2015) and iterative pruning with sparsity scheduling (Frankle & Carbin, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We have utilized the former for our experiments, as although studies have shown that iterative and incremental pruning approaches lead to better generalization performance, they typically require training for high number of epochs, need tuning and selection of optimal sparsity schedules and are computationally resource intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We have therefore not considered them in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For our prune and fine-tune experiments, we have used 8-GPU boxes, with each box having Nvidia V100 (Volta) 32G GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' As described in Section 1, we perform both global and magnitude-based pruning experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For the latency benchmarking, we also perform magnitude-based uniform pruning with a sparse block size of 1 × 4 as explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We have conducted a hyper-parameter tuning for the learning rate parameter, with LR values in the set {4e-5, 8e-5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='6e-4}, as fine-tuning generally admits smaller learning rates than training from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We have found that using the same learning rate for all models, along with the same hyper-parameter settings used for training the seed model is sufficient to obtain pruned networks which are superior to the baseline FBNetV3 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Hence minimal hyper-parameter tuning was required for our experiments and we have used values of settings such as weight decay and momentum to be the same as those used for training the baseline FBNetV3 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' During fine-tuning after pruning, we have used a smoothed validation loss to stop the process early after a convergence tolerance (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='01%) is reached between two consecutive epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Generally, we have observed fine-tuning to converge around ∼250 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='3 LATENCY MEASUREMENTS AND METRICS We are interested not only in the sparsity level of our pruned models and the image recognition performance, but also in metrics which potentially improve due to model sparsity, such as number of parameters, the FLOP count and the model latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For reporting model performance under pruning, we use standard image recognition metrics such as Top-1 and Top-5 test accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We measure overall model sparsity, which is different to the layer sparsity since we only prune 1×1 convolution layers, as explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We report the model FLOPs, because this metric captures the computational footprint of the model and its power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' 5 Last, we record the total latency (in ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=') under pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The sparse kernels used in our experiments are already in open-source and released under the PyTorch sparse quantization library3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Prior to using these kernels, we perform uniform layer-wise block-based pruning with block sizes of 1 × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Magnitude based pruning is implemented at block level, and the model is quantized to 8-bit integers (int8) before latency benchmarking, which is performed on Intel CPUs designed using the Skylake micro-architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' While we would expect sparsity to translate to tangible inference speedups, this is highly dependent on the sparse kernel support provided by hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Current hardware is not well- suited for unstructured randomly sparse matrix multiplications and tend to do better with structured sparsity in models (Anwar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We have utilized block sparsity within the weight matrix for latency experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' However this often tends to come at a decreased level of model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The design of highly performant sparse models under structured sparsity with reasonable inference speedups remains an important research topic outside the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' 4 RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='1 PRUNED FBNETV3 MODEL PERFORMANCE To answer RQ1, we consider the family of FBNetV3 models as baselines and seed models for further pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For each pair of models X, Y in the family, we calculate the amount of sparsity required to prune the larger model X to a model that consumes the same number of FLOPs as the target smaller model Y , via Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' There are 21 potential seed and target model pairs, however we conduct pruning experiments only for a depth of 2 for tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For example, given FBNetV3E as the seed, we only prune it to FLOPs targets corresponding to FBNetV3D and FBNetV3C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Table 1 presents the accuracy and number of parameters of the pruned models at each target FLOPs level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The improvement in performance is apparent even at lower FLOPs targets, where we might expect baseline models such as FBNetV3A to not be over-parameterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For example, pruning FBNetV3C to a target of 356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='6 MFLOPs obtains a network which is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='43% better than FBNetV3A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Figure 1 plots the Top-1 ImageNet testing accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' FLOPs for the best pruned models as seen from Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' This clearly shows that pruning FBNetV3 models with minimal fine-tuning can significantly improve on the state-of-the-art for FLOPs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Top-1 accuracy trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' This analysis is performed for both uniform layer-wise and global magnitude-based prune with fine-tune settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Global pruning ranks the weights of the entire network in contrast to uniform layer-wise pruning, which ranks each layer’s weights to determine the sparsity mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' It would be expected that global pruning performs better than uniform pruning for the same target sparsity level or number of non-sparse parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' However in our experiments we determine the pruning threshold based on FLOPs targets, and find global pruning to require higher sparsity levels, which results in uniform pruning outperforming global pruning in Top-1 ImageNet accuracy in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='2 PRUNING COMPLEXITY In addition to demonstrating the improvement over state-of-the-art obtained by pruning FBNetV3 models, it is also important to quantify the reduction in computational complexity obtained in pruning a larger FBNetV3 model compared to training an FBNetV3 model directly through NAS (Network Architecture Search).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' RQ2 (pruning for efficient model search) asks if the pruning and subsequent fine-tuning approach in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='1 is faster than a full-fledged neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' During pruning and subsequent fine-tuning, we train the pruned networks till the validation loss converges to within a pre-specified tolerance, as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The time needed is generally less than when training the original FBNetV3 models, which runs for 400 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The number of GPU-hours is computed as (number of training GPU nodes) * (number of GPUs per node) * (training time to convergence) for each network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' In Table 2, for each of the best performing uniformly-pruned models in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='1 we report the number of GPU-hours consumed by the prune and fine-tune strategy, along with the GPU-hours consumed when obtaining a FBNetV3 model through architecture search using the method described in Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The results are quite conclusive—we not only obtain pruned models superior in performance to the original neural search optimized models, but also as 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='com/pytorch/pytorch/blob/master/torch/ao/nn/sparse/quantized/linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='py 6 Table 1: Sparsity level (in percentage) and performance of pruned FBNetV3 networks on ImageNet dataset for different target MFLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The best accuracy obtained at each target FLOPs level is highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Seed network FBNetV3_ Target network FBNetV3_ Target MFLOPs Baseline Accuracy Uniform pruning Global pruning Sparsity level(%) Top-1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Gain(%) Sparsity level(%) Top-1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Gain(%) B A 356.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='643 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='8 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='454 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='553 A B C D E F C->A D->B E->C F->D F->E G->F Figure 1: FLOPs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' performance (ImageNet Top-1 acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=') for different pruned FBNetV3 networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For comparison, the existing FBNetV3 networks are also shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' described in Section 1, computational cost is significantly lower when starting from a pre-trained model with higher FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Given the performance improvements obtained with lower computational resources, this approach is beneficial for an experimental setting where researchers have access to open-sourced pre-trained models and limited GPU resources, for example in a small startup or an academic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We observe that the degree of speedup reduces as the network size gets bigger (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', in FBNetV3A vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' FBNetV3C) due to higher training time to convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Nevertheless, we still obtain a speedup of 3-5 times compared to a full NAS (Neural Architecture Search).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='3 LATENCY EXPERIMENTS We also measure the latency-performance tradeoff for the pruned FBNetV3G models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' FBNetV3G is the largest model in the family and so is expected to have the best generalization performance under high sparsity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' As described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='3, we prune the network using block sparsity (where the block size is 1 × 4) to sparsity levels in the set {40%, 50%, 60%}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We have not utilized lower sparsity levels, as we have observed that for the selected kernels we need at least 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='5 LD L FBNetV3 Pruned FBNetV3 (our models) FLOPs (M]Table 2: Computation speedup in term of GPU-hours when comparing NAS (neural Architecture Search) with pruning and fine-tuning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The selected seed networks are drawn from those in Table 1 with the best performance at target FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Target FLOPs (FBNetV3 Model) GPU-hours in NAS GPU-hours in pruning and fine-tuning Computational cost speedup 356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='6 (FBNetV3A) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='7k 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='240k 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='77 557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='0 (FBNetV3C) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='7k 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='496k 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='28 762.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='0 (FBNetV3E) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='7k 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='456k 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='09 (a) Latency vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Top-1 accuracy on ImageNet (b) Layer-wise sparsity pattern for FBNetV3E (c) Layer-wise FLOPs distribution for FBNetV3E (d) Performance distribution for layer type Figure 2: Latency benchmarking on FBNetV3G for different sparsity levels {40%, 50%, 60%} and layer-wise sparsity/FLOPs/accuracy sensitivity for a pruned FBNetV3E network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' 40% sparsity to yield any significant latency benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We have pruned all 1×1 convolution layers uniformly here and subsequently converted them to fully-connected layers for compatibility with the quantized sparse kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' In Figure 2a, we present the Top-1 ImageNet accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' latency curve after pruning the FBNetV3G network for different sparsity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The pruned FBNetV3G models show marked performance reduction with lower latency as expected, with a sparsity level of 60% translating to around 7% absolute accuracy reduction with a latency reduction of 18 ms (16% relative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' While the 1×1 convolution layers account for >80% of FLOPs, they only constitute 25% of overall network latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' This is consistent with previous literature (Dudziak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=', 2020) which shows that computational complexity (ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' FLOPs) and latency are not well-correlated, and indeed the latter is more dependent on layer shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' This result underscores the need to develop more latency-friendly pruning techniques which can potentially improve on the state-of-the-art in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='4 INSIGHTS INTO PRUNING EXPERIMENTS Our pruning experiments demonstrate that we can improve on the state-of-the-art FBNetV3 models in generalization performance for a given FLOPs level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' In this subsection, we obtain an insight into (1) the sparsity pattern under global magnitude-based pruning and (2) the sensitivity of each layer when pruned in isolation under uniform layer-wise magnitude pruning (sparsity level of 95%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For (1) in Figure 2b, we plot the amount of sparsity obtained per 1×1 convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The model being considered is an FBNetV3E network pruned to a sparsity level of 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='6%, to the same FLOPs level as FBNetV3C and subsequently fine-tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We note that the sparsity level in lower layers is lower which is potentially required for maintaining the performance .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Higher sparsity can be admitted in 8 81 80 79 Top-1 (ImageNet) 78 77 Accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' T 76 75 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='5 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='0 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='5 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='0 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='5 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='0 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='5 Latency fms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='8 Layer spars 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='2 20 40 60 80 100 1x1 conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' layer index25 20 Layer MFLOPs 15 10 5 0 20 40 60 80 100 1x1 conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' layer index80 Accuracy (ImageNet) 60 40 80 20 Top-1 0 PW PWL SE Type of 1x1 conv layerupper layers of the network where it has learnt more redundant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' SE (Squeeze and Excitation) 1×1 convolution layers generally tend to get pruned more compared to other layers, with the sparsity being >99% for two such SE layers in stage xif5_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' This indicates that we can also consider revisiting SE layer role in FBNetV3 networks, and even remove entire layers in future work to yield additional latency and FLOPs benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For analysis (2) we prune each 1×1 convolution layer in isolation at a sparsity target of 95% and record the Top-1 test accuracy obtained on ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' For each type of layer, PW:expansion, PWL: bottleneck, SE: Squeeze-Excitation we plot the distribution of accuracies in Figure 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We observe that the PW and PWL layers are most sensitive to high sparsity, while SE layers are able to retain performance adequately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We could also avoid pruning the most sensitive layers (appearing as outliers in the figure) to maintain generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' This observation corroborates findings from analysis (1), and motivates us to revisit the role of squeeze-excitation layers in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' 5 CONCLUSIONS In this paper, we have investigated the problem of improving on the current state-of-the-art FLOPs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' performance trade-off for FBNets which have been pre-optimized by NAS (Neural Architecture Search).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We have employed network pruning techniques, and our results demonstrate that we can further improve on performance over FBNetV3 at a given FLOPs target through global as well as uniform magnitude-based pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' This happens not only for relatively over-parameterized networks such as FBNetV3G, but also smaller networks such as FBNetV3A which have lower computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' On average, the GPU-hours incurred during pruning is about ∼ 4× less than that consumed by a full-scale NAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We have also performed latency measurements on the FBNetV3G model and conducted an analysis to understand the sparsity patterns and 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experiments Baseline No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' of parameters (in millions) MFLOPs Top-1 Accuracy (ImageNet) Top-5 Accuracy (ImageNet) FBNetV3A 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='5 356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='6 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='6 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='7 FBNetV3B 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='5 461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='9 FBNetV3C 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='9 557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='8 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='3 FBNetV3D 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='2 644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='0 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='4 FBNetV3E 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='7 762.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='0 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='3 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='5 FBNetV3F 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='8 1181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='9 FBNetV3G 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='5 2129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='7 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='2 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='3 A APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='1 DATASET Our pruning experiments are conducted on the ImageNet dataset, which is commonly used in the literature to evaluate performance of image classification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' It is a collection of millions of images, where there is a defined taxonomy based on the WordNet hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The taxonomy comprises of approximately 22,000 visual subcategories, making this a large-scale classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' ImageNet was first introduced by Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' (2009) and has been adopted by the machine learning and computer vision communities to benchmark image classification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We use the entire dataset consisting of 14 million images for our experiments, and we utilize both the training set and the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We split the training set to also create a smaller validation set (of 50,000 images evenly distributed across all image categories) for parameter tuning and setting the training convergence criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' The ImageNet validation set is used in our experiments as an testing set for reporting model generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content='2 OBTAINING SPARSITY LEVELS FOR GLOBAL MAGNITUDE-BASED PRUNING While the sparsity level required to prune the 1×1 convolution layers can be obtained from Equation 1 for the uniform pruning case, for global magnitude-based pruning all weights in such layers are ranked globally and then thresholded to determine the sparsity mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' There is no closed form expression we can use to determine the sparsity level given the FLOPs target for this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' To obtain the sparsity level, we have used the pre-trained seed model and pruned it to a sparsity level s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' We can calculate the overall FLOPs consumed due to sparsity s by plugging in the layer-wise shapes and sparsity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' Extrapolating backwards from target FLOPs, we can easily find out which sparsity level corresponds to this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' It is important to note that since we prune and fine-tune with a fixed sparsity mask, the number of FLOPs estimated at sparsity level s does not change even after the network is fine-tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} +page_content=' 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfaApj/content/2301.04502v1.pdf'} diff --git a/UNAzT4oBgHgl3EQf0_5k/vector_store/index.pkl b/UNAzT4oBgHgl3EQf0_5k/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..f534957f7e19bb565f5c7a6e6e241ffbf3f93784 --- /dev/null +++ b/UNAzT4oBgHgl3EQf0_5k/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0e83b846d2ce989291faa6c6f34757fd9280034e599a2d09c0633544e224af50 +size 98567 diff --git 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In this way, motions replace +unit dual quaternions to represent rigid body movements in the 3D space. The infinites- +imal unit is no longer needed. By means of motion addition, we formulate two classical +problems in robot research, i.e., the hand-eye calibration problem and the simultaneous +localization and mapping (SLAM) problem as motion optimization problems, which are +actually real unconstrained optimization problems. In particular, it avoids to go through +the unit dual quaternion operations. +Key words. Motion, motion addition, rotation, rotation addition, hand-eye calibra- +tion, simultaneous localization and mapping. +1 +Introduction +In [12], motions, as six-dimensional real vectors, were introduced to represent +rigid body movements in the 3D space. Operators mapping from motions to unit +dual quaternions, and from unit dual quaternions to motions, were defined. Two +classical problems in robot research, i.e., the hand-eye calibration problem [7, +9, 10, 14, 19] and the simultaneous localization and mapping (SLAM) problem +[1, 2, 3, 4, 17, 18] were formulated as motion optimization problems, which are +actually unconstrained real optimization problems. This approach improved +the dual quaternion optimization approach, studied in [11, 5]. +∗Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018 China; +Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong +Kong (maqilq@polyu.edu.hk). +1 + +The approach in [12] raised one question. Can we define some operations +of motions such that we may use motions to represent rigid body movements +in the 3D space directly, instead of mapping into unit dual quaternions to +make operations, then mapping back? In this way, we may even get rid of +the infinitesimal unit, which somehow downgrades translations too much, and +whose true role is merely an operation symbol. +In this paper, we introduce rotation addition and motion addition to rep- +resent rigid body movements in the 3D space. Two motions were combined +directly without bothering unit dual quaternions. The infinitesimal unit is no +longer to be used. Based on these, we establish the motion optimization formu- +lations for the hand-eye calibration problem and the SLAM problem directly. +The distribution of the remainder of this paper is as follows. In the next +section, we review some basic properties of quaternions. Then we define rota- +tion addition and study its properties in Section 3. We further define motion +addition and study its properties in Section 4. In Section 5 we formulate the +hand-eye calibration problem and the SLAM problem as motion optimization +problems without involving unit dual quaternions. Some final remarks are made +in Section 6. +2 +Quaternions +A quaternion ˜q = [q0, q1, q2, q3] is a real four-dimensional vector. We use a +tilde symbol to distinguish a quaternion. Denote the set of all quaternions by Q. +Suppose that we have two quaternions ˜p = [p0, p1, p2, p3], ˜q = [q0, q1, q2, q3] ∈ Q. +The sum of ˜p and ˜q is defined as +˜p + ˜q = [p0 + q0, p1 + q1, p2 + q2, p3 + q3]. +(1) +The product of ˜p and ˜q is defined by ˜p˜q = +[p0q0−p1q1−p2q2−p3q3, p0q1+p1q0+p2q3−p3q2, p0q2+p2q0−p1q3+p3q1, p0q3+p3q0+p2q3−p3q2]. +(2) +The conjugate of a quaternion ˜q = [q0, q1, q2, q3] ∈ Q is defined as ˜q∗ = +[q0, −q1, −q2, −q3]. +Let ˜1 = [1, 0, 0, 0] ∈ Q. +Then for any ˜q ∈ Q, we have +2 + +˜q˜1 = ˜1˜q = ˜q. For any ˜p, ˜q ∈ Q, we have +(˜p˜q)∗ = ˜q∗˜p∗. +(3) +A quaternion ˜q = [q0, q1, q2, q3] ∈ Q is called a unit quaternion if +q2 +0 + q2 +1 + q2 +2 + q2 +3 = 1. +(4) +Denote the set of all unit quaternions by U. The following proposition can be +verified directly. +Proposition 2.1. If ˜p, ˜q ∈ U, then ˜p˜q ∈ U. For any ˜q ∈ U, we have +˜q˜q∗ = ˜q∗˜q = ˜1, +(5) +i.e., ˜q is invertible and ˜q−1 = ˜q∗. +A quaternion ˜q = [0, q1, q2, q3] ∈ Q is called a vector quaternion. +A +quaternion ˜q ∈ Q is a vector quaternion if and only if ˜q = −˜q∗. +Suppose +that ˜q is a vector quaternion and ˜p is a quaternion, then ˜p∗˜q˜p is still a vector +quaternion. +3 +Rotation Addition +A rotation r is a three-dimensional real vector r = [r1, r2, r3]. Write r = θl, +where l = [l1, l2, l3] is a unit vector, satisfying +l2 +1 + l2 +2 + l2 +3 = 1. +(6) +Call l the rotation axis of r, while θ the rotation angle of r. If 0 ≤ θ < 2π, +then r is called a basic rotation. The set of all basic rotations is denoted as +S. Thuswe have +S = +� +r ∈ V : ∥r∥2 +2 = r2 +1 + r2 +2 + r2 +3 < 4π2� +. +The space of three-dimensional real vectors is denoted as V. +3 + +A mapping U : V → U, called the UQ operator, defined in [12], was as +follows +U(r) = + + + + + +� +cos ∥r∥2 +2 , +r +∥r∥2 +sin ∥r∥2 +2 +� +, if r ̸= 0, +˜1, if r = 0. +(7) +Another mapping R : U → V, called the rotation operator, was also defined +in [12] as follows +R(˜q) = + + + + + +2 cos−1 q0 +� +q2 +1 + q2 +2 + q2 +3 +[q1, q2, q3], if q2 +0 ̸= 1, +0, otherwise, +(8) +where ˜q = [q0, q1, q2, q3] ∈ U. +Then for any ˜q ∈ U \ {−˜1}, we have U(R(˜q)) = ˜q, and for any r ∈ S, we +have R(U(r)) = r. +The above contents can be found in [12]. Suppose that r = [r1, r2, r3], s = +[s1, s2, s3] ∈ V. Define +r ⊕ s = R(U(r)U(s)). +(9) +We call this operation rotation addition. In general it is not commutative, +i.e., we have r ⊕ s ̸= s ⊕ r. +We call it rotation addition, not rotation multiplication, as it is closely re- +lated to the addition operation in space V. For any r ∈ V, we have +r ⊕ 0 = 0 ⊕ r = r = r + 0 = 0 + r. +(10) +Here 0 = [0, 0, 0] is the zero in V. +By the definition of rotation addition, we may prove the following proposi- +tion. +Proposition 3.1. Suppose that r, s ∈ V represent two rotations of a rigid body. +Then r ⊕ s ∈ V represents the combining rotation of r and s for the rigid body. +For r ∈ V, we always have +r ⊕ (−r) = (−r) ⊕ r = r + (−r) = 0. +(11) +4 + +For any r(1), r(2), r(3) ∈ V, +� +r(1) ⊕ r(2)� +⊕ r(3) = r(1) ⊕ +� +r(2) ⊕ r(3)� +. +(12) +We have the following conjecture. +Conjecture 1 For any r, s ∈ V, we have +r ⊕ s = r + s, +(13) +if and only if +r ⊕ s = s ⊕ r. +(14) +4 +Motion Addition +A motion x is a six-dimensional real vector, which has two parts: x = [r, t], +where r is a rotation, while t is a translation. +Both r and t are three- +dimensional real vectors, r, t ∈ V. +Rotations have been studied in the last +section. Denote the space of six-dimensional real vectors by M. +Note that from a vector t ∈ V, we may have a vector quaternion ˜t = [0, t] ∈ Q +and from a vector quaternion ˜t = [0, t] ∈ Q, we have a vector t ∈ V. We use a +tilde symbol to denote the vector quaternion, corresponding to a vector in V. +Suppose that we have two motions x = [r, t], y = [s, u] ∈ M. We now define +the motion addition x ⊕ y as follows. We have +x ⊕ y = [r ⊕ s, v], +(15) +where +˜v = [0, v] = U(s)∗˜tU(s) + ˜u, +(16) +˜t = [0, t] and ˜u = [0, u]. +Theorem 4.1. Suppose that x, y ∈ M represent two motions of a rigid body. +Then x ⊕ y ∈ M represents the combined motion of x and y for the rigid body. +Proof. We know that the motion of a rigid body in the 3D space can be rep- +resented by unit dual quaternions. Then motion x is corresponding to a unit +dual quaternion [12] +ˆp = ˜p + ǫ +2 ˜p˜t, +(17) +5 + +and y is corresponding to a unit dual quaternion [12] +ˆq = ˜q + ǫ +2 ˜q˜u. +(18) +Here, ˜p and ˜q are unit quaternions, ˜p = U(r) and ˜q = U(s), ǫ is the infinitesimal +unit, satisfying ǫ2 = 0. Combining ˆp with ˆq, we have +ˆpˆq = ˜p˜q + ǫ +2 ˜p(˜t˜q + ˜q˜u) += (˜p˜q) + ǫ +2(˜p˜q)(˜q∗˜t˜q + ˜u) += (˜p˜q) + ǫ +2(˜p˜q)˜v. +As ˜q = U(s), we have (16). Since ˜p˜q is corresponding to r ⊕ s, we have the +conclusion. +Theorem 4.2. Suppose that x = [r, t] ∈ M. Let y = [−r, u], where +˜u = [0, u] = −U(r)˜tU(r)∗, +(19) +where ˜t = [0, t]. Then we have +x ⊕ y = y ⊕ x = 0 = [0, 0]. +(20) +Proof. By Theorem 4.1 we have +x ⊕ y = [r ⊕ (−r), v] = [0, v], +where +˜v = U(−r)∗˜tU(−r) + ˜u = U(r)˜tU(r)∗ − U(r)˜tU(r)∗ = 0, +i.e., x ⊕ y = 0. +On the other hand, by Theorem 4.1 we have +y ⊕ x = [(−r) ⊕ r, v] = [0, v], +where +˜v = U(r)∗(−U(r))˜tU(r)∗U(r) + ˜t = 0, +i.e., y ⊕ x = 0. +6 + +In this way, we see that we use motions instead of unit dual quaternions to +represent rigid body movements in the 3D space. +We also have the following proposition. +Proposition 4.3. For any x, y, z ∈ M, +(x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) . +(21) +In general, we have x ⊕ y ̸= y ⊕ x. Also see [12, 13, 16] for the discussion on +the logarithm of unit dual quaternions. +5 +Motion Optimization +As in [12], for a motion x = [r, t] = [r1, r2, r3, t1, t2, t3] ∈ M, define its magnitude +as +|x| = +� +|r1|2 + |r2|2 + |r3|2 + σ|t1|2 + σ|t2|2 + σ|t3|2, +(22) +where σ is a positive number. +We define a motion vector x = [x(1), · · · , x(n)] as an n-component vector such +that its ith components is a motion x(i) = +� +r(i), t(i)� += +� +r(i) +1 , r(i) +2 , r(i) +3 , t(i) +1 , t(i) +2 , t(i) +3 +� +∈ +M, for i = 1, · · · , n. Thus, x may be also regarded as a 6n-dimensional real vec- +tor. We use small bold letters such as x to denote motion vectors, and denote +the space of n-component motion vectors by Mn. The norm in Mn is defined +by +∥x∥ = +� +� +� +� +n +� +i=1 +��x(i)��2 = +� +� +� +� +n +� +i=1 +���r(i) +1 +��� +2 ++ +���r(i) +2 +��� +2 ++ +���r(i) +3 +��� +2 ++ σ +���t(i) +1 +��� +2 ++ σ +���t(i) +2 +��� +2 ++ σ +���t(i) +3 +��� +2 +. +(23) +Then it is a norm. +Assume that z : Mn → Mm. A motion optimization problem is formu- +lated as +min +�1 +2∥z(x)∥2 : x ∈ Mn +� +, +(24) +which is a 6n-dimensional unconstrained optimization problem. +7 + +In [12], the hand-eye calibration problem and the simultaneous localization +and mapping (SLAM) problem were formulated as motion optimization problem +via the UDQ operators and the motion operators. Here, we use motion addition +to establish somewhat different motion optimization formulations. +Example 1 We now consider the 1989 Shiu and Ahmad [14] and Tsai and +Lenz [15] hand-eye calibration model. Then n = 1. We may formulate it as +motion equations +a(i) ⊕ x = x ⊕ b(i), +(25) +where, ˆa(i),ˆb(i), x ∈ M, for i = 1, · · · , m. Here, x is the transformation motion +from the camera (eye) to the gripper (hand), ˆa(i),ˆb(i), for i = 1, · · · , m, are +some data motions from experiments. The aim is to find the best motion x to +satisfy (25). Then, let +zi = +� +a(i) ⊕ x +� +⊕ +� +−x ⊕ b(i)� +, +(26) +and z = [z1, · · · , zm] ∈ Mm. We have the following motion optimization problem +min +�1 +2∥z(x)∥2 : x ∈ M +� +(27) +for this hand-eye calibration model. +This is a 6-dimensional unconstrained +optimization problem. +Example 2 We further consider the 1994 Zhuang, Roth and Sudhaker [19] +hand-eye calibration model. +Then n = 2. +We may formulate it as motion +equations +a(i) ⊕ x = y ⊕ b(i), +(28) +where, a(i), b(i), x, y ∈ M, for i = 1, · · · , m. Here, y is the transformation motion +from the world coordinate system to the robot base. The aim is to find the best +motions x and y to satisfy (28). Then, let +x = [x, y] , +(29) +zi = +� +a(i) ⊕ x +� +⊕ +� +−y ⊕ b(i)� +, +(30) +8 + +and z = [z1, · · · , zm] ∈ Mm. We have the following motion optimization problem +min +�1 +2∥z(x)∥2 : x ∈ M2 +� +(31) +for this hand-eye calibration model. This is a 12-dimensional unconstrained +optimization problem. +Example 3 We now consider the simultaneous localization and mapping +(SLAM) problem. We have a directed graph G = (V, E) [4], where each vertex +i ∈ V corresponds to a robot pose x(i) ∈ M for i = 1, · · · , n, and each directed +edge (arc) (i, j) ∈ E corresponds to a relative measurement y(ij), also a motion. +There are m directed edges in E. The aim is to find the best x(i) for i = 1, · · · , n, +to satisfy +y(ij) = −x(i) ⊕ x(j) +(32) +for (i, j) ∈ E. We now formulate it as a motion optimization problem. Let +x = [x(1), · · · , x(n)]. Then x ∈ Mn. Let +z(ij) = −x(i) ⊕ x(j) ⊕ +� +−y(ij)� +(33) +for (i, j) ∈ E and z = [z(ij) : (i, j) ∈ E] ∈ Mm. Then we have the following +motion optimization problem +min +�1 +2∥z(x)∥2 : x ∈ Mn +� +(34) +for the SLAM problem. This is a 6n-dimensional unconstrained optimization +problem. +6 +Final Remarks +In this paper, we introduced rotation addition and motion addition. We use +them to represent rigid body movements in the 3D space. Unit dual quater- +nions are no longer used. As motions are six-dimensional real vectors, and unit +dual quaternions involve eight-dimensional variables, the infinitesimal unit and +9 + +additional constraints, this simplifies the representations. As unit dual quater- +nions are used widely in engineering, it is worth studying more about motions +and motion addition, and explore their applications. +On the other hand, quaternions are still deeply involved. Thus, it is worth +studying more about rotation addition to see if we may express rotation addition +explicitly, to replace the role of quaternions. +Acknowledgment I am thankful to Chen Ling and Ziyan Luo for their +comments. +References +[1] M. Bryson and S. 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Zheng, “The geometric structure of unit +quaternion with application in kinematic cpntrol”, Journal of Mathemati- +cal Analysis and Applications 389 (2012) 1352-1364. +[17] E. Wei, S. Jin and Q. Zhang, “Autonomous navigation of mars probe using +X-ray pulsars: Modeling and results”, Advances in Space Research 51 +(2013) 849-857. +[18] S.B. Williams, P. Newman and G. Dissanayake, “Autonomous underwater +simultaneous localization and map building”, IEEE International Confer- +ence on Robotics and Automation (ICRA), (2000) 1793-1798. +[19] H.Q. Zhuang, Z. Roth and R. Sudhakar, “Simultaneous robot-world and +tool/flange calibration by solving homogeneous transformation of the form +AX = Y B”, IEEE Transactions on Robotics and Automation 10 (1994) +549-554. +12 + diff --git a/WtE1T4oBgHgl3EQfbgSn/content/tmp_files/load_file.txt b/WtE1T4oBgHgl3EQfbgSn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..664244ad634379ef0ba6b8851013f6c52db4fa18 --- /dev/null +++ b/WtE1T4oBgHgl3EQfbgSn/content/tmp_files/load_file.txt @@ -0,0 +1,260 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf,len=259 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='03174v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='RO] 9 Jan 2023 Motion Addition and Motion Optimization Liqun Qi∗ January 10, 2023 Abstract We introduce rotation addition and motion addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' In this way, motions replace unit dual quaternions to represent rigid body movements in the 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' The infinites- imal unit is no longer needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' By means of motion addition, we formulate two classical problems in robot research, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=', the hand-eye calibration problem and the simultaneous localization and mapping (SLAM) problem as motion optimization problems, which are actually real unconstrained optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' In particular, it avoids to go through the unit dual quaternion operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Motion, motion addition, rotation, rotation addition, hand-eye calibra- tion, simultaneous localization and mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' 1 Introduction In [12], motions, as six-dimensional real vectors, were introduced to represent rigid body movements in the 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Operators mapping from motions to unit dual quaternions, and from unit dual quaternions to motions, were defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Two classical problems in robot research, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=', the hand-eye calibration problem [7, 9, 10, 14, 19] and the simultaneous localization and mapping (SLAM) problem [1, 2, 3, 4, 17, 18] were formulated as motion optimization problems, which are actually unconstrained real optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' This approach improved the dual quaternion optimization approach, studied in [11, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' ∗Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018 China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong (maqilq@polyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='hk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' 1 The approach in [12] raised one question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Can we define some operations of motions such that we may use motions to represent rigid body movements in the 3D space directly, instead of mapping into unit dual quaternions to make operations, then mapping back?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' In this way, we may even get rid of the infinitesimal unit, which somehow downgrades translations too much, and whose true role is merely an operation symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' In this paper, we introduce rotation addition and motion addition to rep- resent rigid body movements in the 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Two motions were combined directly without bothering unit dual quaternions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' The infinitesimal unit is no longer to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Based on these, we establish the motion optimization formu- lations for the hand-eye calibration problem and the SLAM problem directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' The distribution of the remainder of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' In the next section, we review some basic properties of quaternions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Then we define rota- tion addition and study its properties in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We further define motion addition and study its properties in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' In Section 5 we formulate the hand-eye calibration problem and the SLAM problem as motion optimization problems without involving unit dual quaternions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Some final remarks are made in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' 2 Quaternions A quaternion ˜q = [q0, q1, q2, q3] is a real four-dimensional vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We use a tilde symbol to distinguish a quaternion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Denote the set of all quaternions by Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Suppose that we have two quaternions ˜p = [p0, p1, p2, p3], ˜q = [q0, q1, q2, q3] ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' The sum of ˜p and ˜q is defined as ˜p + ˜q = [p0 + q0, p1 + q1, p2 + q2, p3 + q3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' (1) The product of ˜p and ˜q is defined by ˜p˜q = [p0q0−p1q1−p2q2−p3q3, p0q1+p1q0+p2q3−p3q2, p0q2+p2q0−p1q3+p3q1, p0q3+p3q0+p2q3−p3q2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' (2) The conjugate of a quaternion ˜q = [q0, q1, q2, q3] ∈ Q is defined as ˜q∗ = [q0, −q1, −q2, −q3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Let ˜1 = [1, 0, 0, 0] ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Then for any ˜q ∈ Q, we have 2 ˜q˜1 = ˜1˜q = ˜q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' For any ˜p, ˜q ∈ Q, we have (˜p˜q)∗ = ˜q∗˜p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' (3) A quaternion ˜q = [q0, q1, q2, q3] ∈ Q is called a unit quaternion if q2 0 + q2 1 + q2 2 + q2 3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' (4) Denote the set of all unit quaternions by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' The following proposition can be verified directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' If ˜p, ˜q ∈ U, then ˜p˜q ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' For any ˜q ∈ U, we have ˜q˜q∗ = ˜q∗˜q = ˜1, (5) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=', ˜q is invertible and ˜q−1 = ˜q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' A quaternion ˜q = [0, q1, q2, q3] ∈ Q is called a vector quaternion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' A quaternion ˜q ∈ Q is a vector quaternion if and only if ˜q = −˜q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Suppose that ˜q is a vector quaternion and ˜p is a quaternion, then ˜p∗˜q˜p is still a vector quaternion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' 3 Rotation Addition A rotation r is a three-dimensional real vector r = [r1, r2, r3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Write r = θl, where l = [l1, l2, l3] is a unit vector, satisfying l2 1 + l2 2 + l2 3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' (6) Call l the rotation axis of r, while θ the rotation angle of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' If 0 ≤ θ < 2π, then r is called a basic rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' The set of all basic rotations is denoted as S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Thuswe have S = � r ∈ V : ∥r∥2 2 = r2 1 + r2 2 + r2 3 < 4π2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' The space of three-dimensional real vectors is denoted as V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' 3 A mapping U : V → U, called the UQ operator, defined in [12], was as follows U(r) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 � cos ∥r∥2 2 , r ∥r∥2 sin ∥r∥2 2 � , if r ̸= 0, ˜1, if r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' (7) Another mapping R : U → V, called the rotation operator, was also defined in [12] as follows R(˜q) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 2 cos−1 q0 � q2 1 + q2 2 + q2 3 [q1, q2, q3], if q2 0 ̸= 1, 0, otherwise, (8) where ˜q = [q0, q1, q2, q3] ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Then for any ˜q ∈ U \\ {−˜1}, we have U(R(˜q)) = ˜q, and for any r ∈ S, we have R(U(r)) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' The above contents can be found in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Suppose that r = [r1, r2, r3], s = [s1, s2, s3] ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Define r ⊕ s = R(U(r)U(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' (9) We call this operation rotation addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' In general it is not commutative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=', we have r ⊕ s ̸= s ⊕ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We call it rotation addition, not rotation multiplication, as it is closely re- lated to the addition operation in space V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' For any r ∈ V, we have r ⊕ 0 = 0 ⊕ r = r = r + 0 = 0 + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' (10) Here 0 = [0, 0, 0] is the zero in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' By the definition of rotation addition, we may prove the following proposi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Suppose that r, s ∈ V represent two rotations of a rigid body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Then r ⊕ s ∈ V represents the combining rotation of r and s for the rigid body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' For r ∈ V, we always have r ⊕ (−r) = (−r) ⊕ r = r + (−r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' (11) 4 For any r(1), r(2), r(3) ∈ V, � r(1) ⊕ r(2)� ⊕ r(3) = r(1) ⊕ � r(2) ⊕ r(3)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' (12) We have the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Conjecture 1 For any r, s ∈ V, we have r ⊕ s = r + s, (13) if and only if r ⊕ s = s ⊕ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' (14) 4 Motion Addition A motion x is a six-dimensional real vector, which has two parts: x = [r, t], where r is a rotation, while t is a translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Both r and t are three- dimensional real vectors, r, t ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Rotations have been studied in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Denote the space of six-dimensional real vectors by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Note that from a vector t ∈ V, we may have a vector quaternion ˜t = [0, t] ∈ Q and from a vector quaternion ˜t = [0, t] ∈ Q, we have a vector t ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We use a tilde symbol to denote the vector quaternion, corresponding to a vector in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Suppose that we have two motions x = [r, t], y = [s, u] ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We now define the motion addition x ⊕ y as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We have x ⊕ y = [r ⊕ s, v], (15) where ˜v = [0, v] = U(s)∗˜tU(s) + ˜u, (16) ˜t = [0, t] and ˜u = [0, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Suppose that x, y ∈ M represent two motions of a rigid body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Then x ⊕ y ∈ M represents the combined motion of x and y for the rigid body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We know that the motion of a rigid body in the 3D space can be rep- resented by unit dual quaternions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Then motion x is corresponding to a unit dual quaternion [12] ˆp = ˜p + ǫ 2 ˜p˜t, (17) 5 and y is corresponding to a unit dual quaternion [12] ˆq = ˜q + ǫ 2 ˜q˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' (18) Here, ˜p and ˜q are unit quaternions, ˜p = U(r) and ˜q = U(s), ǫ is the infinitesimal unit, satisfying ǫ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Combining ˆp with ˆq, we have ˆpˆq = ˜p˜q + ǫ 2 ˜p(˜t˜q + ˜q˜u) = (˜p˜q) + ǫ 2(˜p˜q)(˜q∗˜t˜q + ˜u) = (˜p˜q) + ǫ 2(˜p˜q)˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' As ˜q = U(s), we have (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Since ˜p˜q is corresponding to r ⊕ s, we have the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Suppose that x = [r, t] ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Let y = [−r, u], where ˜u = [0, u] = −U(r)˜tU(r)∗, (19) where ˜t = [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Then we have x ⊕ y = y ⊕ x = 0 = [0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' (20) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='1 we have x ⊕ y = [r ⊕ (−r), v] = [0, v], where ˜v = U(−r)∗˜tU(−r) + ˜u = U(r)˜tU(r)∗ − U(r)˜tU(r)∗ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=', x ⊕ y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' On the other hand, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='1 we have y ⊕ x = [(−r) ⊕ r, v] = [0, v], where ˜v = U(r)∗(−U(r))˜tU(r)∗U(r) + ˜t = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=', y ⊕ x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' 6 In this way, we see that we use motions instead of unit dual quaternions to represent rigid body movements in the 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We also have the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' For any x, y, z ∈ M, (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' (21) In general, we have x ⊕ y ̸= y ⊕ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Also see [12, 13, 16] for the discussion on the logarithm of unit dual quaternions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' 5 Motion Optimization As in [12], for a motion x = [r, t] = [r1, r2, r3, t1, t2, t3] ∈ M, define its magnitude as |x| = � |r1|2 + |r2|2 + |r3|2 + σ|t1|2 + σ|t2|2 + σ|t3|2, (22) where σ is a positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We define a motion vector x = [x(1), · · · , x(n)] as an n-component vector such that its ith components is a motion x(i) = � r(i), t(i)� = � r(i) 1 , r(i) 2 , r(i) 3 , t(i) 1 , t(i) 2 , t(i) 3 � ∈ M, for i = 1, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Thus, x may be also regarded as a 6n-dimensional real vec- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We use small bold letters such as x to denote motion vectors, and denote the space of n-component motion vectors by Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' The norm in Mn is defined by ∥x∥ = � � � � n � i=1 ��x(i)��2 = � � � � n � i=1 ���r(i) 1 ��� 2 + ���r(i) 2 ��� 2 + ���r(i) 3 ��� 2 + σ ���t(i) 1 ��� 2 + σ ���t(i) 2 ��� 2 + σ ���t(i) 3 ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' (23) Then it is a norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Assume that z : Mn → Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' A motion optimization problem is formu- lated as min �1 2∥z(x)∥2 : x ∈ Mn � , (24) which is a 6n-dimensional unconstrained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' 7 In [12], the hand-eye calibration problem and the simultaneous localization and mapping (SLAM) problem were formulated as motion optimization problem via the UDQ operators and the motion operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Here, we use motion addition to establish somewhat different motion optimization formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Example 1 We now consider the 1989 Shiu and Ahmad [14] and Tsai and Lenz [15] hand-eye calibration model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Then n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We may formulate it as motion equations a(i) ⊕ x = x ⊕ b(i), (25) where, ˆa(i),ˆb(i), x ∈ M, for i = 1, · · · , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Here, x is the transformation motion from the camera (eye) to the gripper (hand), ˆa(i),ˆb(i), for i = 1, · · · , m, are some data motions from experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' The aim is to find the best motion x to satisfy (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Then, let zi = � a(i) ⊕ x � ⊕ � −x ⊕ b(i)� , (26) and z = [z1, · · · , zm] ∈ Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We have the following motion optimization problem min �1 2∥z(x)∥2 : x ∈ M � (27) for this hand-eye calibration model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' This is a 6-dimensional unconstrained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Example 2 We further consider the 1994 Zhuang, Roth and Sudhaker [19] hand-eye calibration model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Then n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We may formulate it as motion equations a(i) ⊕ x = y ⊕ b(i), (28) where, a(i), b(i), x, y ∈ M, for i = 1, · · · , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Here, y is the transformation motion from the world coordinate system to the robot base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' The aim is to find the best motions x and y to satisfy (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Then, let x = [x, y] , (29) zi = � a(i) ⊕ x � ⊕ � −y ⊕ b(i)� , (30) 8 and z = [z1, · · · , zm] ∈ Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We have the following motion optimization problem min �1 2∥z(x)∥2 : x ∈ M2 � (31) for this hand-eye calibration model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' This is a 12-dimensional unconstrained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Example 3 We now consider the simultaneous localization and mapping (SLAM) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We have a directed graph G = (V, E) [4], where each vertex i ∈ V corresponds to a robot pose x(i) ∈ M for i = 1, · · · , n, and each directed edge (arc) (i, j) ∈ E corresponds to a relative measurement y(ij), also a motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' There are m directed edges in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' The aim is to find the best x(i) for i = 1, · · · , n, to satisfy y(ij) = −x(i) ⊕ x(j) (32) for (i, j) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We now formulate it as a motion optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Let x = [x(1), · · · , x(n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Then x ∈ Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Let z(ij) = −x(i) ⊕ x(j) ⊕ � −y(ij)� (33) for (i, j) ∈ E and z = [z(ij) : (i, j) ∈ E] ∈ Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Then we have the following motion optimization problem min �1 2∥z(x)∥2 : x ∈ Mn � (34) for the SLAM problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' This is a 6n-dimensional unconstrained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' 6 Final Remarks In this paper, we introduced rotation addition and motion addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' We use them to represent rigid body movements in the 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Unit dual quater- nions are no longer used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' As motions are six-dimensional real vectors, and unit dual quaternions involve eight-dimensional variables, the infinitesimal unit and 9 additional constraints, this simplifies the representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' As unit dual quater- nions are used widely in engineering, it is worth studying more about motions and motion addition, and explore their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' On the other hand, quaternions are still deeply involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Thus, it is worth studying more about rotation addition to see if we may express rotation addition explicitly, to replace the role of quaternions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Acknowledgment I am thankful to Chen Ling and Ziyan Luo for their comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Bryson and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Sukkarieh, “Building a robust implementation of bearing only inertial SLAM for a UAV”, Journal of Field Robotics 24 (2007) 113- 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfbgSn/content/2301.03174v1.pdf'} +page_content=' Bultmann, K.' metadata={'source': 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Østvold1and Magiel Bruntink2 +1Norwegian Computing Center, Oslo, Norway +2Software Improvement Group, Amsterdam, The Netherlands +{feiyang, bjarte}@nr.no, m.bruntink@sig.eu +Keywords: +Data privacy protection, code review, static analysis +Abstract: +Code review is a critical step in the software development life cycle, which assesses and boosts the code’s +effectiveness and correctness, pinpoints security issues, and raises its quality by adhering to best practices. Due +to the increased need for personal data protection motivated by legislation, code reviewers need to understand +where personal data is located in software systems and how it is handled. Although most recent work on +code review focuses on security vulnerabilities, privacy-related techniques are not easy for code reviewers to +implement, making their inclusion in the code review process challenging. In this paper, we present ongoing +work on a new approach to identifying personal data processing, enabling developers and code reviewers +in drafting privacy analyses and complying with regulations such as the General Data Protection Regulation +(GDPR). +1 +INTRODUCTION +The General Data Protection Regulation (GDPR) lays +the legal foundation for data protection in the EU and +increases individual data protection rights through- +out Europe. It also carries significant fines of up to +4% of yearly worldwide revenue for businesses that +do not comply with the legislation. Many IT system +providers, especially software-producing firms, may +need to alter their systems in order to comply with +the GDPR. This is predicted to require significant ef- +fort (Blume, 2016). As a result, providing software +engineers in the industry with effective and system- +atic ways to build data protection into software is an +essential and beneficial study topic (Lenhard et al., +2017). Organizations are pushing security to the soft- +ware development life cycle, such as code review, +to prevent software security vulnerabilities (Braz and +Bacchelli, 2022). Similarly, to comply with privacy- +by-design and perform privacy analysis tasks, code +reviewers would benefit from similar tools to those +used for security to identify privacy-related patterns +in software. +Developers address privacy concerns using data +security terminology, and this vocabulary confines +their notions of privacy to threats outside of the orga- +nization (Hadar et al., 2018). However, even though +data security is the main prerequisite of data privacy, +privacy protection in software is still very much dif- +ferent from traditional security-related vulnerabilities. +And according to Bambauer: “security and privacy +can and should be treated as distinct concerns” (Bam- +bauer, 2013). Developers struggle to convert legal, +ethical, and social privacy concerns into concrete +technology and solutions (Notario et al., 2015). +Assessing privacy involves not only finding per- +sonal data in the software but also evaluating compli- +ance with the related processing. GDPR defines as +processing: “any operation or set of operations which +is performed on personal data or on sets of personal +data, whether or not by automated means.” The defi- +nition encompasses a vast range of actions performed +on personal data, such as collecting, recording, orga- +nization, structuring, storage, adaption or modifica- +tion, retrieval, transit, etc. Privacy assessment tasks +beg the question: How can we assist code review- +ers and software developers in assessing personal data +processing? By identifying personal data and the rele- +vant processing in the system, code reviewers can un- +cover interesting patterns and utilize them to redesign +the system to be more privacy-friendly or perform pri- +vacy analysis. +In this paper, we present ongoing work on a +novel approach designed to assist developers and code +reviewers in identifying personal data processing, +which can subsequently be used for privacy analy- +sis. This enables developers and code reviewers to as- +sist organizations with a variety of important privacy- +arXiv:2301.01568v1 [cs.SE] 4 Jan 2023 + +related tasks, such as completing a data protection im- +pact assessment (DPIA) and creating a privacy policy. +2 +RELATED WORK +An essential step in the software development pro- +cess, code reviewing incorporates both manual and/or +automated reviews. The main goal of code reviews is +to assess and boost the code’s effectiveness and cor- +rectness, pinpoint security issues, and raise its qual- +ity by adhering to best practices (McIntosh et al., +2014). To automatically evaluate code, a variety of +vulnerability detection tools have been built. They are +also known as source code analyzers or static analysis +tools, as they can analyze a program’s code without +having to execute it (McGraw, 2008). +CodeQL1, and Semgrep (r2c, 2022)2 are two pop- +ular code review tools that utilize static analysis. Cod- +eQL treats code as if it were data, and issues are mod- +eled as queries. +Following the extraction of these +queries from the code, they are executed against a +database. The database is a directory containing data, +a source reference for displaying query results, query +results, and log files. Semgrep matches grammatical +patterns on parsed programs (represented as an Ab- +stract Syntax Tree (AST)) instead of matching string +or regular expression (regex) patterns on the program +as a string. Semgrep makes it considerably simpler +to construct customized rules than CodeQL, which +needs rules to be defined in QL, a declarative object- +oriented query language. +There is relatively little published work that fo- +cuses on code reviews to identify privacy-related vul- +nerabilities, and it is problematic to translate current +security knowledge to privacy, which we will explain +in Section 3. There are studies on the identification +of personal data that are valuable to our research. +Fugkeaw et al. (Fugkeaw et al., 2021) proposed AP2I +to enable organizations to identify and manage per- +sonal data in the local file system automatically. By +monitoring network traffic, ReCon (Ren et al., 2016) +utilized machine learning to identify probable per- +sonal data breaches. van der Plas et al. (van der Plas, +2022) used CodeBERT, a RoBERT-like transformer +model, to identify personal data in Git commits. +1https://codeql.github.com/ +2https://semgrep.dev/ +3 +BACKGROUND AND +CHALLENGES +Data privacy analysis is becoming as crucial as secu- +rity vulnerability discovery and has brought a new di- +mension to the data security dilemma (Bertino, 2016). +It is advantageous for code reviewers to be able to +conduct a similar privacy analysis that they did for se- +curity. +The current state of the art is mostly focused on +security analysis. Although data security is a primary +requirement for data privacy, the analysis domain and +identification process are rather different (Jain et al., +2016). +Simply adopting security mechanisms and +mindsets to analyze privacy can be misguided, and +even harmful (Bambauer, 2013). +Integration of recent studies on assessing software +privacy during code review is challenging. On the +subject of program analysis, three well-known pri- +vacy analysis methods are available. First, static anal- +ysis based on bytecode requires project compilation, +whereas dynamic taint analysis requires project ex- +ecution. This is not practical nor efficient for code +reviewers to implement. A machine learning-based +technique is similarly difficult to implement, as it re- +quires a large and diverse training data set. Obtaining +and generating such data sets requires additional ef- +fort and could be outside the scope of code reviewers’ +capabilities. Lastly, text analysis based on UI widgets +is constrained for privacy by domain-specific UI at- +tributes. A financial web application that employs a +model trained on an Android health mobile applica- +tion is unlikely to benefit. Code reviewers require an +approach that is simple to deploy, efficient, and adapt- +able (Buse and Zimmermann, 2012). +Due to the complex nature of privacy and the flu- +idity of the definition of personal data, identifying the +processing of personal data in the codebase presents +challenges. +In the following paragraphs, we highlight the two +most significant challenges related to the task in the +context of code review. +3.1 +The Ambiguous Definition of +Personal Data +Article 4(1) in GDPR defines personal data as: +any information relating to an identified or +identifiable natural person (‘data subject’); +an identifiable natural person is one who can +be identified, directly or indirectly, in partic- +ular by reference to an identifier such as a +name, an identification number, location data, + +an online identifier or to one or more factors +specific to the physical, physiological, genetic, +mental, economic, cultural or social identity +of that natural person. +The definition of personal data in the GDPR is so +broad that almost any information may qualify as per- +sonal data if it refers to a specific individual, such as +the fact that a person is wearing a red shirt (Berˇciˇc +and George, 2009). The definition is also semanti- +cally ambiguous. +In contrast to the fact that certain data may be +anonymous from the start (such as weather sensor +data without any connection to real people), other data +may initially be personal data but later be success- +fully altered to no longer have any connection to an +identified or identifiable natural person. This empha- +sizes how flexible the categorization of personal data +is (Finck and Pallas, 2020). +The same data point may be personal or non- +personal depending on the context and may thus be +covered by the regulation or not. This implies that +the categories of personal data in the software vary +depending on the software and the processing under- +lying it. For instance, health data such as blood pres- +sure and medical records, for example, are sensitive +for a health application, but location data is sensitive +for navigation software. +Even if we accept that content-wise every item of +information can be considered personal data if it can +be related to an individual, the GDPR’s definition is +still rather vague structurally since it is not always +clear what kind of structure every ‘record’ of an indi- +vidual must have to be considered personal data (Voss +and Houser, 2019). +Due to the ambiguous nature of the definition of +personal data in the relevant legislation, it is practi- +cally difficult for us to have a clear and fixed identifier +to precisely locate personal data in code. +3.2 +What Counts as Sensitive +Processing? +Data subjects may agree to data processing for par- +ticular reasons. This is the usual legal basis but only +counts as one factor. Processing may also be “nec- +essary for the performance of a contract to which the +data subject is a party or in order to take steps at the +request of the data subject prior to entering into a con- +tract.” (Voss and Houser, 2019) +Unfortunately, concerns that arise in principle +about the relationship between contract and consent +tend to be avoided in reality by disregarding consent +requirements (Pormeister, 2017). +We cannot rely on existing privacy policies and +written consent to uncover personal data processing in +the codebase. This requires us to consider all potential +personal data processing in the codebase. Later we +will explain how we define and identify the relevant +processing in software in Section 4.1.2. +4 +APPROACH +We present an approach to identify instances of per- +sonal data processing in the codebase and present +them in a way that facilitates the code review. +The approach has three primary phases: pattern +matching, labeling, and grouping of results. As input, +we take the codebase, which consists of source code +files. Then, a static analyzer will evaluate these source +code files using our rules and patterns. The code snip- +pets discovered by the static analyzer are then labeled +according to the various features they include. Fi- +nally, we allow users to group the results by single +or several labels, allowing a personalized exploration +of the findings. +An illustration of our approach is shown in Fig- +ure 1. +Codebase +Code snippets +Labeled snippets +Results +pattern +matching +labeling +grouping +Figure 1: Approach +4.1 +Design Choices +In the following paragraphs, we discuss our design +choices for implementing the approach. +4.1.1 +Types of Findings +We want to have a basic default list of personal data +that we want to locate, this is mostly personal identi- +fication and characteristics data, such as full name, +email address, gender, sexual orientation, and age. +We call them fixed personal data. +According to different types of software, we cus- +tomize default lists for them. For example, for a bank- +ing/finance application, the list may contain bank ac- + +count numbers, credit scores, and salary information. +This type of personal data is subject to context - the +types of processing in specific software, which we +named contextual personal data. +Depending on how we locate the mentioned per- +sonal data in the software, we can divide their occur- +rences in code into simply two types. +The first is in clear text. This includes all kinds +of locations where personal data appear in clear text. +It is verbatim or direct personal data. For example, +a credit card number appears in an SQL query, or an +email address falls into a log function. +The other type is more common and subtle, where +personal data is stored in a variable or an object. De- +pending on the different types of programming lan- +guages, the object types might vary from a local vari- +able, a class instance, or a prototype. This means we +aim to find the code that processes this type of data. +4.1.2 +Types of Processing +Simply locating every instance of personal data pro- +duces a large number of results. Many of these do not +directly help the code reviewer’s work, which is to +find meaningful processing. We want to use a hybrid +approach to cover as many as processing as possible. +Processing personal data represents a specific +behavior. +This motivates our first approach: +to +use an action name tag to find relevant processing. +We adopted most of the verbs from Section 3 of +DPV (Pandit et al., 2019) 3. These vocabularies help +us to find relevant processes in the software. +The second approach is the identification of exter- +nal libraries. We know that modern applications rely +on various APIs to achieve different goals. Therefore, +obtaining a list of relevant APIs and detecting the ex- +istence of personal data that flows into them helps us +find meaningful patterns. +4.2 +Pattern Matching +The first step is to feed our codebase (consisting of +source code files) to the static analyzer for pattern +matching. We chose Semgrep as our analyzer because +of its user-friendly rules and rapid processing perfor- +mance. Depending on the different syntactic char- +acteristics of personal data, as we discussed in Sec- +tion 4.1.1, we adopt a hybrid approach that combines +two different types of analysis. +• Match personal data in clear text using regular ex- +pression matching. +3https://w3c.github.io/dpv/dpv/ +• Taint analysis to find flows in each file between +a source (where personal data enters the analysis +scope) and a sink (where personal data gets pro- +cessed) that match our criteria. +Our personal data processing rules currently sup- +port Java, JavaScript, and TypeScript as our primary +analysis domains. However, our rules for identifying +clear-text personal data apply to the vast majority of +Semgrep-supported languages. +4.2.1 +Source and Sink +Our prototype classifies the sources into nine separate +categories. As stated in Section 4.1.1, we divide fixed +personal data into four different categories: account, +contact, national ID, and personal ID. Included are +five more contextual personal data categories, such as +location, health, and financial data. In addition, we +provide a template for identifying the processing of +personal data and enable code reviewers and develop- +ers to submit additional personal data simply by enter- +ing the relevant keywords. Then, corresponding rules +will be automatically produced for future use. +Sinks are categorized into five main types. Three +types of action: data manipulation (M), data trans- +portation (T), and data creation/deletion (C/D). An- +other two represent two special types: database (DB) +and encryption (E). +A sink’s name may contain a specific type of +source. For example, setLatitude(100,100) +does not take any source into the method, but +includes a source identifier Latitude and a sink +identifier set, showing that it processes values +directly as a source into a sink. +We call this spe- +cial type of sink a source-specific sink. +When a +source-specific sink invokes anything, we mark this +source-specific sink as the new source but the caller +of the source-specific sink as the new sink. For exam- +ple, +in +gpsTracker.setLatitude(100,100), +setLatitude +becomes +the +new +source +and +gpsTracker is the new sink. +Inspired by how Privado 4 uses regular expres- +sions to identify GDPR-related data in Java applica- +tions, a sample Semgrep rule that matches the pattern +of account data source goes into a transportation (T) +sink is shown below in Figure 2, followed by a sample +code snippet detected in Figure 5. +4.3 +Labeling +The identified findings from Semgrep are in the form +of various lengths of code snippets (consisting of +statements and expressions). Each finding contains +4https://www.privado.ai + +Figure 2: Semgrep rule: find personal data flows from ac- +count data source to transportation sink +Figure 3: Sample code snippet (from ToolJet) detected by +Semgrep showing a flow from account personal data to a +transportation sink. +at least one detected sink and one source (or an object +that received value from a source). We abstract the +structure of possible sources and sinks in each code +snippet using the symbols below. +• O ranges over sources +• I ranges over sinks +• IO ranges over source-specific sinks +We write ¯O as shorthand for a possibly empty se- +quence O1,··· ,On. Here the underscore +represents +a placeholder for an expression that is insignificant in +terms of privacy - it is neither a source nor sink nor +contains a value from a source. +Below is a list of the common flow abstracts be- +tween sources and sinks that we observed in each +code snippet. Each abstract represents a typical flow, +for example, 1 to 3 show that there are values pass- +ing through a sink to a source, from a non-privacy +sensitive value ( 1 ) or from another source ( 2 ) or +from innovating a sink inside another source object +( 3 ). +1 O = .I( ) +2 O2 = .I(O1, ) +3 O2 = .O1.I( ) +4 += .O.I( ) +5 += .I( ¯O, ) +6 +.O.I( ) +7 +.O.I( , ¯O) +8 +.IO( ) +9 +.IO( , ¯O) +10 +.I( ¯O, ) +For each identified code snippet, we label them +with 22 labels (9 types of source, 5 types of sink, 5 +types of source-specific sink, and 3 types of change +in the sensitivity level), which are listed in Table 1. +Besides the definition of source and sinks, we also in- +troduce an important label: sensitivity. The sensitivity +level can increase, decrease, and stay the same in one +identified code snippet. +O +Nine types of source: {O1,O2,...,O9} +I +Five types of sink: {I1,I2,...,I5} +I O +Five types of source-specific sink : +{IO1,IO2,...,IO5} +S +Sensitivity level change: {up, down, equal} +Table 1: Labels to be assigned to each code snippet +Sensitivity Level +Not every result shares the same +level of sensitivity regarding personal data process- +ing. After processing, the data from the source might +remain at a similar sensitivity level, become more sen- +sitive, or become less sensitive. +• S = up: 1 , 4 , 5 +• S = equal: 2 , 3 , 6 , 7 , 8 , 9 +• S = down: 10 +4.4 +Result Presentation +Johnson et al. (Johnson et al., 2013) pointed out +that “because the results are dumped onto a code re- +viewer’s screen with no distinct structure causing him +to spend a lot of time trying to figure out what needs +to be done”. This indicates that developers and code +reviewers may not benefit from ungrouped code snip- +pets from static analysis tools if they are not presented +in a sensible manner. +To tackle this issue, we present a two-phase tech- +nique to process the findings from Semgrep and +present them to code reviewers in a smart way. +After each code snippet is labeled, we start to +group them for presentation using their labels and +other criteria. Criteria for grouping include not only +the labels but also other properties: +• neighboring results will be combined (same file +and within a line number threshold); + +1 this.usersService.updateUser(newUser.id, +2 +{defaultorganizationId: +3 +newUser.organizationId}) +4 +.catch((error) => +5 +6 +organization id', error); +7 +);1 +rules: +2 +- id: account-data-transportation +3 +languages: +4 +- javascript +5 +- java +6 +- typescript +7 +mode: taint +8 +message: Match found +9 +pattern-sinks: +10 +patterns: +11 +- pattern: $sINK(..., $Z, ...) +12 +- metavariable-regex: +13 +metavariable: $sINK +14 +regex: (?i)(.*(sendImovelconnectlescaplstreamlredirectl +eraselquerylsharelstorltransferltransmitlmove).*) +15 +pattern-sources: +16 +- pattern-regex: (?i).*(?:accountluserlcustomerldoctorIpatientl +policyholder|insurer|claimant)[^\ls/(;)I,=!>]{o,3}(idlnumber|nol +num) +17 +- pattern-regex: (?i)(?:facebookltwitterlinstagramllinkedinl +pinterestlbehanceldribble)[^lls/(;)l,=!>]{,2}(?:idlaccountl +usernamelhandle) +18 +- pattern-regex: (?i).*(?:dbldatabaseljiralsqllpostgresImongolaws) +[^//s/(;)/,=!>]{0,3}(pswlpswdlpasswordlpasswrd) +19 +- pattern-regex: (?i)(.*(?:dbldatabaseljiralsqllpostgresImongol +aws)[^lls/(;)/,=!>]{0,3}user[^/ls/(;),=!>]{0,3}name)(.*(account| +customer doctor patient teacher student person organilzsjation +company)[^/s/(;)1,=!>]{0,3}name) +20 +severity: WARNING• same or similar source/sink name; +• same API usage (e.g. every code snippet that is +related to the same API MongoDB). +Figure 4 following provides a straightforward il- +lustration of how we present our results. The results +are presented in two separate sections: plain text re- +sults and flow results. Users have the flexibility to +select any label or label combination to filter the re- +sults. +Plain Text Personal Data +email: 'user@enterprise.org' +queryExecutor('SELECT John Doe FROM userDB') +Flows +Group by: source +Account data +Government ID data +Transportation +Manipulation +Sensitivity level: up +Creation/deletion +Manipulation +createRepository{userId: userId; + pasId: pasId; + ssn: ssn} +Figure 4: Example presentation of the result. Personal data +occurrences is at the top and personal data processing code +is at the bottom. +5 +DEMONSTRATION +We created rules in Semgrep trying to capture as +many useful findings for our analysis. +The soft- +ware we analyzed here is ToolJet5, an open-source +low-code framework for building React-based web +applications. ToolJet’s implementation is mostly in +JavaScript and TypeScript. Users can build internal +tools using ToolJet’s prebuilt UI widgets to connect to +data sources like databases, API endpoints, and exter- +nal services. This means ToolJet has many parts that +process personal data, which makes it a good starting +example. +Our Semgrep rules produce a total of 1,589 results +from ToolJet’s source code. We manually reviewed +each of the results and calculated the precision for +each category. If a single result can clearly demon- +strate the processing of personal data, we consider it +relevant and it could be beneficial for privacy code +review. Surprisingly, most false positives come from +the personal data occurrence detector (with a preci- +sion of only 46.6%), while most personal data pro- +cessing results are relevant (with an average of 90.9% +precision for categories that have more than 50 code +snippets identified). +5https://github.com/ToolJet/ToolJet +Detailed statistics are listed in Tables 2 and Ta- +ble 3. +M +T +C/D +DB +E +L +Account +66 +171 +84 +24 +- +21 +Contact +89 +175 +36 +3 +- +3 +Personal ID +56 +133 +41 +7 +1 +4 +Online ID +6 +26 +1 +- +- +1 +Location +1 +2 +- +- +- +- +Table 2: The code snippet count for each identified source +and sink identified, ‘-’ marks labels for which our approach +detected no code snippet. Sink types are: data manipulation +(M), data transportation (T), data creation/deletion (C/D), +database (DB), encryption (E) and log (L). +M +T +C/D +DB +E +L +Account +90.9 +90.6 +95.2 +91.67 +- +95.2 +Contact +89.9 +94.9 +80.6 +* +- +* +Personal ID +92.9 +81.9 +85.4 +* +* +* +Online ID +* +84.6 +* +- +- +* +Location +* +* +- +- +- +- +Table 3: The precision of code snippet relevance (in %) for +each identified type of source and sink, ‘-’ marks the labels +for which our approach did not detect any code snippet, ‘*’ +marks the labels for which our approach detected less than +10 results. Sink types are: data manipulation (M), data +transportation (T), data creation/deletion (C/D), database +(DB), encryption (E) and log (L). +Figure 5 shows a simple interesting example +of a grouped result showing how personal data +userId is retrieved from a local repository in +app users.service.ts and then utilized to generate +many data structures, such as the app object in +app service.ts. +app.service.ts +app_users.service.ts +Figure +5: +Grouped +example +results +showing +how +organizationUserId flows between functions. + +1 +async create(user: User, appId: string, organizationUserId: string, +2 +role: string): Promise { +3 +const organizationuser = await this.organizationusersRepository. +4 +findone({ where: { id: organizationUserId } }); +5 +6 +return await this.appUsersRepository.save( +7 +this.appusersRepository.create(L +8 +appId, +9 +userId: organizationUser.userId, +10 +role, +11 +createdAt: new Date(), +12 +updatedAt: new Date(), +13 +( +14 +); +15L +async create(user: User): Promise { +2 +const app = await this.appsRepository.save( +w +this.appsRepository.create( +4 +name: 'Untitled app', +5 +createdAt: new Date(), +6 +updatedAt: new Date(), +7 +organizationId: user.organizationId, +8 +userId: user.id. +9 +( +10 +);5.1 +Future work +Since our objective is to identify all relevant process- +ing of personal data in source code, reducing false +negatives is our next primary priority. However, in +our case, false positives are not a major concern. Due +to the subtlety of personal data processing, determin- +ing relevance without human assistance is particularly +challenging. Specifying the analysis to certain spe- +cific patterns would ease manual analysis. This ne- +cessitates the implementation of a privacy taxonomy. +Using Ethyca’s taxonomy (Ethyca, 2022) as an exam- +ple, we may modify our labels to match the technique +with the taxonomy. +As an extension of this article, we propose an au- +tomated mapping of personal data in an unpublished +(under review) manuscript (Tang et al., 2023) to assist +developers and code reviewers in identifying privacy- +related code. The mapping based on static analysis +automatically detects personal data and the code that +processes it, and we offer semantics of personal data +flows. +6 +CONCLUSIONS +This short paper presented ongoing work on a novel, +customizable approach to identify personal data pro- +cessing for code review. This three-phase technique +first uses Semgrep to match patterns in the code based +on rules for sources and sinks, then associates code +snippets generated from pattern matching with a set of +behavioral labels, and finally groups results to reduce +code reviewer workload. Our demonstration shows +the utility and feasibility of this method for gathering +and presenting code snippets related to personal data +processing from a codebase. +Along with the continued development of the ap- +proach architecture (refined rules for source and sink, +more meaningful labels, and additional criteria for +grouping), future work will focus on expanding the +case study to include a larger set of open-source soft- +ware from various domains and conducting a thor- +ough user evaluation of the resulting platform. +ACKNOWLEDGEMENTS +This work is part of the Privacy Matters (PriMa) +project. The PriMa project has received funding from +European Union’s Horizon 2020 research and inno- +vation program under the Marie Skłodowska-Curie +grant agreement No. 860315. +REFERENCES +Bambauer, D. E. (2013). Privacy versus security. J. Crim. +L. & Criminology, 103:667. +Berˇciˇc, B. and George, C. (2009). Identifying personal data +using relational database design principles. 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J., Fern´andez, J. D., Hamed, R. G., +Kiesling, E., Lizar, M., et al. (2019). Creating a vo- +cabulary for data privacy. In OTM Confederated In- +ternational Conferences” On the Move to Meaningful +Internet Systems”, pages 714–730. Springer. +Pormeister, K. (2017). Informed consent to sensitive per- +sonal data processing for the performance of digital +consumer contracts on the example of “23andMe”. +Journal of European Consumer and Market Law, 6(1). +r2c (2022). Semgrep. https://semgrep.dev/. (Accessed on +11/15/2022). +Ren, J., Rao, A., Lindorfer, M., Legout, A., and Choffnes, +D. (2016). Recon: Revealing and controlling pii leaks +in mobile network traffic. In Proceedings of the 14th +Annual International Conference on Mobile Systems, +Applications, and Services, pages 361–374. +Tang, F., Østvold, B. M., and Bruntink, M. (2023). Mapping +personal data in source code for GDPR compliance. +van der Plas, N. (2022). Detecting PII in Git commits. TU +Delft Master’s thesis. +Voss, W. 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American Business Law Journal, +56(2):287–344. + diff --git a/YdAzT4oBgHgl3EQfmv1k/content/tmp_files/load_file.txt b/YdAzT4oBgHgl3EQfmv1k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ab86cdb7e52ff47a9bf2c1eef9d00076a15b19f3 --- /dev/null +++ b/YdAzT4oBgHgl3EQfmv1k/content/tmp_files/load_file.txt @@ -0,0 +1,493 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf,len=492 +page_content='Identifying Personal Data Processing for Code Review Feiyang Tang1, Bjarte M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Østvold1and Magiel Bruntink2 1Norwegian Computing Center, Oslo, Norway 2Software Improvement Group, Amsterdam, The Netherlands {feiyang, bjarte}@nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='no, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='bruntink@sig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='eu Keywords: Data privacy protection, code review, static analysis Abstract: Code review is a critical step in the software development life cycle, which assesses and boosts the code’s effectiveness and correctness, pinpoints security issues, and raises its quality by adhering to best practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Due to the increased need for personal data protection motivated by legislation, code reviewers need to understand where personal data is located in software systems and how it is handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Although most recent work on code review focuses on security vulnerabilities, privacy-related techniques are not easy for code reviewers to implement, making their inclusion in the code review process challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' In this paper, we present ongoing work on a new approach to identifying personal data processing, enabling developers and code reviewers in drafting privacy analyses and complying with regulations such as the General Data Protection Regulation (GDPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 1 INTRODUCTION The General Data Protection Regulation (GDPR) lays the legal foundation for data protection in the EU and increases individual data protection rights through- out Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' It also carries significant fines of up to 4% of yearly worldwide revenue for businesses that do not comply with the legislation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Many IT system providers, especially software-producing firms, may need to alter their systems in order to comply with the GDPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' This is predicted to require significant ef- fort (Blume, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' As a result, providing software engineers in the industry with effective and system- atic ways to build data protection into software is an essential and beneficial study topic (Lenhard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Organizations are pushing security to the soft- ware development life cycle, such as code review, to prevent software security vulnerabilities (Braz and Bacchelli, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Similarly, to comply with privacy- by-design and perform privacy analysis tasks, code reviewers would benefit from similar tools to those used for security to identify privacy-related patterns in software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Developers address privacy concerns using data security terminology, and this vocabulary confines their notions of privacy to threats outside of the orga- nization (Hadar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' However, even though data security is the main prerequisite of data privacy, privacy protection in software is still very much dif- ferent from traditional security-related vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' And according to Bambauer: “security and privacy can and should be treated as distinct concerns” (Bam- bauer, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Developers struggle to convert legal, ethical, and social privacy concerns into concrete technology and solutions (Notario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Assessing privacy involves not only finding per- sonal data in the software but also evaluating compli- ance with the related processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' GDPR defines as processing: “any operation or set of operations which is performed on personal data or on sets of personal data, whether or not by automated means.” The defi- nition encompasses a vast range of actions performed on personal data, such as collecting, recording, orga- nization, structuring, storage, adaption or modifica- tion, retrieval, transit, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Privacy assessment tasks beg the question: How can we assist code review- ers and software developers in assessing personal data processing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' By identifying personal data and the rele- vant processing in the system, code reviewers can un- cover interesting patterns and utilize them to redesign the system to be more privacy-friendly or perform pri- vacy analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' In this paper, we present ongoing work on a novel approach designed to assist developers and code reviewers in identifying personal data processing, which can subsequently be used for privacy analy- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' This enables developers and code reviewers to as- sist organizations with a variety of important privacy- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='01568v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='SE] 4 Jan 2023 related tasks, such as completing a data protection im- pact assessment (DPIA) and creating a privacy policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 2 RELATED WORK An essential step in the software development pro- cess, code reviewing incorporates both manual and/or automated reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' The main goal of code reviews is to assess and boost the code’s effectiveness and cor- rectness, pinpoint security issues, and raise its qual- ity by adhering to best practices (McIntosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' To automatically evaluate code, a variety of vulnerability detection tools have been built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' They are also known as source code analyzers or static analysis tools, as they can analyze a program’s code without having to execute it (McGraw, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' CodeQL1, and Semgrep (r2c, 2022)2 are two pop- ular code review tools that utilize static analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Cod- eQL treats code as if it were data, and issues are mod- eled as queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Following the extraction of these queries from the code, they are executed against a database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' The database is a directory containing data, a source reference for displaying query results, query results, and log files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Semgrep matches grammatical patterns on parsed programs (represented as an Ab- stract Syntax Tree (AST)) instead of matching string or regular expression (regex) patterns on the program as a string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Semgrep makes it considerably simpler to construct customized rules than CodeQL, which needs rules to be defined in QL, a declarative object- oriented query language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' There is relatively little published work that fo- cuses on code reviews to identify privacy-related vul- nerabilities, and it is problematic to translate current security knowledge to privacy, which we will explain in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' There are studies on the identification of personal data that are valuable to our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Fugkeaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' (Fugkeaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=', 2021) proposed AP2I to enable organizations to identify and manage per- sonal data in the local file system automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' By monitoring network traffic, ReCon (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=', 2016) utilized machine learning to identify probable per- sonal data breaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' van der Plas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' (van der Plas, 2022) used CodeBERT, a RoBERT-like transformer model, to identify personal data in Git commits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 1https://codeql.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='com/ 2https://semgrep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='dev/ 3 BACKGROUND AND CHALLENGES Data privacy analysis is becoming as crucial as secu- rity vulnerability discovery and has brought a new di- mension to the data security dilemma (Bertino, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' It is advantageous for code reviewers to be able to conduct a similar privacy analysis that they did for se- curity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' The current state of the art is mostly focused on security analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Although data security is a primary requirement for data privacy, the analysis domain and identification process are rather different (Jain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Simply adopting security mechanisms and mindsets to analyze privacy can be misguided, and even harmful (Bambauer, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Integration of recent studies on assessing software privacy during code review is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' On the subject of program analysis, three well-known pri- vacy analysis methods are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' First, static anal- ysis based on bytecode requires project compilation, whereas dynamic taint analysis requires project ex- ecution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' This is not practical nor efficient for code reviewers to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' A machine learning-based technique is similarly difficult to implement, as it re- quires a large and diverse training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Obtaining and generating such data sets requires additional ef- fort and could be outside the scope of code reviewers’ capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Lastly, text analysis based on UI widgets is constrained for privacy by domain-specific UI at- tributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' A financial web application that employs a model trained on an Android health mobile applica- tion is unlikely to benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Code reviewers require an approach that is simple to deploy, efficient, and adapt- able (Buse and Zimmermann, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Due to the complex nature of privacy and the flu- idity of the definition of personal data, identifying the processing of personal data in the codebase presents challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' In the following paragraphs, we highlight the two most significant challenges related to the task in the context of code review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='1 The Ambiguous Definition of Personal Data Article 4(1) in GDPR defines personal data as: any information relating to an identified or identifiable natural person (‘data subject’);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' an identifiable natural person is one who can be identified, directly or indirectly, in partic- ular by reference to an identifier such as a name, an identification number, location data, an online identifier or to one or more factors specific to the physical, physiological, genetic, mental, economic, cultural or social identity of that natural person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' The definition of personal data in the GDPR is so broad that almost any information may qualify as per- sonal data if it refers to a specific individual, such as the fact that a person is wearing a red shirt (Berˇciˇc and George, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' The definition is also semanti- cally ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' In contrast to the fact that certain data may be anonymous from the start (such as weather sensor data without any connection to real people), other data may initially be personal data but later be success- fully altered to no longer have any connection to an identified or identifiable natural person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' This empha- sizes how flexible the categorization of personal data is (Finck and Pallas, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' The same data point may be personal or non- personal depending on the context and may thus be covered by the regulation or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' This implies that the categories of personal data in the software vary depending on the software and the processing under- lying it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' For instance, health data such as blood pres- sure and medical records, for example, are sensitive for a health application, but location data is sensitive for navigation software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Even if we accept that content-wise every item of information can be considered personal data if it can be related to an individual, the GDPR’s definition is still rather vague structurally since it is not always clear what kind of structure every ‘record’ of an indi- vidual must have to be considered personal data (Voss and Houser, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Due to the ambiguous nature of the definition of personal data in the relevant legislation, it is practi- cally difficult for us to have a clear and fixed identifier to precisely locate personal data in code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='2 What Counts as Sensitive Processing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Data subjects may agree to data processing for par- ticular reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' This is the usual legal basis but only counts as one factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Processing may also be “nec- essary for the performance of a contract to which the data subject is a party or in order to take steps at the request of the data subject prior to entering into a con- tract.” (Voss and Houser, 2019) Unfortunately, concerns that arise in principle about the relationship between contract and consent tend to be avoided in reality by disregarding consent requirements (Pormeister, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' We cannot rely on existing privacy policies and written consent to uncover personal data processing in the codebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' This requires us to consider all potential personal data processing in the codebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Later we will explain how we define and identify the relevant processing in software in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 4 APPROACH We present an approach to identify instances of per- sonal data processing in the codebase and present them in a way that facilitates the code review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' The approach has three primary phases: pattern matching, labeling, and grouping of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' As input, we take the codebase, which consists of source code files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Then, a static analyzer will evaluate these source code files using our rules and patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' The code snip- pets discovered by the static analyzer are then labeled according to the various features they include.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Fi- nally, we allow users to group the results by single or several labels, allowing a personalized exploration of the findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' An illustration of our approach is shown in Fig- ure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Codebase Code snippets Labeled snippets Results pattern matching labeling grouping Figure 1: Approach 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='1 Design Choices In the following paragraphs, we discuss our design choices for implementing the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='1 Types of Findings We want to have a basic default list of personal data that we want to locate, this is mostly personal identi- fication and characteristics data, such as full name, email address, gender, sexual orientation, and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' We call them fixed personal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' According to different types of software, we cus- tomize default lists for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' For example, for a bank- ing/finance application, the list may contain bank ac- count numbers, credit scores, and salary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' This type of personal data is subject to context - the types of processing in specific software, which we named contextual personal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Depending on how we locate the mentioned per- sonal data in the software, we can divide their occur- rences in code into simply two types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' The first is in clear text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' This includes all kinds of locations where personal data appear in clear text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' It is verbatim or direct personal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' For example, a credit card number appears in an SQL query, or an email address falls into a log function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' The other type is more common and subtle, where personal data is stored in a variable or an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' De- pending on the different types of programming lan- guages, the object types might vary from a local vari- able, a class instance, or a prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' This means we aim to find the code that processes this type of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='2 Types of Processing Simply locating every instance of personal data pro- duces a large number of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Many of these do not directly help the code reviewer’s work, which is to find meaningful processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' We want to use a hybrid approach to cover as many as processing as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Processing personal data represents a specific behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' This motivates our first approach: to use an action name tag to find relevant processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' We adopted most of the verbs from Section 3 of DPV (Pandit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=', 2019) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' These vocabularies help us to find relevant processes in the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' The second approach is the identification of exter- nal libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' We know that modern applications rely on various APIs to achieve different goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Therefore, obtaining a list of relevant APIs and detecting the ex- istence of personal data that flows into them helps us find meaningful patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='2 Pattern Matching The first step is to feed our codebase (consisting of source code files) to the static analyzer for pattern matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' We chose Semgrep as our analyzer because of its user-friendly rules and rapid processing perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Depending on the different syntactic char- acteristics of personal data, as we discussed in Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='1, we adopt a hybrid approach that combines two different types of analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Match personal data in clear text using regular ex- pression matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 3https://w3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='io/dpv/dpv/ Taint analysis to find flows in each file between a source (where personal data enters the analysis scope) and a sink (where personal data gets pro- cessed) that match our criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Our personal data processing rules currently sup- port Java, JavaScript, and TypeScript as our primary analysis domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' However, our rules for identifying clear-text personal data apply to the vast majority of Semgrep-supported languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='1 Source and Sink Our prototype classifies the sources into nine separate categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' As stated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='1, we divide fixed personal data into four different categories: account, contact, national ID, and personal ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Included are five more contextual personal data categories, such as location, health, and financial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' In addition, we provide a template for identifying the processing of personal data and enable code reviewers and develop- ers to submit additional personal data simply by enter- ing the relevant keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Then, corresponding rules will be automatically produced for future use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Sinks are categorized into five main types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Three types of action: data manipulation (M), data trans- portation (T), and data creation/deletion (C/D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' An- other two represent two special types: database (DB) and encryption (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' A sink’s name may contain a specific type of source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' For example, setLatitude(100,100) does not take any source into the method, but includes a source identifier Latitude and a sink identifier set, showing that it processes values directly as a source into a sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' We call this spe- cial type of sink a source-specific sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' When a source-specific sink invokes anything, we mark this source-specific sink as the new source but the caller of the source-specific sink as the new sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' For exam- ple, in gpsTracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='setLatitude(100,100), setLatitude becomes the new source and gpsTracker is the new sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Inspired by how Privado 4 uses regular expres- sions to identify GDPR-related data in Java applica- tions, a sample Semgrep rule that matches the pattern of account data source goes into a transportation (T) sink is shown below in Figure 2, followed by a sample code snippet detected in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='3 Labeling The identified findings from Semgrep are in the form of various lengths of code snippets (consisting of statements and expressions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Each finding contains 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='privado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='ai Figure 2: Semgrep rule: find personal data flows from ac- count data source to transportation sink Figure 3: Sample code snippet (from ToolJet) detected by Semgrep showing a flow from account personal data to a transportation sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' at least one detected sink and one source (or an object that received value from a source).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' We abstract the structure of possible sources and sinks in each code snippet using the symbols below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' O ranges over sources I ranges over sinks IO ranges over source-specific sinks We write ¯O as shorthand for a possibly empty se- quence O1,··· ,On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Here the underscore represents a placeholder for an expression that is insignificant in terms of privacy - it is neither a source nor sink nor contains a value from a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Below is a list of the common flow abstracts be- tween sources and sinks that we observed in each code snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Each abstract represents a typical flow, for example, 1 to 3 show that there are values pass- ing through a sink to a source, from a non-privacy sensitive value ( 1 ) or from another source ( 2 ) or from innovating a sink inside another source object ( 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 1 O = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='I( ) 2 O2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='I(O1, ) 3 O2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='O1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='I( ) 4 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='I( ) 5 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='I( ¯O, ) 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='I( ) 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='I( , ¯O) 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='IO( ) 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='IO( , ¯O) 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='I( ¯O, ) For each identified code snippet, we label them with 22 labels (9 types of source, 5 types of sink, 5 types of source-specific sink, and 3 types of change in the sensitivity level), which are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Besides the definition of source and sinks, we also in- troduce an important label: sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' The sensitivity level can increase, decrease, and stay the same in one identified code snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' O Nine types of source: {O1,O2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=',O9} I Five types of sink: {I1,I2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=',I5} I O Five types of source-specific sink : {IO1,IO2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=',IO5} S Sensitivity level change: {up, down, equal} Table 1: Labels to be assigned to each code snippet Sensitivity Level Not every result shares the same level of sensitivity regarding personal data process- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' After processing, the data from the source might remain at a similar sensitivity level, become more sen- sitive, or become less sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' S = up: 1 , 4 , 5 S = equal: 2 , 3 , 6 , 7 , 8 , 9 S = down: 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='4 Result Presentation Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=', 2013) pointed out that “because the results are dumped onto a code re- viewer’s screen with no distinct structure causing him to spend a lot of time trying to figure out what needs to be done”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' This indicates that developers and code reviewers may not benefit from ungrouped code snip- pets from static analysis tools if they are not presented in a sensible manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' To tackle this issue, we present a two-phase tech- nique to process the findings from Semgrep and present them to code reviewers in a smart way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' After each code snippet is labeled, we start to group them for presentation using their labels and other criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Criteria for grouping include not only the labels but also other properties: neighboring results will be combined (same file and within a line number threshold);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 1 this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='usersService.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='updateUser(newUser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='id, 2 {defaultorganizationId: 3 newUser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='organizationId}) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content="catch((error) => 5 6 organization id', error);" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 7 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='1 rules: 2 id: account-data-transportation 3 languages: 4 javascript 5 java 6 typescript 7 mode: taint 8 message: Match found 9 pattern-sinks: 10 patterns: 11 pattern: $sINK(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=', $Z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=') 12 metavariable-regex: 13 metavariable: $sINK 14 regex: (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='i)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' *(sendImovelconnectlescaplstreamlredirectl eraselquerylsharelstorltransferltransmitlmove).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' *) 15 pattern-sources: 16 pattern-regex: (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='*(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' :accountluserlcustomerldoctorIpatientl policyholder|insurer|claimant)[^\\ls/(;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=')I,=!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='>]{o,3}(idlnumber|nol num) 17 pattern-regex: (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='i)(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' :facebookltwitterlinstagramllinkedinl pinterestlbehanceldribble)[^lls/(;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=')l,=!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='>]{,2}(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' :idlaccountl usernamelhandle) 18 pattern-regex: (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='*(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' :dbldatabaseljiralsqllpostgresImongolaws) [^//s/(;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=')/,=!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='>]{0,3}(pswlpswdlpasswordlpasswrd) 19 pattern-regex: (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='i)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='*(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' :dbldatabaseljiralsqllpostgresImongol aws)[^lls/(;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=')/,=!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='>]{0,3}user[^/ls/(;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='),=!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='>]{0,3}name)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' *(account| customer doctor patient teacher student person organilzsjation company)[^/s/(;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=')1,=!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='>]{0,3}name) 20 severity: WARNING• same or similar source/sink name;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' same API usage (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' every code snippet that is related to the same API MongoDB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Figure 4 following provides a straightforward il- lustration of how we present our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' The results are presented in two separate sections: plain text re- sults and flow results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Users have the flexibility to select any label or label combination to filter the re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=" Plain Text Personal Data email: 'user@enterprise." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content="org' queryExecutor('SELECT John Doe FROM userDB') Flows Group by: source Account data Government ID data Transportation Manipulation Sensitivity level: up Creation/deletion Manipulation createRepository{userId: userId;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' pasId: pasId;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' ssn: ssn} Figure 4: Example presentation of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Personal data occurrences is at the top and personal data processing code is at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 5 DEMONSTRATION We created rules in Semgrep trying to capture as many useful findings for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' The soft- ware we analyzed here is ToolJet5, an open-source low-code framework for building React-based web applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' ToolJet’s implementation is mostly in JavaScript and TypeScript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Users can build internal tools using ToolJet’s prebuilt UI widgets to connect to data sources like databases, API endpoints, and exter- nal services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' This means ToolJet has many parts that process personal data, which makes it a good starting example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Our Semgrep rules produce a total of 1,589 results from ToolJet’s source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' We manually reviewed each of the results and calculated the precision for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' If a single result can clearly demon- strate the processing of personal data, we consider it relevant and it could be beneficial for privacy code review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Surprisingly, most false positives come from the personal data occurrence detector (with a preci- sion of only 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='6%), while most personal data pro- cessing results are relevant (with an average of 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='9% precision for categories that have more than 50 code snippets identified).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='com/ToolJet/ToolJet Detailed statistics are listed in Tables 2 and Ta- ble 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' M T C/D DB E L Account 66 171 84 24 21 Contact 89 175 36 3 3 Personal ID 56 133 41 7 1 4 Online ID 6 26 1 1 Location 1 2 Table 2: The code snippet count for each identified source and sink identified, ‘-’ marks labels for which our approach detected no code snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Sink types are: data manipulation (M), data transportation (T), data creation/deletion (C/D), database (DB), encryption (E) and log (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' M T C/D DB E L Account 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='9 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='6 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='67 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='2 Contact 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='9 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='6 Personal ID 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='9 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='9 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='4 Online ID 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='6 Location Table 3: The precision of code snippet relevance (in %) for each identified type of source and sink, ‘-’ marks the labels for which our approach did not detect any code snippet, ‘*’ marks the labels for which our approach detected less than 10 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Sink types are: data manipulation (M), data transportation (T), data creation/deletion (C/D), database (DB), encryption (E) and log (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Figure 5 shows a simple interesting example of a grouped result showing how personal data userId is retrieved from a local repository in app users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='ts and then utilized to generate many data structures, such as the app object in app service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='ts app_users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='ts Figure 5: Grouped example results showing how organizationUserId flows between functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 1 async create(user: User, appId: string, organizationUserId: string, 2 role: string): Promise { 3 const organizationuser = await this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='organizationusersRepository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 4 findone({ where: { id: organizationUserId } });' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 5 6 return await this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='appUsersRepository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='save( 7 this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='appusersRepository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='create(L 8 appId, 9 userId: organizationUser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='userId, 10 role, 11 createdAt: new Date(), 12 updatedAt: new Date(), 13 ( 14 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 15L async create(user: User): Promise { 2 const app = await this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='appsRepository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='save( w this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='appsRepository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content="create( 4 name: 'Untitled app', 5 createdAt: new Date(), 6 updatedAt: new Date(), 7 organizationId: user." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='organizationId, 8 userId: user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 9 ( 10 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content='1 Future work Since our objective is to identify all relevant process- ing of personal data in source code, reducing false negatives is our next primary priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' However, in our case, false positives are not a major concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Due to the subtlety of personal data processing, determin- ing relevance without human assistance is particularly challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Specifying the analysis to certain spe- cific patterns would ease manual analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' This ne- cessitates the implementation of a privacy taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Using Ethyca’s taxonomy (Ethyca, 2022) as an exam- ple, we may modify our labels to match the technique with the taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' As an extension of this article, we propose an au- tomated mapping of personal data in an unpublished (under review) manuscript (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=', 2023) to assist developers and code reviewers in identifying privacy- related code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' The mapping based on static analysis automatically detects personal data and the code that processes it, and we offer semantics of personal data flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 6 CONCLUSIONS This short paper presented ongoing work on a novel, customizable approach to identify personal data pro- cessing for code review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' This three-phase technique first uses Semgrep to match patterns in the code based on rules for sources and sinks, then associates code snippets generated from pattern matching with a set of behavioral labels, and finally groups results to reduce code reviewer workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Our demonstration shows the utility and feasibility of this method for gathering and presenting code snippets related to personal data processing from a codebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Along with the continued development of the ap- proach architecture (refined rules for source and sink, more meaningful labels, and additional criteria for grouping), future work will focus on expanding the case study to include a larger set of open-source soft- ware from various domains and conducting a thor- ough user evaluation of the resulting platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work is part of the Privacy Matters (PriMa) project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' The PriMa project has received funding from European Union’s Horizon 2020 research and inno- vation program under the Marie Skłodowska-Curie grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' 860315.' metadata={'source': 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Revealing and controlling pii leaks in mobile network traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' In Proceedings of the 14th Annual International Conference on Mobile Systems, Applications, and Services, pages 361–374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Tang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=', Østvold, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=', and Bruntink, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Voss, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' and Houser, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' Personal data and the GDPR: providing a competitive advantage for US companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} +page_content=' American Business Law Journal, 56(2):287–344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfmv1k/content/2301.01568v1.pdf'} diff --git a/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf b/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b6bac1b32ce1e0fdd00b14aa90793495228ab6de --- /dev/null +++ b/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c89b2cd51527b6f0ebf62642253ad65478e9cd1d361da0a22f66683f45965fbd +size 3645650 diff --git a/YtE2T4oBgHgl3EQfZAd1/vector_store/index.pkl b/YtE2T4oBgHgl3EQfZAd1/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..51708dc6ccacee5bf03229966e8371e82770c81e --- /dev/null +++ b/YtE2T4oBgHgl3EQfZAd1/vector_store/index.pkl @@ -0,0 +1,3 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Chena,b,* +a Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas 78758, +United State +b Omega Optics, Inc., 8500 Shoal Creek Blvd., Austin, Texas 78757, United State +*Email：chenrt@austin.utexas.edu +Abstract +In the context of continued spread of coronavirus disease 2019 (COVID-19) caused by +SARS-CoV-2 and the emergence of new variants, the demand for rapid, accurate, and +frequent detection is increasing. Besides, the new predominant strain, Omicron variant, +manifests more similar clinical features to those of other common respiratory infections. +The concurrent detection of multiple potential pathogens helps distinguish SARS-CoV-2 +infection from other diseases with overlapping symptoms, which is significant for patients +to receive tailored treatment and containing the outbreak. Here, we report a lab-on-a-chip +biosensing platform for SARS-CoV-2 detection based on subwavelength grating micro-ring +resonator. The sensing surface is functionalized by specific antibody against SARS-CoV-2 +spike protein, which could produce redshifts of resonant peaks by antigen-antibody +combination, thus achieving quantitative detection. Additionally, the sensor chip is +integrated with a microfluidic chip with an anti-backflow Y-shaped structure that enables +the concurrent detection of two analytes. In this study, we realized the detection and +differentiation of COVID-19 and influenza A H1N1. Experimental results show that the +limit of detection of our device reaches 100 fg/mL (1.31 fM) within 15 min detecting time, +and cross-reactivity tests manifest the specificity of the optical diagnostic assay. Further, +the integrated packaging and streamlined workflow facilitate its use for clinical +applications. Thus, the biosensing platform offers a promising solution to achieve +ultrasensitive, selective, multiplexed, and quantitative point-of-care detection of COVID- +19. + +Keywords: COVID-19, SARS-CoV-2 spike protein, biosensor, subwavelength, micro-ring resonator, +influenza + + + +2 + +1. Introduction +Since a novel coronavirus disease (COVID-19) caused by severe acute respiratory syndrome +coronavirus 2 (SARS-CoV-2) was reported in late 2019, the world is continuously threatened +by the potentially fatal infectious disease.1,2 The highly contagious virus quickly spread to most +continents within a few weeks and has infected more than 620 million people, including 6.5 +million deaths by November 2022.3,4 During the COVID-19 pandemic, genetic variants of +SARS-COV-2 are constantly emerging and spreading as new epidemic strains.5,6 Despite the +tremendous advance in epidemiological studies and vaccine developments curbing the progress +of epidemic, the variant viruses are more contagious, and may generate immune escape from +innate or acquired immune responses, resulting in continued transmission around the world.7,9 +The early diagnose of suspected cases is still regarded as the best viable solution to slow down +the pandemic without guaranteed preventive measures.10 +The Omicron variant was officially named by the WHO on November 26, 2021, and quickly +replaced Delta variant as the predominant strain.11 Compared with the Alpha or Delta subvariant, +Omicron has lower disease severity, hospitalization and death rates.11-13 On the other hand, +COVID-19 has become more atypical due to the mild symptoms of Omicron and thus difficult +to distinguish from other infectious diseases with similar symptoms.13,14 Among common +respiratory infections, influenza presents many overlapping clinical manifestations with +COVID-19, including fever, cough, sore throat, headache, fatigue, and myalgia.15,16 However, +the basic reproduction number (R0) of COVID-19 (9.5 for Omicron) is much higher than +influenza (0.9–2.1), which means that the SARS‐CoV‐2 virus is more contagious.15,17 Hence, +timely detection is significant for patients to receive tailored treatment and curbing the epidemic, +considering the long-term co-existing of COVID-19 and other respiratory infectious diseases +with overlapping symptoms. +In clinical practice, the primary method for COVID-19 diagnosis relies on real-time reverse +transcription−polymerase chain reaction (RT-PCR). PCR-based detection shows high accuracy +even in the early stages of infection, which makes it the “gold standard” in diagnosis.18 +However, RT-PCR needs advanced laboratories, expensive equipment, and medically trained +personnel. Besides, RT-PCR assay requires the amplification for viral RNA, which is time- +consuming and may delay the diagnosis.19,20 Because of the increasing demand for testing and +the difficulty of large-scale RT-PCR testing, reliable and rapid diagnostic methods for COVID- +19 are necessary. To overcome the challenges, some lab-on-a-chip (LOC) platforms for point- +of-care (POC) COVID-19 diagnosis have been developed.9,21-23 Due to short detection time, +convenient operation, and low sample requirements, these LOC techniques show great potential +in clinical applications. Among these reported techniques, optical biosensors utilize the change +of optical properties results from photon-matter interaction to realize the detection of analytes. +Optical sensing shows several advantages in the biomedical scenario, including label-free + +3 + +detection, multiplexing capability, instantaneous measurements, etc.10,24 In the past decades, the +maturity of silicon photonics and photonic integrated circuits (PICs) technology promoted the +development of optical sensors.25 Furthermore, the compatibility with microfluidic systems +offers more opportunities for LOC sensing.26-28 Micro-ring resonators have been extensively +investigated in PIC optical sensors because of their high packing density and ease of fabrication. +Fig. 1. Working mechanism and design of SUMIRR-based biosensor. a SARS-CoV-2 spike protein in phosphate buffered +saline (PBS) is the target analyte for detection. b SARS-CoV-2 spike antibodies (blue) are conjugated on SUMIRRs as specific +probes. The SUMIRR-based biosensor could quantitatively detect spike protein (red) by tracking the redshift of resonate peaks +caused by antigen-antibody combination. C Optical micrograph of the silicon sensing chip which supports concurrent detection +of two analytes. There are eight independent sensing channels, each of which has a pair of grating couplers as the input and +output. Six channels are divided into two groups for two analytes, while the remaining two channels are treated as dummy +group and reference. d An exploded view of the LOC biosensing platform. e Photograph of the biosensing platform with a +quarter dollar for scale. f SEM images of the SUMIRR fabricated by E-beam lithography. + +a +b +SpikeProtein +Peak Shift +Intensity +M +M +Time +COVID-19 patient +SARS-CoV-2 Virus +Subwavelength Grating +SARS-CoV-2 +Anti-S Protein +SpikeProtein +Antibody +C +d +Grating Coupler +Top Microfluidic Chip +FiberArray +Dummy +Sensing Chip +GroupB +GroupA +Bottom Microfluidic +Chip +250 μm +Chip Holder +e +2μm4 + +However, the limited sensitivity of micro-ring resonator impedes the application in clinical +diagnostic assays that require low limit of detection (LOD).25,29 +In this paper, we demonstrate an optical biosensing platform for the rapid detection of SARS- +CoV-2 using subwavelength grating micro-ring resonator (SUMIRR) as shown in Fig. 1. +Compared with conventional micro-ring resonator with strip waveguides, the subwavelength +grating (SWG) structure (Fig. 1.b & f) extends the photon-matter interaction region, thus +improving sensitivity.30-33 The SUMIRRs are functionalized by SARS-CoV-2 spike antibody +for the detection of SARS-CoV-2 spikes proteins (Fig. 1.b). To address the challenges posed by +the untypical and diverse clinical manifestations of new epidemic strains, the LOC sensing +platform enables the concurrent detection of another pathogen (influenza A H1N1 in this study) +with two parallel detection groups (Fig. 1.c). To facilitate operation and improve reliability of +device, we design and fabricate a double-layer microfluidic chip with an anti-backflow Y- +shaped structure, which has two operating modes for surface functionalization and concurrent +detection, respectively. Additionally, a three-dimensional (3D)-printed holder and specialized +photonic packaging are presented to realize system-level integration. In the past few years, +various diagnostic assays for COVID-19 were widely available (summarized in Table S1). +However, a particularly promising solution, including packaging, testing and data processing, +for POC use that can achieve concurrent quantitative detection of multiple analytes is not yet +available. Our SUMIRR-based sensor detects target SARS-CoV-2 antigen with a conservative +LOD of 100 fg/mL (1.31 fM). Furthermore, cross-reactivity tests for SARS-CoV-2 and +influenza indicate the specificity of the optical diagnostic assay. The integrated device and +auxiliary portable terminal, which offers real-time data processing and the potential to interface +with electronic medical records, make the platform promising for POC diagnosis. +2. Device design +2.1 Design of the sensing platform +In considering the design of a POC platform for quantitative detection, we aimed to develop +a device with clinical practicality, high accuracy and the capacity to integrate with digital +systems. Toward this goal, we designed a microfluidic chip that supports dual-channel +concurrent detection with a 3D-printing polylactic acid (PLA) chip holder (Fig. 1.d and e). +For concurrent detection, two parallel channels need to be placed on the silicon sensing chip +with limited area (5mm × 5mm), thus the microfluidic device was designed as a double-layer +structure for reliability and ease of operation (Fig. 1.d &Fig. 2.a). As shown in Fig. 2.b, the +bottom microfluidic chip was designed with two independent microchannels, and the unilateral +channel covers three SUMIRRs as a sensing group. Each channel had an inlet and an outlet +connected to the top microfluidic chip through via holes. In addition to the left and right ports +connected to the two channels in the bottom layer respectively, the top microfluidic chip has a + +5 + +common port connected to both channels. Besides, the common port is connected to a special +Y-shape anti-counterflow splitter that can operate in two modes. In single-channel mode, the +left/right port works as inlet, while the common port works as outlet. When the sample flows +toward downstream (common port) from one branch of the Y-shaped structure, the significant +difference in flow resistance of downstream and the other branch will “block” the other branch, +thus avoiding the cross-contamination caused by counterflow (Fig. 2.c and Video S1). The other +mode is the dual-channel mode, where the common port works as inlet while left port and right +port are both outlets. In this mode, the structure is a splitter that bifurcates the upstream fluids +towards two sensing groups (Fig. 2.d and Video S1). The anti-counterflow splitter is simulated +by the finite element method to optimize chip functionality.34-36 As shown in Fig. 2.e, the Y- +shape structure can avoid counterflow in the single-channel mode without losing the bifurcating +function for dual-channel mode. +For better packaging and integration, we designed a 3D-printed holder. The upper surface of +holder is slotted, and the depth equals the thickness of sensing chip (0.75 mm). The slot area is +larger than that of sensing chip, which aims to utilize the limited elastic deformation of PDMS +to eliminate manufacturing errors from 3D printing while providing enough support for the +microfluidic chip. Besides, we designed a comb structure with a height slightly less than slot +Fig. 2. Overview of microfluidics design. a A double-layer PDMS microfluidic chip was designed for dual-channel concurrent +detection. Three ports on the top layer play different roles in different operation modes. b The optical micrograph illustrates +that the bottom microfluidic chip bonded on the sensing chip has two independent channels, and each channel covers three +resonators as one sensing group. c & d The working mechanisms of Y-shape anti-counterflow structure works in two different +modes. Red pigment was added to DI water for demonstration purpose. e Hydromechanical simulation results demonstrate the +velocity field distribution of two modes. In single-channel mode, the flow rate is 5 μL/min, and the right port works as inlet. +While the common port is inlet in dual-channel mode with a flow rate of 10 μL/min. Details of the numerical simulation are in +the SI. + +a +b +2mm +Sensing +Chip +LeftPort +Right Port +Group B +GroupA +Common +Port +200μm +C +d +e +Single-channel Mode +Dual-channelMode +Single-channel Mode +Dual-channel Mode +12 +06 + +depth, which serves two purposes. First, the comb structure helps to alignment; meanwhile, the +fiber array is fixed on the holder by UV light adhesives, and the comb structure could provide +a larger contact area resulting in higher bond strength. +2.2 Design of optical biosensor +The SUMIRR consists of a micro-ring and a bus waveguide constructed by periodic pillars with +a period much smaller than the operating wavelength.30,31 One advantage of the subwavelength +structure is that the optical properties, including effective index, loss, guiding capabilities, etc., +can be modulated by geometric topological designs.31,32 Importantly, SUMIRRs show good +potential for biosensing because the periodic structure increases the effective sensing region, +including not only the top and the side of pillars, which leads to more significant photon-matter +interaction and higher sensitivity.37,38 +In this study, SUMIRRs are fabricated on silicon on insulator (SOI) wafers using E-beam +lithography.32,39 Specifically, the silicon pattern sits on the SiO2 buried layer (Fig. 1.f and Fig. +S7), and SUMIRRs are covered by aqueous cladding as sensing units. To ensure that guided +mode exists in SWG waveguides, the effective refractive index of SWG waveguide (neff) must +be bigger than the refractive index of bottom buried layer (nSiO2 = 1.45) and cladding layer (nclad +≈ 1.35).40 The optimized design shall provide large overlap integral with analytes while +maintaining decent waveguide propagation loss.41 The resonant wavelength λres can be +expressed as: + +2 +eff +res +R n +m +π +λ +⋅ +⋅ += + +(1) +where R is the radius of micro-ring and m is a positive integer denoting the mode order. In the +biosensing scenario, biochemical reactions, such as antigen-antibody combination and DNA +hybridization,42,43 could affect the photon-matter interaction and thus change neff according to +the Lorentz-Lorenz relation, which manifests in the shift of resonant peaks (Fig. 1.b).44 +To evaluate the device's performance, we use quality factor Q and bulk sensitivity Sbulk to +quantify the sensing properties of device. A higher Q means that light has a longer lifetime in +the resonator, and thus provides stronger interaction with analyte.37 Q is defined as Eq (2). + +1 +[ +] +2 +g +res +res +res +m +n +E +Q +dE dt +FWHM +π +λ +ω +λ +α +− +⋅ += += +≈ +⋅ + +(2) +Here ωres is the resonant frequency, E is the mode’s electric field intensity, α represents the total +loss in the resonator and ng is the group index of the mode which can be approximated to neff +within a slight shift of λres.45 From the experimental point of view, Q can be approximated by +the ratio of λres to the full width at half maximum (FWHM) bandwidth of resonant peak.46,47 A +higher Q is desirable because of sharper peaks which are easier to detect. The bulk sensitivity +of SUMIRR Sbulk was defined as the slope of peak shift versus change in nclad,48 i.e., + +7 + + +eff +res +res +bulk +clad +g +clad +n +S +n +n +n +λ +λ +∂ +∆ += += +⋅ +∆ +∂ + +(3) +where Sbulk is often written as nm/Refractive Index Units (nm/RIU). Considering the influence +of Q and Sbulk on sensing performance, the inherent limit of detection (ILOD) of ring resonator +can be define as follows:49 + +ILOD +res +bulk +Q S +λ += +⋅ + +(4) +Taking ILOD as an evaluation criteria, the optimal geometric parameters of SUMIRRs are +determined by full 3D finite difference time domain method (Lumerical FDTD Solutions) and +the testing after fabrication. The details of simulation and design parameters are shown in SI. +3. Materials and methods +3.1 Device fabrication and packaging +The SUMIRR-based sensing chip was custom manufactured by Applied Nanotools Inc. using +E-Beam lithography on an SOI wafer with a 220 nm active layer and a 2 μm buried oxide layer +(Soitec). Before use, the chip was soaked in piranha for 30 min, followed by deionized (DI) +water (W4502, Sigma-Aldrich) and isopropanol (200440, CMC Materials) washing, then dried +under nitrogen (N2) stream. +Microfluidic microchannels were fabricated from an SU-8 photoresist (SU-8 2035, Kayaku +Advanced Materials) male mold patterned on a 4-inch silicon wafer (71893-07, Electron +Microscopy Sciences) using standard soft-lithography processes.50 The height of microchannel +determined by the thickness of SU-8 coating was 50 μm. Microfluidic channels were cast from +the male mold with a 10:1 mixture of PDMS base and curing agent (SYLGARD 184 silicone +elastomer kit, Dow) and cured at 90 ℃ for 40 min. The bottom and top layer thicknesses are +3mm and 6mm, respectively. Because of the small chip size, the spacing between two +independent microchannels in the bottom layer should be small to ensure sufficient space for +packaging. A 240 μm spacing between two 300μm-width channels showed good stability at a +flow rate ≤ 100 μL/min without leakage. The via holes on bottom chip and 3 ports of top layer +were punched by 0.75 mm and 1 mm punchers (PT-T983, Darwin Microfluidics), respectively. +The holder for integration was designed by CAD software (AutoCAD 2023, Autodesk) and +printed by a 3D printer (FLOW, Craftbot) with PLA polymers. The printing thickness of single +PLA layer was 100 μm. After cleaning with IPA and DI water, the bonding surface of sensing +chip and PDMS were treated with UV rays in a UV ozone cleaner (T10X10/OES, UVOCS Inc.) +for 8min and 5 min, respectively. Then, sensing chip and bottom PDMS were aligned manually +according to the pre-designed alignment marks with IPA lubrication and baked in an oven at +90℃ for 1 h. The bonding for bottom and top PDMS layers was the same as the above step, + +8 + +except that the UV treatment time was 5min. For the precise alignments of fibers and grating +couplers, the packaging holder and fiber array (pitch 127um 8 degrees, Gloriole Electroptic +Technology Corp) were anchored to a mechanical stage, as shown in Fig. S1. The input fiber +was connected to a broadband LED (DL-BX9-CS5403A, DenseLight), and the output was +connected to a C-band optical spectrum analyzer (OM-1C2MM353, Optoplex). The position of +fiber array is fine-tuned to maximize the output power without losing sufficient responses from +other channels. When the optimal position was determined, dropped UV light adhesives glue +(37-322, Edmund Optics) on the comb structure of holder, then treated with UV light overnight +for curing. +3.2 Surface functionalization of micro-ring resonators +The packaging device was treated by UV rays for 8 min to form hydroxyl groups (-OH) on +the SUMIRR surface and remove organic contaminants. 2%* organosilane reagent (3- +aminopropyl) triethoxysilane (APTES) diluted in 95% ethanol solution was pumped to +microfluidic channels via a PTFE tubing (0.6 mm ID x 1 mm OD, Uxcell) at 5 μL/min. The +flow rate was controlled by a syringe pump (70-4504, Harvard Apparatus). APTES condensed +with -OH results in the formation of siloxane bonds (Si-O-Si) on UV-treated silicon +surfaces.51,52 Notably, it is the oxide layer on Si surface that participates in silanization. The +following experiment and related studies confirmed that the natural silicon oxide layer is +sufficient for silanization.53-56 The unbounded APTES was removed by 95% ethanol washing +at 10 μL/min for 20 min, following by drying under N2 stream. Then the device was baked at +95℃ for 1 h to enhance bonding stability. After silanization, an aqueous solution of 2.5% +glutaraldehyde was introduced at 5 μL/min for 1 h, and then the chip was washed with PBS +solution (J61196AP, Thermo Fisher Scientific) at 10 μL/min for 20 min. After silanization, an +aqueous solution of 2.5% glutaraldehyde (GA) was introduced at 5 μL/min for 1 h. One +aldehyde group bound to the surface expressing -NH2 (from APTES) and the other aldehyde +group for further crosslink with bioreceptor protein. Then the chip was washed with PBS +solution at 10 μL/min for 20 min. + For antibody immobilization, 10 μg/mL antibody in PBS buffer was introduced for 40 min, +followed by 20 min PBS washing. There is a potential issue with crosslinking by GA since the +aldehyde groups are nonspecific to proteins. Hence, bovine serum albumin (BSA) was used to +block the aldehyde sites that did not combine with antibodies to avoid nonspecific bonding and +ensure the peak shift is entirely due to specific antigen-antibody combination.57,58 0.4 mg/ml +BSA (B8667, Sigma-Aldrich) was added to block the remaining sites without antibody coating. +In this work, we used two antibodies, i.e., SARS-CoV-2 spike antibody (40150-D003, Sino +Biological, Inc.) and pan influenza A nucleoprotein antibody (40205-R063, Sino Biological, +Inc.). Notably, the immobilization for two antibodies was independent. Specifically, the SARS- + +* All concentrations expressed as percentages in this paper are volume ratios. + +9 + +CoV-2 antibody was introduced from one port (left port or right port), followed by BSA +blocking, while the immobilization of influenza antibody was realized through the other port +with the same process. The SARS-CoV-2 antibody with green fluorescence was purchased from +Thermo Fisher Scientific (53-6491-82). +3.3 Sample preparation for antigen detection +The SARS-CoV-2 Spike S1-His recombinant protein (40591-V08H, Sino Biological, Inc.) +and influenza A H1N1 nucleoprotein (40205-V08B, Sino Biological, Inc.) were diluted in PBS +buffer to given concentrations depending on experimental requirements. +3.4 Sensing measurement +Measurements were conducted on an optical table. The fiber array was connected to a +broadband LED and an OSA which communicated with PC through serial communication. The +output spectrum signals were filtered to remove background noise. The real-time peak tracking +and analysis were realized with custom programs (LabView, National Instruments). Plots and +histograms were created and analyzed in Origin (Origin 2022, OriginLab). +Optical micrograph and videos were taken on the viewing stage of a microscope (BX51, +Olympus) under 5× magnification. Fluorescent images were captured with a spinning disk +confocal system (CSU-W1, Nikon) and supporting commercial software for analysis. +4. Results and Discussion +4.1 Bulk Sensitivity Analysis +Before detecting biological samples, we first measured the bulk sensitivity of fabricated +SUMIRRs and tested the sensing performance after packaging. Seven samples, including DI +water and six PBS solutions with different concentrations, were introduced to the device in +order of concentration from low to high at 10 μL/min. The transmission spectra of SUMIRRs +are shown in Fig. 3.a. The free spectral range (FSR) around 1550 nm is 14 nm, and the FWHM +of resonate peaks is ~ 0.93 nm, corresponding to a Q ≈ 1650. According to transmission spectra, +the signal amplitude slightly decreases with 0.3 dB/nm as wavelength increases. Therefore, the +resonate peaks near 1535 nm were selected as analytic targets for higher signal to noise ratio +(SNR). In Fig. 3.b, we took the resonate wavelength with DI water cladding as a baseline and +tracked peak shifts in real time. The relationship between peak shift and refractive index change +was drawn and fitted linearly in Fig. 3.c. The result shows that a linear function fits well with a +regression correlation coefficient R2 = 0.999, and the slope, i.e., the bulk sensitivity is 437.2 +nm/RIU. The noise level is ~ 3 pm, and the ILOD is ~ 2.1×10-3 RIU by Eq (4). + +10 + +4.2 Detection of SARS-CoV-2 Antigen +The viral envelope of SARS-CoV-2 consists of three structural proteins, including membrane +protein (M), envelope protein (E), and spike protein (S).60 Among these proteins, the spike +protein is crucial in penetrating host cells as the major transmembrane protein.60,61 Besides, the +spike protein exhibits diversity and specificity among coronaviruses, contributing to the most +immune recognition in the human body.62 Therefore, the S protein represents an ideal target for +the specific detection of SARS-CoV-2. In this study, we utilized the specific combination of S +protein and SARS-CoV-2 spike antibody to realize quantitative detection. The antibody we used +shows cross-reactivities with most of sub-variants of SARS-CoV-2, including Delta and +Omicron.51 After introducing samples, antibodies could combine with S proteins and form +antigen-antibody complex leading to peak shifts. +We developed a protocol for surface functionalization for the diagnostic assay on the +SUMIRR-based sensor platform (Fig. 4.a). The shift of resonate peak was monitored in real +time throughout the above process and used to represent the extent of reaction. As shown in Fig. +4.b, the trend of peak shift in each step is similar, i.e., shift increases rapidly at the beginning +and then slows down until becomes flat. To further confirm that the SUMIRR surface was +functionalized, we immobilized the SARS-CoV-2 spike antibody with green fluorescence and +took fluorescence images for characterization. Fig. 4.c indicates that antibodies with green +fluorescent coat uniformly on the waveguide, and the SWG structure manifests higher +fluorescence intensity due to a larger surface area. +To investigate the performance of the SUMIRR-based biosensing platform for COVID-19 +detection, we first evaluated the dynamic response of the sensor to SARS-CoV-2 spike protein +with different concentrations (Fig. 5). Because we only detected one antigen in this section, the +left and right sensing groups had the same function. Therefore, the reagents for surface +functionalization were introduced from the common port, divided into two tributaries by splitter, +and flowed to two parallel sensing groups, as shown in Fig. S8.a. +Fig. 3. Bulk sensitivity analysis of the fabricated SUMIRRs. The multiple represents the relative concentration compared +with the standard PBS solution. a The transmission spectra of ring resonator in different solutions. b The real-time peak shift +tracking with increasing solution concentration and refractive index. c The linear regression shows a good linear relationship +between peak shift refractive index change, and the slope indicates the bulk sensitivity of device. Details about refractive +indexes of PBS solution are shown in Table S4.50 + +a +b +c +DIWater +-35 - +FWHM = 0.93 nm +-33 +7 +DI Wat +7 1 +Experimental data +6 +Linear fitting result +-38 J +-40 +-43 +(dBr +-45 - +Amplitude ( +50- +Q~ 1653 +-58 - +1 +R2 = 0.999 +-55 - +S = 437.2 nm/RIU +89- +FSR = 14 nm +F 0 +1530 +1535 +1540 +1545 +1550 +1555 +1560 +1536 1537 1538 +300 +600 +900 +1200 +1500 +1800 +2100 +2 +6 +8 +10 +12 +Wavelength (nm) +Wavelength (nm) +Time (s) +Refractive Index Change An (RIU)11 + +We prepared samples with various concentrations of SARS-CoV-2 spike protein (from 10 +pg/mL to 1 μg/mL) and introduced samples successively in order of concentration from low to +high at 5 μL/min. The response to different samples was monitored in real time, and the dynamic +tracking of peak shift was shown in Fig. 5.a. The stepwise change indicates that the peak shift +within the same reaction time shows a positive correlation with antigen concentration. Besides, +the negative control group without antibody coating only shows a slight response (grey line in +Fig. 5.a), indicating that the reason for shift is antigen-antigen combination rather than the +refractive index change due to increasing concentration. In addition, the peak shift increases +rapidly at the beginning of reaction, and the increase slows down gradually. This general trend, +also seen in surface functionalization, manifests that the peak shifts over a certain period of +time can characterize the extent of antigen-antibody combination, thereby realizing the +quantitative detection of the antigen of interest. +After demonstrating the sensing performance of the platform, we further examined the +quantitative relationship between peak shift and antigen concentration for clinical detection and +explored the LOD of the device. To this end, we prepared samples containing SARS-CoV-2 +spike protein ranging from 100 fg/mL to 1 μg/mL. In practical detections, one sensing group +can test only one antigen in the sample because the antigen cannot be removed completely once +attached to the antibody. Hence, samples with different concentrations were introduced to +different sensing groups instead of successively introduced to one group. The reaction time and +flow rate of each sample is 10 min with a flow rate of 5 μL/min. In order to remove the antigen +that was not bound to antibody and keep the refractive index of aqueous cladding layer +consistent with that before the test, we used PBS solution to wash sensing surface for 5 min at +Fig. 4. The surface functionalization of SUMIRRs. a Schematic illustrating for surface functionalization and SARS-CoV-2 +spike protein detection. b Real-time peak shift tracking for the surface functionalization workflow. Since the solvents for +APTES and GA are ethanol and DI water, respectively, instead of standard PBS solution for the following reagents, the real- +time tracking for APTES/GA coating is demonstrated individually in the inserts with different baselines. c Fluorescence image +of the SUMIRR modified by SARS-CoV-2 spike antibody with green fluorescent. + +a +b +10 min +1 h +10 min +20 min +10 min +PBS +Anti-SAntibody +PBS +BSA +PBS +APTES-GA +2.0- +Peak Shift △入 (nm) +1.6 +1.2 +0.5 +APTES +Antibody +0.8 +900 +1800 +2700 +1.0 +0.4 - +0.5 +GA +900 +1800 +2700 +3600 +BSA +0 +006 +1800 +2700 +3600 +4500 +5400 +6300 +Time (s) +C +Spikeprotein +20μm12 + +10 μL/min. The real-time response of the SUMIRR-based biosensor to a specific concentration +of SARS-CoV-2 antigen is shown in Fig. 5.b. Additionally, we analyzed the peak shift for each +concentration before and after washing (Fig. 5.c). The concentration-dependent response after +washing fits well with the Hill model, as shown in Fig. 5.d.63,64 These results further indicate +the positive correlation between peak shift and antigen concentration, and the antigen +concentration ≥ 100fg/mL could bring a remarkable response much higher than noise level. +4.3 Specificity Analysis and Concurrent Detection +This study aimed to realize the concurrent detedction of SARS-CoV-2 and influenza viurs, +so two sensing groups need to be functionalized separately. Specifically, the SARS-CoV-2 spike +antibody and pan influenza A nucleoprotein antibody were introduced from the left port and +right port surface for functionalization,65 respectively, while the common port worked as the +inlet during testing (Fig. S8.b). Before demonstrating the concurrent detection, we first +examined the specificity of the optical assay based on antibody-antigen combination. Samples +Fig. 5. Real-time detection of SARS-CoV-2 antigen. a Real-time response of the SUMIRR-based biosensor to SARS-CoV-2 +spike protein was introduced successively by increasing order of concentration. The grey line represents the negative control +group without anti-S protein antibody coating. The interval for sample replacement and PBS washing is not shown in the figure, +so there are step changes. b Real-time response of the SUMIRR biosensor to a specific concentration of SARS-CoV-2 spike +protein. c Resonate peak shifts due to spike protein with different concentrations before and after PBS washing. d +Concentration-dependent response curve based on the peak shifts after PBS washing (details of the curve fitting are in the SI). + +a +b +10 min +5 min +PBS +10 pg/mL +100 pg/mL + 1 ng/mL +10 ng/mL +100 ng/mL1 μg/mL +PBS +COVID S Protein +PBS +0.7 +1 μg/mL +0.42 +With Antibody +100 ng/mL +F9'0 +Without Antibody +10 ng/mL +0.36 +1 ng/mL +0.5 +100 pg/mL +(uu) +(uu) V +0.30 +10 pg/mL + Shift △入 ( +1 pg/mL +0.4 +0.24 - +100 fg/mL +Shift +0.3 +0.18 - +eak +Peak +0.2 - +0.12 +0.1 +0.06 } +F0 +F0 +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +0 +150 +300 +450 +600 +750 +900 +1050 +Time (s) +Time (s) +c +d +0.42 7 +0.45 +Experimental data +Before Wash +0.40 - +0.36 +After Wash +Fitting result +0.35 - + Shift △入 (nm) +0.30 - + Shift △入 (nm) +0.30 +0.24 +0.25 . +0.20 - +0.18 . +Peak +0.15 - +0.12 +0.10 - +F 90'0 +R2 = 0.992 +0.05 - +to +0. +10-13 +10-11 +10-10 +10-9 +10-8 +10-7 +10-6 +10-13 +10-12 +10-11 +10-10 +10-9 +10-8 +10-7 +10-6 +Concentration (g/mL) +Concentration (g/mL)13 + +containing influenza nucleoprotein and SARS-CoV-2 spike protein were successively +introduced into the sensing group functionalized by SARS-CoV-2 spike antibody, and the +verification for influenza nucleoprotein antibody was taken with the same logic. The flow rate +and reaction time for each step were consistent with those in the previous section. Besides, +according to the antigen concentration in nasopharyngeal swab specimens from COVID-19 +patients, we selected 100 pg/mL as the antigen concentration for this experiment.23 As shown +in Fig. 6.a & b, the optical assay shows a clear signal difference between positive and negative. +However, the antigen that does not match the antibody also causes a slight shift (~ 15 pm), +which may be related to nonspecific interaction or incomplete BSA blocking. This undesired +response requires more careful determination of the LOD of sensing system. The concurrent +detection results further indicate the specificity of antibody-antigen combination (Fig. 6.c). +Besides, SARS-CoV-2 spike protein samples with concentrations above 100 fg/mL can result +in significant peak shifts, which are distinct from nonspecific responses. Therefore, taking 100 +fg/mL as the LOD of system is conservative and reliable. +5. Conclusions +We have developed a POC biosensor that supports the concurrent detection and +differentiation of two analytes using SUMIRRs. A microfluidic chip and an advanced packaging +are developed to integrate the sensing unit, which is more convenient for clinical tests and +improves reliability. The antibody immobilized on SUMIRRs could bring resonate peak shifts +after combining to the antigen in sample. By analyzing the redshifts, we demonstrated the +ability to quantitatively detect the concentration of SARS-CoV-2 spike protein with a LOD of +100 fg/mL. Furthermore, the cross-validation of SARS-CoV-2/influenza antigen and +corresponding antibody indicates the high specificity of the optical assay based on antibody- +antigen combination. Therefore, the SUMIRR-based LOC biosensor for the quantitative +detection of COVID-19 provides a promising solution to overcome challenges in the rapid +Fig. 6. Concurrent detection for SARS-CoV-2 spike protein and influenza nucleoprotein. a & b Specific response of +SUMIRR biosensor to different antigens. The real-time responses shown in a and b were collected from the sensing groups +functionalized by SARS-CoV-2 and influenza antibody, respectively. c Cross-reactivity tests for SARS-CoV-2 and influenza. +Samples that contain SARS-CoV-2 spike protein/ influenza nucleoprotein test positive for the corresponding antibody while +negative for the other antibody. + +b +a +c +10 min +5 min +10 min +5 min +10 min + 5 min +10 min +5 min +NP Protein + S Protein +NP Protein + S Protein +PBS +NP Protein +PBS +COVID S Protein +PBS +PBS +COVID S Protein +PBS +NP Protein +PBS +(-) +(+) +(+) +(-) +0.18 - +0.21 - +0.21 } +0.18 J +0.15 +0.18 } +0.15 +(wu) +0.12 - +Shift +nift +0.09 - + 0.09 手 +100 fg/mL +0.03 - +0.03 } +0.03 +01 +F0 +mw E 0 +300 +600 +900 +1200 +1500 +1800 +300 +600 +900 +1200 +1500 +1800 +Anti-S protein Antibody +Anti-NP protein Antibody +Time (s) +Time (s)14 + +identification of pathogenic virus with similar symptoms, and promotes the development of +POC diagnostic tools. +Conflict of interest +The authors declare that they have no competing financial interest or associative interest that +represents a conflict of interest in connection with the work reported. +CRediT authorship contribution statement +Shupeng Ning: Conceptualization, Methodology, Formal analysis, Investigation, Writing +Original Draft, Visualization Hao-Chen Chang: Methodology, Formal analysis, Supervision, +Resource Kang-Chieh Fan: Software Po-yu Hsiao: Methodology, Investigation Chenghao +Feng: Methodology, Investigation, Validation Devan Shoemaker: Methodology Ray T. Chen: +Conceptualization, Writing - Review & Editing, Supervision, Project administration, Funding +acquisition +Acknowledgments +This research program is supported by Omega Optics, NIH, AFOSR MURI and NSF and the +Texas State Endowment from the University of Texas, Austin. +Appendix A. Supplementary Information +Reference +1 +Zhao, S., Lin, Q., Ran, J., Musa, S.S., Yang, G., Wang, W., Lou, Y., Gao, D., Yang, L., He, D., 2020. Preliminary +estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in China, from 2019 to 2020: +A data-driven analysis in the early phase of the outbreak. Int. J. Infect. Dis. 92, 214-217. +2 +Ren, L.-L., Wang, Y.-M., Wu, Z.-Q., Xiang, Z.-C., Guo, L., Xu, T., Jiang, Y.-Z., Xiong, Y., Li, Y.-J., Li, X.-W., 2020. +Identification of a novel coronavirus causing severe pneumonia in human: a descriptive study. Chin. Med. J. +133(09), 1015-1024. +3 +Khan, M., Adil, S.F., Alkhathlan, H.Z., Tahir, M.N., Saif, S., Khan, M., Khan, S.T., 2020. 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Lancet 395(10224), 565-574. +61 +Shi, Y., Wang, Y., Shao, C., Huang, J., Gan, J., Huang, X., Bucci, E., Piacentini, M., Ippolito, G., Melino, G., 2020. +COVID-19 infection: the perspectives on immune responses. Cell Death Differ. 27(5), 1451-1454. +62 +Zhou, G., Zhao, Q., 2020. Perspectives on therapeutic neutralizing antibodies against the Novel Coronavirus +SARS-CoV-2. Int. J. Biol. Sci. 16(10), 1718-1723. +63 +Ligiero, C., Fernandes, T., D'Amato, D., Gaspar, F., Duarte, P., Strauch, M., Fonseca, J., Meirelles, L., da Silva, P.B., +Azevedo, R., 2022. Influence of particle size on the SARS-CoV-2 spike protein detection using IgG-capped gold +nanoparticles and dynamic light scattering. Mater. Today Chem. 25, 100924. +64 +Ruiz, G., Ryan, N., Rutschke, K., Awotunde, O., Driskell, J.D., 2019. Antibodies irreversibly adsorb to gold +nanoparticles and resist displacement by common blood proteins. Langmuir 35(32), 10601-10609. +65 +Leirs, K., Tewari Kumar, P., Decrop, D., Pérez-Ruiz, E., Leblebici, P., Van Kelst, B., Compernolle, G., Meeuws, H., +Van Wesenbeeck, L., Lagatie, O., 2016. Bioassay development for ultrasensitive detection of influenza a +nucleoprotein using digital ELISA. Anal. Chem. 88(17), 8450-8458. + diff --git a/ZtE3T4oBgHgl3EQf2QvF/content/tmp_files/load_file.txt b/ZtE3T4oBgHgl3EQf2QvF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1fdb3e4df636d158f102b03b858c12a26dfb1893 --- /dev/null +++ b/ZtE3T4oBgHgl3EQf2QvF/content/tmp_files/load_file.txt @@ -0,0 +1,1258 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf,len=1257 +page_content='1 A Point-of-Care Biosensor for Rapid Detection and Differentiation of COVID-19 Virus (SARS-CoV-2) and Influenza Virus Using Subwavelength Grating Micro-ring Resonator Shupeng Ning,a Hao-Chen Chang,b Kang-Chieh Fan,a Po-yu Hsiao,a Chenghao Feng,a Devan Shoemaker,a and Ray T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Chena,b,* a Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas 78758, United State b Omega Optics, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=', 8500 Shoal Creek Blvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=', Austin, Texas 78757, United State Email：chenrt@austin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='edu Abstract In the context of continued spread of coronavirus disease 2019 (COVID-19) caused by SARS-CoV-2 and the emergence of new variants, the demand for rapid, accurate, and frequent detection is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Besides, the new predominant strain, Omicron variant, manifests more similar clinical features to those of other common respiratory infections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The concurrent detection of multiple potential pathogens helps distinguish SARS-CoV-2 infection from other diseases with overlapping symptoms, which is significant for patients to receive tailored treatment and containing the outbreak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Here, we report a lab-on-a-chip biosensing platform for SARS-CoV-2 detection based on subwavelength grating micro-ring resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The sensing surface is functionalized by specific antibody against SARS-CoV-2 spike protein, which could produce redshifts of resonant peaks by antigen-antibody combination, thus achieving quantitative detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Additionally, the sensor chip is integrated with a microfluidic chip with an anti-backflow Y-shaped structure that enables the concurrent detection of two analytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' In this study, we realized the detection and differentiation of COVID-19 and influenza A H1N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Experimental results show that the limit of detection of our device reaches 100 fg/mL (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='31 fM) within 15 min detecting time, and cross-reactivity tests manifest the specificity of the optical diagnostic assay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Further, the integrated packaging and streamlined workflow facilitate its use for clinical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Thus, the biosensing platform offers a promising solution to achieve ultrasensitive, selective, multiplexed, and quantitative point-of-care detection of COVID- 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Keywords: COVID-19, SARS-CoV-2 spike protein, biosensor, subwavelength, micro-ring resonator, influenza 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Introduction Since a novel coronavirus disease (COVID-19) caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) was reported in late 2019, the world is continuously threatened by the potentially fatal infectious disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='1,2 The highly contagious virus quickly spread to most continents within a few weeks and has infected more than 620 million people, including 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='5 million deaths by November 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='3,4 During the COVID-19 pandemic, genetic variants of SARS-COV-2 are constantly emerging and spreading as new epidemic strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='5,6 Despite the tremendous advance in epidemiological studies and vaccine developments curbing the progress of epidemic, the variant viruses are more contagious, and may generate immune escape from innate or acquired immune responses, resulting in continued transmission around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='7,9 The early diagnose of suspected cases is still regarded as the best viable solution to slow down the pandemic without guaranteed preventive measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='10 The Omicron variant was officially named by the WHO on November 26, 2021, and quickly replaced Delta variant as the predominant strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='11 Compared with the Alpha or Delta subvariant, Omicron has lower disease severity, hospitalization and death rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='11-13 On the other hand, COVID-19 has become more atypical due to the mild symptoms of Omicron and thus difficult to distinguish from other infectious diseases with similar symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='13,14 Among common respiratory infections, influenza presents many overlapping clinical manifestations with COVID-19, including fever, cough, sore throat, headache, fatigue, and myalgia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='15,16 However, the basic reproduction number (R0) of COVID-19 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='5 for Omicron) is much higher than influenza (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='9–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='1), which means that the SARS‐CoV‐2 virus is more contagious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='15,17 Hence, timely detection is significant for patients to receive tailored treatment and curbing the epidemic, considering the long-term co-existing of COVID-19 and other respiratory infectious diseases with overlapping symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' In clinical practice, the primary method for COVID-19 diagnosis relies on real-time reverse transcription−polymerase chain reaction (RT-PCR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' PCR-based detection shows high accuracy even in the early stages of infection, which makes it the “gold standard” in diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='18 However, RT-PCR needs advanced laboratories, expensive equipment, and medically trained personnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Besides, RT-PCR assay requires the amplification for viral RNA, which is time- consuming and may delay the diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='19,20 Because of the increasing demand for testing and the difficulty of large-scale RT-PCR testing, reliable and rapid diagnostic methods for COVID- 19 are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' To overcome the challenges, some lab-on-a-chip (LOC) platforms for point- of-care (POC) COVID-19 diagnosis have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='9,21-23 Due to short detection time, convenient operation, and low sample requirements, these LOC techniques show great potential in clinical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Among these reported techniques, optical biosensors utilize the change of optical properties results from photon-matter interaction to realize the detection of analytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Optical sensing shows several advantages in the biomedical scenario, including label-free 3 detection, multiplexing capability, instantaneous measurements, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='10,24 In the past decades, the maturity of silicon photonics and photonic integrated circuits (PICs) technology promoted the development of optical sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='25 Furthermore, the compatibility with microfluidic systems offers more opportunities for LOC sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='26-28 Micro-ring resonators have been extensively investigated in PIC optical sensors because of their high packing density and ease of fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Working mechanism and design of SUMIRR-based biosensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' a SARS-CoV-2 spike protein in phosphate buffered saline (PBS) is the target analyte for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' b SARS-CoV-2 spike antibodies (blue) are conjugated on SUMIRRs as specific probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The SUMIRR-based biosensor could quantitatively detect spike protein (red) by tracking the redshift of resonate peaks caused by antigen-antibody combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' C Optical micrograph of the silicon sensing chip which supports concurrent detection of two analytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' There are eight independent sensing channels, each of which has a pair of grating couplers as the input and output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Six channels are divided into two groups for two analytes, while the remaining two channels are treated as dummy group and reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' d An exploded view of the LOC biosensing platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' e Photograph of the biosensing platform with a quarter dollar for scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' f SEM images of the SUMIRR fabricated by E-beam lithography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' a b SpikeProtein Peak Shift Intensity M M Time COVID-19 patient SARS-CoV-2 Virus Subwavelength Grating SARS-CoV-2 Anti-S Protein SpikeProtein Antibody C d Grating Coupler Top Microfluidic Chip FiberArray Dummy Sensing Chip GroupB GroupA Bottom Microfluidic Chip 250 μm Chip Holder e 2μm4 However, the limited sensitivity of micro-ring resonator impedes the application in clinical diagnostic assays that require low limit of detection (LOD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='25,29 In this paper, we demonstrate an optical biosensing platform for the rapid detection of SARS- CoV-2 using subwavelength grating micro-ring resonator (SUMIRR) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Compared with conventional micro-ring resonator with strip waveguides, the subwavelength grating (SWG) structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='b & f) extends the photon-matter interaction region, thus improving sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='30-33 The SUMIRRs are functionalized by SARS-CoV-2 spike antibody for the detection of SARS-CoV-2 spikes proteins (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' To address the challenges posed by the untypical and diverse clinical manifestations of new epidemic strains, the LOC sensing platform enables the concurrent detection of another pathogen (influenza A H1N1 in this study) with two parallel detection groups (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' To facilitate operation and improve reliability of device, we design and fabricate a double-layer microfluidic chip with an anti-backflow Y- shaped structure, which has two operating modes for surface functionalization and concurrent detection, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Additionally, a three-dimensional (3D)-printed holder and specialized photonic packaging are presented to realize system-level integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' In the past few years, various diagnostic assays for COVID-19 were widely available (summarized in Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' However, a particularly promising solution, including packaging, testing and data processing, for POC use that can achieve concurrent quantitative detection of multiple analytes is not yet available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Our SUMIRR-based sensor detects target SARS-CoV-2 antigen with a conservative LOD of 100 fg/mL (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='31 fM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Furthermore, cross-reactivity tests for SARS-CoV-2 and influenza indicate the specificity of the optical diagnostic assay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The integrated device and auxiliary portable terminal, which offers real-time data processing and the potential to interface with electronic medical records, make the platform promising for POC diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Device design 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='1 Design of the sensing platform In considering the design of a POC platform for quantitative detection, we aimed to develop a device with clinical practicality, high accuracy and the capacity to integrate with digital systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Toward this goal, we designed a microfluidic chip that supports dual-channel concurrent detection with a 3D-printing polylactic acid (PLA) chip holder (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='d and e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' For concurrent detection, two parallel channels need to be placed on the silicon sensing chip with limited area (5mm × 5mm), thus the microfluidic device was designed as a double-layer structure for reliability and ease of operation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='d &Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='b, the bottom microfluidic chip was designed with two independent microchannels, and the unilateral channel covers three SUMIRRs as a sensing group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Each channel had an inlet and an outlet connected to the top microfluidic chip through via holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' In addition to the left and right ports connected to the two channels in the bottom layer respectively, the top microfluidic chip has a 5 common port connected to both channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Besides, the common port is connected to a special Y-shape anti-counterflow splitter that can operate in two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' In single-channel mode, the left/right port works as inlet, while the common port works as outlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' When the sample flows toward downstream (common port) from one branch of the Y-shaped structure, the significant difference in flow resistance of downstream and the other branch will “block” the other branch, thus avoiding the cross-contamination caused by counterflow (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='c and Video S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The other mode is the dual-channel mode, where the common port works as inlet while left port and right port are both outlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' In this mode, the structure is a splitter that bifurcates the upstream fluids towards two sensing groups (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='d and Video S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The anti-counterflow splitter is simulated by the finite element method to optimize chip functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='34-36 As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='e, the Y- shape structure can avoid counterflow in the single-channel mode without losing the bifurcating function for dual-channel mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' For better packaging and integration, we designed a 3D-printed holder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The upper surface of holder is slotted, and the depth equals the thickness of sensing chip (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='75 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The slot area is larger than that of sensing chip, which aims to utilize the limited elastic deformation of PDMS to eliminate manufacturing errors from 3D printing while providing enough support for the microfluidic chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Besides, we designed a comb structure with a height slightly less than slot Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Overview of microfluidics design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' a A double-layer PDMS microfluidic chip was designed for dual-channel concurrent detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Three ports on the top layer play different roles in different operation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' b The optical micrograph illustrates that the bottom microfluidic chip bonded on the sensing chip has two independent channels, and each channel covers three resonators as one sensing group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' c & d The working mechanisms of Y-shape anti-counterflow structure works in two different modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Red pigment was added to DI water for demonstration purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' e Hydromechanical simulation results demonstrate the velocity field distribution of two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' In single-channel mode, the flow rate is 5 μL/min, and the right port works as inlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' While the common port is inlet in dual-channel mode with a flow rate of 10 μL/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Details of the numerical simulation are in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' a b 2mm Sensing Chip LeftPort Right Port Group B GroupA Common Port 200μm C d e Single-channel Mode Dual-channelMode Single-channel Mode Dual-channel Mode 12 06 depth, which serves two purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' First, the comb structure helps to alignment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' meanwhile, the fiber array is fixed on the holder by UV light adhesives, and the comb structure could provide a larger contact area resulting in higher bond strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='2 Design of optical biosensor The SUMIRR consists of a micro-ring and a bus waveguide constructed by periodic pillars with a period much smaller than the operating wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='30,31 One advantage of the subwavelength structure is that the optical properties, including effective index, loss, guiding capabilities, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=', can be modulated by geometric topological designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='31,32 Importantly, SUMIRRs show good potential for biosensing because the periodic structure increases the effective sensing region, including not only the top and the side of pillars, which leads to more significant photon-matter interaction and higher sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='37,38 In this study, SUMIRRs are fabricated on silicon on insulator (SOI) wafers using E-beam lithography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='32,39 Specifically, the silicon pattern sits on the SiO2 buried layer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='f and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' S7), and SUMIRRs are covered by aqueous cladding as sensing units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' To ensure that guided mode exists in SWG waveguides, the effective refractive index of SWG waveguide (neff) must be bigger than the refractive index of bottom buried layer (nSiO2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='45) and cladding layer (nclad ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='40 The optimized design shall provide large overlap integral with analytes while maintaining decent waveguide propagation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='41 The resonant wavelength λres can be expressed as: 2 eff res R n m π λ ⋅ ⋅ = (1) where R is the radius of micro-ring and m is a positive integer denoting the mode order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' In the biosensing scenario, biochemical reactions, such as antigen-antibody combination and DNA hybridization,42,43 could affect the photon-matter interaction and thus change neff according to the Lorentz-Lorenz relation, which manifests in the shift of resonant peaks (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content="44 To evaluate the device's performance, we use quality factor Q and bulk sensitivity Sbulk to quantify the sensing properties of device." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' A higher Q means that light has a longer lifetime in the resonator, and thus provides stronger interaction with analyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='37 Q is defined as Eq (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 1 [ ] 2 g res res res m n E Q dE dt FWHM π λ ω λ α − ⋅ = = ≈ ⋅ (2) Here ωres is the resonant frequency, E is the mode’s electric field intensity, α represents the total loss in the resonator and ng is the group index of the mode which can be approximated to neff within a slight shift of λres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='45 From the experimental point of view, Q can be approximated by the ratio of λres to the full width at half maximum (FWHM) bandwidth of resonant peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='46,47 A higher Q is desirable because of sharper peaks which are easier to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The bulk sensitivity of SUMIRR Sbulk was defined as the slope of peak shift versus change in nclad,48 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=', 7 eff res res bulk clad g clad n S n n n λ λ ∂ ∆ = = ⋅ ∆ ∂ (3) where Sbulk is often written as nm/Refractive Index Units (nm/RIU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Considering the influence of Q and Sbulk on sensing performance, the inherent limit of detection (ILOD) of ring resonator can be define as follows:49 ILOD res bulk Q S λ = ⋅ (4) Taking ILOD as an evaluation criteria, the optimal geometric parameters of SUMIRRs are determined by full 3D finite difference time domain method (Lumerical FDTD Solutions) and the testing after fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The details of simulation and design parameters are shown in SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Materials and methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='1 Device fabrication and packaging The SUMIRR-based sensing chip was custom manufactured by Applied Nanotools Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' using E-Beam lithography on an SOI wafer with a 220 nm active layer and a 2 μm buried oxide layer (Soitec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Before use, the chip was soaked in piranha for 30 min, followed by deionized (DI) water (W4502, Sigma-Aldrich) and isopropanol (200440, CMC Materials) washing, then dried under nitrogen (N2) stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Microfluidic microchannels were fabricated from an SU-8 photoresist (SU-8 2035, Kayaku Advanced Materials) male mold patterned on a 4-inch silicon wafer (71893-07, Electron Microscopy Sciences) using standard soft-lithography processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='50 The height of microchannel determined by the thickness of SU-8 coating was 50 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Microfluidic channels were cast from the male mold with a 10:1 mixture of PDMS base and curing agent (SYLGARD 184 silicone elastomer kit, Dow) and cured at 90 ℃ for 40 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The bottom and top layer thicknesses are 3mm and 6mm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Because of the small chip size, the spacing between two independent microchannels in the bottom layer should be small to ensure sufficient space for packaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' A 240 μm spacing between two 300μm-width channels showed good stability at a flow rate ≤ 100 μL/min without leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The via holes on bottom chip and 3 ports of top layer were punched by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='75 mm and 1 mm punchers (PT-T983, Darwin Microfluidics), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The holder for integration was designed by CAD software (AutoCAD 2023, Autodesk) and printed by a 3D printer (FLOW, Craftbot) with PLA polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The printing thickness of single PLA layer was 100 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' After cleaning with IPA and DI water, the bonding surface of sensing chip and PDMS were treated with UV rays in a UV ozone cleaner (T10X10/OES, UVOCS Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=') for 8min and 5 min, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Then, sensing chip and bottom PDMS were aligned manually according to the pre-designed alignment marks with IPA lubrication and baked in an oven at 90℃ for 1 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The bonding for bottom and top PDMS layers was the same as the above step, 8 except that the UV treatment time was 5min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' For the precise alignments of fibers and grating couplers, the packaging holder and fiber array (pitch 127um 8 degrees, Gloriole Electroptic Technology Corp) were anchored to a mechanical stage, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The input fiber was connected to a broadband LED (DL-BX9-CS5403A, DenseLight), and the output was connected to a C-band optical spectrum analyzer (OM-1C2MM353, Optoplex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The position of fiber array is fine-tuned to maximize the output power without losing sufficient responses from other channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' When the optimal position was determined, dropped UV light adhesives glue (37-322, Edmund Optics) on the comb structure of holder, then treated with UV light overnight for curing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='2 Surface functionalization of micro-ring resonators The packaging device was treated by UV rays for 8 min to form hydroxyl groups (-OH) on the SUMIRR surface and remove organic contaminants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 2%* organosilane reagent (3- aminopropyl) triethoxysilane (APTES) diluted in 95% ethanol solution was pumped to microfluidic channels via a PTFE tubing (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='6 mm ID x 1 mm OD, Uxcell) at 5 μL/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The flow rate was controlled by a syringe pump (70-4504, Harvard Apparatus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' APTES condensed with -OH results in the formation of siloxane bonds (Si-O-Si) on UV-treated silicon surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='51,52 Notably, it is the oxide layer on Si surface that participates in silanization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The following experiment and related studies confirmed that the natural silicon oxide layer is sufficient for silanization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='53-56 The unbounded APTES was removed by 95% ethanol washing at 10 μL/min for 20 min, following by drying under N2 stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Then the device was baked at 95℃ for 1 h to enhance bonding stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' After silanization, an aqueous solution of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='5% glutaraldehyde was introduced at 5 μL/min for 1 h, and then the chip was washed with PBS solution (J61196AP, Thermo Fisher Scientific) at 10 μL/min for 20 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' After silanization, an aqueous solution of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='5% glutaraldehyde (GA) was introduced at 5 μL/min for 1 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' One aldehyde group bound to the surface expressing -NH2 (from APTES) and the other aldehyde group for further crosslink with bioreceptor protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Then the chip was washed with PBS solution at 10 μL/min for 20 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' For antibody immobilization, 10 μg/mL antibody in PBS buffer was introduced for 40 min, followed by 20 min PBS washing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' There is a potential issue with crosslinking by GA since the aldehyde groups are nonspecific to proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Hence, bovine serum albumin (BSA) was used to block the aldehyde sites that did not combine with antibodies to avoid nonspecific bonding and ensure the peak shift is entirely due to specific antigen-antibody combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='57,58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='4 mg/ml BSA (B8667, Sigma-Aldrich) was added to block the remaining sites without antibody coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' In this work, we used two antibodies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=', SARS-CoV-2 spike antibody (40150-D003, Sino Biological, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=') and pan influenza A nucleoprotein antibody (40205-R063, Sino Biological, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Notably, the immobilization for two antibodies was independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Specifically, the SARS- All concentrations expressed as percentages in this paper are volume ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 9 CoV-2 antibody was introduced from one port (left port or right port), followed by BSA blocking, while the immobilization of influenza antibody was realized through the other port with the same process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The SARS-CoV-2 antibody with green fluorescence was purchased from Thermo Fisher Scientific (53-6491-82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='3 Sample preparation for antigen detection The SARS-CoV-2 Spike S1-His recombinant protein (40591-V08H, Sino Biological, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=') and influenza A H1N1 nucleoprotein (40205-V08B, Sino Biological, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=') were diluted in PBS buffer to given concentrations depending on experimental requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='4 Sensing measurement Measurements were conducted on an optical table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The fiber array was connected to a broadband LED and an OSA which communicated with PC through serial communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The output spectrum signals were filtered to remove background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The real-time peak tracking and analysis were realized with custom programs (LabView, National Instruments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Plots and histograms were created and analyzed in Origin (Origin 2022, OriginLab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Optical micrograph and videos were taken on the viewing stage of a microscope (BX51, Olympus) under 5× magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Fluorescent images were captured with a spinning disk confocal system (CSU-W1, Nikon) and supporting commercial software for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Results and Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='1 Bulk Sensitivity Analysis Before detecting biological samples, we first measured the bulk sensitivity of fabricated SUMIRRs and tested the sensing performance after packaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Seven samples, including DI water and six PBS solutions with different concentrations, were introduced to the device in order of concentration from low to high at 10 μL/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The transmission spectra of SUMIRRs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The free spectral range (FSR) around 1550 nm is 14 nm, and the FWHM of resonate peaks is ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='93 nm, corresponding to a Q ≈ 1650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' According to transmission spectra, the signal amplitude slightly decreases with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='3 dB/nm as wavelength increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Therefore, the resonate peaks near 1535 nm were selected as analytic targets for higher signal to noise ratio (SNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='b, we took the resonate wavelength with DI water cladding as a baseline and tracked peak shifts in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The relationship between peak shift and refractive index change was drawn and fitted linearly in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The result shows that a linear function fits well with a regression correlation coefficient R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='999, and the slope, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=', the bulk sensitivity is 437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='2 nm/RIU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The noise level is ~ 3 pm, and the ILOD is ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='1×10-3 RIU by Eq (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='2 Detection of SARS-CoV-2 Antigen The viral envelope of SARS-CoV-2 consists of three structural proteins, including membrane protein (M), envelope protein (E), and spike protein (S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='60 Among these proteins, the spike protein is crucial in penetrating host cells as the major transmembrane protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='60,61 Besides, the spike protein exhibits diversity and specificity among coronaviruses, contributing to the most immune recognition in the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='62 Therefore, the S protein represents an ideal target for the specific detection of SARS-CoV-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' In this study, we utilized the specific combination of S protein and SARS-CoV-2 spike antibody to realize quantitative detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The antibody we used shows cross-reactivities with most of sub-variants of SARS-CoV-2, including Delta and Omicron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='51 After introducing samples, antibodies could combine with S proteins and form antigen-antibody complex leading to peak shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' We developed a protocol for surface functionalization for the diagnostic assay on the SUMIRR-based sensor platform (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The shift of resonate peak was monitored in real time throughout the above process and used to represent the extent of reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='b, the trend of peak shift in each step is similar, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=', shift increases rapidly at the beginning and then slows down until becomes flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' To further confirm that the SUMIRR surface was functionalized, we immobilized the SARS-CoV-2 spike antibody with green fluorescence and took fluorescence images for characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='c indicates that antibodies with green fluorescent coat uniformly on the waveguide, and the SWG structure manifests higher fluorescence intensity due to a larger surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' To investigate the performance of the SUMIRR-based biosensing platform for COVID-19 detection, we first evaluated the dynamic response of the sensor to SARS-CoV-2 spike protein with different concentrations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Because we only detected one antigen in this section, the left and right sensing groups had the same function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Therefore, the reagents for surface functionalization were introduced from the common port, divided into two tributaries by splitter, and flowed to two parallel sensing groups, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Bulk sensitivity analysis of the fabricated SUMIRRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The multiple represents the relative concentration compared with the standard PBS solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' a The transmission spectra of ring resonator in different solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' b The real-time peak shift tracking with increasing solution concentration and refractive index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' c The linear regression shows a good linear relationship between peak shift refractive index change, and the slope indicates the bulk sensitivity of device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Details about refractive indexes of PBS solution are shown in Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='50 a b c DIWater 35 - FWHM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='93 nm 33 7 DI Wat 7 1 Experimental data 6 Linear fitting result 38 J 40 43 (dBr 45 - Amplitude ( 50- Q~ 1653 58 - 1 R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='999 55 - S = 437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='2 nm/RIU 89- FSR = 14 nm F 0 1530 1535 1540 1545 1550 1555 1560 1536 1537 1538 300 600 900 1200 1500 1800 2100 2 6 8 10 12 Wavelength (nm) Wavelength (nm) Time (s) Refractive Index Change An (RIU)11 We prepared samples with various concentrations of SARS-CoV-2 spike protein (from 10 pg/mL to 1 μg/mL) and introduced samples successively in order of concentration from low to high at 5 μL/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The response to different samples was monitored in real time, and the dynamic tracking of peak shift was shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The stepwise change indicates that the peak shift within the same reaction time shows a positive correlation with antigen concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Besides, the negative control group without antibody coating only shows a slight response (grey line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='a), indicating that the reason for shift is antigen-antigen combination rather than the refractive index change due to increasing concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' In addition, the peak shift increases rapidly at the beginning of reaction, and the increase slows down gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' This general trend, also seen in surface functionalization, manifests that the peak shifts over a certain period of time can characterize the extent of antigen-antibody combination, thereby realizing the quantitative detection of the antigen of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' After demonstrating the sensing performance of the platform, we further examined the quantitative relationship between peak shift and antigen concentration for clinical detection and explored the LOD of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' To this end, we prepared samples containing SARS-CoV-2 spike protein ranging from 100 fg/mL to 1 μg/mL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' In practical detections, one sensing group can test only one antigen in the sample because the antigen cannot be removed completely once attached to the antibody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Hence, samples with different concentrations were introduced to different sensing groups instead of successively introduced to one group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The reaction time and flow rate of each sample is 10 min with a flow rate of 5 μL/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' In order to remove the antigen that was not bound to antibody and keep the refractive index of aqueous cladding layer consistent with that before the test, we used PBS solution to wash sensing surface for 5 min at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The surface functionalization of SUMIRRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' a Schematic illustrating for surface functionalization and SARS-CoV-2 spike protein detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' b Real-time peak shift tracking for the surface functionalization workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Since the solvents for APTES and GA are ethanol and DI water, respectively, instead of standard PBS solution for the following reagents, the real- time tracking for APTES/GA coating is demonstrated individually in the inserts with different baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' c Fluorescence image of the SUMIRR modified by SARS-CoV-2 spike antibody with green fluorescent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' a b 10 min 1 h 10 min 20 min 10 min PBS Anti-SAntibody PBS BSA PBS APTES-GA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='0- Peak Shift △入 (nm) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='5 APTES Antibody 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='8 900 1800 2700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='5 GA 900 1800 2700 3600 BSA 0 006 1800 2700 3600 4500 5400 6300 Time (s) C Spikeprotein 20μm12 10 μL/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The real-time response of the SUMIRR-based biosensor to a specific concentration of SARS-CoV-2 antigen is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Additionally, we analyzed the peak shift for each concentration before and after washing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The concentration-dependent response after washing fits well with the Hill model, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='63,64 These results further indicate the positive correlation between peak shift and antigen concentration, and the antigen concentration ≥ 100fg/mL could bring a remarkable response much higher than noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='3 Specificity Analysis and Concurrent Detection This study aimed to realize the concurrent detedction of SARS-CoV-2 and influenza viurs, so two sensing groups need to be functionalized separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Specifically, the SARS-CoV-2 spike antibody and pan influenza A nucleoprotein antibody were introduced from the left port and right port surface for functionalization,65 respectively, while the common port worked as the inlet during testing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Before demonstrating the concurrent detection, we first examined the specificity of the optical assay based on antibody-antigen combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Samples Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Real-time detection of SARS-CoV-2 antigen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' a Real-time response of the SUMIRR-based biosensor to SARS-CoV-2 spike protein was introduced successively by increasing order of concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The grey line represents the negative control group without anti-S protein antibody coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The interval for sample replacement and PBS washing is not shown in the figure, so there are step changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' b Real-time response of the SUMIRR biosensor to a specific concentration of SARS-CoV-2 spike protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' c Resonate peak shifts due to spike protein with different concentrations before and after PBS washing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' d Concentration-dependent response curve based on the peak shifts after PBS washing (details of the curve fitting are in the SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' a b 10 min 5 min PBS 10 pg/mL 100 pg/mL 1 ng/mL 10 ng/mL 100 ng/mL1 μg/mL PBS COVID S Protein PBS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='7 1 μg/mL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content="42 With Antibody 100 ng/mL F9'0 Without Antibody 10 ng/mL 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='36 1 ng/mL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='5 100 pg/mL (uu) (uu) V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='30 10 pg/mL Shift △入 ( 1 pg/mL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='24 - 100 fg/mL Shift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='18 - eak Peak 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='06 } F0 F0 0 500 1000 1500 2000 2500 3000 3500 4000 0 150 300 450 600 750 900 1050 Time (s) Time (s) c d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='42 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='45 Experimental data Before Wash 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='40 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='36 After Wash Fitting result 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='35 - Shift △入 (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='30 - Shift △入 (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Peak 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='15 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content="10 - F 90'0 R2 = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='05 - to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 10-13 10-11 10-10 10-9 10-8 10-7 10-6 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 Concentration (g/mL) Concentration (g/mL)13 containing influenza nucleoprotein and SARS-CoV-2 spike protein were successively introduced into the sensing group functionalized by SARS-CoV-2 spike antibody, and the verification for influenza nucleoprotein antibody was taken with the same logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The flow rate and reaction time for each step were consistent with those in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Besides, according to the antigen concentration in nasopharyngeal swab specimens from COVID-19 patients, we selected 100 pg/mL as the antigen concentration for this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='23 As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='a & b, the optical assay shows a clear signal difference between positive and negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' However, the antigen that does not match the antibody also causes a slight shift (~ 15 pm), which may be related to nonspecific interaction or incomplete BSA blocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' This undesired response requires more careful determination of the LOD of sensing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The concurrent detection results further indicate the specificity of antibody-antigen combination (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Besides, SARS-CoV-2 spike protein samples with concentrations above 100 fg/mL can result in significant peak shifts, which are distinct from nonspecific responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Therefore, taking 100 fg/mL as the LOD of system is conservative and reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Conclusions We have developed a POC biosensor that supports the concurrent detection and differentiation of two analytes using SUMIRRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' A microfluidic chip and an advanced packaging are developed to integrate the sensing unit, which is more convenient for clinical tests and improves reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The antibody immobilized on SUMIRRs could bring resonate peak shifts after combining to the antigen in sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' By analyzing the redshifts, we demonstrated the ability to quantitatively detect the concentration of SARS-CoV-2 spike protein with a LOD of 100 fg/mL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Furthermore, the cross-validation of SARS-CoV-2/influenza antigen and corresponding antibody indicates the high specificity of the optical assay based on antibody- antigen combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Therefore, the SUMIRR-based LOC biosensor for the quantitative detection of COVID-19 provides a promising solution to overcome challenges in the rapid Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Concurrent detection for SARS-CoV-2 spike protein and influenza nucleoprotein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' a & b Specific response of SUMIRR biosensor to different antigens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' The real-time responses shown in a and b were collected from the sensing groups functionalized by SARS-CoV-2 and influenza antibody, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' c Cross-reactivity tests for SARS-CoV-2 and influenza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Samples that contain SARS-CoV-2 spike protein/ influenza nucleoprotein test positive for the corresponding antibody while negative for the other antibody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' b a c 10 min 5 min 10 min 5 min 10 min 5 min 10 min 5 min NP Protein S Protein NP Protein S Protein PBS NP Protein PBS COVID S Protein PBS PBS COVID S Protein PBS NP Protein PBS (-) (+) (+) (-) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='18 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='21 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='21 } 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='18 J 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='18 } 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='15 (wu) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='12 - Shift nift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='09 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='09 手 100 fg/mL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='03 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='03 } 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content='03 01 F0 mw E 0 300 600 900 1200 1500 1800 300 600 900 1200 1500 1800 Anti-S protein Antibody Anti-NP protein Antibody Time (s) Time (s)14 identification of pathogenic virus with similar symptoms, and promotes the development of POC diagnostic tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Conflict of interest The authors declare that they have no competing financial interest or associative interest that represents a conflict of interest in connection with the work reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' CRediT authorship contribution statement Shupeng Ning: Conceptualization, Methodology, Formal analysis, Investigation, Writing Original Draft, Visualization Hao-Chen Chang: Methodology, Formal analysis, Supervision, Resource Kang-Chieh Fan: Software Po-yu Hsiao: Methodology, Investigation Chenghao Feng: Methodology, Investigation, Validation Devan Shoemaker: Methodology Ray T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Chen: Conceptualization, Writing - Review & Editing, Supervision, Project administration, Funding acquisition Acknowledgments This research program is supported by Omega Optics, NIH, AFOSR MURI and NSF and the Texas State Endowment from the University of Texas, Austin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Supplementary Information Reference 1 Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=', Lin, Q.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=', Van Wesenbeeck, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=', Lagatie, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Bioassay development for ultrasensitive detection of influenza a nucleoprotein using digital ELISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQf2QvF/content/2301.04754v1.pdf'} +page_content=' Chem.' metadata={'source': 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Eddegdag1, A. Naamane2, M. Radouani3 + + +Abstract – This study aims to establish and compare different methods allowing to describe +precisely the viscous supersonic laminar flow behaviour and the Shock Wave- Laminar Boundary +Layer Interaction around the F-16 laminar NACA 6-series airfoil NACA 64A204. This study’s +particularity is carried out for a laminar viscous supersonic flow. The approach adopted to describe +these phenomena is by the design and production and experimental testing of a mock-up F-16 airfoil +with pressure taps for the supersonic burst wind tunnel AF300, followed by a numerical validation +with Ansys Fluent and a theoretical validation based on our previously established analytical +SWBLI model. The comparison and analysis of obtained experimental, numerical and analytical +results validate our designed NACA 64A204 mock-up to a great extent. + +Keywords: Airfoil Mock-up, Laminar Viscous Supersonic Flow, NACA 64A204, Supersonic Burst +Wind Tunnel AF300, SWBLI, + +Nomenclature +NACA +National Advisory Committee for +Aeronautics +SWBLI +Shock wave-boundary layer interaction +SBWT AF300 +Supersonic burst wind tunnel AF300 +SW +Shock wave +BL +Boundary layer +VDAS +Versatile data acquisition system +ABS +Acrylonitrile Butadiene Styrene +𝛽 +Ratio of maximum thickness and chord. +𝑅𝑒 +Reynold number +𝜑 +Velocity disturbance potential +𝛿 +Similarity ration +Cli +Optimal lift coefficient +𝜎𝑒𝑥𝑡 +Experimental angle of the curved detached +shock wave in extrados +𝜎𝑖𝑛𝑡 +Experimental angle of the curved detached +shock wave in intrados +𝜔𝑒𝑥𝑡 +Angle of Mach line in extrados +𝜔𝑖𝑛𝑡 +Angle of Mach line in intrados +Δ +Distance between the curved detached shock +and the leading edge +AOA +Angle of incidence +𝑀0 +Mach number upstream of the airfoil +𝛽𝑒𝑥𝑡 +Numerical angle of the curved detached +shock wave in extrados +𝛽𝑖𝑛𝑡 +Numerical angle of the curved detached +shock wave in intrados +𝜇 +Mach angle +𝑙0 +Boundary layer thickness +𝛾𝑒𝑥𝑡 +Angle of oblique shock wave at trailing edge +η +Normal coordinate in the Frenet coordinate +system +𝜂̄ +Non-dimensional normal coordinate in the +Frenet coordinate system +m +Local incompressible Mach number linking +the Lower Deck to the Main Deck +c +Airfoil chord + + + + + + + + +a +Mean-line designation, fraction of chord +from the leading edge over which design +load is uniform +𝑦𝑐 +Mean-line ordinate +𝐶𝑙𝑖𝑚𝑜𝑑 +NACA 6A-series optimal lift coefficient +x +Distance along the chord +y +Distance perpendicular to the chord +𝑦𝑡 +Ordinate of symmetrical thickness +distribution +t +Airfoil thickness ratio +𝜀 +Airfoil parameter +𝑦(𝑥) +Wing airfoil equation +R +Radius of the leading edge +𝑦𝑐 +𝐹16(𝑥) +NACA64A204 mean-line ordinate +𝑦𝑡 +𝐹16(𝑥) +NACA64A204 Ordinate of symmetrical +thickness distribution +I. +Introduction +The understanding of the supersonic domain, a priori +hostile, is crucial and essential from an operational point +of view to master it and therefore ensure the achievement +of various operational missions. It is also imperative in +order to be able to respect the safety and integrity of +aircrew and equipment. +The phenomena that appear due to the evolution of an +aircraft in a supersonic flow, therefore, continue to be +among the subjects of study and research, due to their +complexity and lack of research in this particular field of +study. One of the main key phenomena that govern and +take place in supersonic flow are SWBLIs. In our +previously published articles and studies, we tackled the +modelling of laminar steady irrotational SWBLI around +a thin airfoil located in the standard atmosphere. The +modelling +process +initiated +from +Navier-Stokes + + +N. Eddegdag et al. + + International Review of Aerospace Engineering, Vol. xx, n. x +equations adapted to our assumptions. We applied the +asymptotic +methods, +adimensionnalisation +and +linearization, to obtain the equation governing our +problem [1]. + Following that, we respectively applied asymptotic +expansions on these governing equations following +physical parameters governing our case of study [𝛽 & +𝑅𝑒−1], and finally to obtain the formulas of perturbation +velocity potential 𝜑 near the airfoil body far from the +leading edge, both inside the boundary layer and in the +undisturbed flow outside the boundary layer, we applied +the singular perturbations methods in respect to the +principle of least degeneracy. +The resolution process consisted of the adoption of the +triple Deck Technic which divides the laminar boundary +layer surrounding our obstacle [airfoil] into three major +decks [Lower Deck, Main Deck and Upper Deck], in +order to identify the constants in the formulas of +perturbation velocity potential.[2] +The analytical model was validated numerically with +Ansys Fluent and experimentally in the supersonic burst +wind tunnel AF300 with a produced reduced airfoil +NACA 5-series with pressure taps NACA 43013.[1]-[2] +However, the previous choice of airfoil wasn’t ideal, +since in our case study we started from the assumption of +a laminar flow, which isn’t accurate, especially when +approaching the trailing edge of the chosen airfoil. Hence +the results of the analytical-experimental-numerical +comparison presented important differences and relative +errors when approaching the trailing edge. +Thus, in this study we focus on a NACA 6-series +laminar airfoil NACA 64A204, to perfectly describe the +behavior of the flow around the airfoil and the different +physical phenomena that take action in supersonic fields. +Hence, in the first part, we will tackle the different +methods of design and production process of the F-16 +airfoil mock-up with pressure taps in extrados for the +Supersonic Burst Wind Tunnel AF300. +Secondly, we address the numerical approach in Ansys +Fluent for the study of the problem under the same +assumptions as in the analytical part [1]. +Furthermore, the experimentation takes parts for the +designed mock-up airfoil in the Supersonic Burst Wind +Tunnel AF300, followed by an analysis and comparison +of the experimental, numerical and analytical results for +the NACA 64A204. +The aim of the experimental-analytical-numerical +confrontation isn’t only to validate our designed F-16 +airfoil mock-up, but also to further refine our analytical +model by detecting the different parameters that play a +role in the behavior of the flow around supersonic laminar +airfoils, that weren’t addressed in the previous studies. +In addition, we aim to open the doors for several other +airfoil mock-ups to be designed and to ensure maximum +usage of the supersonic wind tunnel in the field of +scientific research in order to be able to study the +phenomena which interest the aeronautical field, and not +only in pedagogical education, given that the latter is only +equipped with a dielectric airfoil mock-up. +II. +Design of the NACA 64A204 F-16 +Airfoil Mock-up +In itself, designing a scale model is not an easy task. +Its size cannot be determined randomly, because the +interpretation of the phenomena generated must also be +verified on the real-size model. To do so, the reduced +model must respect the notions of similarities. Otherwise, +the "scale effect" claims that phenomena appearing with +the model do not necessarily occur with the object in full +size, and even if they reproduce this raises the question of +how to transpose the results. [3] +II.1. Similarities Between Models +The first of the similarities to follow is geometrical +similarity. It is carried out only when the dimensions of +the reduced and real models are linearly homologous +according to the same ratio 𝛿, called the “similarity ratio”. +The models are then geometrically similar. In addition, +this ratio is the basis of the industrial design, since it +concretely represents the scale and then makes it possible +to transpose any point from one model to another.[4] +In this project, this similarity is respected, as all +dimensions are in per cent of chord length. Thus, the +similarity ratio 𝛿 is described by the ratio of the reduced +model chord to the real model chord: + +𝛿 = +𝑐𝑚𝑜𝑐𝑘−𝑢𝑝 +𝑐𝑟𝑒𝑎𝑙 (1) + +With 𝑐𝑚𝑜𝑐𝑘−𝑢𝑝 is the chord for the designed airfoil +mock-up and 𝑐𝑟𝑒𝑎𝑙 is the chord for the real aircraft airfoil. +II.2. Definition of the F-16 NACA 64A204 Airfoil +The three-dimensional wing of an aircraft is +established from its two-dimensional section, called the +airfoil. There are different types of airfoil categories in +aeronautics. +In this specific case, it is a NACA airfoil. These +airfoils are found on the majority of aircraft. They are +airfoil designs for aircraft wings developed by NACA. +They thus describe the shape of the airfoil thanks to a +series of figures, which represent parameters in the +equations generating precisely the section of the wing. +For the F-16, this is in particular the NACA 64A204 +airfoil (see Fig. 1), described by 5 numbers and a letter. +This follows a specific convention. All of the following +% dimensions are in % chord length 𝑐. +The explanation of the numbers and the letter of the +NACA 64A204 airfoil is as follows [5]: +− +6: indicates the series, this airfoil belongs to the +NACA series 6 airfoils, which corresponds to +the supercritical airfoils, with double camber. +− +4: corresponds to the position of the zone of +minimum pressure, i.e., the position of the zone +of maximum thickness, in tens of per cent. + + +N. Eddegdag et al. + + International Review of Aerospace Engineering, Vol. xx, n. x +− +A: reflects the fact that from approximately 85% +to the trailing edge, the trajectory of the middle +line retains a constant steering coefficient [6]. +− +2: indicates the optimum lift coefficient Cli, in +tenths of a per cent. +− +04: describe the maximum thickness 𝑡, in per +cent. +So, in other words, this supercritical airfoil has an +optimal lift coefficient Cli of 0.2 and a maximum +thickness 𝑡 of 4% to 40%. +Moreover, the belonging of a NACA airfoil to serie-6 +means that its geometry is deduced from its pressure +distribution established in the wind tunnel. The 6-Serie +also maximizes laminar flow by reducing drag. +The advantage of such an airfoil is that it has a very +thin relative thickness, which leads to a high critical +Mach number. On the contrary, this same small thickness +close to the trailing edge can also become a drawback +from the point of view of the design of the structure and +of its manufacturing.[7] + + + +Fig. 1. F-16 NACA 64A204 Airfoil and F-16 schematics +II.3. Design of the F-16 airfoil Mock-up for the AF300 +Wind Tunnel +The AF300 Rafale supersonic wind tunnel is equipped +with a 100mm x 25mm dimension test section in which +the 'scale' model of the airfoil is integrated. +II.3.1. +Airfoil Modeling on CATIA V5 + +For this part, we introduce the airfoil parameters in the +GNacaLt software (see Fig. 2). Then, we choose the +maximum of points proposed, that is to say, 200. In order +to be able to use this software correctly, the decimal +symbol must be the point. + + +Fig. 2. GNacaLt Software + + Secondly, we regroup these coordinates for extrados +and intrados in per cent of the chord in a .dat extension +file, +which +we +transfer +to +the +GSD_PointSplineLoftFromExcel +spreadsheet +in +Microsoft Office Excel. + Furthermore, we transfer the Excel spreadsheet to +CATIA V5 via the StarLoft option while activating +Macros in the excel file, after interpreting the +coordinates, CATIA attach the different uploaded points +forming the airfoil (see Fig. 3). + + +Fig. 3. NACA 64A204 Airfoil, Chord 60mm in CATIA V5 Software + + Finally, we create a 3D mock-up of the NACA 64A204 +with 60mm Chord, 25mm wingspan and two pressure +taps in extrados with separate canalisation for air to enter +and thus measure the exact experimental local pressure +and Mach values in these pressure taps in the Supersonic +wind tunnel AF300, in addition to that slots in the test +stream of the wind tunnel AF300 to support the mock-up +(see Fig. 4) [8] + + +Fig. 4. NACA 64A204 Airfoil 3D Mock-up, Chord 60mm, width +25mm with two pressure taps with canalisation and mock-up supports +for the slots, in CATIA V5 Software + + Generation des profils NACA (GNacalt) +一 +口 +X +Naca 4-digit +Naca 5-digit +Naca Mod 4-dgit +Naca Mod 5-digi +NACA Serie +NACA Serie 6A +NACA Serie 16 +NACA Serie 6A +Nom du profl : : 64A204 +Nombre de points du profl: : 200 +ide + Calcul Naca Serie 6A +口 +A propos de .. +chfte 6 +6 Non modifiable +Eermer/Quitter +GNacaLt Version : 0.1.0 BozoSoft 02/2015 +Deuxieme chifie +[4 +De3a5 +Francais +A +Troitieme chffre +2 +Entre 0 et 9 +Deux dermiers chiffres +[04.00 +De 1.00 a 30.00 +Nombre de points du profil +[200 +De 40 a 200 +Fichier creo : naca_64A204.dat +Aide +[Creer un profi +FeimerCATAVS-Proll,GeacaCATPot +ProflGnac +Puanxy +Plann +netunnintrunecnmmnde +N. Eddegdag et al. + + International Review of Aerospace Engineering, Vol. xx, n. x +II.3.2. Manufacturing of the NACA 64A204 Mock-up +For the manufacturing of the airfoil NACA 64A204 +mock-up, two methods were proposed: +− +1st method: by 3D printing +− +2nd method: A mock-up of the airfoil by laminate +bronze, followed by a canalization by pipes and +filling with polyester putty +a. 3D Printing of the NACA 64A204 Mock-up +3D printing did not fully meet expectations. Indeed, as +explained before, the design of airfoils with a very low +relative thickness encounters difficulties in achieving the +trailing edge. Here, the dimensions of the models are too +thin for the 3D printer, which is therefore unable to finish +the trailing edge from about 90-95% of the chord. Thus, +all scale models are approximately 5 𝑚𝑚 missing. +In addition, the models seemed fragile and had +imperfections in their surface condition., and the pressure +taps weren’t doable due to the fragility of the used +material and the small diameter of the pressure taps’ +canalization (see Fig. 5). + + +Fig. 5. NACA 64A204 Airfoil 3D printed mock-up with 100mm +chord +b. Manually Manufactured NACA 64A204 Mock-up +Note that this method will allow us to have pressure +taps on the extrados and the intrados (simultaneously). +Thus, we can evaluate the variations of the Mach number +along the airfoil (unlike the dihedral airfoils where we +only need the pressure taps on the extrados given the +symmetry of the airfoil). +Our model is made in three parts: the two sides of the +airfoil and a closed box with a stressed coating (see Fig. +6). For the two static pressure taps on the extrados, a +channel was designed using pipes with a radius of 1.5mm. +Then, the airfoil mock-up is filled with polyester putty +(material that resists high pressures and temperatures to +prevent deformation and give weight to the mock-up). +Finally, the assembly is soldered by a silver gun. + + +Fig. 6. Images of the surfaces and piping constituting the +manufactured airfoil mock-up + +Finally, it was possible to produce the airfoil mock-up +NACA 64A204 (F-16) with two pressure taps on the +extrados (see Fig. 7), with the following properties: +− +Chord 100mm +− +Maximum thickness = 100 x 4% = 4 mm +− +Pressure taps at the leading edge at 30%c +x1=30mm +− +Pressure taps at the trailing edge at 60%c +x2=60mm + + + +Fig. 7. NACA 64A204 Airfoil mock-up with two pressure taps in the +extrados and a 100mm chord +III. Experimentation in the Supersonic +Burst Wind Tunnel AF300 +In the aeronautical field, wind tunnel testing remains +an essential element. However, they are not trivial. +Indeed, the AF300 wind tunnel has a particular operating +regime, because it is a supersonic burst wind tunnel. On +the other hand, it should be noted that the flow created by +this machine is also subject to the scale effect and must +therefore respect similarity conditions. +The complete similarity concerns the flows and adds +dynamic similarity to the geometrical similarity. For the +flow of the experiment and that under real conditions to +be similar, they must respond to their balance equations +in dimensionless form with an identical solution, thanks +to initial conditions and limits of the same order. This is + + +N. Eddegdag et al. + + International Review of Aerospace Engineering, Vol. xx, n. x +possible only when their dimensionless characteristic +numbers, determined according to the principle of +dimensional analysis, are equal.[8] +However, this last principle is in reality difficult to +obtain. It is then necessary to concentrate on the +characteristic number translating the phenomenon which +plays the most important role, to validate a restricted +similarity. +In particular for this study, concerning the flow of a +compressible fluid like air, it is necessary to take into +account the effects of compressibility in addition to the +effects of viscosity. So, the characteristic numbers, in this +case, correspond to the Mach number 𝑀 and the Reynolds +number 𝑅𝑒. +III.1. Experimental Tests and Results +The AF300 supersonic wind tunnel is a Göttingen type +return wind tunnel. More specifically, it is an economical +scaled-down +laboratory +wind +tunnel +allowing +experiments to be carried out at air speeds of Mach 1.4 +and 1.8. It has three different parts: the wind tunnel duct, +the test section and the instrumentation panel. +For our two conducted experimental studies, the +operating conditions (see Table I) are presented as +follows: + +TABLE I +OPERATING CONDITIONS FOR NACA 64A204 IN SBWT AF300 +Time +Operating conditions +Time +Liner +ATM Press +Model Angel +AOA +[s] + +[mbar] +[degree] +0.00 +Mach 1.8 +915 +-1.1 + +III.1.1. Experiments for the 3D-printed mock-ups + +The experiment aims to determine via VDAS the +Mach number at a point on the upper surface and lower +surface, thanks to the new pressure taps 26 and 27. It can +then be compared with those established by theoretical +and numerical models. In a second step, with the +Schlieren device, the shock waves at the leading and +trailing edges can be visualized, possibly with the +expansion waves. +However, the fragility of the mock-ups, which are +made of ABS, made the experiments complex. At the end +of the experiment, all the mock-ups were destroyed in the +SBWT AF300, because the hooks could not withstand the +supersonic gusts, as shown in Fig 8. + + +Fig. 8. Destroyed 3D-printed NACA 64A204 mock-ups + +The mock-up with a 30𝑚𝑚 chord, reduced to 25𝑚𝑚 +after 3D printing, was the only one that held in place long +enough to visualize, albeit faintly, the curved detached +shock waves, below, the leading edge with the Mach 1.4 +SBWT AF300 nozzle. (See Fig.9) + + +Fig. 9. Detached curved shock wave visualization for the 25mm +chord’s NACA 64A204 mock-up + +Photo [a] was taken at the beginning of the experiment, +the second half through. Thus, the shock on [b] is tighter +on the walls because the speed of the flow has increased. +Also likely, the mock-up is subjected to very virulent +vibrations which make it move, this could in another way +explain the fact that the shock waves are different +between these two photos. +Photo [b] also shows two expansion waves visible on + +[a] +[b]T +N. Eddegdag et al. + + International Review of Aerospace Engineering, Vol. xx, n. x +the extrados and the intrados. They seem to correspond to +Mach lines and are also more stuck to the walls. +In the two photos, the oblique shock wave at the +trailing edge is not visible, certainly because the +vibrations were the most intense there. +By graphic measurements on the second photo, which +is the most usable, the angles of the curved detached +shock wave are: 𝜎𝑒𝑥𝑡=55° and 𝜎𝑖𝑛𝑡=58°. Those of the +expansion waves measure: 𝜔𝑒𝑥𝑡=34.5° and 𝜔𝑖𝑛𝑡=36.5°. +Then, the distance Δ between the curved detached shock +and the leading edge is estimated to equal Δ =0.87 𝑐𝑚. +The angles of the shocks on the lower surface should +be lower than those on the upper surface because the +angles of deflection are greater there. This may come +from the approximation of the representation of the shock +waves around the airfoil since the measured angles’ data +are unreliable but remain consistent with the order of +magnitude. +These parameters are the only ones that could be +evaluated during the various experiments of the project. +They nevertheless transcribe experimental phenomena on +which this project can be based. In particular, these +empirical values will be used when compared to rational +values. + +III.1.2. Experiments for the manually manufactured +mock-ups + +After focusing on the smooth running of the +experiments, we, therefore, began to extract several +readings of values of local static pressure from the +pressure taps in the extrados, with the Mach 1.8 nozzle +airfoil, so we obtained the following results for the +reduced wing airfoil NACA 64A204 (see Table II): + +TABLE II +RECORDING OF LOCAL MACH NUMBER VALUES IN THE PRESSURE +TAPS FOR NACA 64A204 FROM VDAS +Mach +Upstream +Pressure Tap +Local Experimental +Mach Number +1.55 +Pressure Tap N26 +1.42 +1.55 +Pressure Tap N27 +1.47 +1.44 +Pressure Tap N26 +1.33 +1.44 +Pressure Tap N27 +1.38 + +Thus, we see that the Mach number in extrados +increases from the 1st pressure tap to the 2nd due to a +supersonic expansion fan. Through such an expansion +[Prandtl–Meyer expansion fan], the flow remains +isentropic, and our hypotheses then remain verified. +There is a decrease in pressure and an acceleration of the +fluid upstream. + + +IV. Numerical Simulation of the flow +around NACA 64A204 + +In this part, the numerical approach for the study will +be presented, under the same assumptions as in the +theoretical part [1]. Indeed, it will make it possible to +obtain the precise behaviour of a viscous supersonic +laminar airflow around the NACA 64A204 airfoil after +setting the Ansys Fluent 2022 R2 simulation software +parameters.[9] + +IV.1. Definition of Geometry + +The +geometry +definition, +by +the +ANSYS +DesignModeler software, concerns two elements. We +must first create the object, therefore our wing airfoil. +Then, it is just as important to establish the calculation +domain, including of course the airfoil. + +IV.1.1. The Airfoil + +Regarding the object, it can be modelled using +different methods. For example, the airfoil coordinates +can be entered directly into ANSYS Fluent, then the +software generates the surface itself. In this project, it was +rather preferred to draw it with SOLIDWORKS software +and then import it later into the digital simulation +software, therefore, requires investment before actually +using the software. +To do this, using the GNacaLt software, a freely +downloadable airfoil generator, it was possible to recover +a file (.txt) containing 200 exact coordinates of the +NACA 64A204 airfoil. Then, the CAD software can +interpret them in space and thus propose a geometry of +the airfoil. +Once this preparatory work has been done, the digital +simulation software can be launched. +In the interface assigned to geometry, ANSYS +DesignModeler, the drawing under CATIA must be +imported and generated, ensuring that it is represented in +the correct plane. + +IV.1.2. The Study Domain +In this area, the calculations will be applied and the +equations will be solved. It is concretely a viewing +window through which the flow is studied. +In this study, after several dimensional tests, the +calculation domain below (see Fig.10.) was selected +because it dispenses with the airfoil of the phenomena +mentioned just above. + + +Fig. 10. Proposed Study Domaine for the airfoil + +A +B +R12. +2 +F +Airfoil +G +E +D +20c +N. Eddegdag et al. + + International Review of Aerospace Engineering, Vol. xx, n. x +There are now two distinct surface-type objects saved +in the software. But, in this state, it could not take into +account the effects of the airfoil on the flow in the +domain. It will then be necessary to merge these two +surfaces, or rather subtract them. Indeed, it is necessary +to create a new surface corresponding to that of the +calculation domain above from which that of the airfoil +must be removed. (See Fig.11.) + + +Fig. 11. Study Domain in Ansys Fluent 2022 R2 for Meshing + +It must be considered that the airfoil is now fully +integrated into the final calculation domain.[10] +Consequently, this last domain is the one that must be +meshed, since the calculations taking place there take into +account the presence of the airfoil in the flow. +IV.2. Meshing + +Under ANSYS Meshing, the Finite Element Method +(FEM), used by ANSYS Fluent, requires a division of the +final domain through a mesh. It is concretely about its +spatial discretization. In other words, the entire domain is +geometrically modelled by many smaller domains. The +interest of meshing is to obtain the simplification of the +simulations of calculations.[10]-[11] +The mesh quality has great importance on the results +obtained by a numerical calculation. To do this, we first +generate a coarse mesh (automatic) which we then +improve by Face Sizing technic in meshing based on a +Sphere Influence option to furthermore refine the mesh +around the airfoil, followed by an All-Triangle Mesh +Control Method, then we insert an Inflation Option to +take into account the effect of the boundary layer, and +last, we perform a Pinch Control to remove small features +at the mesh level to generate better quality elements +around those features. Finally, a second modification is +to introduce a progressive refinement as the airfoil moves +away. +Hence, we end up with a Meshing of 1,689,093 +Elements and 847,438 Nodes. (See Fig.12.) + + + +Fig. 12. Final Refined Mesh in Ansys Mesher 2022 R2 + +The numerical resolution procedure can therefore be +initiated. + +IV.3. Calculation Process +IV.3.1. Configuration Of Solver Settings + +Before +launching +calculations, +the +following +parameters must be configured: +• An adopted model (for our problem we will take the +laminar viscous model with energy equation) +• Fluid parameters (air): +− Air density = 1.225 kg.m-3 +− Air viscosity = 1.7894.10-5 kg. (m.s)-1 +− Sound speed in air = 347.092 m. s-1 +For the boundary conditions, it is possible to establish +the conditions for the evolution of the flow on the walls +of the domain, then on the extrados and the intrados. +On the four outer walls of the calculation domain, the +initial conditions are established as if the domain were +open to immerse the airfoil in it. It is necessary to specify +the value 𝑀0 of the Mach number upstream of the airfoil +according to the direction of the flow [for our case since +in the experimental study we have an AOA=-1.1º, thus we +have the flow in a direction of +𝑓𝑙𝑜𝑤𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 +⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ = 𝑐𝑜𝑠(−1. 1∘) 𝑥 + 𝑠𝑖𝑛(−1. 1∘) 𝑦 (2) +], and to enter the atmospheric pressure, as well as the +temperature. Moreover, in the case where the fluid is +viscous, it is necessary to impose a condition of a “sliding +wall” on the upper and lower walls of the domain to force +the flow to remain straight there without friction. Then, +for the extrados and the intrados, the solver must be +informed that these are simple walls immersed in the +flow. +Thus, once the parameters have been saved, we can +start the calculation process. + +Graphics +Ansys +2022R2 +STUDENT +0.00 +200.00 +400.00 (mm) +300.00 +Model ViewPrint Preview +No Selection +Millimeter DegreeDutinr +Clpboard-[Empty] Elend-oSdetEy +Name + Project" +hadd (s) +ansys +2022R2 +Gecmetry Inport (A2) +STUDENT +Infisso +NeTedSeects +Dnpln +Detaults +Dlsplsy Shie +he Geotn +Firpilcs Prelerene +SoerPreferenre +CFD +Fluent +Elenent Order +Ernemi Sies +Linear +Drad1 y6425 mml +Eoport Forrat + Qually +suinAnsys +2022R2 +N. Eddegdag et al. + + International Review of Aerospace Engineering, Vol. xx, n. x +IV.3.2. Calculation Process + +The solver starts from the initial solution and, thanks +to an iterative algorithm for solving the matrix system +obtained by discretization, it will perform iterations of the +problem. Each iteration must modify the current solution +and replace it with a solution closer to the exact solution +sought. The solver gives at each iteration and for each +equation the residual. +Moreover, during a simulation, it is interesting to focus +on the convergence of the calculation residuals. Indeed, +at each iteration and for each equation, an error, called +“residual”, is evaluated concerning an exact solution to +the problem. And, it is said that a computation converges +if the residuals decrease during the iterations. Thus, a +result is only valid if the residuals converge (See Fig.13.). +If the residuals do not converge sufficiently, the operation +must be repeated until they converge properly. + + +Fig. 13. Evolution of residuals according to iterations + +IV.4. Visualization And Analysis of Results + +The numerical resolution will initially allow checking +if the geometry and the mesh have been correctly +parameterized thanks to the consistency of the results, +and will allow a second time to visualize the behavior of +the flow according to the solver settings. The simulation +step is very important because it allows us to determine +the domain of validity of our model. + + +V.1.1. NACA 64A204 for M0=1.44 + +For 𝑀0=1.44, with the parameterization established +above and after convergence of the residuals, Figure 14 +below is obtained. It represents the behavior of the flow +through the evolution of the speed and the Mach number +in the final computational domain. The different velocity +values are displayed by colour. It also reveals three +particular phenomena due to the presence of the airfoil in +the supersonic flow. +First of all, it allows visualizing the variation of the +velocity from the leading edge and all along the airfoil. In +particular, it is observed that on the upper surface the +velocity keeps increasing from the leading edge to the +trailing +edge +where +𝑀𝐸𝑥𝑡𝑟𝑎𝑑𝑜𝑠𝑇𝑟𝑎𝑖𝑙𝑖𝑛𝑔𝐸𝑑𝑔𝑒 +1.44 += +1.5305While on the lower surface, it grows faster from +the leading edge up to 50% of the chord where +𝑀𝐼𝑛𝑡𝑟𝑎𝑑𝑜𝑠50% +1.44 += 1.5004, then decreases to the trailing +edge where 𝑀𝐼𝑛𝑡𝑟𝑎𝑑𝑜𝑠𝑇𝑟𝑎𝑖𝑙𝑖𝑛𝑔 𝐸𝑑𝑔𝑒 +1.44 += 1.4497, which +translates due to the double camber of the NACA64A204 +airfoil. + + +Fig. 14. Evolution of velocity along the NACA64A204 airfoil for +M0=1.44 + +Secondly, the figure 15 depicts the curved detached +shock wave on the leading edge, with the presence of a +blue-green sonic stagnation bubble, where 𝑀<1.00, +where +the +flow +is +locally +rotational +subsonic +incompressible and that is due to the curved quasi- +hemispherical nature of the airfoil and the supersonic +nature of the flow. As for the vicinity of the trailing edge, +only the shock wave attached obliquely to the upper +surface is visible. Finally, the expansion waves in form of +Mach Lines appear clearly on the airfoil in form of a +Prandtl-Meyer expansion wave that is due to the convex +and concave forms of the airfoil curves along its extrados +and intrados + + +Fig. 15. Visualization of SW and Sonic Bubble for NACA64A204 + +Finally, by graphical measurements in Figure 15, the +shock angles of the curved detached wave are worth at +the leading edge: 𝛽𝑒𝑥𝑡=62° and 𝛽𝑖𝑛𝑡=60°. These angles are +between 90° and the Mach angle of 𝜇=𝑎𝑟𝑐𝑠𝑖𝑛(1/M0) = +43.983°. Similarly, the angle at the trailing edge is equal +to 𝛾𝑒𝑥𝑡=40°. + +1e+00 +Ansys +2022R2 +STUDENT +1e-01- +1e-02- +1e-03- +0 +500 +1000 +1500 +Iterations +-continuity---x-velocity y-velocity ---energyVelocity +Ansys +airtiow +5287e+02 +2022R2 +4.993e+02 +4.700e+02 +STUDENT +4.406e+02 +1.112e+02 +.525e+02 +818et ++02 +102 +937e+02 +644 ++02 +.875e+0 +937 +0.000e+0 +etl +[ms^-1] +0.150 +0.300(m) +0.075 +0.225Velocity +Ansys +airtlow +5.287e+02 +2022R2 +993e+02 +700e+02 +STUDENT +406e+0 +02 +Be +25e +e +-12 +056e+ +t69e- +875e +00+a000 +[msA-1] +0.01 +0.02(m) +0.005 +0.015 +N. Eddegdag et al. + + International Review of Aerospace Engineering, Vol. xx, n. x +The angles 𝛽𝑒𝑥𝑡 and 𝛽𝑖𝑛𝑡 are not the same because the +airfoil is not symmetrical. Thus, it happens 𝛽𝑒𝑥𝑡>𝛽𝑖𝑛𝑡 +since the angle of deflection of the flow on the upper +surface is greater than that on the lower surface. +On the other hand, the distance Δ𝑛𝑢𝑚 between the +curved shock wave and the leading edge is graphically +worth: Δ𝑛𝑢𝑚=2.687 𝑐𝑚. +Finally, the third observed phenomenon is the shock +wave-boundary layer interaction, which is the result of +viscous supersonic flow. According to figure 16, we note +the existence of a boundary layer all along the wall of the +thickness airfoil 𝑙0=0.3076mm where the Mach is +subsonic and the speed tends towards a zero-value +approaching the surface of the airfoil. Thus, the fluid +particle (air) adheres to the wall (no-slip condition). It is +also clear the three sub-layers of the boundary layer +(viscous sub-layer in dark blue, main layer in sky blue +and the upper layer in green) highlight our triple-deck +model chosen during the theoretical part. We also note +the existence of shock waves after the boundary layer (in +height, Prandtl-Meyer expansion fan) where the Mach is +supersonic, which highlights the phenomenon of the +shock/boundary layer interaction. [12] +Thus, the local Mach number goes from a subsonic +value in the boundary layer (M<1) to a supersonic value +during shock waves where the external flow is supersonic +(M>1), by a sonic line where the local Mach number +along this line is equal to 1, to characterize and allow the +subsonic-transonic-supersonic transition. [13] + + + +Fig. 16. Visualization of the Triple Deck Boundary Layer around the +NACA 64A204 airfoil by Simulation on Ansys Fluent + +For comparison, it is interesting to run the calculation +again for a different upstream Mach number 𝑀0. + +V.1.2. NACA 64A204 for M0=1.55 + +For 𝑀0=1.55, with the same parameterization and still, +after convergence of the residuals, results are obtained. +The same phenomena of SWBLI and Curved Detached +SW with the clear appearance of the triple-layered BL +appear with increasing width of l0’=0.4262mm. +Similarly, it is observed that on the upper surface the +Mach number keeps increasing from the leading edge to +the trailing edge where 𝑀𝐸𝑥𝑡𝑟𝑎𝑑𝑜𝑠𝑇𝑟𝑎𝑖𝑙𝑖𝑛𝑔𝐸𝑑𝑔𝑒 +1.55 += 1.606 +While on the intrados, it also grows faster from the +leading edge to the same position at 50% of the chord +where this time 𝑀𝐼𝑛𝑡𝑟𝑎𝑑𝑜𝑠50% +1.55 += 1.579, to then decrease +to the trailing edge where 𝑀𝐼𝑛𝑡𝑟𝑎𝑑𝑜𝑠𝑇𝑟𝑎𝑖𝑙𝑖𝑛𝑔𝐸𝑑𝑔𝑒 +1.55 += +1.545. +Additionally, the curved detached SW, here tighter on +the leading edge, with the presence of a blue-green sonic +bubble, where 𝑀′<1.00, with a larger radius. Then, near +the trailing edge, the oblique attached shock wave is still +only visible on the upper surface. Finally, the expansion +waves always appear clearly on the profile. +In the same way as for the previous simulation, by +graphic measurements, the shock angles of the curved +wave are worth in this case: 𝛽’𝑒𝑥𝑡=55° and 𝛽’𝑖𝑛𝑡=52°. +These +angles +are, +likewise, +between +90° +and +𝜇′=𝑎𝑟𝑐𝑠𝑖𝑛(1/𝑀0) =40.1778°<𝜇, corresponding to the +minimum shock angle. Moreover, that at the trailing edge +is equal to 𝛾𝑒𝑥𝑡′=30°<𝛾𝑒𝑥𝑡. +For the distance Δ𝑛𝑢𝑚′, it finally comes graphically +here: Δ𝑛𝑢𝑚′=1.124 𝑐𝑚<Δ𝑛𝑢𝑚. +Consequently, it is deduced that the higher the velocity +of the upstream flow, the smaller the angle of the curved +detached shock wave for the same profile, and the greater +the distance between this shock is the edge of attack is +small. +The first simulations are thus completed and analysed. +However, it was expected to obtain a great finesse of the +solutions according to the mesh and the established +settings. Now, from these figures, the results look coarse +along the different shock waves, especially for a high 𝑀0. +We must therefore find a way to be even more precise in +post-processing simulations. + +V.1.3. From Simulation to Analytical Model +Improvement + +Based on our previous published studies concerning +the modelling of the laminar SWBLI around a thin airfoil +[1]. And, as a reflection and application of our soon-to- +be-published +study +entitled +‘Improved +Analytical/Statistical Modelling of the Shock Wave- +Laminar Boundary Layer Around a Thin Airfoil: +Standard Atmosphere Case’ [2]. We tend similarly for +our object of this study F-16 airfoil NACA64A204, to +improve Analytically/Statistically the analytical model +for the F-16 airfoil to obtain the most accurate analytical +results for the confrontation and validation of our mock- +up and exact description of the multitude of phenomena +that govern the supersonic laminar viscous flow around +F-16 laminar thin airfoil. +For the NACA 64A204 series-6 airfoil with coefficient +β=0.04, the Local Mach numbers are compared on the +upper surface and lower surface in the wall far from the +leading edge, thus the position of the maximum relative +camber characterizes the end of the zone from the leading +edge. It is known for every NACA wing profile. +Generally, it is defined by the second digit of the series, +for the NACA 64A204 profile it is 4% of the chord, so + +Velocity +Ansys +airfiow +5.2878+02 +2022R2 +993e+02 +STUDENT +000 ++00 +ms +0 +0.0002 +0.0004 (m) +0.0001 +000'0 +N. Eddegdag et al. + + International Review of Aerospace Engineering, Vol. xx, n. x +we perform our calculations and extract our results for x +greater than 0.04. In addition, we take different points in +the 3 parts of the profile function as well as different +altitudes of η (in the boundary layer and outside the +boundary layer), to be able to check our model. Similarly, +we have that 0 ≤ 𝜂̄ ≤ 0.13257143, as well as y must be +between 0 and 0.132 for x between 0.006158 and 1. +Thus, for M0=1.44, we have that the equation for the +Link Mach Number m (Local incompressible Mach +number linking the Lower Deck to the Main Deck) is the +following: +𝑚𝑁𝐴𝐶𝐴64𝐴204 +1.44 +(𝜂̄, 𝑠̄, 𝑀∞) = 21.0474𝜂̄ − 2.5406𝑠̄ − 0.0722 (3) + +Same for M0=1.55, we have that the equation for the +Link Mach Number is the following: + +𝑚𝑁𝐴𝐶𝐴64𝐴204 +1.55 +(𝜂̄, 𝑠̄, 𝑀∞) = −35.6199𝜂̄ + 2.8276𝑠̄ + 0.7234(4) +V. Confrontation Of Analytical-Numerical- +Experimental Results +The aim of this chapter is to compare our experimental +results of the Local Mach numbers in Pressure Taps 1 and +2, to the results of the Local Mach numbers in the same +coordinates from simulation in Ansys Fluent and the +analytical results based on our improved analytical model +[2]. The comparison will lead not only to the validation +of the designed mock-up of the F-16 airfoil, but also to +further elaborate the phenomena that take place in +SWBLI around the F-16 wing. +V.1 Analytical results + +Our analytical results for the Local Mach numbers for +NACA64A204 airfoil in the extrados at 30% and 60% +of the chord (c=100mm) are calculated from our +previously established velocity formulas from our +previous studies [1]-[2]. However, since these formulas +are in the Frenet coordinate system, it is imperative to +establish these formulas in the cartesian system by first +and foremost establishing the NACA64A204 airfoil +equation. + +V.1.1 NACA 64A204 Wing Section Equation + +The NACA airfoils are constructed by combining a +thickness envelope with a camber or mean line. The 6- +series mean lines were designed using Thin Airfoil +Theory to produce a constant loading from the leading +edge back to x/c = a, after which the loading decreases +linearly to zero at the trailing edge. The term 𝑎, not to be +confused with the speed of sound in air, denotes a +particular position on the string. [14] +Indeed, the camber line of a Series-6 NACA airfoil +produces a uniform loading along the chord from the +leading edge at 𝑥/𝑐=0 to the point 𝑥/𝑐=𝑎, then produces a +linearly decreasing load from that point to the trailing +edge at 𝑥/𝑐=1. Generally, 𝑎 is greater than or equal to the +position of maximum thickness. For a series 6A-NACA +airfoil, this parameter is approximately equal to: 𝑎=0.8. +And for this particular airfoil, it is equal to: 𝑎=0.831. +The equation of the mean line for the 6-series NACA +airfoil is the following [15], which is a simplification of +the original expression for mean-lines ordinates given in +[16] + +{ + + + + + + + + +𝑦𝑐 +𝑐 = +𝐶𝑙𝑖𝑚𝑜𝑑 +2𝜋(𝑎+1) +{ + + + + 1 +1−𝑎[ +1 +2(𝑎−𝑥 +𝑐) +2 +𝑙𝑜𝑔𝑒|𝑎−𝑥 +𝑐|−1 +2(1−𝑥 +𝑐) +2 +𝑙𝑜𝑔𝑒|1−𝑥 +𝑐|+1 +4(1−𝑥 +𝑐) +2 +−1 +4(𝑎−𝑥 +𝑐) +2] +−𝑥 +𝑐 𝑙𝑜𝑔𝑒(𝑥 +𝑐)+𝑔−ℎ𝑥 +𝑐 +} + + + + +𝑔 = +1 +1−𝑎 [𝑎2 ( +1 +2 𝑙𝑜𝑔𝑒 𝑎 − +1 +4) + +1 +4] +ℎ = +1 +1−𝑎 [ +1 +2 (1 − 𝑎)2 𝑙𝑜𝑔𝑒(1 − 𝑎) − +1 +4 (1 − 𝑎)2] + 𝑔 +𝐶𝑙𝑖𝑚𝑜𝑑 = +𝐶𝑙𝑖 +1.0209 𝑓𝑜𝑟 0 ≺ [ +𝑥 +𝑐 = 𝑎 = 0.831] ≺ 0.87437 + (5) + +Hence, for our chosen airfoil NACA 64A204, we have +the following expression + +𝑦𝑐 +𝐹16(𝑥) = +[ + + + 3.479 𝑙𝑛(|−0.831 + 0.01𝑥|) − 0.084 𝑙𝑛(|−0.831 + 0.01𝑥|) 𝑥 ++5.038 ∗ 10−4 𝑙𝑛(|−0.831 + 0.01𝑥|) 𝑥2 − 5.038 ∗ 10−4 𝑙𝑛(|−1 + 0.01𝑥|) 𝑥2 ++1.007 ∗ 10−1 𝑙𝑛(|−1 + 0.01𝑥|) 𝑥 − 5.038 𝑛(|−1 + 0.01𝑥|) + 0.644 + 0.107𝑥 +−1.703 ∗ 10−2𝑥 𝑙𝑛(𝑥) +] + + + + +(6) +As for thickness distributions, there is no exact +analytical definition for 6-series NACA airfoils, mostly +done through a result of numerical methods which +produced tabulated coordinates. However, the closest +approximation for an analytical expression for thickness +distribution (characterizing the half-thickness of the +airfoil for x) is the following [17]: +𝑦𝑡 +𝑐 = +𝑡 +0.2[0.2969√ +𝑥 +𝑐 − 0.126 +𝑥 +𝑐 − 0.3516 ( +𝑥 +𝑐) +2 ++ 0.2843 ( +𝑥 +𝑐) +3 +− 0.1036 ( +𝑥 +𝑐) +4 +](7) + +Similarly, NACA 64A204 the expression of thickness +distribution is + +𝑦𝑡 +𝐹16(𝑥) = [593.8√𝑥 − 20.72 ∗ 10−6𝑥4 + 5.69 ∗ 10−3𝑥3 +−703.2 ∗ 10−3𝑥2 − 25.2𝑥 +] (8) + +Nevertheless, for the rest of the calculations, since the +profile is supposed to be thin, it is necessary to show the +relative thickness in the equations of the extrados, in the +form [18]: +𝑦 = 𝜀ℎ(𝑥) (9) + + +Fig. 17. Coordinate Projection +Following the projection (See Fig.17.), the formula +used to determine the value of the height of the extrados +for a selected point 𝑄(𝑥𝑄, 𝑦𝑄) is of the following form: + +Q +Ya +Yt +α +Tangent +ye +S +LP +Yt +0 +Xa x +YR +R +N. Eddegdag et al. + + International Review of Aerospace Engineering, Vol. xx, n. x + +{𝑥𝑄 = 𝑥 − 𝑦𝑡 𝑠𝑖𝑛(𝛼) +𝑦𝑄 = 𝑦𝑐 + 𝑦𝑡 𝑐𝑜𝑠(𝛼) with 𝛼 = 𝑎𝑟𝑐𝑡𝑎𝑛 (𝑑𝑦𝑐 +𝑑𝑥 ) (10) + +With 𝑦𝑄 = 𝜀ℎ𝑄(𝑥) for ℎ𝑄(𝑥) = +1 +𝜀 [𝑦𝑐(𝑥) + 𝑦𝑡(𝑥) 𝑐𝑜𝑠(𝛼)] (11) + +Finally, with these extrados Thickness equations, the +NACA 64A204 airfoil can be accurately modelled. +Interestingly, these formulas are actually valid for any +thin airfoil. Hence, after calculations we got the equation +of the extrados curve in 𝑦(𝑥) as following: + +{ + + + + + + + + + + + + + + + + 𝑦1(𝑥) ≅ 1.4574𝑥 − 148.4546𝑥2 +→ 𝐹𝑜𝑟 0.002219 ≤ 𝑥 ≺ 0.0039426 +𝑦2(𝑥) ≅ +[ +7.8283∗10−13+1.0178∗10−1𝑥8−2.3723∗10−2𝑥7 +−5.0395∗10−4𝑥6+4.1529∗10−4𝑥5 +−3.5898∗10−5𝑥4+1.293∗10−6𝑥3− +2.0646∗10−8𝑥2+9.1687∗10−11𝑥 +] +[ +3.934∗10−10+𝑥7−3.5901∗10−1𝑥6+4.1242∗10−2𝑥5 +−1.8993∗10−3𝑥4+1.6406∗10−5𝑥3 ++1.33312∗10−6𝑥2−4.175∗10−8𝑥 +] +→ For 0.0061558 ≤ 𝑥 ≺ 0.1049225 +𝑦3(𝑥) ≅ 8.3094 ∗ 10−3 + 1.2213 ∗ 10−1𝑥 − 1.6721 ∗ 10−1𝑥2 ++3.6633 ∗ 10−2𝑥3 +→ For 0.1049225 ≤ 𝑥 ≤ 1 + (12) + +Now we can calculate Analytical Local Mach numbers +at 30%c and 60%c for NACA64A204 based on our +previously established analytical model [1]-[2]. + +V.1.2. Analytical Local Mach Number for +NACA64A204 + +For F-16 airfoil NACA 64A204, at 30%c and 60%c +Pressure Taps, we get the following analytical results +(See Table III): + +TABLE III +ANALYTICAL LOCAL MACH NUMBER VALUES IN THE PRESSURE TAPS +FOR NACA 64A204 AFTER CALCULATIONS +Mach Upstream +Pressure Tap +Local Analytical Mach +Number +1.55 +Pressure Tap +N26 +1.413532608 +1.55 +Pressure Tap +N27 +1.516532226 +1.44 +Pressure Tap +N26 +1.332598661 +1.44 +Pressure Tap +N27 +1.466334279 +V.2. Results Comparison +In this part, we are interested in comparing the Local +experimental, analytical and numerical Mach number +values at 30%c and 60%c, additionally, we will be +comparing the angles of the detached curved SW and its +distance from the leading edge both numerically and +experimentally. +The purpose of this comparison is to find to which +extent our analytical model [1]-[2] is valid for the case of +a supersonic laminar thin airfoil, as well as to validate our +designed F-16 airfoil mock-ups. + +V.2.1. Local Mach Number Comparison + +After comparing numerical-analytical-experimental +values for Local Mach number at 30%c for NACA +64A204 we can find that the results are very similar, with +relative errors below 𝜀30%𝑐 = 2%. +However, at 60%c when approaching the trailing edge, +we can see that the relative errors slightly increase while +always maintaining a relative error below 𝜀60%𝑐 = 6%. +(See Table IV for data). + + +TABLE IV +Numerical-Experimental-Analytical Local Mach Number at 30%c & 60%c for NACA 64A204 with Relative Errors for Values +Mach +Upstream +Pressure Tap +Local Analytical +Mach Number +Local Numerical +Mach Number +Local Experimental +Mach Number +Relative Error +in % + +1.55 +Pressure Tap N26 +1.413532608 +1.42307 +1.42 +1.18968 +Num-Exp +1.87243 +Num-Analytical +1.55 +Pressure Tap N27 +1.516532226 +1.5296 +1.47 +3.89644 +Num-Exp +3.06833 +Num-Analytical +1.44 +Pressure Tap N26 +1.332598661 +1.3219 +1.33 +1.36924 +Num-Exp +0.55541 +Num-Analytical +1.44 +Pressure Tap N27 +1.466334279 +1.4257 +1.38 +3.20544 +Num-Exp +5.88776 +Num-Analytical +V.2.2. Comparison of Angles of Detached Curved SW +and Its Distance from The Leading Edge + +Taking into consideration the angles of the detached +curved SW from experimentation in SBWT AF300 and +numerical simulations in Ansys Fluent 2022 R2, the +errors between the values 𝛽𝑒𝑥𝑡 and 𝜎𝑒𝑥𝑡, also between + +𝛽𝑖𝑛𝑡 and 𝜎𝑖𝑛𝑡 for an M0=1.44 are as follows: +𝜀𝑒𝑥𝑡 = 12.72% 𝑎𝑛𝑑 𝜀𝑖𝑛𝑡 = 11.11% +As for the distance between the Detached SW and the +leading edge of the NACA 64A204 airfoil, the analytical- +numerical-experimental values for M0=1.44 are as +follows (See Table V): + + + + + +N. Eddegdag et al. + + International Review of Aerospace Engineering, Vol. xx, n. x +TABLE V +Numerical-Experimental-Analytical Distance between Detached +SW and Leading edge for NACA 64A204 +Mach +Upstream +Distance +for +Experiment +Distance for +Simulation +Distance +analytically +M0 +𝛥𝑒𝑥𝑝 +𝛥𝑛𝑢𝑚 +𝛥𝑎𝑛𝑙𝑦 +1.44 +3,48 𝑐𝑚 +2.687 𝑐𝑚 +4.51416 cm +See Appendix for calculation of analytical distance between detached +SW and the leading edge of the airfoil + +Thus, it is possible to determine the relative errors of +the +analytical +distance +𝛥𝑎𝑛𝑙𝑦 +then +numerical +𝛥𝑛𝑢𝑚 for 𝑀0=1.44 with this experimental value: +𝜀𝑛𝑢𝑚 +𝛥 += 22.787% and 𝜀𝑎𝑛𝑎𝑙𝑦 +𝛥 += 29.7172% +These deviations seem large when the distances are +compared to each other, but compared to the overall size +of the airfoil, therefore to the chord, they are negligible. +They correspond to a difference of half a centimetre. It is +assumed that the viscosity of the air which cannot be +neglected in the wind tunnel is the cause. +VI. Discussion & Conclusion +In this study, we managed to prove the validity of our +designed 3D-printed and manually manufactured F-16 +NACA64A204 airfoil mock-ups. The mock-ups were the +subject of experimentation in the SBWT AF300 for a +chosen tunnel section of M0=1.4. The numerical +simulation in Ansys Fluent 2022 R2 and the theoretical +approach [1]-[2] has helped not only to describe +numerically the different phenomena governing SWBLI +around the thin airfoil but also to confirm the validity and +effectiveness of the technics adopted for our mock-up, +first for our 3D-printed mock-up by comparing the angles +of the detached curved SW and its distance from the +leading edge of the airfoil numerically, experimentally +and analytically, the relative errors between values were +of average approx. 26% for the angles and 12% for the +distance, the reason for these errors being moderately +high is due to the 3D-printed mock-up was too fragile and +not resisting vibrations and supersonic gusts, and the +visualization of the shock waves was approximate in the +SBWT. Nevertheless, the precision of the theoretical and +numerical models slightly catches up with the +experimental hazards, since the data of the angles +measured are not reliable but remain consistent to the +order of magnitude. Additionally, for the comparison of +the local Mach number at 30%c and 60%c for NACA +64A204, with our manually manufactured mock-up, +firstly for the 1st Pressure Tap, the error between the +analytical-numerical-experimental results was negligible +at an average below 2%, which greatly validates, at a first +instance, our designed model and improved theoretical +model. As for the 2nd Pressure Tap, the average error +between the analytical-numerical-experimental results +increases slightly but always stays under 6%, which also +validates greatly of mock-up and analytical model. The +reason for which the errors are all under 6% is thanks to +the airfoil we opted for in our study which is a laminar +airfoil, helping to maintain a laminar flow up to 80%c and +then a quasi-transitional flow until the trailing edge due +to the benefit of the eliminated cusp in NACA 6A-series +airfoils. However, the slight increase can be given to the +fact that the flow at the exit of the nozzle of the wind +tunnel which could be not perfectly horizontal because of +the shape of the profile of the nozzle, so the flow will take +a form similar to the shape of the profile. This could +therefore result in a streamline that is not perfectly +horizontal at the level of the experimental wing airfoil, +which would influence the incidence of the fluid +upstream and therefore the acquisitions of pressure +tapping. +Finally, this study not only allowed us to validate our +designed mock-ups, allowing us to open the door for +several other airfoils to be tested and ensure maximum +exploitation of SBWT AF300 in the field of scientific +research, additionally we managed to prove once again +the validity of our previously established theoretical +model at a high percentage of the chord thanks to the +choice of a laminar supersonic airfoil, and last but not +least, we were able to study and dissect different +phenomena that govern viscous supersonic flows around +NACA 6A-Series airfoil. +Appendix +In theory, the distance Δanaly between the curved shock +wave and the leading edge is found using the following +formula: +𝛥𝑎𝑛𝑙𝑦 = 𝑅 × 0.386𝑒 +4.67 +𝑀02 +Where R is the radius of the leading edge of NACA +64A204, for a chord equal to one meter it equals: 𝑅=1.23 +c𝑚. +Acknowledgements +The authors would like to express profound gratitude +and +respect +to +the +late +Professor +Mohammed +HASNAOUI, who passed away due to Covid 19 during +this study, in December 2020. His outstanding morality, +astonishing expertise in the field of fluid mechanics and +asymptotical modelling, as well as supervising role were +very enriching; we, as a team, are honoured to have been +his pupils. +Additionally, we would like to express our deepest +sentiments of respect for our colleague and research team +member Mr Omar El-Aajine for his great commitment +and work that helped lead us to this study. +References +[1] El-Aajine, O., N. Eddegdag, A. Naamane, M. Radouani, and B. El +Fahime. "Asymptotic Modeling of a Viscous Laminar Flow +Around Thin Airfoils: Resolution and Experimental Treatment in + + +N. Eddegdag et al. + + International Review of Aerospace Engineering, Vol. xx, n. x +Case of Supersonic Flow." International Review of Mechanical +Engineering (I.RE.M.E.), Vol. 16, n. 1 (2022). +[2] N. Eddegdag, El-Aajine, O., A. Naamane and M. Radouani. 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Akhmari, Asymptotic modeling the +aerodynamic coefficients of the NACA Airfoil. Modeling, IIETA +Journals, Measurement and Control B Vol. 88, No. 2-4, Page: 58- +66 (2019) +[12] Billig, Frederick S. "Shock-wave shapes around spherical-and +cylindrical-nosed bodies." Journal of Spacecraft and Rockets 4, +no. 6 (1967): 822-823. +[13] Zeytounian, Radyadour Kh. "Singular Coupling and the Triple- +Deck Model." Asymptotic Modelling of Fluid Flow Phenomena +(2002): 471-525. +[14] Abbott IH, Von Doenhoff AE. Theory of wing sections: including +a summary of airfoil data. Courier Corporation; 2012 Apr 26. +[15] Loftin Jr, Laurence K. Theoretical and experimental data for a +number of NACA 6A-series airfoil sections. No. NACA-RM- +L6J01. 1946. +[16] Jacobs, Eastman N. Preliminary report on laminar-flow airfoils +and new methods adopted for airfoil and boundary-layer +investigations. National Aeronautics and Space Admin Langley +Research Center Hampton VA, 1949 +[17] Jacobs, Eastman Nixon, Kenneth Edwards Ward, and Robert +McLean Pinkerton. The Characteristics of 78 related airfoil +section from tests in the Variable-Density Wind Tunnel. No. 460. +US Government Printing Office, 1933. +[18] Series, NACA Four-Digit. "The NACA airfoil series." (2012). +[19] James Campbell & Rade Vignjevic. Artificial Viscosity Methods +for Modeling Shock Wave Propagation (June 2009). +Doi: 10.1007/978-1-4419-0727-1_19 +[20] José Pontes, Norberto Mangiavacchi, Gustavo Rabello dos Anjos, +An Introduction to Compressible Flows with Applications Quasi +One Dimensional Approximation and General Formulation for +Subsonic, Transonic, and Supersonic Flows p. 43 (Springer +International Publishing, 2019). +[21] Boin, J-Ph, J. Ch Robinet, Ch Corre, and H. Deniau. "3d steady +and unsteady bifurcations in a shock-wave/laminar boundary layer +interaction: a numerical study." Theoretical and Computational +Fluid Dynamics 20, no. 3 (2006): 163-180. +[22] Winslow, Justin, Hikaru Otsuka, Bharath Govindarajan, and +Inderjit Chopra. "Basic understanding of airfoil characteristics at +low Reynolds numbers (10 4–10 5)." Journal of Aircraft 55, no. 3 +(2018): 1050-1061. +Authors’ information +1Moulay Ismail University of Meknes, Meknes, Morocco +2Royal Air School, Marrakesh, Morocco +3Moulay Ismail University of Meknes, Meknes, Morocco + +Nasser Eddegdag is a PhD student, and an +engineer in aeronautical systems. He has been +working in the aeronautical industry field since +his graduation in 2020, additionally has been +interested in many research fields concerning +aerodynamics and aeroacoustics, asymptotical +modelling, CFD, boundary layer control, fluid +mechanics and airfoil parametrization and +optimization since 2018. He graduated from the Royal Air Force +Academy (Marrakesh - Morocco) in 2020 as an engineer in aeronautical +systems and is a part of the Multidisciplinary Engineering and +Mechatronic Systems Research Team at Moulay Ismail University of +Meknes. +E-mail: n.eddegdag@edu.umi.ac.ma + + +Aze-eddine Naamane is a professor at the Royal +Air Force Academy (Marrakech - Morocco). He +obtained his PhD thesis in Mechanical +Engineering from Crafts and Technologies, +ENSAM Meknès - Moulay Ismail University, +Morocco. He is a member of the Laboratory of +Mechanics, +mechatronic +and +commands +Meknès, Morocco. His research work is dealing +with the specification and inspection of mechanical systems according +to ISO standards. He is also interested in product numerical engineering +E-mail: azeddine.naamane@gmail.com + + +Mohammed Radouani is a professor at the +National Higher School of Engineering (Crafts +and Technologies, ENSAM Meknès - Moulay +Ismail University, Morocco). He obtained his +PhD thesis in Mechanical Engineering from a +Prestigious training college for teachers and +researchers in Technics (ENS of Cachan, +University of Paris-south XI France, in 2003) +and his Habilitation of supervising scientific research Dissertation from +the Faculty of Sciences of Meknes. His research work is dealing with +the specification and inspection of mechanical systems according to +ISO standards. He is also interested in product numerical engineering +E-mail: M.RADOUANI@ensam-umi.ac.ma + + + + + + + + + + + + + + diff --git a/g9AzT4oBgHgl3EQfMvsC/content/tmp_files/load_file.txt b/g9AzT4oBgHgl3EQfMvsC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5e5798796eaded636babedb2bff47b48959b51e4 --- /dev/null +++ b/g9AzT4oBgHgl3EQfMvsC/content/tmp_files/load_file.txt @@ -0,0 +1,923 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf,len=922 +page_content='International Review of Aerospace Engineering (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='AS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='E), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' xx, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' x XXXX 2023 Manuscript received XXXX 2023, revised XXXX 2023 Design of F-16 Airfoil Mock-ups for Supersonic Wind Tunnel: Study, Production, Testing and Validation N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Eddegdag1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Naamane2, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Radouani3 Abstract – This study aims to establish and compare different methods allowing to describe precisely the viscous supersonic laminar flow behaviour and the Shock Wave- Laminar Boundary Layer Interaction around the F-16 laminar NACA 6-series airfoil NACA 64A204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' This study’s particularity is carried out for a laminar viscous supersonic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The approach adopted to describe these phenomena is by the design and production and experimental testing of a mock-up F-16 airfoil with pressure taps for the supersonic burst wind tunnel AF300, followed by a numerical validation with Ansys Fluent and a theoretical validation based on our previously established analytical SWBLI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The comparison and analysis of obtained experimental, numerical and analytical results validate our designed NACA 64A204 mock-up to a great extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Keywords: Airfoil Mock-up, Laminar Viscous Supersonic Flow, NACA 64A204, Supersonic Burst Wind Tunnel AF300, SWBLI, Nomenclature NACA National Advisory Committee for Aeronautics SWBLI Shock wave-boundary layer interaction SBWT AF300 Supersonic burst wind tunnel AF300 SW Shock wave BL Boundary layer VDAS Versatile data acquisition system ABS Acrylonitrile Butadiene Styrene 𝛽 Ratio of maximum thickness and chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='Lower ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='Deck ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='Main ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='Deck ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='Airfoil ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='chord ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='a Mean-line designation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' fraction of chord from the leading edge over which design load is uniform 𝑦𝑐 Mean-line ordinate 𝐶𝑙𝑖𝑚𝑜𝑑 NACA 6A-series optimal lift coefficient x Distance along the chord y Distance perpendicular to the chord 𝑦𝑡 Ordinate of symmetrical thickness distribution t Airfoil thickness ratio 𝜀 Airfoil parameter 𝑦(𝑥) Wing airfoil equation R Radius of the leading edge 𝑦𝑐 𝐹16(𝑥) NACA64A204 mean-line ordinate 𝑦𝑡 𝐹16(𝑥) NACA64A204 Ordinate of symmetrical thickness distribution I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Introduction The understanding of the supersonic domain, a priori hostile, is crucial and essential from an operational point of view to master it and therefore ensure the achievement of various operational missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' It is also imperative in order to be able to respect the safety and integrity of aircrew and equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The phenomena that appear due to the evolution of an aircraft in a supersonic flow, therefore, continue to be among the subjects of study and research, due to their complexity and lack of research in this particular field of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' One of the main key phenomena that govern and take place in supersonic flow are SWBLIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In our previously published articles and studies, we tackled the modelling of laminar steady irrotational SWBLI around a thin airfoil located in the standard atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The modelling process initiated from Navier-Stokes N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Eddegdag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' International Review of Aerospace Engineering, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' xx, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' x equations adapted to our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' We applied the asymptotic methods, adimensionnalisation and linearization, to obtain the equation governing our problem [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Following that, we respectively applied asymptotic expansions on these governing equations following physical parameters governing our case of study [𝛽 & 𝑅𝑒−1], and finally to obtain the formulas of perturbation velocity potential 𝜑 near the airfoil body far from the leading edge, both inside the boundary layer and in the undisturbed flow outside the boundary layer, we applied the singular perturbations methods in respect to the principle of least degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The resolution process consisted of the adoption of the triple Deck Technic which divides the laminar boundary layer surrounding our obstacle [airfoil] into three major decks [Lower Deck, Main Deck and Upper Deck], in order to identify the constants in the formulas of perturbation velocity potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [2] The analytical model was validated numerically with Ansys Fluent and experimentally in the supersonic burst wind tunnel AF300 with a produced reduced airfoil NACA 5-series with pressure taps NACA 43013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [1]-[2] However, the previous choice of airfoil wasn’t ideal, since in our case study we started from the assumption of a laminar flow, which isn’t accurate, especially when approaching the trailing edge of the chosen airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Hence the results of the analytical-experimental-numerical comparison presented important differences and relative errors when approaching the trailing edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Thus, in this study we focus on a NACA 6-series laminar airfoil NACA 64A204, to perfectly describe the behavior of the flow around the airfoil and the different physical phenomena that take action in supersonic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Hence, in the first part, we will tackle the different methods of design and production process of the F-16 airfoil mock-up with pressure taps in extrados for the Supersonic Burst Wind Tunnel AF300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Secondly, we address the numerical approach in Ansys Fluent for the study of the problem under the same assumptions as in the analytical part [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Furthermore, the experimentation takes parts for the designed mock-up airfoil in the Supersonic Burst Wind Tunnel AF300, followed by an analysis and comparison of the experimental, numerical and analytical results for the NACA 64A204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The aim of the experimental-analytical-numerical confrontation isn’t only to validate our designed F-16 airfoil mock-up, but also to further refine our analytical model by detecting the different parameters that play a role in the behavior of the flow around supersonic laminar airfoils, that weren’t addressed in the previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In addition, we aim to open the doors for several other airfoil mock-ups to be designed and to ensure maximum usage of the supersonic wind tunnel in the field of scientific research in order to be able to study the phenomena which interest the aeronautical field, and not only in pedagogical education, given that the latter is only equipped with a dielectric airfoil mock-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Design of the NACA 64A204 F-16 Airfoil Mock-up In itself, designing a scale model is not an easy task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Its size cannot be determined randomly, because the interpretation of the phenomena generated must also be verified on the real-size model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' To do so, the reduced model must respect the notions of similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Otherwise, the "scale effect" claims that phenomena appearing with the model do not necessarily occur with the object in full size, and even if they reproduce this raises the question of how to transpose the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [3] II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Similarities Between Models The first of the similarities to follow is geometrical similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' It is carried out only when the dimensions of the reduced and real models are linearly homologous according to the same ratio 𝛿, called the “similarity ratio”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The models are then geometrically similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In addition, this ratio is the basis of the industrial design, since it concretely represents the scale and then makes it possible to transpose any point from one model to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [4] In this project, this similarity is respected, as all dimensions are in per cent of chord length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Thus, the similarity ratio 𝛿 is described by the ratio of the reduced model chord to the real model chord: 𝛿 = 𝑐𝑚𝑜𝑐𝑘−𝑢𝑝 𝑐𝑟𝑒𝑎𝑙 (1) With 𝑐𝑚𝑜𝑐𝑘−𝑢𝑝 is the chord for the designed airfoil mock-up and 𝑐𝑟𝑒𝑎𝑙 is the chord for the real aircraft airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Definition of the F-16 NACA 64A204 Airfoil The three-dimensional wing of an aircraft is established from its two-dimensional section, called the airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' There are different types of airfoil categories in aeronautics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In this specific case, it is a NACA airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' These airfoils are found on the majority of aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' They are airfoil designs for aircraft wings developed by NACA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' They thus describe the shape of the airfoil thanks to a series of figures, which represent parameters in the equations generating precisely the section of the wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' For the F-16, this is in particular the NACA 64A204 airfoil (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 1), described by 5 numbers and a letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' This follows a specific convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' All of the following % dimensions are in % chord length 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The explanation of the numbers and the letter of the NACA 64A204 airfoil is as follows [5]: − 6: indicates the series, this airfoil belongs to the NACA series 6 airfoils, which corresponds to the supercritical airfoils, with double camber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' − 4: corresponds to the position of the zone of minimum pressure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=', the position of the zone of maximum thickness, in tens of per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Eddegdag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' International Review of Aerospace Engineering, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' xx, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' x − A: reflects the fact that from approximately 85% to the trailing edge, the trajectory of the middle line retains a constant steering coefficient [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' − 2: indicates the optimum lift coefficient Cli, in tenths of a per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' − 04: describe the maximum thickness 𝑡, in per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' So, in other words, this supercritical airfoil has an optimal lift coefficient Cli of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2 and a maximum thickness 𝑡 of 4% to 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Moreover, the belonging of a NACA airfoil to serie-6 means that its geometry is deduced from its pressure distribution established in the wind tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The 6-Serie also maximizes laminar flow by reducing drag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The advantage of such an airfoil is that it has a very thin relative thickness, which leads to a high critical Mach number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' On the contrary, this same small thickness close to the trailing edge can also become a drawback from the point of view of the design of the structure and of its manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [7] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' F-16 NACA 64A204 Airfoil and F-16 schematics II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=" Design of the F-16 airfoil Mock-up for the AF300 Wind Tunnel The AF300 Rafale supersonic wind tunnel is equipped with a 100mm x 25mm dimension test section in which the 'scale' model of the airfoil is integrated." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Airfoil Modeling on CATIA V5 For this part, we introduce the airfoil parameters in the GNacaLt software (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Then, we choose the maximum of points proposed, that is to say, 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In order to be able to use this software correctly, the decimal symbol must be the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' GNacaLt Software Secondly, we regroup these coordinates for extrados and intrados in per cent of the chord in a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='dat extension file, which we transfer to the GSD_PointSplineLoftFromExcel spreadsheet in Microsoft Office Excel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Furthermore, we transfer the Excel spreadsheet to CATIA V5 via the StarLoft option while activating Macros in the excel file, after interpreting the coordinates, CATIA attach the different uploaded points forming the airfoil (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' NACA 64A204 Airfoil, Chord 60mm in CATIA V5 Software Finally, we create a 3D mock-up of the NACA 64A204 with 60mm Chord, 25mm wingspan and two pressure taps in extrados with separate canalisation for air to enter and thus measure the exact experimental local pressure and Mach values in these pressure taps in the Supersonic wind tunnel AF300, in addition to that slots in the test stream of the wind tunnel AF300 to support the mock-up (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 4) [8] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' NACA 64A204 Airfoil 3D Mock-up, Chord 60mm, width 25mm with two pressure taps with canalisation and mock-up supports for the slots, in CATIA V5 Software Generation des profils NACA (GNacalt) 一 口 X Naca 4-digit Naca 5-digit Naca Mod 4-dgit Naca Mod 5-digi NACA Serie NACA Serie 6A NACA Serie 16 NACA Serie 6A Nom du profl : : 64A204 Nombre de points du profl: : 200 ide Calcul Naca Serie 6A 口 A propos de .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='. chfte 6 6 Non modifiable Eermer/Quitter GNacaLt Version : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='0 BozoSoft 02/2015 Deuxieme chifie [4 De3a5 Francais A Troitieme chffre 2 Entre 0 et 9 Deux dermiers chiffres [04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='00 De 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='00 a 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='00 Nombre de points du profil [200 De 40 a 200 Fichier creo : naca_64A204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='dat Aide [Creer un profi FeimerCATAVS-Proll,GeacaCATPot ProflGnac Puanxy Plann netunnintrunecnmmnde N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Eddegdag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' International Review of Aerospace Engineering, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' xx, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' x II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Manufacturing of the NACA 64A204 Mock-up For the manufacturing of the airfoil NACA 64A204 mock-up, two methods were proposed: − 1st method: by 3D printing − 2nd method: A mock-up of the airfoil by laminate bronze, followed by a canalization by pipes and filling with polyester putty a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 3D Printing of the NACA 64A204 Mock-up 3D printing did not fully meet expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Indeed, as explained before, the design of airfoils with a very low relative thickness encounters difficulties in achieving the trailing edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Here, the dimensions of the models are too thin for the 3D printer, which is therefore unable to finish the trailing edge from about 90-95% of the chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Thus, all scale models are approximately 5 𝑚𝑚 missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In addition, the models seemed fragile and had imperfections in their surface condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=', and the pressure taps weren’t doable due to the fragility of the used material and the small diameter of the pressure taps’ canalization (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' NACA 64A204 Airfoil 3D printed mock-up with 100mm chord b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Manually Manufactured NACA 64A204 Mock-up Note that this method will allow us to have pressure taps on the extrados and the intrados (simultaneously).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Thus, we can evaluate the variations of the Mach number along the airfoil (unlike the dihedral airfoils where we only need the pressure taps on the extrados given the symmetry of the airfoil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Our model is made in three parts: the two sides of the airfoil and a closed box with a stressed coating (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' For the two static pressure taps on the extrados, a channel was designed using pipes with a radius of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='5mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Then, the airfoil mock-up is filled with polyester putty (material that resists high pressures and temperatures to prevent deformation and give weight to the mock-up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Finally, the assembly is soldered by a silver gun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Images of the surfaces and piping constituting the manufactured airfoil mock-up Finally, it was possible to produce the airfoil mock-up NACA 64A204 (F-16) with two pressure taps on the extrados (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 7), with the following properties: − Chord 100mm − Maximum thickness = 100 x 4% = 4 mm − Pressure taps at the leading edge at 30%c x1=30mm − Pressure taps at the trailing edge at 60%c x2=60mm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' NACA 64A204 Airfoil mock-up with two pressure taps in the extrados and a 100mm chord III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Experimentation in the Supersonic Burst Wind Tunnel AF300 In the aeronautical field, wind tunnel testing remains an essential element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' However, they are not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Indeed, the AF300 wind tunnel has a particular operating regime, because it is a supersonic burst wind tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' On the other hand, it should be noted that the flow created by this machine is also subject to the scale effect and must therefore respect similarity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The complete similarity concerns the flows and adds dynamic similarity to the geometrical similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' For the flow of the experiment and that under real conditions to be similar, they must respond to their balance equations in dimensionless form with an identical solution, thanks to initial conditions and limits of the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' This is N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Eddegdag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' International Review of Aerospace Engineering, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' xx, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' x possible only when their dimensionless characteristic numbers, determined according to the principle of dimensional analysis, are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [8] However, this last principle is in reality difficult to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' It is then necessary to concentrate on the characteristic number translating the phenomenon which plays the most important role, to validate a restricted similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In particular for this study, concerning the flow of a compressible fluid like air, it is necessary to take into account the effects of compressibility in addition to the effects of viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' So, the characteristic numbers, in this case, correspond to the Mach number 𝑀 and the Reynolds number 𝑅𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Experimental Tests and Results The AF300 supersonic wind tunnel is a Göttingen type return wind tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' More specifically, it is an economical scaled-down laboratory wind tunnel allowing experiments to be carried out at air speeds of Mach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' It has three different parts: the wind tunnel duct, the test section and the instrumentation panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' For our two conducted experimental studies, the operating conditions (see Table I) are presented as follows: TABLE I OPERATING CONDITIONS FOR NACA 64A204 IN SBWT AF300 Time Operating conditions Time Liner ATM Press Model Angel AOA [s] [mbar] [degree] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='00 Mach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='8 915 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Experiments for the 3D-printed mock-ups The experiment aims to determine via VDAS the Mach number at a point on the upper surface and lower surface, thanks to the new pressure taps 26 and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' It can then be compared with those established by theoretical and numerical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In a second step, with the Schlieren device, the shock waves at the leading and trailing edges can be visualized, possibly with the expansion waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' However, the fragility of the mock-ups, which are made of ABS, made the experiments complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' At the end of the experiment, all the mock-ups were destroyed in the SBWT AF300, because the hooks could not withstand the supersonic gusts, as shown in Fig 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Destroyed 3D printed NACA 64A204 mock ups The mock-up with a 30𝑚𝑚 chord, reduced to 25𝑚𝑚 after 3D printing, was the only one that held in place long enough to visualize, albeit faintly, the curved detached shock waves, below, the leading edge with the Mach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='4 SBWT AF300 nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='9) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Detached curved shock wave visualization for the 25mm chord’s NACA 64A204 mock-up Photo [a] was taken at the beginning of the experiment, the second half through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Thus, the shock on [b] is tighter on the walls because the speed of the flow has increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Also likely, the mock-up is subjected to very virulent vibrations which make it move, this could in another way explain the fact that the shock waves are different between these two photos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Photo [b] also shows two expansion waves visible on [a] [b]T N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Eddegdag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' International Review of Aerospace Engineering, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' xx, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' x the extrados and the intrados.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' They seem to correspond to Mach lines and are also more stuck to the walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In the two photos, the oblique shock wave at the trailing edge is not visible, certainly because the vibrations were the most intense there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' By graphic measurements on the second photo, which is the most usable, the angles of the curved detached shock wave are: 𝜎𝑒𝑥𝑡=55° and 𝜎𝑖𝑛𝑡=58°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Those of the expansion waves measure: 𝜔𝑒𝑥𝑡=34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='5° and 𝜔𝑖𝑛𝑡=36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='5°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Then, the distance Δ between the curved detached shock and the leading edge is estimated to equal Δ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='87 𝑐𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The angles of the shocks on the lower surface should be lower than those on the upper surface because the angles of deflection are greater there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' This may come from the approximation of the representation of the shock waves around the airfoil since the measured angles’ data are unreliable but remain consistent with the order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' These parameters are the only ones that could be evaluated during the various experiments of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' They nevertheless transcribe experimental phenomena on which this project can be based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In particular, these empirical values will be used when compared to rational values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Experiments for the manually manufactured mock-ups After focusing on the smooth running of the experiments, we, therefore, began to extract several readings of values of local static pressure from the pressure taps in the extrados, with the Mach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='8 nozzle airfoil, so we obtained the following results for the reduced wing airfoil NACA 64A204 (see Table II): TABLE II RECORDING OF LOCAL MACH NUMBER VALUES IN THE PRESSURE TAPS FOR NACA 64A204 FROM VDAS Mach Upstream Pressure Tap Local Experimental Mach Number 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='55 Pressure Tap N26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='55 Pressure Tap N27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44 Pressure Tap N26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44 Pressure Tap N27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='38 Thus, we see that the Mach number in extrados increases from the 1st pressure tap to the 2nd due to a supersonic expansion fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Through such an expansion [Prandtl–Meyer expansion fan], the flow remains isentropic, and our hypotheses then remain verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' There is a decrease in pressure and an acceleration of the fluid upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Numerical Simulation of the flow around NACA 64A204 In this part, the numerical approach for the study will be presented, under the same assumptions as in the theoretical part [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Indeed, it will make it possible to obtain the precise behaviour of a viscous supersonic laminar airflow around the NACA 64A204 airfoil after setting the Ansys Fluent 2022 R2 simulation software parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [9] IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Definition of Geometry The geometry definition, by the ANSYS DesignModeler software, concerns two elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' We must first create the object, therefore our wing airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Then, it is just as important to establish the calculation domain, including of course the airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The Airfoil Regarding the object, it can be modelled using different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' For example, the airfoil coordinates can be entered directly into ANSYS Fluent, then the software generates the surface itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In this project, it was rather preferred to draw it with SOLIDWORKS software and then import it later into the digital simulation software, therefore, requires investment before actually using the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' To do this, using the GNacaLt software, a freely downloadable airfoil generator, it was possible to recover a file (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='txt) containing 200 exact coordinates of the NACA 64A204 airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Then, the CAD software can interpret them in space and thus propose a geometry of the airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Once this preparatory work has been done, the digital simulation software can be launched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In the interface assigned to geometry, ANSYS DesignModeler, the drawing under CATIA must be imported and generated, ensuring that it is represented in the correct plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The Study Domain In this area, the calculations will be applied and the equations will be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' It is concretely a viewing window through which the flow is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In this study, after several dimensional tests, the calculation domain below (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=') was selected because it dispenses with the airfoil of the phenomena mentioned just above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Proposed Study Domaine for the airfoil A B R12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 2 F Airfoil G E D 20c N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Eddegdag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' International Review of Aerospace Engineering, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' xx, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' x There are now two distinct surface-type objects saved in the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' But, in this state, it could not take into account the effects of the airfoil on the flow in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' It will then be necessary to merge these two surfaces, or rather subtract them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Indeed, it is necessary to create a new surface corresponding to that of the calculation domain above from which that of the airfoil must be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=') Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Study Domain in Ansys Fluent 2022 R2 for Meshing It must be considered that the airfoil is now fully integrated into the final calculation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [10] Consequently, this last domain is the one that must be meshed, since the calculations taking place there take into account the presence of the airfoil in the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Meshing Under ANSYS Meshing, the Finite Element Method (FEM), used by ANSYS Fluent, requires a division of the final domain through a mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' It is concretely about its spatial discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In other words, the entire domain is geometrically modelled by many smaller domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The interest of meshing is to obtain the simplification of the simulations of calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [10]-[11] The mesh quality has great importance on the results obtained by a numerical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' To do this, we first generate a coarse mesh (automatic) which we then improve by Face Sizing technic in meshing based on a Sphere Influence option to furthermore refine the mesh around the airfoil, followed by an All-Triangle Mesh Control Method, then we insert an Inflation Option to take into account the effect of the boundary layer, and last, we perform a Pinch Control to remove small features at the mesh level to generate better quality elements around those features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Finally, a second modification is to introduce a progressive refinement as the airfoil moves away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Hence, we end up with a Meshing of 1,689,093 Elements and 847,438 Nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=') Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Final Refined Mesh in Ansys Mesher 2022 R2 The numerical resolution procedure can therefore be initiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Calculation Process IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Configuration Of Solver Settings Before launching calculations, the following parameters must be configured: • An adopted model (for our problem we will take the laminar viscous model with energy equation) • Fluid parameters (air): − Air density = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='225 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='m-3 − Air viscosity = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='7894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='10-5 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='s)-1 − Sound speed in air = 347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='092 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' s-1 For the boundary conditions, it is possible to establish the conditions for the evolution of the flow on the walls of the domain, then on the extrados and the intrados.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' On the four outer walls of the calculation domain, the initial conditions are established as if the domain were open to immerse the airfoil in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' It is necessary to specify the value 𝑀0 of the Mach number upstream of the airfoil according to the direction of the flow [for our case since in the experimental study we have an AOA=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1º, thus we have the flow in a direction of 𝑓𝑙𝑜𝑤𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ = 𝑐𝑜𝑠(−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 1∘) 𝑥 + 𝑠𝑖𝑛(−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 1∘) 𝑦 (2) ], and to enter the atmospheric pressure, as well as the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Moreover, in the case where the fluid is viscous, it is necessary to impose a condition of a “sliding wall” on the upper and lower walls of the domain to force the flow to remain straight there without friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Then, for the extrados and the intrados, the solver must be informed that these are simple walls immersed in the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Thus, once the parameters have been saved, we can start the calculation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Graphics Ansys 2022R2 STUDENT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='00 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='00 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='00 (mm) 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='00 Model ViewPrint Preview No Selection Millimeter DegreeDutinr Clpboard [Empty] Elend oSdetEy Name Project" hadd (s) ansys 2022R2 Gecmetry Inport (A2) STUDENT Infisso NeTedSeects Dnpln Detaults Dlsplsy Shie he Geotn Firpilcs Prelerene SoerPreferenre CFD Fluent Elenent Order Ernemi Sies Linear Drad1 y6425 mml Eoport Forrat Qually suinAnsys 2022R2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Eddegdag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' International Review of Aerospace Engineering, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' xx, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' x IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Calculation Process The solver starts from the initial solution and, thanks to an iterative algorithm for solving the matrix system obtained by discretization, it will perform iterations of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Each iteration must modify the current solution and replace it with a solution closer to the exact solution sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The solver gives at each iteration and for each equation the residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Moreover, during a simulation, it is interesting to focus on the convergence of the calculation residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Indeed, at each iteration and for each equation, an error, called “residual”, is evaluated concerning an exact solution to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' And, it is said that a computation converges if the residuals decrease during the iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Thus, a result is only valid if the residuals converge (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' If the residuals do not converge sufficiently, the operation must be repeated until they converge properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Evolution of residuals according to iterations IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Visualization And Analysis of Results The numerical resolution will initially allow checking if the geometry and the mesh have been correctly parameterized thanks to the consistency of the results, and will allow a second time to visualize the behavior of the flow according to the solver settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The simulation step is very important because it allows us to determine the domain of validity of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' NACA 64A204 for M0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44 For 𝑀0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44, with the parameterization established above and after convergence of the residuals, Figure 14 below is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' It represents the behavior of the flow through the evolution of the speed and the Mach number in the final computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The different velocity values are displayed by colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' It also reveals three particular phenomena due to the presence of the airfoil in the supersonic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' First of all, it allows visualizing the variation of the velocity from the leading edge and all along the airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In particular, it is observed that on the upper surface the velocity keeps increasing from the leading edge to the trailing edge where 𝑀𝐸𝑥𝑡𝑟𝑎𝑑𝑜𝑠𝑇𝑟𝑎𝑖𝑙𝑖𝑛𝑔𝐸𝑑𝑔𝑒 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='5305While on the lower surface, it grows faster from the leading edge up to 50% of the chord where 𝑀𝐼𝑛𝑡𝑟𝑎𝑑𝑜𝑠50% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='5004, then decreases to the trailing edge where 𝑀𝐼𝑛𝑡𝑟𝑎𝑑𝑜𝑠𝑇𝑟𝑎𝑖𝑙𝑖𝑛𝑔 𝐸𝑑𝑔𝑒 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='4497, which translates due to the double camber of the NACA64A204 airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Evolution of velocity along the NACA64A204 airfoil for M0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44 Secondly, the figure 15 depicts the curved detached shock wave on the leading edge, with the presence of a blue-green sonic stagnation bubble, where 𝑀<1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='00, where the flow is locally rotational subsonic incompressible and that is due to the curved quasi- hemispherical nature of the airfoil and the supersonic nature of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' As for the vicinity of the trailing edge, only the shock wave attached obliquely to the upper surface is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Finally, the expansion waves in form of Mach Lines appear clearly on the airfoil in form of a Prandtl-Meyer expansion wave that is due to the convex and concave forms of the airfoil curves along its extrados and intrados Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Visualization of SW and Sonic Bubble for NACA64A204 Finally, by graphical measurements in Figure 15, the shock angles of the curved detached wave are worth at the leading edge: 𝛽𝑒𝑥𝑡=62° and 𝛽𝑖𝑛𝑡=60°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' These angles are between 90° and the Mach angle of 𝜇=𝑎𝑟𝑐𝑠𝑖𝑛(1/M0) = 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='983°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Similarly, the angle at the trailing edge is equal to 𝛾𝑒𝑥𝑡=40°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 1e+00 Ansys 2022R2 STUDENT 1e 01 1e 02 1e 03 0 500 1000 1500 Iterations continuity-- x velocity y velocity -- energyVelocity Ansys airtiow 5287e+02 2022R2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='993e+02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='700e+02 STUDENT 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='406e+02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='112e+02 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='525e+02 818et +02 102 937e+02 644 +02 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='875e+0 937 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='000e+0 etl [ms^ 1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='300(m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='225Velocity Ansys airtlow 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='287e+02 2022R2 993e+02 700e+02 STUDENT 406e+0 02 Be 25e e 12 056e+ t69e 875e 00+a000 [msA 1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='02(m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='015 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Eddegdag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' International Review of Aerospace Engineering, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' xx, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' x The angles 𝛽𝑒𝑥𝑡 and 𝛽𝑖𝑛𝑡 are not the same because the airfoil is not symmetrical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Thus, it happens 𝛽𝑒𝑥𝑡>𝛽𝑖𝑛𝑡 since the angle of deflection of the flow on the upper surface is greater than that on the lower surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' On the other hand, the distance Δ𝑛𝑢𝑚 between the curved shock wave and the leading edge is graphically worth: Δ𝑛𝑢𝑚=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='687 𝑐𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Finally, the third observed phenomenon is the shock wave-boundary layer interaction, which is the result of viscous supersonic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' According to figure 16, we note the existence of a boundary layer all along the wall of the thickness airfoil 𝑙0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='3076mm where the Mach is subsonic and the speed tends towards a zero-value approaching the surface of the airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Thus, the fluid particle (air) adheres to the wall (no-slip condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' It is also clear the three sub-layers of the boundary layer (viscous sub-layer in dark blue, main layer in sky blue and the upper layer in green) highlight our triple-deck model chosen during the theoretical part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' We also note the existence of shock waves after the boundary layer (in height, Prandtl-Meyer expansion fan) where the Mach is supersonic, which highlights the phenomenon of the shock/boundary layer interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [12] Thus, the local Mach number goes from a subsonic value in the boundary layer (M<1) to a supersonic value during shock waves where the external flow is supersonic (M>1), by a sonic line where the local Mach number along this line is equal to 1, to characterize and allow the subsonic-transonic-supersonic transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [13] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Visualization of the Triple Deck Boundary Layer around the NACA 64A204 airfoil by Simulation on Ansys Fluent For comparison, it is interesting to run the calculation again for a different upstream Mach number 𝑀0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' NACA 64A204 for M0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='55 For 𝑀0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='55, with the same parameterization and still, after convergence of the residuals, results are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The same phenomena of SWBLI and Curved Detached SW with the clear appearance of the triple-layered BL appear with increasing width of l0’=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='4262mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Similarly, it is observed that on the upper surface the Mach number keeps increasing from the leading edge to the trailing edge where 𝑀𝐸𝑥𝑡𝑟𝑎𝑑𝑜𝑠𝑇𝑟𝑎𝑖𝑙𝑖𝑛𝑔𝐸𝑑𝑔𝑒 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='55 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='606 While on the intrados, it also grows faster from the leading edge to the same position at 50% of the chord where this time 𝑀𝐼𝑛𝑡𝑟𝑎𝑑𝑜𝑠50% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='55 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='579, to then decrease to the trailing edge where 𝑀𝐼𝑛𝑡𝑟𝑎𝑑𝑜𝑠𝑇𝑟𝑎𝑖𝑙𝑖𝑛𝑔𝐸𝑑𝑔𝑒 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='55 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Additionally, the curved detached SW, here tighter on the leading edge, with the presence of a blue-green sonic bubble, where 𝑀′<1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='00, with a larger radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Then, near the trailing edge, the oblique attached shock wave is still only visible on the upper surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Finally, the expansion waves always appear clearly on the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In the same way as for the previous simulation, by graphic measurements, the shock angles of the curved wave are worth in this case: 𝛽’𝑒𝑥𝑡=55° and 𝛽’𝑖𝑛𝑡=52°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' These angles are, likewise, between 90° and 𝜇′=𝑎𝑟𝑐𝑠𝑖𝑛(1/𝑀0) =40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1778°<𝜇, corresponding to the minimum shock angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Moreover, that at the trailing edge is equal to 𝛾𝑒𝑥𝑡′=30°<𝛾𝑒𝑥𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' For the distance Δ𝑛𝑢𝑚′, it finally comes graphically here: Δ𝑛𝑢𝑚′=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='124 𝑐𝑚<Δ𝑛𝑢𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Consequently, it is deduced that the higher the velocity of the upstream flow, the smaller the angle of the curved detached shock wave for the same profile, and the greater the distance between this shock is the edge of attack is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The first simulations are thus completed and analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' However, it was expected to obtain a great finesse of the solutions according to the mesh and the established settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Now, from these figures, the results look coarse along the different shock waves, especially for a high 𝑀0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' We must therefore find a way to be even more precise in post-processing simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' From Simulation to Analytical Model Improvement Based on our previous published studies concerning the modelling of the laminar SWBLI around a thin airfoil [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' And, as a reflection and application of our soon-to- be-published study entitled ‘Improved Analytical/Statistical Modelling of the Shock Wave- Laminar Boundary Layer Around a Thin Airfoil: Standard Atmosphere Case’ [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' We tend similarly for our object of this study F-16 airfoil NACA64A204, to improve Analytically/Statistically the analytical model for the F-16 airfoil to obtain the most accurate analytical results for the confrontation and validation of our mock- up and exact description of the multitude of phenomena that govern the supersonic laminar viscous flow around F-16 laminar thin airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' For the NACA 64A204 series-6 airfoil with coefficient β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='04, the Local Mach numbers are compared on the upper surface and lower surface in the wall far from the leading edge, thus the position of the maximum relative camber characterizes the end of the zone from the leading edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' It is known for every NACA wing profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Generally, it is defined by the second digit of the series, for the NACA 64A204 profile it is 4% of the chord, so Velocity Ansys airfiow 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2878+02 2022R2 993e+02 STUDENT 000 +00 ms 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='0004 (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content="0001 000'0 N." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Eddegdag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' International Review of Aerospace Engineering, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' xx, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' x we perform our calculations and extract our results for x greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' In addition, we take different points in the 3 parts of the profile function as well as different altitudes of η (in the boundary layer and outside the boundary layer), to be able to check our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Similarly, we have that 0 ≤ 𝜂̄ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='13257143, as well as y must be between 0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='132 for x between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='006158 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Thus, for M0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44, we have that the equation for the Link Mach Number m (Local incompressible Mach number linking the Lower Deck to the Main Deck) is the following: 𝑚𝑁𝐴𝐶𝐴64𝐴204 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44 (𝜂̄, 𝑠̄, 𝑀∞) = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='0474𝜂̄ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='5406𝑠̄ − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='0722 (3) Same for M0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='55, we have that the equation for the Link Mach Number is the following: 𝑚𝑁𝐴𝐶𝐴64𝐴204 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='55 (𝜂̄, 𝑠̄, 𝑀∞) = −35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='6199𝜂̄ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='8276𝑠̄ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='7234(4) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Confrontation Of Analytical-Numerical- Experimental Results The aim of this chapter is to compare our experimental results of the Local Mach numbers in Pressure Taps 1 and 2, to the results of the Local Mach numbers in the same coordinates from simulation in Ansys Fluent and the analytical results based on our improved analytical model [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The comparison will lead not only to the validation of the designed mock-up of the F-16 airfoil, but also to further elaborate the phenomena that take place in SWBLI around the F-16 wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1 Analytical results Our analytical results for the Local Mach numbers for NACA64A204 airfoil in the extrados at 30% and 60% of the chord (c=100mm) are calculated from our previously established velocity formulas from our previous studies [1]-[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' However, since these formulas are in the Frenet coordinate system, it is imperative to establish these formulas in the cartesian system by first and foremost establishing the NACA64A204 airfoil equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1 NACA 64A204 Wing Section Equation The NACA airfoils are constructed by combining a thickness envelope with a camber or mean line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The 6- series mean lines were designed using Thin Airfoil Theory to produce a constant loading from the leading edge back to x/c = a, after which the loading decreases linearly to zero at the trailing edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The term 𝑎, not to be confused with the speed of sound in air, denotes a particular position on the string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [14] Indeed, the camber line of a Series-6 NACA airfoil produces a uniform loading along the chord from the leading edge at 𝑥/𝑐=0 to the point 𝑥/𝑐=𝑎, then produces a linearly decreasing load from that point to the trailing edge at 𝑥/𝑐=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Generally, 𝑎 is greater than or equal to the position of maximum thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' For a series 6A-NACA airfoil, this parameter is approximately equal to: 𝑎=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' And for this particular airfoil, it is equal to: 𝑎=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='831.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The equation of the mean line for the 6-series NACA airfoil is the following [15], which is a simplification of the original expression for mean-lines ordinates given in [16] { 𝑦𝑐 𝑐 = 𝐶𝑙𝑖𝑚𝑜𝑑 2𝜋(𝑎+1) { 1 1−𝑎[ 1 2(𝑎−𝑥 𝑐) 2 𝑙𝑜𝑔𝑒|𝑎−𝑥 𝑐|−1 2(1−𝑥 𝑐) 2 𝑙𝑜𝑔𝑒|1−𝑥 𝑐|+1 4(1−𝑥 𝑐) 2 −1 4(𝑎−𝑥 𝑐) 2] −𝑥 𝑐 𝑙𝑜𝑔𝑒(𝑥 𝑐)+𝑔−ℎ𝑥 𝑐 } 𝑔 = 1 1−𝑎 [𝑎2 ( 1 2 𝑙𝑜𝑔𝑒 𝑎 − 1 4) + 1 4] ℎ = 1 1−𝑎 [ 1 2 (1 − 𝑎)2 𝑙𝑜𝑔𝑒(1 − 𝑎) − 1 4 (1 − 𝑎)2] + 𝑔 𝐶𝑙𝑖𝑚𝑜𝑑 = 𝐶𝑙𝑖 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='0209 𝑓𝑜𝑟 0 ≺ [ 𝑥 𝑐 = 𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='831] ≺ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='87437 (5) Hence, for our chosen airfoil NACA 64A204, we have the following expression 𝑦𝑐 𝐹16(𝑥) = [ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='479 𝑙𝑛(|−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='831 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='01𝑥|) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='084 𝑙𝑛(|−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='831 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='01𝑥|) 𝑥 +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='038 ∗ 10−4 𝑙𝑛(|−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='831 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='01𝑥|) 𝑥2 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='038 ∗ 10−4 𝑙𝑛(|−1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='01𝑥|) 𝑥2 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='007 ∗ 10−1 𝑙𝑛(|−1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='01𝑥|) 𝑥 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='038 𝑛(|−1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='01𝑥|) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='644 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='107𝑥 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='703 ∗ 10−2𝑥 𝑙𝑛(𝑥) ] (6) As for thickness distributions, there is no exact analytical definition for 6-series NACA airfoils, mostly done through a result of numerical methods which produced tabulated coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' However, the closest approximation for an analytical expression for thickness distribution (characterizing the half-thickness of the airfoil for x) is the following [17]: 𝑦𝑡 𝑐 = 𝑡 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2969√ 𝑥 𝑐 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='126 𝑥 𝑐 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='3516 ( 𝑥 𝑐) 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2843 ( 𝑥 𝑐) 3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1036 ( 𝑥 𝑐) 4 ](7) Similarly, NACA 64A204 the expression of thickness distribution is 𝑦𝑡 𝐹16(𝑥) = [593.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='8√𝑥 − 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='72 ∗ 10−6𝑥4 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='69 ∗ 10−3𝑥3 −703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2 ∗ 10−3𝑥2 − 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2𝑥 ] (8) Nevertheless, for the rest of the calculations, since the profile is supposed to be thin, it is necessary to show the relative thickness in the equations of the extrados, in the form [18]: 𝑦 = 𝜀ℎ(𝑥) (9) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Coordinate Projection Following the projection (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' ), the formula used to determine the value of the height of the extrados for a selected point 𝑄(𝑥𝑄, 𝑦𝑄) is of the following form: Q Ya Yt α Tangent ye S LP Yt 0 Xa x YR R N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Eddegdag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' International Review of Aerospace Engineering, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' xx, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' x {𝑥𝑄 = 𝑥 − 𝑦𝑡 𝑠𝑖𝑛(𝛼) 𝑦𝑄 = 𝑦𝑐 + 𝑦𝑡 𝑐𝑜𝑠(𝛼) with 𝛼 = 𝑎𝑟𝑐𝑡𝑎𝑛 (𝑑𝑦𝑐 𝑑𝑥 ) (10) With 𝑦𝑄 = 𝜀ℎ𝑄(𝑥) for ℎ𝑄(𝑥) = 1 𝜀 [𝑦𝑐(𝑥) + 𝑦𝑡(𝑥) 𝑐𝑜𝑠(𝛼)] (11) Finally, with these extrados Thickness equations, the NACA 64A204 airfoil can be accurately modelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Interestingly, these formulas are actually valid for any thin airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Hence, after calculations we got the equation of the extrados curve in 𝑦(𝑥) as following: { 𝑦1(𝑥) ≅ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='4574𝑥 − 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='4546𝑥2 → 𝐹𝑜𝑟 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='002219 ≤ 𝑥 ≺ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='0039426 𝑦2(𝑥) ≅ [ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='8283∗10−13+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='0178∗10−1𝑥8−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='3723∗10−2𝑥7 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='0395∗10−4𝑥6+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1529∗10−4𝑥5 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='5898∗10−5𝑥4+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='293∗10−6𝑥3− 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='0646∗10−8𝑥2+9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1687∗10−11𝑥 ] [ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='934∗10−10+𝑥7−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='5901∗10−1𝑥6+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1242∗10−2𝑥5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='8993∗10−3𝑥4+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='6406∗10−5𝑥3 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='33312∗10−6𝑥2−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='175∗10−8𝑥 ] → For 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='0061558 ≤ 𝑥 ≺ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1049225 𝑦3(𝑥) ≅ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='3094 ∗ 10−3 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2213 ∗ 10−1𝑥 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='6721 ∗ 10−1𝑥2 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='6633 ∗ 10−2𝑥3 → For 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1049225 ≤ 𝑥 ≤ 1 (12) Now we can calculate Analytical Local Mach numbers at 30%c and 60%c for NACA64A204 based on our previously established analytical model [1]-[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Analytical Local Mach Number for NACA64A204 For F-16 airfoil NACA 64A204, at 30%c and 60%c Pressure Taps, we get the following analytical results (See Table III): TABLE III ANALYTICAL LOCAL MACH NUMBER VALUES IN THE PRESSURE TAPS FOR NACA 64A204 AFTER CALCULATIONS Mach Upstream Pressure Tap Local Analytical Mach Number 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='55 Pressure Tap N26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='413532608 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='55 Pressure Tap N27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='516532226 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44 Pressure Tap N26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='332598661 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44 Pressure Tap N27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='466334279 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Results Comparison In this part, we are interested in comparing the Local experimental, analytical and numerical Mach number values at 30%c and 60%c, additionally, we will be comparing the angles of the detached curved SW and its distance from the leading edge both numerically and experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The purpose of this comparison is to find to which extent our analytical model [1]-[2] is valid for the case of a supersonic laminar thin airfoil, as well as to validate our designed F-16 airfoil mock-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Local Mach Number Comparison After comparing numerical-analytical-experimental values for Local Mach number at 30%c for NACA 64A204 we can find that the results are very similar, with relative errors below 𝜀30%𝑐 = 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' However, at 60%c when approaching the trailing edge, we can see that the relative errors slightly increase while always maintaining a relative error below 𝜀60%𝑐 = 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' (See Table IV for data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' TABLE IV Numerical-Experimental-Analytical Local Mach Number at 30%c & 60%c for NACA 64A204 with Relative Errors for Values Mach Upstream Pressure Tap Local Analytical Mach Number Local Numerical Mach Number Local Experimental Mach Number Relative Error in % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='55 Pressure Tap N26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='413532608 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='42307 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='18968 Num-Exp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='87243 Num-Analytical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='55 Pressure Tap N27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='516532226 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='5296 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='89644 Num-Exp 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='06833 Num-Analytical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44 Pressure Tap N26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='332598661 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='3219 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='36924 Num-Exp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='55541 Num-Analytical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44 Pressure Tap N27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='466334279 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='4257 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='20544 Num-Exp 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='88776 Num-Analytical V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Comparison of Angles of Detached Curved SW and Its Distance from The Leading Edge Taking into consideration the angles of the detached curved SW from experimentation in SBWT AF300 and numerical simulations in Ansys Fluent 2022 R2, the errors between the values 𝛽𝑒𝑥𝑡 and 𝜎𝑒𝑥𝑡, also between 𝛽𝑖𝑛𝑡 and 𝜎𝑖𝑛𝑡 for an M0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44 are as follows: 𝜀𝑒𝑥𝑡 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='72% 𝑎𝑛𝑑 𝜀𝑖𝑛𝑡 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='11% As for the distance between the Detached SW and the leading edge of the NACA 64A204 airfoil, the analytical- numerical-experimental values for M0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44 are as follows (See Table V): N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Eddegdag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' International Review of Aerospace Engineering, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' xx, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' x TABLE V Numerical-Experimental-Analytical Distance between Detached SW and Leading edge for NACA 64A204 Mach Upstream Distance for Experiment Distance for Simulation Distance analytically M0 𝛥𝑒𝑥𝑝 𝛥𝑛𝑢𝑚 𝛥𝑎𝑛𝑙𝑦 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44 3,48 𝑐𝑚 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='687 𝑐𝑚 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='51416 cm See Appendix for calculation of analytical distance between detached SW and the leading edge of the airfoil Thus, it is possible to determine the relative errors of the analytical distance 𝛥𝑎𝑛𝑙𝑦 then numerical 𝛥𝑛𝑢𝑚 for 𝑀0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='44 with this experimental value: 𝜀𝑛𝑢𝑚 𝛥 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='787% and 𝜀𝑎𝑛𝑎𝑙𝑦 𝛥 = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='7172% These deviations seem large when the distances are compared to each other, but compared to the overall size of the airfoil, therefore to the chord, they are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' They correspond to a difference of half a centimetre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' It is assumed that the viscosity of the air which cannot be neglected in the wind tunnel is the cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Discussion & Conclusion In this study, we managed to prove the validity of our designed 3D-printed and manually manufactured F-16 NACA64A204 airfoil mock-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The mock-ups were the subject of experimentation in the SBWT AF300 for a chosen tunnel section of M0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The numerical simulation in Ansys Fluent 2022 R2 and the theoretical approach [1]-[2] has helped not only to describe numerically the different phenomena governing SWBLI around the thin airfoil but also to confirm the validity and effectiveness of the technics adopted for our mock-up,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' first for our 3D-printed mock-up by comparing the angles of the detached curved SW and its distance from the leading edge of the airfoil numerically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' experimentally and analytically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' the relative errors between values were of average approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 26% for the angles and 12% for the distance, the reason for these errors being moderately high is due to the 3D-printed mock-up was too fragile and not resisting vibrations and supersonic gusts, and the visualization of the shock waves was approximate in the SBWT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Nevertheless, the precision of the theoretical and numerical models slightly catches up with the experimental hazards, since the data of the angles measured are not reliable but remain consistent to the order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Additionally, for the comparison of the local Mach number at 30%c and 60%c for NACA 64A204, with our manually manufactured mock-up, firstly for the 1st Pressure Tap, the error between the analytical-numerical-experimental results was negligible at an average below 2%, which greatly validates, at a first instance, our designed model and improved theoretical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' As for the 2nd Pressure Tap, the average error between the analytical-numerical-experimental results increases slightly but always stays under 6%, which also validates greatly of mock-up and analytical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The reason for which the errors are all under 6% is thanks to the airfoil we opted for in our study which is a laminar airfoil, helping to maintain a laminar flow up to 80%c and then a quasi-transitional flow until the trailing edge due to the benefit of the eliminated cusp in NACA 6A-series airfoils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' However, the slight increase can be given to the fact that the flow at the exit of the nozzle of the wind tunnel which could be not perfectly horizontal because of the shape of the profile of the nozzle, so the flow will take a form similar to the shape of the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' This could therefore result in a streamline that is not perfectly horizontal at the level of the experimental wing airfoil, which would influence the incidence of the fluid upstream and therefore the acquisitions of pressure tapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' this study not only allowed us to validate our designed mock-ups,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' allowing us to open the door for several other airfoils to be tested and ensure maximum exploitation of SBWT AF300 in the field of scientific research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' additionally we managed to prove once again the validity of our previously established theoretical model at a high percentage of the chord thanks to the choice of a laminar supersonic airfoil,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' and last but not least,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' we were able to study and dissect different phenomena that govern viscous supersonic flows around NACA 6A-Series airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Appendix In theory, the distance Δanaly between the curved shock wave and the leading edge is found using the following formula: 𝛥𝑎𝑛𝑙𝑦 = 𝑅 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='386𝑒 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='67 𝑀02 Where R is the radius of the leading edge of NACA 64A204, for a chord equal to one meter it equals: 𝑅=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='23 c𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Acknowledgements The authors would like to express profound gratitude and respect to the late Professor Mohammed HASNAOUI, who passed away due to Covid 19 during this study, in December 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' His outstanding morality, astonishing expertise in the field of fluid mechanics and asymptotical modelling, as well as supervising role were very enriching;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' we, as a team, are honoured to have been his pupils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Additionally, we would like to express our deepest sentiments of respect for our colleague and research team member Mr Omar El-Aajine for his great commitment and work that helped lead us to this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' References [1] El-Aajine, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Eddegdag, A.' metadata={'source': 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https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='org/submit/4661338/view [3] Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=', Zhou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=', Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=', Shuai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=', Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=', & Cai, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The design of a small lab-scale wind turbine model with high performance similarity to its utility-scale prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Renewable Energy, 149, 435-444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [4] He, Shun, Shijun Guo, Ying Liu, and Wukui Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' "Passive gust alleviation of a flying-wing aircraft by analysis and wind-tunnel test of a scaled model in dynamic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='" Aerospace Science and Technology 113 (2021): 106689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [5] Ladson, Charles L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=', Cuyler W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Brooks Jr, Acquilla S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Hill, and Darrell W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Sproles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Computer program to obtain ordinates for NACA airfoils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' L-17509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [6] Abbott, Ira H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=', Albert E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' von Doenhoff, and Louis Stivers Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Summary of airfoil data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' NACA-TR-824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 1945.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [7] Patterson, Elizabeth W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=', and Albert L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Braslow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Ordinates and Theoretical Pressure-distribution Data for NACA 6-and 6A-series Airfoil Sections with Thicknesses from 2 to 21 and from 2 to 15 Percent Chord, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' National Aeronautics and Space Administration, 1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [8] TecQuipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' AF300 Supersonic Burst Wind Tunnel User Manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [9] Ives, Rob, Edet BASSEY, and Faik A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' HAMAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' "Investigation of the flow around an aircraft wing of section NACA 2412 utilising ANSYS fluent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='" INCAS Bulletin 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 1 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [10] Naamane, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Hasnaoui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' "Supersonic Flow around a Dihedral Airfoil: Modeling and Experimentation Investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='" International Journal of Aerospace and Mechanical Engineering 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 6 (2019): 413-417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='Hasnaoui, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Naamane, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Akhmari, Asymptotic modeling the aerodynamic coefficients of the NACA Airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Modeling, IIETA Journals, Measurement and Control B Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 88, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 2-4, Page: 58- 66 (2019) [12] Billig, Frederick S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' "Shock-wave shapes around spherical-and cylindrical-nosed bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='" Journal of Spacecraft and Rockets 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 6 (1967): 822-823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [13] Zeytounian, Radyadour Kh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' "Singular Coupling and the Triple- Deck Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='" Asymptotic Modelling of Fluid Flow Phenomena (2002): 471-525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [14] Abbott IH, Von Doenhoff AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Theory of wing sections: including a summary of airfoil data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Courier Corporation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 2012 Apr 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [15] Loftin Jr, Laurence K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Theoretical and experimental data for a number of NACA 6A-series airfoil sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' NACA-RM- L6J01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 1946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [16] Jacobs, Eastman N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Preliminary report on laminar-flow airfoils and new methods adopted for airfoil and boundary-layer investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' National Aeronautics and Space Admin Langley Research Center Hampton VA, 1949 [17] Jacobs, Eastman Nixon, Kenneth Edwards Ward, and Robert McLean Pinkerton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' The Characteristics of 78 related airfoil section from tests in the Variable-Density Wind Tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' US Government Printing Office, 1933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [18] Series, NACA Four-Digit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' "The NACA airfoil series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='" (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [19] James Campbell & Rade Vignjevic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Artificial Viscosity Methods for Modeling Shock Wave Propagation (June 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='1007/978-1-4419-0727-1_19 [20] José Pontes, Norberto Mangiavacchi, Gustavo Rabello dos Anjos, An Introduction to Compressible Flows with Applications Quasi One Dimensional Approximation and General Formulation for Subsonic, Transonic, and Supersonic Flows p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 43 (Springer International Publishing, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [21] Boin, J-Ph, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Ch Robinet, Ch Corre, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Deniau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' "3d steady and unsteady bifurcations in a shock-wave/laminar boundary layer interaction: a numerical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='" Theoretical and Computational Fluid Dynamics 20, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 3 (2006): 163-180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' [22] Winslow, Justin, Hikaru Otsuka, Bharath Govindarajan, and Inderjit Chopra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' "Basic understanding of airfoil characteristics at low Reynolds numbers (10 4–10 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='" Journal of Aircraft 55, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' 3 (2018): 1050-1061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' Authors’ information 1Moulay Ismail University of Meknes, Meknes, Morocco 2Royal Air School, Marrakesh, Morocco 3Moulay Ismail University of Meknes, Meknes, Morocco Nasser Eddegdag is a PhD student, and an engineer in aeronautical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' He has been working in the aeronautical industry field since his graduation in 2020, additionally has been interested in many research fields concerning aerodynamics and aeroacoustics, asymptotical modelling, CFD, boundary layer control, fluid mechanics and airfoil parametrization and optimization since 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' He graduated from the Royal Air Force Academy (Marrakesh - Morocco) in 2020 as an engineer in aeronautical systems and is a part of the Multidisciplinary Engineering and Mechatronic Systems Research Team at Moulay Ismail University of Meknes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' E-mail: n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='eddegdag@edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='umi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='ma Aze-eddine Naamane is a professor at the Royal Air Force Academy (Marrakech - Morocco).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' He obtained his PhD thesis in Mechanical Engineering from Crafts and Technologies, ENSAM Meknès - Moulay Ismail University, Morocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' He is a member of the Laboratory of Mechanics, mechatronic and commands Meknès, Morocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' His research work is dealing with the specification and inspection of mechanical systems according to ISO standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' He is also interested in product numerical engineering E-mail: azeddine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='naamane@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='com Mohammed Radouani is a professor at the National Higher School of Engineering (Crafts and Technologies, ENSAM Meknès - Moulay Ismail University, Morocco).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' He obtained his PhD thesis in Mechanical Engineering from a Prestigious training college for teachers and researchers in Technics (ENS of Cachan, University of Paris-south XI France, in 2003) and his Habilitation of supervising scientific research Dissertation from the Faculty of Sciences of Meknes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' His research work is dealing with the specification and inspection of mechanical systems according to ISO standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content=' He is also interested in product numerical engineering E-mail: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='RADOUANI@ensam-umi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} +page_content='ma' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfMvsC/content/2301.01135v1.pdf'} diff --git a/gdAyT4oBgHgl3EQfj_hk/content/tmp_files/2301.00424v1.pdf.txt b/gdAyT4oBgHgl3EQfj_hk/content/tmp_files/2301.00424v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d7987b763fcec24fe7c23bee118ef91367ab036e --- /dev/null +++ b/gdAyT4oBgHgl3EQfj_hk/content/tmp_files/2301.00424v1.pdf.txt @@ -0,0 +1,1797 @@ +GoogLe2Net: Going Transverse with Convolutions +Yuanpeng Hea,b +aKey Laboratory of High Confidence Software Technologies, Peking +University, Peking, 100871, China +bSchool of Computer Science, Peking University, Peking, 100871, China +Abstract +Capturing feature information effectively is of great importance in vision +tasks. With the development of convolutional neural networks (CNNs), con- +cepts like residual connection and multiple scales promote continual perfor- +mance gains on diverse deep learning vision tasks. However, the existing +methods do not organically combined advantages of these valid ideas. In +this paper, we propose a novel CNN architecture called GoogLe2Net, it con- +sists of residual feature-reutilization inceptions (ResFRI) or split residual +feature-reutilization inceptions (Split-ResFRI) which create transverse pas- +sages between adjacent groups of convolutional layers to enable features flow +to latter processing branches and possess residual connections to better pro- +cess information. Our GoogLe2Net is able to reutilize information captured +by foregoing groups of convolutional layers and express multi-scale features +at a fine-grained level, which improves performances in image classification. +And the inception we proposed could be embedded into inception-like net- +works directly without any migration costs. Moreover, in experiments based +on popular vision datasets, such as CIFAR10 (97.94%), CIFAR100 (85.91%) +and Tiny Imagenet (70.54%), we obtain better results on image classification +task compared with other modern models. +Keywords: +Feature-reutilization Transverse passages Inception +1. Introduction +In recent years, we’ve witnessed a rapid advance of CNNs and this field +is attracting more and more attention from researchers around the world. +Noticeably, in order to meet demands of different vision tasks such as im- +age classification, target tracking, image segmentation, skeleton extraction +Preprint submitted to Elsevier +January 3, 2023 +arXiv:2301.00424v1 [cs.CV] 1 Jan 2023 + +, facial recognition and image description, a large number of vision neural +network models have been proposed [1, 2, 3, 4, 5, 6]. And how to effectively +extract information to satisfy demands of different kinds of vision tasks is +still an open issue. Remarkably, capturing features from multiple scales to +obtain more information has been a hot spot in computer vision-related fields +which boosts performances of models. +The concept of multi-scale has already been introduced into deep learning- +related fields [7, 8, 9, 10] and its superiority was fully demonstrated by various +applications. As a general rule, CNNs may acquire features utilizing convo- +lutional kernels with different sizes from roughness to detail. Therefore, the +key to boost performance of vision models is to devise a more efficient and +effective policy of capturing features. And recently, on the basis of common +residual block [11], a multi-scale architecture called Res2Net [12] is devised +to better obtain and aggregate information at different scales. The idea of it +resembles the one of Pyramid networks [13] and the Res2Net block can con- +tinually enlarge the receptive filed through stacking 3×3 convolutional layers. +Besides, the effectiveness of it is proved by the outstanding performance in +diverse vision tasks. +11 +11 +11 +11 Max +Pooling +33 +55 +Previous Layer +11 +Filter Concatenation +Figure 1: Original Inception from GoogLeNet +2 + +Enlightened by the concept of Pyramid network and compositions of +Res2Net block, we intend to generalize the idea of them to other networks +which own relatively small parameter amount and similar architecture to +ensure that the newly proposed block of network is efficient and modifica- +tions on it are straightforward. In order to fuse information more efficiently +and acquire multi-scale features in larger receptive fields, we propose a novel +GoogLe2Net based on GoogLeNet [14]. The proposed GoogLe2Net has two +versions which consists of residual feature-reutilization inception (ResFRI) +and split-residual feature-reutilization inception (Split-ResFRI) respectively. +About the model architecture, firstly, for the input layer, we adopt two dis- +parate policies. For the first one, we utilize the original input layer from +GoogLeNet without any changes; with respect to the second one, we split +the input features into four different parts according to ratio of numbers of +channels designed in GoogLeNet. The operation of split will significantly +reduce the number of parameters and decrease training time a lot, how- +ever, which will also lead to a slight accuracy loss under some circumstances. +For convolutional layers, different from existing inceptions with residual con- +nections [15], we utilize the original structure of multi-scale of inceptions +contained in GoogLeNet, which replaces the role of 3×3 convolutional layers +in Res2Net to enhance the ability of network to extract more features from +different scales. And the usage of 1×1 convolutional layers enables the model +to capture stronger non-linearity in the same receptive field and avoids in- +creasing calculation complexity too much. Therefore, we choose to remain +consistent with GoogLeNet on the layout of convolutional layers. But for the +improvement of performance, we construct transverse passages from the first +to the last convolutional layer group, then information being processed can +flow to next groups of convolutional layers. This operation enables informa- +tion to be reutilized, in other words, the changes on the structure provide +multi-scale feature extraction with a larger receptive field with respect to lat- +ter groups of convolutional layers, which makes up for the problem that the +original structure does not utilize larger receptive field. Besides, in transverse +passages, we adopt 1×1 convolutional layer to match features from channels +between different groups of convolutional layers, which not only realizes the +goal of construction of passages between groups of convolutional layers, but +also reduces amount of parameters in comparison with 3×3 convolutional +layer used in the structure of Res2Net. Besides, a residual connection is also +added to the proposed inception to reduce difficulty of network optimiza- +tion. Synthesizing the peculiarities mentioned before, the proposed network +3 + +achieves relatively smaller model size and higher performance simultaneously. +As a result, the ResFRI structure can be regarded as a satisfying solution in +image classification task and innovation in CNN architecture. +All in all, GoogLe2Net combines features of multiple models and possess +considerable advantages compared with other modern models. And the de- +tails of inception of GoogLeNet and ResFRI is provided in Fig.1 and Fig.2, +3. The main contribution of the ResFRI can be can be summed up in four +points which are listed as below: +1. GoogLe2Net explores influences brought by segmentation of informa- +tion, which leads to reduction of parameter amount and training time. +Besides, the loss of accuracy is also acceptable. +2. The transverse passages in ResFRI enable the model to extract infor- +mation in larger receptive fields and to fully utilize multi-scale features +at fine-grained levels. +3. The newly added residual connection in ResFRI could help GoogLe2Net +optimize the whole network and gain better performance. +4. GoogLe2Net investigates the effect of pruning and pruning ratio on +the performance of this model, which inherits the idea provided by +CondenseNet [16]. +2. Related work +With the popularity of vision tasks, CNNs have made great progress +[17, 18, 19, 20, 21, 22] and all of them contribute to the development of com- +puter vision a lot. In order to improve performance of networks, researchers +focus on adjusting depth and width of CNNs to better capture and process +information. From the pioneering appearance of LeNet [23] to some inspiring +modern networks like AlexNet [24] and VGG [25], both of them accelerate +the advance of applications of neural networks. AlexNet [24] first adopts +ReLu as activation function and utilizes dropout to ignore a part of neurons +so that model overfitting can be avoided to some extent. Besides, AlexNet +and its variant [26] also achieve breakthroughs on network performance with +respect to vision tasks, which is an outstanding progress compared with the +methods proposed previously. And it’s worth noting that there are lots of +potentials on depth, width and receptive field of network which are also fo- +cuses in the future researches. In recent years, VGG-like networks [25, 27] +concentrate on stacking convolutional layers with small kernel size to enlarge +4 + +11 +11 +11 +11 Max +Pooling +33 +55 +Previous Layer +11 +11 +11 +11 +Filter Concatenation +11 +11 +11 Max +Pooling +BN +Relu +Figure 2: Residual Feature-Reutilization Inception from GoogLe2Net +11 +11 +11 +11 Max +Pooling +33 +55 + 3/8 3/8 1/8 1/8 +11 +11 +11 +11 +Filter Concatenation +11 +11 +11 Max +Pooling +BN +Relu +Figure 3: Split-Residual Feature-Reutilization Inception from GoogLe2Net +5 + +size of receptive field and obtain information at a larger scale. And the work +[27] also introduces residual-like connections into framework of network to +further enhance performance on vision tasks. More importantly, VGG out- +performs AlexNet with less parameter amount, which mainly benefits from +its ability to capture features at large scales. Compared with the proposed +method in this paper, the receptive field of the two classical framework of +networks are relatively fixed, which restricts their capability on processing +information at diverse scales. Moreover, at that time, researchers also found +that networks may encounter obstacles of overfitting, gradient vanishing and +explosion while they’re going deeper, which are difficulties need to be solved +urgently. +Then, a classical neural network called GoogLeNet [14] which was pro- +posed by Christian Szegedy in 2014. The module presented in Fig.1 is the +basic structure of it. +In order to avoid problem of overfitting and large +calculation consumption, the inceptions contained in GoogLeNet improve +performance of network and reduce parameter amount through combining +convolutional layers on different magnitudes, which enhances its ability of +more efficient utilization of computation resources and capture of more fea- +tures at multi-scales. In the next year, another kind of network with residual +connections called Resnet [11] was proposed by Kaiming He to solve prob- +lem of network degradation and maintain accuracy when network becomes +deeper. +Following works like ResNext [28], PreActResNet [29], DenseNet +[30] and Wide Residual Networks [31] prove the effectiveness and validity of +residual connection, and as a result, the performances of networks are also +guaranteed. With respect to vision task object detection, an efficient model +called Pyramid networks [13] was proposed and the concept of feature reuti- +lization is introduced into modern neural network systems. And it can be +roughly explained as that the high-level feature map will send the features +back down and build the feature pyramid in reverse. +Then the low-level +feature map contains more location information, while the high-level feature +map contains better classification information, combining the two level, the +dual requirements of information for detection tasks can be satisfied. All in +all, different models of networks contribute the development of CNNs through +adjusting structures of them according to one or more specific properties of +the networks. +6 + +11 +11 +11 +11 Max +Pooling +33 +55 +Previous Layer +11 +11 +11 +11 +Filter Concatenation +11 +Concatenation +Or +Addition +Figure 4: Details of transverse passages of GoogLe2Net. +Note: The Split-ResFRI also adopts the same information interaction strategy +as ResFRI +3. GoogLe2Net +3.1. Brief Introduction of Structure of GoogLe2Net +The detail of ResFRI and Split-ResPRI are presented in Fig.2 and 3. Sup- +pose information from previous layer as ξPre and the operations of convolu- +tional layers as Conv, the main difference of ResFRI (RI) and Split-ResFRI +(SRI) in processing of information input can be defined as: +� +� +� +� +� +� +� +Conv(ξPre, ξPre, ξPre, ξPre), +RI +Conv(γ1, γ2, γ3, γ4) +γ1,2 = 3 ∗ ξPre//8 +γ3,4 = ξPre//8, +SRI +(1) +Compared with the original structure of inception contained in GoogLeNet, +7 + +residual connection and passages of information interaction between different +groups of convolutional layers are added into ResFRI and Split-ResFRI. In +order to reuse information, we construct transverse passages between adja- +cent groups of convolutional layers. Moreover, a residual connection is also +devised to reduce difficulty of network optimization and to avoid problems +like overfitting and abnormal gradients. Besides, to match feature channels +between groups of convolutional layers and residual connection to final out- +put, a structure consists of layers of 1×1 Convolutional layers, 3×3 MaxPool, +BatchNorm and ReLu (cmbr) is utilized. It also further enhances extrac- +tion of information and realizes cross channel information combination and +non-linear feature transference. And it’s worth noting that the information +combination is mainly achieved by adding or concatenating features and the +operation is described in Fig.4. Suppose the information processed by former +group of convolutional layer as δ and the information input to this group as +κ, then the fusion of information between groups of convolutional layer can +be defined as: +F = +� Addition(cmbr(δ), κ) +Concat(cmbr(δ), κ) +(2) +Moreover, the comparison of performance and resource consumption be- +tween these methods can be found in the ablation study based on ResFRI. +To reduce consumption of computation resources, we discard the 3×3 +convolutional layers designed by Res2Net and comply with the original de- +sign of inception of GoogLeNet. And we notice that the idea of connections +between different groups of convolutional layers is very similar to the one of +DenseNet [30], the extra passages may help improve performance of network. +However, [16] points out that the dense connections are actually redundant +under certain circumstances and this phenomenon may reduce accuracy and +efficiency of network. +As a result, we prune newly-added passages of in- +formation transference except the residual connection in ResFRI to avoid +unnecessary calculations and obtain higher accuracy. More specifically, we +adopt unstructured pruning which trims the single weight and does not re- +quire a whole row of pruning. The advantage is that the original accuracy +can be maintained, because structured pruning is easy to cut out those im- +portant weights. The tools of pruning is provided by Pytorch and unstruc- +tured pruning will abandon a part of weight parameters using mask matrices +without changing the original size of models. For the filter concatenation +(FC) and synthesizing the operations defined above, suppose Conv consists +8 + +of [C1, C2, C3, C4], it can be defined as: +FC = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Concat(C1(ξPre), C2(F(C1(ξPre)), ξPre), +C3(F(C2(F(C1(ξPre)), ξPre)), ξPre), +C4(F(C3(F(C2(F(C1(ξPre)), ξPre)), ξPre)), ξPre)), RI +Concat(C1(γ1), C2(F(C1(γ1)), γ2), +C3(F(C2(F(C1(γ1)), γ2)), γ3), +C4(F(C3(F(C2(F(C1(γ1)), γ2)), γ3)), γ4)), SRI +(3) +The results of experiments in the following will prove the validity of prun- +ing on diverse vision datasets. +3.2. Other Important Settings of GoogLe2Net +To ensure fair comparisons, the rest of settings of the whole network +generally follow the principle formulated in GoogLeNet. +And during the +process of experiment, we notice that the MaxPool layers may hamper the +network to capture information effectively and weaken performance of it, +we argue that the MaxPool layers may destruct information contained in +the low-resolution pictures instead of being helpful in extraction of features. +Verified by experiments, we change the MaxPool layer into AvgPool layer +eventually. +Argued by [16], the dense connections may have negative impact on the +process of learning and decrease accuracy of models. Therefore, we try to can- +cel some transverse passages to avoid too dense connections between adjacent +groups of convolutional layers contained in the two version of GoogLe2Net +utilizing different pruning ratio. +Eventually we set the drop rate of pas- +sages of information transference to 0.7 and 0 on addition and concatenation +version of ResFRI respectively, which can be defined as: +Pruning Ratio = +� 0.7, +Addition, RI +0, +Concatenation, RI +(4) +With respect to Split-ResFRI, because of underlying performance loss +which may be brought by segmentation of information, we set the pruning +rate uniformly to 0 in order to strengthen information interaction among +groups of convolutional layers. And it is worth noting that when the classes +contained in datasets are becoming more, we are supposed to reduce the +amount of pruning to better promote information transference for the version +9 + +of addition of ResFRI, which can be illustrated in the following experiments +on vision datasets. In the last, the results in the part of ablation study will +prove the effectiveness of these modifications based on ResFRI. +4. EXPERIMENTS +0 +200 +Epoch +0.0 +0.5 +1.0 +1.5 +Loss on CIFAR10 +Train +Test +0 +200 +400 +Epoch +1 +2 +3 +Loss on CIFAR100 +Train +Test +0 +200 +400 +Epoch +1 +2 +3 +4 +Loss on Tiny Imagenet +Train +Test +0 +100 +200 +Epoch +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Loss on MNSIT +Train +Test +0 +50 +100 +Epoch +0.0 +0.2 +0.4 +0.6 +Loss on FashionMNIST +Train +Test +0 +100 +200 +Epoch +0.0 +0.5 +1.0 +1.5 +Loss on SVHN +Train +Test +0 +200 +Epoch +0.2 +0.4 +0.6 +0.8 +1.0 +Accuracy on CIFAR10 +Train +Test +0 +200 +400 +Epoch +0.0 +0.2 +0.4 +0.6 +0.8 +Accuracy on CIFAR100 +Train +Test +0 +200 +400 +Epoch +0.0 +0.2 +0.4 +0.6 +Accuracy on Tiny Imagenet +Train +Test +0 +100 +200 +Epoch +0.80 +0.85 +0.90 +0.95 +1.00 +Accuracy on MNSIT +Train +Test +0 +50 +100 +Epoch +0.7 +0.8 +0.9 +1.0 +Accuracy on FashionMNIST +Train +Test +0 +100 +200 +Epoch +0.2 +0.4 +0.6 +0.8 +1.0 +Accuracy on SVHN +Train +Test +Figure 5: Loss and Accuracy of ResFRI-addition on Datasets +4.1. Implementation Details +We implement the whole framework of GoogLe2Net utilizing code frame- +work provided by Pytorch. And in order to ensure fairness of comparison +among different methods, we select experiment results of classical and newly +proposed models without pre-training. Due to our limited computation re- +sources, apart from necessary ablation experiments, we choose the task of +image classification on the common datasets, such as CIFAR10, CIFAR100, +Tiny Imagenet, MNIST, FashionMNIST and SVHN. Besides, in the process +of training on one RTX 3060 GPU, we use the optimizer SGD with momen- +tum 0.9, weight decay 0.0005, batch size 64 and data augmentation tools +provided in packages of torchvision. Moreover, the initial learning rate is set +to 0.01 and it is reduced by half if validation loss does not decrease within 10 +epochs. And tendency of accuracy and loss in the training process of ResFRI +is given in Fig.5. +4.2. Experiments on CIFAR-10 +The CIFAR10 dataset contains 50k training images and 10k testing im- +ages from 10 classes whose resolution is 32×32. And the detail results of +10 + +comparisons of different models will be clearly provided in Table 1 and Fig.6. +10 +20 +30 +40 +50 +60 +70 +Parameter Amount (M) +1 +2 +3 +4 +5 +6 +7 +8 +Top-1 error(%) +GoogLeNet +ResNet-101 +Shake-Shake +(26 2�96d) +ResNeXt-29, +16�32d +PyramidNet +PyramidNet ++ +ShakeDrop +WRN-28-10 +ResNeXt-29, +8�64d +ResNeXt-29, +16�64d +MixNet-190 +Split- +ResFRI- +addition +Split- +ResFRI- +concatenation +ResFRI- +addition +ResFRI- +concatenation +CIFRA-10 +1.53 GFlops +1.99 GFlops +2.06 GFlops +2.52 GFlops +2.76 GFlops +3.04 GFlops +3.79 GFlops +4.05 GFlops +4.55 GFlops +4.73 GFlops +5.25 GFlops +5.41 GFlops +10.73 GFlops +17.30 GFlops +Figure 6: Comparisons of models on CIFAR10 Dataset +It can be obtained that the ResFRI and Split-ResFRI achieve relatively +satisfying performance on image classification task on CIFAR-10 dataset. +Compared with traditional models like ResNet-101 and ResNeXt-29, Res- +FRI and Split-ResFRI have much better performance with much lower pa- +rameter amount. Although ResFRI-addition has 0.24 GFlops and ResFRI- +concatenation has 0.52 GFlops higher than ResNet-101, we have a remarkable +3.44% and 3.46% performance gain on top-1 err while parameter amounts +reduce by 32.4M and 31.1M. For Split-ResFRI, the version of addition has +0.53 GFlops and 36.1M parameters lower than ResNet-101, but we get 3.17% +performance improvement. +Besides, Split-ResFRI-concatenation has 0.46 +GFlops and 35.7M parameters lower than ResNet-101. Both of the Split- +ResFRIs have lower Flops and parameter amounts and achieve better results +than ResNet-101. Compared with two versions of ResFRI, Split-ResFRIs +sacrifice a little bit of precision in exchange for a considerable reduction in +Flops and parameter amount. For ResNeXt-29, it outperforms ResNet-101 +using larger model scales, but it still trails by at least 1.23% in compar- +ison with ResFRI and Split-ResFRI. And with respect to GoogLeNet, no +matter it is ResFRI or Split-ResFRI, we all have achieved performance lead- +11 + +Table 1: Error rate (%) and Model Size on the CIFAR-10 Dataset +Model +Flops +Params +top-1 err. +ResNet-101 [11] +2.52 GFlops +44.5M +5.52 +GoogLeNet [14] +1.53 GFlops +6.6M +5.16 +ResNeXt-29, 16×32d [28] +4.05 GFlops +25.2M +3.87 +ResNeXt-29, 8×64d [28] +5.41 GFlops +34.4M +3.65 +ResNeXt-29, 16×64d [28] +10.73 GFlops +68.1M +3.58 +CapsNet [32] +- +- +10.6 +DropConnect [33] +- +- +9.32 +NIN + Dropout + Data Augmentation [34] +- +0.96M +8.81 +RMDL [35] +- +- +8.74 +FractalNet [36] +- +38.6M +7.27 +FitNet-LSUV [37] +- +0.3M +6.06 +SOPCNN [38] +- +4.2MB +5.71 +DenseNet-BC (k=24) [30] +- +15.3M +5.19 +DPN-28-10 [39] +- +47.8M +3.65 +NASNet-A [39] +- +3.3M +3.41 +AmoebaNet-A [39] +- +4.6M +3.34 +AOGNet [39] +- +24.8M +3.27 +MixNet-190 [39] +17.3 GFlops +48.5M +3.13 +AmoebaNet-B [39] +- +34.9M +2.98 +OR-WideResNet [40] +- +18.2M +2.98 +WRN-28-10 [41] +5.25 GFlops +36.5M +2.6 +PyramidNet [13] +4.55 GFlops +26.2M +2.5 +Shake-Shake (26 2x96d) [42] +3.79 GFlops +26.2M +2.3 +PyramidNet+ShakeDrop [42] +4.73 GFlops +28.4M +2.1 +ResFRI-addition +2.76 GFlops +12.1M +2.08 +Split-ResFRI-addition +1.99 GFlops +8.4M +2.35 +ResFRI-concatenation +3.04 GFlops +13.4M +2.06 +Split-ResFRI-concatenation +2.06 GFlops +8.7M +2.28 +12 + +ership. It is worth noting that both versions of Split-ResFRIs have similar +flops and parameter amounts to GoogLeNet, but still achieve a performance +lead of over 2.8 percentage. And with respect to CapsNet, DropConnect, +NIN and RMDL, the four models reach a fairly satisfying level on small-size +datasets like MNIST utilizing very small model scales, which partly outper- +forms many classical and novel methods including GoogLe2Net. However, +all of the four models are not as good a performance as before in the more +popular vision dataset, CIFAR-10, other modern models have overwhelm- +ing advantages compared with their results. Especially, the series of models +belonging to ResFRI achieve at least a 6.39% performance lead. +Moreover, when encountering some newly proposed models, ResFRI and +Split-ResFRI still prove their superiority on classification task. For DenseNet, +it has a similar model scale to ResFRI-concatenation, but it has a 3.13% +performance disadvantage in the final result. Besides, with respect to OR- +WideResNet, it achieves a relatively satisfying results with acceptable model +size. Compared with ResFRI and Split-ResFRI, its disadvantage is still sig- +nificant with performance trailing by at least 0.63%. +Then, WRN-28-10, +PyramidNet and Shake-Shake(26 2×96d) all of them have higher flops and +parameter amount than ResFRI and Split-ResFRI, but all of them achieve +better accuracy except for Split-ResFRI-addition meanwhile. However, we +want to point out that Split-ResFRI-addition has far less GFlops and param- +eter amount than the above model for comparison. Moreover, we notice that +PyramidNet+ShakeDrop has a a very approximate performance (−0.04%) to +ResFRI-concatenation, which is a is a very competitive opponent. However, +the cost of the combination of PyramidNet and ShakeDrop is 71.3% higher +flops and 134.7% larger parameter amount than Res-FRI-concatenation. We +think this comparison also illustrates the advantage of the proposed method +when considering differences on computing resources consumption of the two +models. In sum, the experiment on CIFAR-10 dataset strongly proves the +effectiveness and validity of GoogLe2Net on image classification task and +Split-ResFRI also has greatly competitive results when considering the re- +duction on GFlops and the number of parameters by a significant amount. +4.3. Experiments on CIFAR-100 +The CIFAR100 dataset consists of 50k training images and 10k testing +images from 100 classes and their resolution is 32×32. And the detail results +of comparisons of different models will be clearly provided in Table 2 and +Fig.7. +13 + +Table 2: Top-1, Top-5 Test Error (%) and Model Size on the CIFAR-100 Dataset +Model +Flops +Params +top-1 err. +top-5 err. +ResNet-101 [11] +2.52 GFlops +42.7M +22.22 +5.61 +ResNeXt-50 [28] +- +14.8M +22.23 +6.00 +ResNeXt-101 [28] +- +25.3M +22.22 +5.99 +ResNeXt-152 [28] +- +33.3M +22.40 +5.58 +DenseNet (k=12, depth=40) [30] +- +1.0M +27.55 +- +DenseNet (k=12, depth=100)[30] +- +7.0M +23.79 +- +DenseNet (k=24, depth=100)[30] +- +27.2M +23.42 +- +DenseNet-BC (k=24) [30] +- +15.3M +19.64 +- +GoogLeNet [14] +1.53 GFlops +6.6M +21.97 +5.94 +Inception v3 [43] +- +22.3M +22.81 +6.39 +Inception v4 [15] +- +41.3M +24.14 +6.90 +InceptionResnet v2 [15] +- +65.4M +27.51 +9.11 +Xception [44] +- +21.0M +25.07 +7.32 +WRN-40-10 [31] +8.08 GFlops +55.9M +21.25 +5.77 +NIN + Dropout [34] +- +0.96M +35.68 +- +FitNet-LSUV [37] +- +0.3M +29.96 +- +FractalNet [36] +- +38.6M +29.05 +- +SOPCNN [38] +- +4.2M +27.04 +- +WRN-28-10 [41] +5.25 GFlops +36.5M +16.9 +- +Res2NeXt-29, 6c×24w×6s [12] +- +36.7M +16.79 +- +Res2NeXt-29, 6c×24w×6s-SE [12] +- +36.9M +16.56 +- +PyramidNet [45] +4.55 GFlops +26.2M +16.4 +- +OR-WideResNet [40] +- +18.2M +16.15 +2.98 +NASNet-A [39] +- +50.9M +16.03 +- +HCGNet-A3 [39] +2.0 GFlops +11.4M +15.96 +- +Shake-Shake (26 2×96d) [42] +3.79 GFlops +26.1M +15.7 +- +PyramidNet+ShakeDrop [42] +4.73 GFlops +28.4M +14.5 +- +ResFRI-addition +2.76 GFlops +12.2M +14.09 +2.42 +Split-ResFRI-addition +1.99 GFlops +8.5M +14.10 +2.48 +ResFRI-concatenation +3.04 GFlops +13.5M +14.31 +2.71 +Split-ResFRI-concatenation +2.06 GFlops +8.8M +14.13 +2.32 +14 + +10 +20 +30 +40 +50 +Parameter Amount (M) +14 +16 +18 +20 +22 +24 +26 +Top-1 error(%) +GoogLeNet +HCGNet-A3 +ResNet-101 +Shake-Shake +(26 2�96d) +PyramidNet +PyramidNet ++ +ShakeDrop +WRN-28-10 +WRN-40-10 +Split- +ResFRI- +addition +Split- +ResFRI- +concatenation +ResFRI- +addition +ResFRI- +concatenation +CIFRA-100 +1.53 GFlops +1.99 GFlops +2.00 GFlops +2.06 GFlops +2.52 GFlops +2.76 GFlops +3.04 GFlops +3.79 GFlops +4.55 GFlops +4.73 GFlops +5.25 GFlops +8.08 GFlops +Figure 7: Comparisons of models on CIFAR100 Dataset +By checking the results given in Table 2, some conclusions can be made. +ResNet-101 has a performance lag of around 8% compared with the pro- +posed method and it utilizes approximate flops and nearly three times pa- +rameter amount of ResFRI. For ResNext-series models, all of them achieves +analogous performance as ResNet-101 with much less flops and parameter +amounts. The situation of DenseNets is also similar, they further reduces +the size and computational complexity of the model, but the accuracy of it +is still at a comparatively low level. The best accuracy of them has at least +a performance disadvantage of more than 5% compared with ResFRI-series +models. +Besides, the inception-series models also have a relatively excel- +lent performance. Particularly, GoogLeNet possesses only 6.6M parameter +amount but achieves an effect that ranks at the top of many models. For +NIN, FitNet and SOPCNN, all of the three models can obtain better re- +sults on smaller datasets, but they can not acquire desirable results on more +convincing datasets like CIFAR-100. Considering the results of WRN-28-10 +provided in [41], it achieves a performance leap with a top-1 error rate of +about 16% and dose not increase flops and parameters amount too much +compared with the previous models. And it can be obtained that Res2NeXt +can reach a similar performance with roughly the same number of parameters +as WRN-28-10. Certainly, ResFRI and Split-ResFRI have higher accuracy +15 + +with much lower flops and parameter amounts compared with the two cate- +gories of models we just discussed. +Moreover, when considering other modern models, the HCGNet-A3 has +a very approxmate flops and parameter amount with GoogLe2Net which +realizes nearly two more percent accuracy improvement on classification +tasks. For PyramidNet, NASNet-A, Shake-Shake (26 2×96d) and Pyramid- +Net+ShakeDrop, ResFRI and Split-ResFRI still achieve better performances +while using less flops and parameter amount. +The most light one, Split- +ResFRI-addition, can achieve almost the best performance with less than +9M parameter amount and 2 Gflops which are between a half and a third +of the scales of the four models mentioned before. +Especially, Pyramid- +Net+ShakeDrop has the closest effect to the proposed method while possess- +ing 55% higher parameter amount and 110% more flops than the proposed +models at least. Compared with the original PyramidNet, the combination of +PyramidNet+ShakeDrop obtains a performance improvement of about 2%, +which illustrates the possibility of follow-up work using this technology and +the effectiveness of ShakeDrop. In sum, based on experimental results pro- +vided in Table 2, it can be concluded that the proposed method possesses a +far better precision on classification task when compared with classical net- +works. Except for GoogLeNet and DenseNet, all of the other models have +larger parameter amount than the proposed model but could not reach a +similar level of accuracy, which demonstrates the efficiency and effectiveness +of GoogLe2Net. Although GoogLeNet and DenseNet with specific settings +own smaller model scale than ResFRI and Split-ResFRI, but our proposed +method has a huge advantage in accuracy. Concretely, the version of ad- +dition of ResFRI reaches a top-1 error rate 14.09 and top-5 error rate 2.42 +on CIFAR-100 dataset, in the meantime, Split-ResFRI could achieve very +similar performance with at most 37% reduction of parameter amount and +34.5% curtailment on flops. In one word, all of the comparisons proves the +superiority of GoogLe2Net on classification tasks which can be regarded as +a satisfying solution in choices among CNN architectures. +4.4. Experiments on Tiny Imagenet +The Tiny Imagenet dataset consists of 100k training images and 10k +testing images from 200 classes and their resolution is 64×64. And the results +of comparisons are given in Table 3. And it is worth noting that flops and +parameter amounts of ResFRI and Split-ResFRI are evaluated using a tensor +16 + +matrix of 3 × 64 × 64 and the model is subtly adjusted to fit the different +type of data, so the number of them will also variate accordingly. +Table 3: Top-1 Test Error (%) and Model Size on the Tiny Imagenet Dataset +Model +Flops +Params +top-1 err. +ResNet-18+Mixup+DM [46] +- +11.1M +34.93 +ResNet-18+CutMix+DM [46] +- +11.1M +33.55 +ResNet-18+ManifoldMix+DM [46] +- +11.1M +34.55 +ResNet-18+ResizeMix+DM [46] +- +11.1M +35.67 +ResNet-18+PuzzleMix+DM [46] +- +11.1M +33.48 +ResNeXt-50+Mixup+DM [46] +- +23.3M +32.30 +ResNeXt-50+CutMix+DM [46] +- +23.3M +32.54 +ResNeXt-50+ManifoldMix+DM [46] +- +23.3M +31.52 +ResNeXt-50+ResizeMix+DM [46] +- +23.3M +31.44 +ResNeXt-50+PuzzleMix+DM [46] +- +23.3M +31.96 +WaveMixLite-144/7 [47] +- +3.01 M +47.62 +DenseNet + Residual Networks [48] +- +- +40.0 +ResNet18 + AutoMix [49] +- +11.1M +32.67 +UPANets [50] +- +24.4M +32.33 +ResNet18 + SAMix [51] +- +11.1M +31.11 +PreActResNet-18-3 + MixMo [52] +- +11.1M +29.76 +ResFRI-addition (pruning ratio 0.7) +3.13 GFlops +12.4M +31.5 +ResFRI-addition (pruning ratio 0) +3.13 GFlops +12.4M +29.60 +Split-ResFRI-addition +2.37 GFlops +8.5M +31.93 +ResFRI-concatenation +3.4 GFlops +13.7M +29.46 +Split-ResFRI-concatenation +2.44 GFlops +9.0M +32.04 +The experiments on the Tiny Imagenet show that the proposed method +achieves a considerably satisfying classification accuracy. +For ResNet-18, +it has nearly the same as many parameters as ResFRI, but achieves far +weaker performance than ResFRI. Besides, compared with Split-ResFRI, +the Split-ResFRI can obtain higher accuracy using less parameter amounts, +which clearly demonstrates the efficiency of the proposed model. Moreover, +17 + +ResNext-50 possesses two to three times as many as parameters as ResFRI +and Split-ResFRI, it is able to get approximate performance to the pro- +posed models but still falls behind in the best model accuracy. And with +respect to WaveMixLite-144/7, it reaches a similar performance to ResNet- +50 utilizing only 3M parameters. But its actual model accuracy is still not +satisfactory. Compared with the methods like ResNet18 and PreActResNet, +ResFRIs provide a best performance exceeding 70% accuracy which is a re- +markable improvement. It is worth noting that Split-ResFRIs are also able +to achieve a similar tier of accuracy utilizing less parameter amounts. In +sum, GoogLe2Net reaches a high level of performance on image classification +task without consuming too many computing resources in comparison with +other models. +4.5. Experiments on MNIST +The MNIST dataset contains 60k training images and 10k testing images +from 10 classes whose resolution is 28×28. And the detail results of com- +parisons of different models will be clearly provided in Table 4. It is worth +noting that parameter amounts of ResFRI and Split-ResFRI are calculated +using a tensor matrix of 3×32×32, because the images of MNIST are resized +into 32 before being inputting proposed models for process of training. +Based on MNIST dataset, there exist many very light models which still +reach a great level of accuracy. The proposed model falls behind by approx- +imately 0.1 to 0.18 percent and consumes much more computing resources. +Nevertheless, FitNet-LSUV and NiN encounter more than 4 and 15 percent +performance loss on CIFAR-10 and CIFAR-100 dataset provided in Table 1 +and 2 respectively compared with GoogLe2Net, which demonstrates that the +relatively lower level of precision of GoogLe2Net on MNIST dataset is com- +pletely acceptable. Besides, the remaining methods like CapsNet, RMDL +and SOPCNN also have similar situations. Thus, the proposed method is +more comprehensive and universal in handling classification tasks. And with +respect to the performances of the two version of ResFRI, we argue that +because the features contained in MNIST are simpler comparatively, the +operation of concatenation is helpful to strengthen features instead of con- +structing too dense connection between convolutional layers. Moreover, for +Split-ResFRIs, the performances of them become a little weaker in compar- +ison with the versions without split, which may be caused by reduction of +feature extraction operations. +18 + +Table 4: Test Accuracy (%) and Model Size on the MNIST Dataset +Model +Params +top-1 err. +PCANET-1 [53] +- +0.62 +FitNet-LSUV [37] +0.3M +0.46 +NiN [34] +0.96M +0.45 +VGG8B [54] +7.3M +0.26 +CapsNet [32] +- +0.25 +DropConnect [33] +- +0.21 +RMDL [35] +- +0.18 +SOPCNN [38] +1.4M +0.17 +ResFRI-addition +12.1M +0.35 +Split-ResFRI-addition +8.4M +0.39 +ResFRI-concatenation +13.4M +0.31 +Split-ResFRI-concatenation +8.8M +0.35 +4.6. Experiments on FashionMNIST +The FashionMNIST dataset consists of 60k training images and 10k test- +ing images from 10 classes and their resolution is 28×28. And the results +of comparisons are given in Table 5. +It is worth noting that parameter +amounts of ResFRI and Split-ResFRI are calculated using a tensor matrix of +3 × 32 × 32, because the images of FashionMNIST are resized into 32 before +being inputting proposed models for process of training. +As shown in the Table 5, ResFRI and Split-ResFRI reach a satisfying level +of accuracy on FashionMNIST dataset. And ResFRI-addition and ResFRI- +concatenation make 1.56% and 1.28% percent performance gains compared +with the Inception v3, which proves the efficiency and effectiveness of ResFRI +compared with other Inception-like architecture. Besides, ResFRI and Split- +ResFRI also outstrips these traditional models such as WideResNet, VGG8B +and DenseNet utilizing much less parameter amount. And it is worth noting +that Split-ResFRI outperforms ResFRI on FashionMNIST dataset, which +is very interesting and probably tells us that extraction of picture features +like simple objects don’t require a deep and dense neural network. All in +all, by checking the results of comparison, it can be concluded that the pro- +posed method guarantees a enough precision on a relatively small and simple +19 + +Table 5: Test Accuracy (%) and Model Size on the FashionMNIST Dataset +Model +Params +top-1 err. +Inception v3 [55] +24.7M +5.56 +SeResNeXt101-STD [56] +- +4.59 +VGG8B(2x) [54] +28M +4.33 +PreAct-ResNet18 [29] +11.1M +4.30 +WideResNet-28-10 [54] +37M +4.16 +DenseNet-BC-190 [30] +25.6M +4.06 +ResFRI-addition +12.1M +4.00 +Split-ResFRI-addition +8.4M +3.80 +ResFRI-concatenation +13.4M +4.29 +Split-ResFRI-concatenation +8.8M +3.87 +dataset and splitting features may be helpful in improving performance in +analogous tasks. +4.7. Experiments on SVHN +The SVHN dataset contains 73257 training images and 26032 testing +images from 10 classes whose resolution is 32×32. And the detail results of +comparisons of different models will be clearly provided in Table 6. +By analyzing the experimental results on SVHN dataset, the GoogLe2Net +also achieves relatively satisfying accuracy. Compared with the classical mod- +els like NiN, FractalNet and DenseNet, the proposed method utilizes much +less parameter amount to reach a similar level of precision. Especially, the +FractalNet possesses 219% higher parameter amount than GoogLe2Net while +falling behind by 0.15 percent accuracy compared with ResFRI. And it’s +worth noting that the performance of proposed model also exceeds Fractal- +Net and DenseNet on CIFAR-10 and CIFAR-100 dataset. +4.8. Ablation Experiment +In this section, we conduct the ablation experiment from two main aspects +which are addition and concatenation version of ResFRI. In the preliminary +stage of our experiment, we notice that for the addition version of ResFRI, +a proper ration of pruning may help to promote the accuracy of the model. +20 + +Table 6: Test Accuracy (%) and Model Size on the SVHN Dataset +Model +Params +top-1 err. +FitNet [57] +- +2.42 +NiN [34] +0.96M +2.35 +FractalNet [36] +38.6M +2.01 +DropConnect [33] +- +1.94 +Deeply Supervised Net [58] +- +1.92 +FractalNet with Dropout/Drop-path [36] +38.6M +1.87 +ResNet with Stochastic Depth [59] +1.7M +1.75 +DenseNet-BC [30] +15.3M +1.74 +ResFRI-addition +12.1M +1.72 +Split-ResFRI-addition +8.4M +1.84 +ResFRI-concatenation +13.4M +1.75 +Split-ResFRI-concatenation +8.8M +1.82 +And in the version of concatenation, no pruning may further enhance per- +formance of the network. Therefore, all of the ablation experiments are not +only involved with adjustment of structure of networks, but also the ratios +of pruning. And all of the results are provided in the following Table 7. +In detail, we remove three key components of ResFRI, namely AvgPool- +ing layer, Residual connection and transverse passages between groups of +convolutional layers respectively, to verify their influence on performance of +the proposed network. And we can notice that when each of them is removed, +the performance will encounter a precision loss to some extent. It strongly +demonstrates that when all of those components are synthesized, the lowest +top1-error can be reached. Moreover, we also test effects of different pruning +ratio on precision of the proposed model, which also proves the rationality +of our settings of ResFRI on CIFAR-10. +5. CONCLUSIONS +In this paper, we first review the architectures of traditional neural net- +works and state importance of multi-scale design in CNNs. For the structure +of incpetion-like networks, we notice that construction of transverse passages +21 + +Table 7: Comparison among ResFRI variants on CIFAR10 dataset +Variants +Params +top-1 err. +ResFRI +(addition, pruning ratio 0.7) +12.1M +2.08 +ResFRI +(addition, pruning ratio 0.35) +12.1M +2.29 +ResFRI +(addition, pruning ratio 0) +12.1M +2.13 +ResFRI +(concatenation, pruning ratio 0.7) +13.4M +2.14 +ResFRI +(concatenation, pruning ratio 0.35) +13.4M +2.23 +ResFRI +(concatenation, pruning ratio 0) +13.4M +2.06 +ResFRI without AvgPooling layer +(addition, pruning ratio 0.7) +12.1M +2.25 +ResFRI without residual connection +(addition, pruning ratio 0.7) +8.9M +2.43 +ResFRI without transverse passages +(addition, pruning ratio 0.7) +9.4M +2.12 +ResFRI without AvgPooling layer +(concatenation, pruning ratio 0)) +13.4M +2.30 +ResFRI without residual connection +(concatenation, pruning ratio 0) +10.2M +2.37 +ResFRI without transverse passages +(concatenation, pruning ratio 0) +9.4M +2.60 +22 + +between adjacent groups of convolutional layers may boost performance of +the network compared with original inception frameworks. Besides, referring +the concept of ResNet, we also adopt a policy that a residual connection +is added to lower difficulty in network optimization. In detail, transverse +passages between adjacent groups of convolutional layers realize feature re- +utilization in groups of convolutional layers and further enhance the ability +of expression and generalization of original inception. Besides, residual con- +nection reduces overfitting and gradient disappearance. They are the main +reasons that GoogLe2Net is able to reach a satisfactory level of accuracy +on mainstream vision datasets under such a light and efficient inception-like +framework. And all the experiments in this paper confirm this perspective. +Moreover, in the future, we believe the organic combination of the concept of +multi-scale and CNNs will be a hot spot in boosting performances on various +vision tasks. +References +[1] Y. Dong, Q. Liu, B. Du, L. 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Welling +(Eds.), Computer Vision - ECCV 2016 - 14th European Conference, +Amsterdam, The Netherlands, October 11-14, 2016, Proceedings, Part +IV, Vol. 9908 of Lecture Notes in Computer Science, Springer, 2016, pp. +646–661. doi:10.1007/978-3-319-46493-0\_39. +URL https://doi.org/10.1007/978-3-319-46493-0_39 +33 + diff --git a/gdAyT4oBgHgl3EQfj_hk/content/tmp_files/load_file.txt b/gdAyT4oBgHgl3EQfj_hk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2bd87f83b3dcedd5f22f8d7cf358d55d16c4d469 --- /dev/null +++ b/gdAyT4oBgHgl3EQfj_hk/content/tmp_files/load_file.txt @@ -0,0 +1,1559 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf,len=1558 +page_content='GoogLe2Net: Going Transverse with Convolutions Yuanpeng Hea,b aKey Laboratory of High Confidence Software Technologies, Peking University, Peking, 100871, China bSchool of Computer Science, Peking University, Peking, 100871, China Abstract Capturing feature information effectively is of great importance in vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' With the development of convolutional neural networks (CNNs), con- cepts like residual connection and multiple scales promote continual perfor- mance gains on diverse deep learning vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' However, the existing methods do not organically combined advantages of these valid ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' In this paper, we propose a novel CNN architecture called GoogLe2Net, it con- sists of residual feature-reutilization inceptions (ResFRI) or split residual feature-reutilization inceptions (Split-ResFRI) which create transverse pas- sages between adjacent groups of convolutional layers to enable features flow to latter processing branches and possess residual connections to better pro- cess information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Our GoogLe2Net is able to reutilize information captured by foregoing groups of convolutional layers and express multi-scale features at a fine-grained level, which improves performances in image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And the inception we proposed could be embedded into inception-like net- works directly without any migration costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Moreover, in experiments based on popular vision datasets, such as CIFAR10 (97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='94%), CIFAR100 (85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='91%) and Tiny Imagenet (70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='54%), we obtain better results on image classification task compared with other modern models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Keywords: Feature-reutilization Transverse passages Inception 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Introduction In recent years, we’ve witnessed a rapid advance of CNNs and this field is attracting more and more attention from researchers around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Noticeably, in order to meet demands of different vision tasks such as im- age classification, target tracking, image segmentation, skeleton extraction Preprint submitted to Elsevier January 3, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='00424v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='CV] 1 Jan 2023 , facial recognition and image description, a large number of vision neural network models have been proposed [1, 2, 3, 4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And how to effectively extract information to satisfy demands of different kinds of vision tasks is still an open issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Remarkably, capturing features from multiple scales to obtain more information has been a hot spot in computer vision-related fields which boosts performances of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' The concept of multi-scale has already been introduced into deep learning- related fields [7, 8, 9, 10] and its superiority was fully demonstrated by various applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' As a general rule, CNNs may acquire features utilizing convo- lutional kernels with different sizes from roughness to detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Therefore, the key to boost performance of vision models is to devise a more efficient and effective policy of capturing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And recently, on the basis of common residual block [11], a multi-scale architecture called Res2Net [12] is devised to better obtain and aggregate information at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' The idea of it resembles the one of Pyramid networks [13] and the Res2Net block can con- tinually enlarge the receptive filed through stacking 3×3 convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Besides, the effectiveness of it is proved by the outstanding performance in diverse vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 1\uf0cd1 1\uf0cd1 1\uf0cd1 1\uf0cd1 Max Pooling 3\uf0cd3 5\uf0cd5 Previous Layer 1\uf0cd1 Filter Concatenation Figure 1: Original Inception from GoogLeNet 2 Enlightened by the concept of Pyramid network and compositions of Res2Net block, we intend to generalize the idea of them to other networks which own relatively small parameter amount and similar architecture to ensure that the newly proposed block of network is efficient and modifica- tions on it are straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' In order to fuse information more efficiently and acquire multi-scale features in larger receptive fields, we propose a novel GoogLe2Net based on GoogLeNet [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' The proposed GoogLe2Net has two versions which consists of residual feature-reutilization inception (ResFRI) and split-residual feature-reutilization inception (Split-ResFRI) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' About the model architecture, firstly, for the input layer, we adopt two dis- parate policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' For the first one, we utilize the original input layer from GoogLeNet without any changes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' with respect to the second one, we split the input features into four different parts according to ratio of numbers of channels designed in GoogLeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' The operation of split will significantly reduce the number of parameters and decrease training time a lot, how- ever, which will also lead to a slight accuracy loss under some circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' For convolutional layers, different from existing inceptions with residual con- nections [15], we utilize the original structure of multi-scale of inceptions contained in GoogLeNet, which replaces the role of 3×3 convolutional layers in Res2Net to enhance the ability of network to extract more features from different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And the usage of 1×1 convolutional layers enables the model to capture stronger non-linearity in the same receptive field and avoids in- creasing calculation complexity too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Therefore, we choose to remain consistent with GoogLeNet on the layout of convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' But for the improvement of performance, we construct transverse passages from the first to the last convolutional layer group, then information being processed can flow to next groups of convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' This operation enables informa- tion to be reutilized, in other words, the changes on the structure provide multi-scale feature extraction with a larger receptive field with respect to lat- ter groups of convolutional layers, which makes up for the problem that the original structure does not utilize larger receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Besides, in transverse passages, we adopt 1×1 convolutional layer to match features from channels between different groups of convolutional layers, which not only realizes the goal of construction of passages between groups of convolutional layers, but also reduces amount of parameters in comparison with 3×3 convolutional layer used in the structure of Res2Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Besides, a residual connection is also added to the proposed inception to reduce difficulty of network optimiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Synthesizing the peculiarities mentioned before, the proposed network 3 achieves relatively smaller model size and higher performance simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' As a result, the ResFRI structure can be regarded as a satisfying solution in image classification task and innovation in CNN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' All in all, GoogLe2Net combines features of multiple models and possess considerable advantages compared with other modern models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And the de- tails of inception of GoogLeNet and ResFRI is provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' The main contribution of the ResFRI can be can be summed up in four points which are listed as below: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' GoogLe2Net explores influences brought by segmentation of informa- tion, which leads to reduction of parameter amount and training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Besides, the loss of accuracy is also acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' The transverse passages in ResFRI enable the model to extract infor- mation in larger receptive fields and to fully utilize multi-scale features at fine-grained levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' The newly added residual connection in ResFRI could help GoogLe2Net optimize the whole network and gain better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' GoogLe2Net investigates the effect of pruning and pruning ratio on the performance of this model, which inherits the idea provided by CondenseNet [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Related work With the popularity of vision tasks, CNNs have made great progress [17, 18, 19, 20, 21, 22] and all of them contribute to the development of com- puter vision a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' In order to improve performance of networks, researchers focus on adjusting depth and width of CNNs to better capture and process information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' From the pioneering appearance of LeNet [23] to some inspiring modern networks like AlexNet [24] and VGG [25], both of them accelerate the advance of applications of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' AlexNet [24] first adopts ReLu as activation function and utilizes dropout to ignore a part of neurons so that model overfitting can be avoided to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Besides, AlexNet and its variant [26] also achieve breakthroughs on network performance with respect to vision tasks, which is an outstanding progress compared with the methods proposed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And it’s worth noting that there are lots of potentials on depth, width and receptive field of network which are also fo- cuses in the future researches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' In recent years,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' VGG-like networks [25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 27] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='concentrate on stacking convolutional layers with small kernel size to enlarge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 Max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='Pooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3\uf0cd3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5\uf0cd5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='Previous Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='Filter Concatenation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 Max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='Pooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='BN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='Relu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='Figure 2: Residual Feature-Reutilization Inception from GoogLe2Net ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 Max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='Pooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3\uf0cd3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5\uf0cd5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3/8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3/8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1/8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1/8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='Filter Concatenation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1\uf0cd1 Max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='Pooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='BN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='Relu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='Figure 3: Split-Residual Feature-Reutilization Inception from GoogLe2Net ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='size of receptive field and obtain information at a larger scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And the work [27] also introduces residual-like connections into framework of network to further enhance performance on vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' More importantly, VGG out- performs AlexNet with less parameter amount, which mainly benefits from its ability to capture features at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Compared with the proposed method in this paper, the receptive field of the two classical framework of networks are relatively fixed, which restricts their capability on processing information at diverse scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Moreover, at that time, researchers also found that networks may encounter obstacles of overfitting, gradient vanishing and explosion while they’re going deeper, which are difficulties need to be solved urgently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Then, a classical neural network called GoogLeNet [14] which was pro- posed by Christian Szegedy in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' The module presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1 is the basic structure of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' In order to avoid problem of overfitting and large calculation consumption, the inceptions contained in GoogLeNet improve performance of network and reduce parameter amount through combining convolutional layers on different magnitudes, which enhances its ability of more efficient utilization of computation resources and capture of more fea- tures at multi-scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' In the next year, another kind of network with residual connections called Resnet [11] was proposed by Kaiming He to solve prob- lem of network degradation and maintain accuracy when network becomes deeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Following works like ResNext [28], PreActResNet [29], DenseNet [30] and Wide Residual Networks [31] prove the effectiveness and validity of residual connection, and as a result, the performances of networks are also guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' With respect to vision task object detection, an efficient model called Pyramid networks [13] was proposed and the concept of feature reuti- lization is introduced into modern neural network systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And it can be roughly explained as that the high-level feature map will send the features back down and build the feature pyramid in reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Then the low-level feature map contains more location information, while the high-level feature map contains better classification information, combining the two level, the dual requirements of information for detection tasks can be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' All in all, different models of networks contribute the development of CNNs through adjusting structures of them according to one or more specific properties of the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 6 1\uf0cd1 1\uf0cd1 1\uf0cd1 1\uf0cd1 Max Pooling 3\uf0cd3 5\uf0cd5 Previous Layer 1\uf0cd1 1\uf0cd1 1\uf0cd1 1\uf0cd1 Filter Concatenation 1\uf0cd1 Concatenation Or Addition Figure 4: Details of transverse passages of GoogLe2Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Note: The Split-ResFRI also adopts the same information interaction strategy as ResFRI 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' GoogLe2Net 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Brief Introduction of Structure of GoogLe2Net The detail of ResFRI and Split-ResPRI are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Sup- pose information from previous layer as ξPre and the operations of convolu- tional layers as Conv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' the main difference of ResFRI (RI) and Split-ResFRI (SRI) in processing of information input can be defined as: � � � � � � � Conv(ξPre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' ξPre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' ξPre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' ξPre),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' RI Conv(γ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' γ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' γ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' γ4) γ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2 = 3 ∗ ξPre//8 γ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4 = ξPre//8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' SRI (1) Compared with the original structure of inception contained in GoogLeNet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 7 residual connection and passages of information interaction between different groups of convolutional layers are added into ResFRI and Split-ResFRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' In order to reuse information, we construct transverse passages between adja- cent groups of convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Moreover, a residual connection is also devised to reduce difficulty of network optimization and to avoid problems like overfitting and abnormal gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Besides, to match feature channels between groups of convolutional layers and residual connection to final out- put, a structure consists of layers of 1×1 Convolutional layers, 3×3 MaxPool, BatchNorm and ReLu (cmbr) is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' It also further enhances extrac- tion of information and realizes cross channel information combination and non-linear feature transference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And it’s worth noting that the information combination is mainly achieved by adding or concatenating features and the operation is described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Suppose the information processed by former group of convolutional layer as δ and the information input to this group as κ, then the fusion of information between groups of convolutional layer can be defined as: F = � Addition(cmbr(δ), κ) Concat(cmbr(δ), κ) (2) Moreover, the comparison of performance and resource consumption be- tween these methods can be found in the ablation study based on ResFRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' To reduce consumption of computation resources, we discard the 3×3 convolutional layers designed by Res2Net and comply with the original de- sign of inception of GoogLeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And we notice that the idea of connections between different groups of convolutional layers is very similar to the one of DenseNet [30], the extra passages may help improve performance of network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' However, [16] points out that the dense connections are actually redundant under certain circumstances and this phenomenon may reduce accuracy and efficiency of network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' As a result, we prune newly-added passages of in- formation transference except the residual connection in ResFRI to avoid unnecessary calculations and obtain higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' More specifically, we adopt unstructured pruning which trims the single weight and does not re- quire a whole row of pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' The advantage is that the original accuracy can be maintained, because structured pruning is easy to cut out those im- portant weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' The tools of pruning is provided by Pytorch and unstruc- tured pruning will abandon a part of weight parameters using mask matrices without changing the original size of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' For the filter concatenation (FC) and synthesizing the operations defined above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' suppose Conv consists 8 of [C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' C4],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' it can be defined as: FC = � � � � � � � � � � � � � � � � � � � Concat(C1(ξPre),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' C2(F(C1(ξPre)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' ξPre),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' C3(F(C2(F(C1(ξPre)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' ξPre)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' ξPre),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' C4(F(C3(F(C2(F(C1(ξPre)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' ξPre)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' ξPre)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' ξPre)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' RI Concat(C1(γ1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' C2(F(C1(γ1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' γ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' C3(F(C2(F(C1(γ1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' γ2)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' γ3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' C4(F(C3(F(C2(F(C1(γ1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' γ2)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' γ3)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' γ4)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' SRI (3) The results of experiments in the following will prove the validity of prun- ing on diverse vision datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Other Important Settings of GoogLe2Net To ensure fair comparisons, the rest of settings of the whole network generally follow the principle formulated in GoogLeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And during the process of experiment, we notice that the MaxPool layers may hamper the network to capture information effectively and weaken performance of it, we argue that the MaxPool layers may destruct information contained in the low-resolution pictures instead of being helpful in extraction of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Verified by experiments, we change the MaxPool layer into AvgPool layer eventually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Argued by [16], the dense connections may have negative impact on the process of learning and decrease accuracy of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Therefore, we try to can- cel some transverse passages to avoid too dense connections between adjacent groups of convolutional layers contained in the two version of GoogLe2Net utilizing different pruning ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Eventually we set the drop rate of pas- sages of information transference to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7 and 0 on addition and concatenation version of ResFRI respectively, which can be defined as: Pruning Ratio = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7, Addition, RI 0, Concatenation, RI (4) With respect to Split-ResFRI, because of underlying performance loss which may be brought by segmentation of information, we set the pruning rate uniformly to 0 in order to strengthen information interaction among groups of convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And it is worth noting that when the classes contained in datasets are becoming more, we are supposed to reduce the amount of pruning to better promote information transference for the version 9 of addition of ResFRI, which can be illustrated in the following experiments on vision datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' In the last, the results in the part of ablation study will prove the effectiveness of these modifications based on ResFRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' EXPERIMENTS 0 200 Epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5 Loss on CIFAR10 Train Test 0 200 400 Epoch 1 2 3 Loss on CIFAR100 Train Test 0 200 400 Epoch 1 2 3 4 Loss on Tiny Imagenet Train Test 0 100 200 Epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5 Loss on MNSIT Train Test 0 50 100 Epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6 Loss on FashionMNIST Train Test 0 100 200 Epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5 Loss on SVHN Train Test 0 200 Epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0 Accuracy on CIFAR10 Train Test 0 200 400 Epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='8 Accuracy on CIFAR100 Train Test 0 200 400 Epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6 Accuracy on Tiny Imagenet Train Test 0 100 200 Epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='00 Accuracy on MNSIT Train Test 0 50 100 Epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0 Accuracy on FashionMNIST Train Test 0 100 200 Epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0 Accuracy on SVHN Train Test Figure 5: Loss and Accuracy of ResFRI-addition on Datasets 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Implementation Details We implement the whole framework of GoogLe2Net utilizing code frame- work provided by Pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And in order to ensure fairness of comparison among different methods, we select experiment results of classical and newly proposed models without pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Due to our limited computation re- sources, apart from necessary ablation experiments, we choose the task of image classification on the common datasets, such as CIFAR10, CIFAR100, Tiny Imagenet, MNIST, FashionMNIST and SVHN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Besides, in the process of training on one RTX 3060 GPU, we use the optimizer SGD with momen- tum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='9, weight decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0005, batch size 64 and data augmentation tools provided in packages of torchvision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Moreover, the initial learning rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='01 and it is reduced by half if validation loss does not decrease within 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And tendency of accuracy and loss in the training process of ResFRI is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Experiments on CIFAR-10 The CIFAR10 dataset contains 50k training images and 10k testing im- ages from 10 classes whose resolution is 32×32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And the detail results of 10 comparisons of different models will be clearly provided in Table 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 10 20 30 40 50 60 70 Parameter Amount (M) 1 2 3 4 5 6 7 8 Top-1 error(%) GoogLeNet ResNet-101 Shake-Shake (26 2�96d) ResNeXt-29, 16�32d PyramidNet PyramidNet + ShakeDrop WRN-28-10 ResNeXt-29, 8�64d ResNeXt-29, 16�64d MixNet-190 Split- ResFRI- addition Split- ResFRI- concatenation ResFRI- addition ResFRI- concatenation CIFRA-10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='53 GFlops 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='99 GFlops 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='06 GFlops 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='52 GFlops 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='76 GFlops 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='04 GFlops 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='79 GFlops 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='05 GFlops 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='55 GFlops 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='73 GFlops 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='25 GFlops 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='41 GFlops 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='73 GFlops 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='30 GFlops Figure 6: Comparisons of models on CIFAR10 Dataset It can be obtained that the ResFRI and Split-ResFRI achieve relatively satisfying performance on image classification task on CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Compared with traditional models like ResNet-101 and ResNeXt-29, Res- FRI and Split-ResFRI have much better performance with much lower pa- rameter amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Although ResFRI-addition has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='24 GFlops and ResFRI- concatenation has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='52 GFlops higher than ResNet-101, we have a remarkable 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='44% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='46% performance gain on top-1 err while parameter amounts reduce by 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M and 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' For Split-ResFRI, the version of addition has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='53 GFlops and 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M parameters lower than ResNet-101, but we get 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='17% performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Besides, Split-ResFRI-concatenation has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='46 GFlops and 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7M parameters lower than ResNet-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Both of the Split- ResFRIs have lower Flops and parameter amounts and achieve better results than ResNet-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Compared with two versions of ResFRI, Split-ResFRIs sacrifice a little bit of precision in exchange for a considerable reduction in Flops and parameter amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' For ResNeXt-29, it outperforms ResNet-101 using larger model scales, but it still trails by at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='23% in compar- ison with ResFRI and Split-ResFRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And with respect to GoogLeNet, no matter it is ResFRI or Split-ResFRI, we all have achieved performance lead- 11 Table 1: Error rate (%) and Model Size on the CIFAR-10 Dataset Model Flops Params top-1 err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' ResNet-101 [11] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='52 GFlops 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5M 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='52 GoogLeNet [14] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='53 GFlops 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6M 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='16 ResNeXt-29, 16×32d [28] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='05 GFlops 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='87 ResNeXt-29, 8×64d [28] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='41 GFlops 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='65 ResNeXt-29, 16×64d [28] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='73 GFlops 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='58 CapsNet [32] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6 DropConnect [33] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='32 NIN + Dropout + Data Augmentation [34] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='96M 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='81 RMDL [35] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='74 FractalNet [36] 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6M 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='27 FitNet-LSUV [37] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='06 SOPCNN [38] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2MB 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='71 DenseNet-BC (k=24) [30] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='19 DPN-28-10 [39] 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='8M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='65 NASNet-A [39] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='41 AmoebaNet-A [39] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='34 AOGNet [39] 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='8M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='27 MixNet-190 [39] 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3 GFlops 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='13 AmoebaNet-B [39] 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='9M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='98 OR-WideResNet [40] 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='98 WRN-28-10 [41] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='25 GFlops 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6 PyramidNet [13] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='55 GFlops 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5 Shake-Shake (26 2x96d) [42] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='79 GFlops 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3 PyramidNet+ShakeDrop [42] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='73 GFlops 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1 ResFRI-addition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='76 GFlops 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='08 Split-ResFRI-addition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='99 GFlops 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='35 ResFRI-concatenation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='04 GFlops 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='06 Split-ResFRI-concatenation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='06 GFlops 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='28 12 ership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' It is worth noting that both versions of Split-ResFRIs have similar flops and parameter amounts to GoogLeNet, but still achieve a performance lead of over 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='8 percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And with respect to CapsNet, DropConnect, NIN and RMDL, the four models reach a fairly satisfying level on small-size datasets like MNIST utilizing very small model scales, which partly outper- forms many classical and novel methods including GoogLe2Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' However, all of the four models are not as good a performance as before in the more popular vision dataset, CIFAR-10, other modern models have overwhelm- ing advantages compared with their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Especially, the series of models belonging to ResFRI achieve at least a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='39% performance lead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Moreover, when encountering some newly proposed models, ResFRI and Split-ResFRI still prove their superiority on classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' For DenseNet, it has a similar model scale to ResFRI-concatenation, but it has a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='13% performance disadvantage in the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Besides, with respect to OR- WideResNet, it achieves a relatively satisfying results with acceptable model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Compared with ResFRI and Split-ResFRI, its disadvantage is still sig- nificant with performance trailing by at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='63%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Then, WRN-28-10, PyramidNet and Shake-Shake(26 2×96d) all of them have higher flops and parameter amount than ResFRI and Split-ResFRI, but all of them achieve better accuracy except for Split-ResFRI-addition meanwhile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' However, we want to point out that Split-ResFRI-addition has far less GFlops and param- eter amount than the above model for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Moreover, we notice that PyramidNet+ShakeDrop has a a very approximate performance (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='04%) to ResFRI-concatenation, which is a is a very competitive opponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' However, the cost of the combination of PyramidNet and ShakeDrop is 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3% higher flops and 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7% larger parameter amount than Res-FRI-concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' We think this comparison also illustrates the advantage of the proposed method when considering differences on computing resources consumption of the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' In sum, the experiment on CIFAR-10 dataset strongly proves the effectiveness and validity of GoogLe2Net on image classification task and Split-ResFRI also has greatly competitive results when considering the re- duction on GFlops and the number of parameters by a significant amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Experiments on CIFAR-100 The CIFAR100 dataset consists of 50k training images and 10k testing images from 100 classes and their resolution is 32×32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And the detail results of comparisons of different models will be clearly provided in Table 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 13 Table 2: Top-1, Top-5 Test Error (%) and Model Size on the CIFAR-100 Dataset Model Flops Params top-1 err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' top-5 err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' ResNet-101 [11] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='52 GFlops 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7M 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='61 ResNeXt-50 [28] 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='8M 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='23 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='00 ResNeXt-101 [28] 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='99 ResNeXt-152 [28] 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='40 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='58 DenseNet (k=12, depth=40) [30] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0M 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='55 DenseNet (k=12, depth=100)[30] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0M 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='79 DenseNet (k=24, depth=100)[30] 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2M 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='42 DenseNet-BC (k=24) [30] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='64 GoogLeNet [14] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='53 GFlops 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6M 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='97 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='94 Inception v3 [43] 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='81 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='39 Inception v4 [15] 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='90 InceptionResnet v2 [15] 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='51 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='11 Xception [44] 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0M 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='07 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='32 WRN-40-10 [31] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='08 GFlops 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='9M 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='77 NIN + Dropout [34] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='96M 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='68 FitNet-LSUV [37] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='96 FractalNet [36] 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6M 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='05 SOPCNN [38] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2M 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='04 WRN-28-10 [41] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='25 GFlops 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5M 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='9 Res2NeXt-29, 6c×24w×6s [12] 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7M 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='79 Res2NeXt-29, 6c×24w×6s-SE [12] 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='9M 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='56 PyramidNet [45] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='55 GFlops 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2M 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4 OR-WideResNet [40] 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2M 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='98 NASNet-A [39] 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='9M 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='03 HCGNet-A3 [39] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0 GFlops 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='96 Shake-Shake (26 2×96d) [42] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='79 GFlops 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7 PyramidNet+ShakeDrop [42] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='73 GFlops 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5 ResFRI-addition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='76 GFlops 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2M 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='42 Split-ResFRI-addition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='99 GFlops 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5M 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='48 ResFRI-concatenation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='04 GFlops 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5M 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='71 Split-ResFRI-concatenation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='06 GFlops 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='8M 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='32 14 10 20 30 40 50 Parameter Amount (M) 14 16 18 20 22 24 26 Top-1 error(%) GoogLeNet HCGNet-A3 ResNet-101 Shake-Shake (26 2�96d) PyramidNet PyramidNet + ShakeDrop WRN-28-10 WRN-40-10 Split- ResFRI- addition Split- ResFRI- concatenation ResFRI- addition ResFRI- concatenation CIFRA-100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='53 GFlops 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='99 GFlops 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='00 GFlops 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='06 GFlops 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='52 GFlops 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='76 GFlops 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='04 GFlops 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='79 GFlops 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='55 GFlops 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='73 GFlops 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='25 GFlops 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='08 GFlops Figure 7: Comparisons of models on CIFAR100 Dataset By checking the results given in Table 2, some conclusions can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' ResNet-101 has a performance lag of around 8% compared with the pro- posed method and it utilizes approximate flops and nearly three times pa- rameter amount of ResFRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' For ResNext-series models, all of them achieves analogous performance as ResNet-101 with much less flops and parameter amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' The situation of DenseNets is also similar, they further reduces the size and computational complexity of the model, but the accuracy of it is still at a comparatively low level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' The best accuracy of them has at least a performance disadvantage of more than 5% compared with ResFRI-series models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Besides, the inception-series models also have a relatively excel- lent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Particularly, GoogLeNet possesses only 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6M parameter amount but achieves an effect that ranks at the top of many models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' For NIN, FitNet and SOPCNN, all of the three models can obtain better re- sults on smaller datasets, but they can not acquire desirable results on more convincing datasets like CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Considering the results of WRN-28-10 provided in [41], it achieves a performance leap with a top-1 error rate of about 16% and dose not increase flops and parameters amount too much compared with the previous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And it can be obtained that Res2NeXt can reach a similar performance with roughly the same number of parameters as WRN-28-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Certainly, ResFRI and Split-ResFRI have higher accuracy 15 with much lower flops and parameter amounts compared with the two cate- gories of models we just discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Moreover, when considering other modern models, the HCGNet-A3 has a very approxmate flops and parameter amount with GoogLe2Net which realizes nearly two more percent accuracy improvement on classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' For PyramidNet, NASNet-A, Shake-Shake (26 2×96d) and Pyramid- Net+ShakeDrop, ResFRI and Split-ResFRI still achieve better performances while using less flops and parameter amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' The most light one, Split- ResFRI-addition, can achieve almost the best performance with less than 9M parameter amount and 2 Gflops which are between a half and a third of the scales of the four models mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Especially, Pyramid- Net+ShakeDrop has the closest effect to the proposed method while possess- ing 55% higher parameter amount and 110% more flops than the proposed models at least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Compared with the original PyramidNet, the combination of PyramidNet+ShakeDrop obtains a performance improvement of about 2%, which illustrates the possibility of follow-up work using this technology and the effectiveness of ShakeDrop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' In sum, based on experimental results pro- vided in Table 2, it can be concluded that the proposed method possesses a far better precision on classification task when compared with classical net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Except for GoogLeNet and DenseNet, all of the other models have larger parameter amount than the proposed model but could not reach a similar level of accuracy, which demonstrates the efficiency and effectiveness of GoogLe2Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Although GoogLeNet and DenseNet with specific settings own smaller model scale than ResFRI and Split-ResFRI, but our proposed method has a huge advantage in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Concretely, the version of ad- dition of ResFRI reaches a top-1 error rate 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='09 and top-5 error rate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='42 on CIFAR-100 dataset, in the meantime, Split-ResFRI could achieve very similar performance with at most 37% reduction of parameter amount and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5% curtailment on flops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' In one word, all of the comparisons proves the superiority of GoogLe2Net on classification tasks which can be regarded as a satisfying solution in choices among CNN architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Experiments on Tiny Imagenet The Tiny Imagenet dataset consists of 100k training images and 10k testing images from 200 classes and their resolution is 64×64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And the results of comparisons are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And it is worth noting that flops and parameter amounts of ResFRI and Split-ResFRI are evaluated using a tensor 16 matrix of 3 × 64 × 64 and the model is subtly adjusted to fit the different type of data, so the number of them will also variate accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Table 3: Top-1 Test Error (%) and Model Size on the Tiny Imagenet Dataset Model Flops Params top-1 err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' ResNet-18+Mixup+DM [46] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='93 ResNet-18+CutMix+DM [46] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='55 ResNet-18+ManifoldMix+DM [46] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='55 ResNet-18+ResizeMix+DM [46] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='67 ResNet-18+PuzzleMix+DM [46] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='48 ResNeXt-50+Mixup+DM [46] 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='30 ResNeXt-50+CutMix+DM [46] 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='54 ResNeXt-50+ManifoldMix+DM [46] 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='52 ResNeXt-50+ResizeMix+DM [46] 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='44 ResNeXt-50+PuzzleMix+DM [46] 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='96 WaveMixLite-144/7 [47] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='01 M 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='62 DenseNet + Residual Networks [48] 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0 ResNet18 + AutoMix [49] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='67 UPANets [50] 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='33 ResNet18 + SAMix [51] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='11 PreActResNet-18-3 + MixMo [52] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='76 ResFRI-addition (pruning ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='13 GFlops 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5 ResFRI-addition (pruning ratio 0) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='13 GFlops 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='60 Split-ResFRI-addition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='37 GFlops 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5M 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='93 ResFRI-concatenation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4 GFlops 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7M 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='46 Split-ResFRI-concatenation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='44 GFlops 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='0M 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='04 The experiments on the Tiny Imagenet show that the proposed method achieves a considerably satisfying classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' For ResNet-18, it has nearly the same as many parameters as ResFRI, but achieves far weaker performance than ResFRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Besides, compared with Split-ResFRI, the Split-ResFRI can obtain higher accuracy using less parameter amounts, which clearly demonstrates the efficiency of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Moreover, 17 ResNext-50 possesses two to three times as many as parameters as ResFRI and Split-ResFRI, it is able to get approximate performance to the pro- posed models but still falls behind in the best model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And with respect to WaveMixLite-144/7, it reaches a similar performance to ResNet- 50 utilizing only 3M parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' But its actual model accuracy is still not satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Compared with the methods like ResNet18 and PreActResNet, ResFRIs provide a best performance exceeding 70% accuracy which is a re- markable improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' It is worth noting that Split-ResFRIs are also able to achieve a similar tier of accuracy utilizing less parameter amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' In sum, GoogLe2Net reaches a high level of performance on image classification task without consuming too many computing resources in comparison with other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Experiments on MNIST The MNIST dataset contains 60k training images and 10k testing images from 10 classes whose resolution is 28×28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And the detail results of com- parisons of different models will be clearly provided in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' It is worth noting that parameter amounts of ResFRI and Split-ResFRI are calculated using a tensor matrix of 3×32×32, because the images of MNIST are resized into 32 before being inputting proposed models for process of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Based on MNIST dataset, there exist many very light models which still reach a great level of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' The proposed model falls behind by approx- imately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='18 percent and consumes much more computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Nevertheless, FitNet-LSUV and NiN encounter more than 4 and 15 percent performance loss on CIFAR-10 and CIFAR-100 dataset provided in Table 1 and 2 respectively compared with GoogLe2Net, which demonstrates that the relatively lower level of precision of GoogLe2Net on MNIST dataset is com- pletely acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Besides, the remaining methods like CapsNet, RMDL and SOPCNN also have similar situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Thus, the proposed method is more comprehensive and universal in handling classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And with respect to the performances of the two version of ResFRI, we argue that because the features contained in MNIST are simpler comparatively, the operation of concatenation is helpful to strengthen features instead of con- structing too dense connection between convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Moreover, for Split-ResFRIs, the performances of them become a little weaker in compar- ison with the versions without split, which may be caused by reduction of feature extraction operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 18 Table 4: Test Accuracy (%) and Model Size on the MNIST Dataset Model Params top-1 err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' PCANET-1 [53] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='62 FitNet-LSUV [37] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='46 NiN [34] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='96M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='45 VGG8B [54] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='26 CapsNet [32] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='25 DropConnect [33] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='21 RMDL [35] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='18 SOPCNN [38] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='17 ResFRI-addition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='35 Split-ResFRI-addition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='39 ResFRI-concatenation 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='31 Split-ResFRI-concatenation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='8M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Experiments on FashionMNIST The FashionMNIST dataset consists of 60k training images and 10k test- ing images from 10 classes and their resolution is 28×28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And the results of comparisons are given in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' It is worth noting that parameter amounts of ResFRI and Split-ResFRI are calculated using a tensor matrix of 3 × 32 × 32, because the images of FashionMNIST are resized into 32 before being inputting proposed models for process of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' As shown in the Table 5, ResFRI and Split-ResFRI reach a satisfying level of accuracy on FashionMNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And ResFRI-addition and ResFRI- concatenation make 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='56% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='28% percent performance gains compared with the Inception v3, which proves the efficiency and effectiveness of ResFRI compared with other Inception-like architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Besides, ResFRI and Split- ResFRI also outstrips these traditional models such as WideResNet, VGG8B and DenseNet utilizing much less parameter amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And it is worth noting that Split-ResFRI outperforms ResFRI on FashionMNIST dataset, which is very interesting and probably tells us that extraction of picture features like simple objects don’t require a deep and dense neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' All in all, by checking the results of comparison, it can be concluded that the pro- posed method guarantees a enough precision on a relatively small and simple 19 Table 5: Test Accuracy (%) and Model Size on the FashionMNIST Dataset Model Params top-1 err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Inception v3 [55] 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7M 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='56 SeResNeXt101-STD [56] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='59 VGG8B(2x) [54] 28M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='33 PreAct-ResNet18 [29] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='30 WideResNet-28-10 [54] 37M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='16 DenseNet-BC-190 [30] 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='06 ResFRI-addition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='00 Split-ResFRI-addition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='80 ResFRI-concatenation 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='29 Split-ResFRI-concatenation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='8M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='87 dataset and splitting features may be helpful in improving performance in analogous tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Experiments on SVHN The SVHN dataset contains 73257 training images and 26032 testing images from 10 classes whose resolution is 32×32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And the detail results of comparisons of different models will be clearly provided in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' By analyzing the experimental results on SVHN dataset, the GoogLe2Net also achieves relatively satisfying accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Compared with the classical mod- els like NiN, FractalNet and DenseNet, the proposed method utilizes much less parameter amount to reach a similar level of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Especially, the FractalNet possesses 219% higher parameter amount than GoogLe2Net while falling behind by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='15 percent accuracy compared with ResFRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And it’s worth noting that the performance of proposed model also exceeds Fractal- Net and DenseNet on CIFAR-10 and CIFAR-100 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Ablation Experiment In this section, we conduct the ablation experiment from two main aspects which are addition and concatenation version of ResFRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' In the preliminary stage of our experiment, we notice that for the addition version of ResFRI, a proper ration of pruning may help to promote the accuracy of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 20 Table 6: Test Accuracy (%) and Model Size on the SVHN Dataset Model Params top-1 err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' FitNet [57] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='42 NiN [34] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='96M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='35 FractalNet [36] 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='01 DropConnect [33] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='94 Deeply Supervised Net [58] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='92 FractalNet with Dropout/Drop-path [36] 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='6M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='87 ResNet with Stochastic Depth [59] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='75 DenseNet-BC [30] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='3M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='74 ResFRI-addition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='72 Split-ResFRI-addition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='84 ResFRI-concatenation 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='75 Split-ResFRI-concatenation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='8M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='82 And in the version of concatenation, no pruning may further enhance per- formance of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Therefore, all of the ablation experiments are not only involved with adjustment of structure of networks, but also the ratios of pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And all of the results are provided in the following Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' In detail, we remove three key components of ResFRI, namely AvgPool- ing layer, Residual connection and transverse passages between groups of convolutional layers respectively, to verify their influence on performance of the proposed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And we can notice that when each of them is removed, the performance will encounter a precision loss to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' It strongly demonstrates that when all of those components are synthesized, the lowest top1-error can be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Moreover, we also test effects of different pruning ratio on precision of the proposed model, which also proves the rationality of our settings of ResFRI on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' CONCLUSIONS In this paper, we first review the architectures of traditional neural net- works and state importance of multi-scale design in CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' For the structure of incpetion-like networks, we notice that construction of transverse passages 21 Table 7: Comparison among ResFRI variants on CIFAR10 dataset Variants Params top-1 err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' ResFRI (addition, pruning ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='08 ResFRI (addition, pruning ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='35) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='29 ResFRI (addition, pruning ratio 0) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='13 ResFRI (concatenation, pruning ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='14 ResFRI (concatenation, pruning ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='35) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='23 ResFRI (concatenation, pruning ratio 0) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='06 ResFRI without AvgPooling layer (addition, pruning ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='25 ResFRI without residual connection (addition, pruning ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='9M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='43 ResFRI without transverse passages (addition, pruning ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='7) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='12 ResFRI without AvgPooling layer (concatenation, pruning ratio 0)) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='30 ResFRI without residual connection (concatenation, pruning ratio 0) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='2M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='37 ResFRI without transverse passages (concatenation, pruning ratio 0) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='4M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='60 22 between adjacent groups of convolutional layers may boost performance of the network compared with original inception frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Besides, referring the concept of ResNet, we also adopt a policy that a residual connection is added to lower difficulty in network optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' In detail, transverse passages between adjacent groups of convolutional layers realize feature re- utilization in groups of convolutional layers and further enhance the ability of expression and generalization of original inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Besides, residual con- nection reduces overfitting and gradient disappearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' They are the main reasons that GoogLe2Net is able to reach a satisfactory level of accuracy on mainstream vision datasets under such a light and efficient inception-like framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' And all the experiments in this paper confirm this perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' Moreover, in the future, we believe the organic combination of the concept of multi-scale and CNNs will be a hot spot in boosting performances on various vision 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+page_content='1007/978-3-319-46493-0\\_39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} +page_content='1007/978-3-319-46493-0_39 33' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfj_hk/content/2301.00424v1.pdf'} diff --git a/idAyT4oBgHgl3EQf-_pB/content/tmp_files/2301.00900v1.pdf.txt b/idAyT4oBgHgl3EQf-_pB/content/tmp_files/2301.00900v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ff553b8e837292acd83c737d4530d9056ed096be --- /dev/null +++ b/idAyT4oBgHgl3EQf-_pB/content/tmp_files/2301.00900v1.pdf.txt @@ -0,0 +1,5939 @@ +State and parameter learning with PARIS particle Gibbs +Gabriel Cardoso†, Yazid Janati El Idrissi‡, Sylvain Le Corff⋆, ´Eric Moulines†, and +Jimmy Olsson⊤ +†CMAP, ´Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau. +‡Samovar, T´el´ecom SudParis, d´epartement CITI, TIPIC, Institut Polytechnique de Paris, Palaiseau. +⋆LPSM, Sorbonne Universit´e, UMR CNRS 8001, 4 Place Jussieu, 75005 Paris. +⊤Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden. +Abstract +Non-linear state-space models, also known as general hidden Markov models, are ubiquitous +in statistical machine learning, being the most classical generative models for serial data and +sequences in general. The particle-based, rapid incremental smoother (PARIS) is a sequential Monte +Carlo (SMC) technique allowing for efficient online approximation of expectations of additive +functionals under the smoothing distribution in these models. Such expectations appear naturally +in several learning contexts, such as likelihood estimation (MLE) and Markov score climbing +(MSC). PARIS has linear computational complexity, limited memory requirements and comes with +non-asymptotic bounds, convergence results and stability guarantees. Still, being based on self- +normalised importance sampling, the PARIS estimator is biased. Our first contribution is to design +a novel additive smoothing algorithm, the Parisian particle Gibbs (PPG) sampler, which can be +viewed as a PARIS algorithm driven by conditional SMC moves, resulting in bias-reduced estimates +of the targeted quantities. We substantiate the PPG algorithm with theoretical results, including +new bounds on bias and variance as well as deviation inequalities. Our second contribution is +to apply PPG in a learning framework, covering MLE and MSC as special examples. +In this +context, we establish, under standard assumptions, non-asymptotic bounds highlighting the value +of bias reduction and the implicit Rao–Blackwellization of PPG. These are the first non-asymptotic +results of this kind in this setting. We illustrate our theoretical results with numerical experiments +supporting our claims. +1. +Introduction +Sequential Monte Carlo (SMC) methods, or particle filters, are simulation-based approaches used for +the online approximation of posterior distributions in the context of Bayesian inference in state space +models. +In nonlinear hidden Markov models (HMM), they have been successfully applied for ap- +proximating online the typically intractable posterior distributions of sequences of unobserved states +(Xs1, . . . , Xs2) given observations (Yt1, . . . , Yt2) for 0 ≤ s1 ≤ s2 and 0 ≤ t1 ≤ t2. Standard SMC +methods use Monte Carlo samples generated recursively by means of sequential importance sampling +and resampling steps. A particle filter approximates the flow of marginal posteriors by a sequence of +occupation measures associated with a sequence {ξi +t}N +i=1, t ∈ N, of Monte Carlo samples, each parti- +cle ξi +t being a random draw in the state space of the hidden process. Particle filters revolve around +two operations: a selection step duplicating/discarding particles with large/small importance weights, +respectively, and a mutation step evolving randomly the selected particles in the state space. Ap- +plying alternatingly and iteratively selection and mutation results in swarms of particles being both +temporally and spatially dependent. The joint state posteriors of an HMM can also be interpreted +as laws associated with a certain kind of Markovian backward dynamics; this interpretation is useful, +1 +arXiv:2301.00900v1 [stat.ME] 2 Jan 2023 + +for instance, when designing backward-sampling-based particle algorithms for nonlinear smoothing +[Douc et al., 2011, Del Moral et al., 2010]. +Throughout the years, several convergence results as the number N of particles tends to infinity +have been established; see, e.g., [Del Moral, 2004, Douc and Moulines, 2008, Capp´e et al., 2005] and +the references therein. In addition, a number of non-asymptotic results have been established, including +time-uniform bounds on the SMC Lp error and bias as well as bounds describing the propagation of +chaos among the particles. Extensions to the backward-sampling-based particle algorithms can also +be found for instance in [Douc et al., 2011, Del Moral et al., 2010, Dubarry and Le Corff, 2013]. +In this paper, we focus on the problem of recursively computing smoothed expectations η0:tht = +E[ht(X0:t) | Y0:t] for additive functionals ht in the form +ht(x0:t) := +t−1 +� +s=0 +˜hs(xs:s+1), +(1.1) +where X0:n and Y0:n denote vectors of states and observations (see below for precise definitions). Such +expectations appear frequently in the context of maximum-likelihood parameter estimation in nonlin- +ear HMMs, for instance, when computing the score function (the gradient of the log-likelihood function) +or the Expectation Maximization intermediate quantity; see [Capp´e, 2001, Andrieu and Doucet, 2003, +Poyiadjis et al., 2005, Capp´e, 2011, Poyiadjis et al., 2011]. The particle-based, rapid incremental smoother +(PARIS) proposed in [Olsson and Westerborn, 2017] is tailored for solving online this additive smooth- +ing problem. +When the transition density of the latent states is lower and upper bounded, this +algorithm can be shown to have a linear computational complexity in the number N of particles and +limited memory requirements. An interesting feature of the PARIS, which samples on-the-fly from the +backward dynamics induced by the particle filter, is that it requires two or more backward draws per +particle to cope with the degeneracy of the sampled trajectories and remain numerically stable in the +long run, with an asymptotic variance that grows only linearly with time. +In this paper, we introduce a method to reduce the bias of the PARIS estimator of η0:tht. The +idea is to mix—by introducing a conditional PARIS algorithm—the PARIS algorithm with a backward- +sampling-based version of the particle Gibbs sampler [Andrieu et al., 2010b, Lindsten et al., 2014a, +Chopin and Singh, 2015a, Del Moral et al., 2016, Del Moral and Jasra, 2018]. This leads to a batch +mode PARIS particle Gibbs (PPG) sampler, which we furnish with an upper bound of the bias that +decreases inversely proportionally to the number N of particles and exponentially fast with the particle +Gibbs iteration index (under the assumption that the particle Gibbs sampler is uniformly ergodic). +As an application we consider the problem of likelihood maximization with stochastic gradient. +In this specific context, where the smoothing estimator is employed repeatedly to produce mean-field +estimates, controlling the bias becomes critical. Thus, it is natural to aim at minimizing the bias +for a fixed computational budget, provided that the variance does not explode. For this reason, bias +reduction in stochastic simulation has been the subject of extensive research during the last decades +[Jacob et al., 2020, Glynn and Rhee, 2014]. The present paper contributes to this line of research. In +particular, we show that stochastic approximation (SA) with PPG achieves a O(log(n)/√n) rate, where +n is the number of SA steps. This improves on a previous result of [?], which establishes the almost +sure convergence (to a stationary point of the likelihood) of an SA Expectation Maximization (EM) +algorithm based on particle Gibbs with ancestor sampling (PGAS). +The paper is structured as follows. In Section 2, we recall the hidden Markov model framework, the +particle filter and the PARIS algorithm. In Section 3, we lay out the PPG algorithm and present the first +central result of this paper, an upper bound on the bias of our estimator as a function of the number +of particles and the iteration index of the Gibbs algorithm. In addition, we provide an upper bound +on the mean-squared error (MSE). In Section 4, we undertake the learning problem and present the +second result of this paper, a O(log(n)/√n) non-asymptotic bound on the expectation of the squared +gradient norm taken at a random index K. In Section 5.1, we illustrate our results through numerical +experiments. All the proofs are collected in the supplementary material. +2 + +Notation. +For a given measurable space (X, X), where X is a countably generated σ-algebra, we +denote by F(X) the set of bounded X/B(R)-measurable functions on X. For any h ∈ F(X), we let +∥h∥∞ := supx∈X |h(x)| and osc(h) := sup(x,x′)∈X2 |h(x) − h(x′)| denote the supremum and oscillator +norms of h, respectively. Let M(X) be the set of σ-finite measures on (X, X) and M1(X) ⊂ M(X) +the probability measures. For any h ∈ F(X) and µ ∈ M(X) we write µ(f) = +� +h(x)µ(dx). For a +Markov kernel K from (X, X) to another measurable space (Y, Y), we define the measurable function +Kh : X ∋ x �→ +� +h(y)K(x, dy). The composition µK is a probability measure on (Y, Y) such that +µK : X ∋ A �→ +� +µ(dx)K(x, dy)1A(y). For all sequences {au}u∈Z and {bu}u∈Z, and all s ≤ t we write +as:t = {as, . . . , at} and bs:t = {bs, . . . , bt}. +2. +Background +2.1 +Hidden Markov models +Hidden Markov models consist of an unobserved state process {Xt}t∈N and observations {Yt}t∈N, +where, at each time t ∈ N, the unobserved state Xt and the observation Yt are assumed to take values +in some general measurable spaces (Xt, Xt) and (Yt, Yt), respectively. It is assumed that {Xt}t∈N is a +Markov chain with transition kernels {Mt+1}t∈N and initial distribution η0. Given the states {Xt}t∈N, +the observations {Yt}t∈N are assumed to be independent and such that for all t ∈ N, the conditional +distribution of the observation Yt depends only on the current state Xt. This distribution is assumed +to admit a density gt(Xt, ·) with respect to some reference measure. In the following we assume that +we are given a fixed sequence {yt}t∈N of observations and define, abusing notations, gt(·) = gt(·, yt) +for each t ∈ N. We denote, for 0 ≤ s ≤ t, Xs:t := �t +u=s Xu and Xs:t := �t +u=s Xu. Consider the +unnormalized transition kernel +Qs : Xs × Xs+1 ∋ (x, A) �→ gs(x)Ms(x, A) +(2.2) +and let +γ0:t : X0:t ∋ A �→ +� +1A(x0:t) η0(dx0) +t−1 +� +s=0 +Qs(xs, dxs+1). +(2.3) +Using these quantities, we may define the joint-smoothing and predictor distributions at time t ∈ N as +η0:t : X0:t ∋ A �→ +γ0:t(A) +γ0:t(X0:t), +(2.4) +ηt : Xt ∋ A �→ η0:t(X0:t−1 × A), +(2.5) +respectively. +It can be shown (see [Capp´e et al., 2005, Section 3]) that η0:t and ηt are the condi- +tional distributions of X0:t and Xt given Y0:t−1 respectively, evaluated at y0:t−1. Unfortunately, these +distributions, which are vital in Bayesian smoothing and filtering as they enable the estimation of +hidden states through the observed data stream, are available in a closed form only in the cases of +linear Gaussian models or models with finite state spaces; see [Capp´e et al., 2009] for a comprehensive +coverage. +2.2 +Particle filters +For most models of interest in practice, the joint smoothing and predictor distributions are intractable, +and so are also any expectation associated with these distributions. +Still, such expectations can +typically be efficiently estimated using particle methods, which are based on the predictor recursion +ηt+1 = ηtQt/ηtgt. At time t, if we assume that we have at hand a consistent particle approximation +of ηt, formed by N random draws {ξi +t}N +i=1, so-called particles, in Xt and given by ηN +t = N −1 �N +i=1 δξi +t, +plugging ηN +t +into the recursion tying ηt+1 and ηt yields the mixture ηN +t Qt, from which a sample of N +new particles can be drawn in order to construct ηN +t+1. To do so, we sample, for all 1 ≤ i ≤ N, ancestor +3 + +indices αi +t ∼ Categorical({gt(ξℓ +t)}N +ℓ=1) and then propagate ξi +t+1 ∼ Mt(ξαi +t +t , ·). This procedure, which is +initialized by sampling the initial particles {ξi +0}N +i=1 independently from η0, describes the particle filter +with multinomial resampling and produces consistent estimators such that for every h ∈ F(Xt), ηN +t (h) +converges almost surely to ηt(h) as the number N of particles tends to infinity. +This procedure can also be extended to produce particle approximations of the joint-smoothing +distributions {η0:t}t∈N. Note that the successive ancestor selection steps described previously generates +an ancestor line for each terminal particle ξi +t, which we denote by ξi +0:t. It can then be easily shown +that ηN +0:t = N −1 �N +i=1 δξi +0:t forms a particle approximation of the joint-smoothing distribution η0:t. +However, it is well known that the same selection operation also depletes the ancestor lines, since, +at each step, two different particles are likely to originate from the same parent in the previous +generation. Thus, eventually, all the particles end up having a large portion of their initial ancestry +in common. This means that in practice, this naive approach, which we refer to as the poor man’s +smoother, suffers generally from high variance when used for estimating joint-smoothing expectations +of objective functionals depending on the whole state trajectory. +2.3 +Backward smoothing and the PARIS algorithm +We now discuss how to avoid the problem of particle degeneracy relative to the smoothing problem by +means of so-called backward sampling. While this line of research has broader applicability, we restrict +ourselves for the sake of simplicity to the case of additive state functionals in the form +ht(x0:t) := +t−1 +� +s=0 +˜hs(xs:s+1), +x0:t ∈ X0:t. +(2.6) +Appealingly, using the poor man’s smoother described in the previous section, smoothing of additive +functionals can be performed online alongside the particle filter by letting, for each s, +ηN +0:shs := N −1 +N +� +i=1 +βi +s, +(2.7) +where the statistics {βi +s}N +i=1 satisfy the recursion +βi +s+1 = βαi +s +s ++ ˜hs(ξαi +s +s , ξi +s+1), +(2.8) +where αi +s is, as described, the ancestor at time s of particle ξi +s+1. +As mentioned above, the previous estimator suffers from high variance when s is relatively large +with respect to N. However, assume now that the model is fully dominated in the sense that each +state process kernel Ms has a transition density ms with respect to some reference measure; then, +interestingly, it is easily seen that the conditional probability that αi +s = j given the offspring ξi +s+1 and +the ancestors {ξℓ +s}N +ℓ=1 is given by +Λs(i, j) := +ωj +sms(ξj +s, ξi +s+1) +�N +ℓ=1 ωℓsms(ξℓs, ξi +s+1) +. +(2.9) +Here Λs forms a backward Markov transition kernel on �1, N� × �1, N�. Using this observation, we +may avoid completely the particle-path degeneracy of the poor man’s smoother by simply replacing +the naive update (2.8) by the Rao–Blackwellized counterpart +βi +s+1 = +N +� +j=1 +Λs(i, j){βj +s + ˜hs(ξj +s, ξi +s+1)}. +(2.10) +This approach, proposed in [Del Moral et al., 2010], avoids elegantly the path degeneracy as is elimi- +nates the ancestral connection between the particles by means of averaging. Furthermore, it is entirely +4 + +online since at step s only the particle populations ξ1:N +s +and ξ1:N +s+1 are needed to perform the update. +Still, a significant drawback is the overall O(N 2) complexity for the computation of β1:N +t +, since the +calculation of each βi +s+1 in (2.10) involves the computation of N 2 terms, which can be prohibitive +when the number N of particles is large. Thus, in [Olsson and Westerborn, 2017], the authors propose +to sample M ≪ N conditionally independent indices {Ji,j +s }M +j=1 from the distribution Λs(i, ·) and to +update the statistics according to +βi +s+1 = M −1 +M +� +j=1 +� +βJi,j +s +s ++ ˜hs(ξJi,j +s +s +, ξi +s+1) +� +. +(2.11) +If the transition density ms is uniformly bounded from above and below, an accept-reject approach +allows the sampling-based update (2.11) to be performed for i ∈ �1, N� at an O(N(M + 1)) overall +complexity if a pre-initialized multinomial sampler is used. A key aspect of this approach is that +the number M of sampled indices at each step can be very small; indeed, for any fixed M ≥ 2, the +algorithm, which is referred to as the PARIS, can be shown to be stochastically stable with an O(t) +variance (see [Olsson and Westerborn, 2017, Section 1] for details), and setting M to 2 or 3 yields +typically fully satisfying results. +The PARIS estimator can be viewed as an alternative to the FFBSm, rather than the FFBSi. Even +if the PARIS and FFBSi are both randomised versions of the FFBSm estimator, the PARIS is of a +fundamentally different nature than the FFBSi. The PARIS approximates the forward-only FFBSm +online in the context of additive functionals by approximating each updating step by additional Monte +Carlo sampling. The sample size M is an accuracy parameter that determines the precision of this +approximation, and by increasing M the statistical properties of the PARIS approaches those of the +forward-only FFBSm. On the other hand, as shown in [Douc et al., 2011, Corollary 9], the asymptotic +variance of FFBSi is always larger than that of the FFBSm, with a gap given by the variance of the +state functional under the joint-smoothing distribution. Thus, we expect, especially in the case of a +low signal-to-noise ratio, the PARIS to be more accurate than the FFBSi for a given computational +budget. Another important reason to focus on the PARIS estimator rather than the FFBSi is the +appealing online properties of the latter, whose interplay with and relevance to the particle MCMC +methodology is to be explored. Our results can be naturally extended to the FFBSi and PGAS but +since the PARIS has a practical edge, we chose to center our contribution around it although the main +idea behind our paper is more general. +3. +PARIS particle Gibbs +3.1 +Particle Gibbs methods +The conditional particle filter (CPF) introduced in [Andrieu et al., 2010a] serves the basis of a particle- +based MCMC algorithm targeting the joint-smoothing distribution η0:t. Let ℓ ∈ N∗ be an iteration +index and ζ0:t[ℓ] a conditional path used at iteration ℓ of the CPF to construct a particle approximation +of η0:t as follows. At step s ∈ �1, t� of the CPF, a randomly selected particle, with uniform probability +1/N, is set to ζs[ℓ], whereas the remaining N − 1 particles are all drawn from the mixture ηN +s−1Qs−1. +At the final step, a new particle path ζ0:t[ℓ + 1] is drawn either: +• by selecting randomly, again with uniform probability 1/N, a genealogical trace from the an- +cestral tree of the particles {ξ1:N +s +}t +s=0 produced by the CPF, as in the vanilla particle Gibbs +sampler; +• or by generating the path by means of backward sampling, i.e., by drawing indices J0:t backwards +in time according to Jt ∼ Categorical({1/N}N +i=1) and, conditionally to Js+1, Js ∼ Λs(Js+1, ·), +s ∈ �0, t−1�, and letting ζ0:t[ℓ+1] = (ξJ0 +0 , . . . ξJt +t ) (where the transition kernels {Λs}t +s=0, defined +by (2.9), are induced by the particles produced by the CPF), as proposed in [Whiteley, 2010]. +5 + +The theoretical properties of the different versions of the particle Gibbs sampler are well studied +[Singh et al., 2017, Chopin and Singh, 2015b, Andrieu et al., 2018]. In short, the produced conditional +paths (ζ0:t[ℓ])ℓ∈N form a Markov chain whose marginal law converges geometrically fast in total vari- +ation to the target distribution η0:t. As it is the case for smoothing algorithms, the vanilla particle +Gibbs sampler suffers from bad mixing due to particle path degeneracy while its backward-sampling +counterpart exhibits superior performance as t increases [Lee et al., 2020]. +3.2 +The PPG algorithm +Remarkably, in order for the standard particle Gibbs samplers to output a single conditional path, a +whole particle filter is run and then discarded, resulting in significant waste of computational work. +Thus, we now introduce a variant of the PARIS algorithm, coined the PARIS particle Gibbs (PPG), in +which the conditional path of particle Gibbs with backward sampling is merged with the intermediate +particles, ensuring less computational waste and reduced bias with respect to the vanilla PARIS. +In the following we let t ∈ N be a fixed time horizon, and describe in detail how the PPG ap- +proximates iteratively η0:tht, where ht is an additive functional in the form (2.6). +Using a given +conditional path ζ0:t[ℓ − 1] as input, the ℓ-th iteration of the PPG outputs a many-body system +υt[ℓ] = ((ξ1 +0:t, β1 +t ), . . . , (ξN +0:t, βN +t )) comprising N backward particle paths {ξi +0:t}N +i=1 with associated PARIS +statistics {βi +t}N +i=1. This is the so-called conditional PARIS update detailed in Algorithm 1. After this, +an updated conditional path is selected with probability 1/N among the N particle paths {ξi +0:t}N +i=1 +and used as input in the next conditional PARIS operation. At each iteration, the produced statistics +{βi +t}N +i=1 provide an approximation of η0:tht according to (2.7). The overall algorithm is summarized +in Algorithm 2. The function CPFs describes one step of the conditional particle filter and is given +in the supplementary material. In addition, the PPG algorithm defines a Markov chain with Markov +transition kernel denoted by Kt and detailed in (A.41). +Algorithm 1 One conditional PARIS update (CondPaRIS) +Input: {(ξi +0:s, βi +s)}N +i=1, ζs+1, ˜hs−1 +Result: {(ξi +0:s+1, βi +s+1)}N +i=1 +1 draw ξ1:N +s+1 ∼ CPFs(ζs+1, ξ1:N +s +) +for i ← 1 to N do +2 +draw {Ji,ℓ +s }M +ℓ=1 ∼ Λ(i, ·)�M +3 +set βi +s+1 ← M −1 �M +ℓ=1 +� +βi,Ji,ℓ +s +s ++ ˜hs(ξi,Ji,ℓ +s +s +, ξi +s+1) +� +4 +set ξi +0:s+1 ← (ξi,Ji,1 +s +0:s +, ξi +s+1) +Algorithm 2 One iteration of PPG +Input: Initial path ζ0:t, {˜hs}t−1 +s=0 +Result: {βi +t}N +i=1, ζ′ +0:t +5 draw ξ1:N +0 +∼ CPF0(ζ0) +6 set βi +0 ← 0 for i ∈ �1, N� +7 for s ← 0 to t − 1 do +8 +set {(ξi +0:s+1, βi +s+1)}N +i=1 ← CondPaRIS({(ξi +0:s, βi +s)}N +i=1, ζs+1, ˜hs) +9 draw ζ′ +0:t ∼ N −1 �N +i=1 δξi +0:t +As performing k steps of the PPG results in k many-body systems, it is natural to consider the +6 + +following roll-out estimator which combines the backward statistics from step k0 < k to k: +Π(k0,k),N(ht) = [N(k − k0)]−1 +k +� +ℓ=k0+1 +N +� +j=1 +βj +t [ℓ]. +(3.12) +The total number of particles used in this estimator is C = (N − 1)k per time step. We denote by +υ = (k−k0)/k the ratio of the number of particles used in the estimator to the total number of sampled +particles. +We now state the first main results of the present paper, in the form of theoretical bounds +on the bias and mean-squared error (MSE) of the roll-out estimator (3.12). These results are ob- +tained under the following strong mixing assumptions, which are now standard in the literature (see +[Del Moral, 2004, Douc and Moulines, 2008, Del Moral, 2013, Del Moral et al., 2016]). It is crucial for +obtaining quantitative bounds for particle smoothing algorithms, see [Olsson and Westerborn, 2017] or +[Gloaguen et al., 2022] but also for the coupled conditional backward sampling particle filter [Lee et al., 2020]. +A 3.1 (strong mixing). For every s ∈ N there exist ¯τs, ¯τs, ¯σs, and ¯σs in R∗ ++ such that +(i) ¯τs ≤ gs(xs) ≤ ¯τs for every xs ∈ Xs, +(ii) ¯σs ≤ ms(xs, xs+1) ≤ ¯σs for every (xs, xs+1) ∈ Xs:s+1. +Under A 3.1, define, for every s ∈ N, +ρs := +max +m∈�0,s� +¯τm¯σm +¯τm¯σm +(3.13) +and, for every N ∈ N∗ and t ∈ N such that N > Nt := (1 + 5ρ2 +t/2) ∨ 2t(1 + ρ2 +t), +κN,t := 1 − 1 − (1 + 5tρ2 +t/2)/N +1 + 4t(1 + 2ρ2 +t)/N . +(3.14) +Note that κN,t ∈ (0, 1) for all N and t as above. +Theorem 1. Assume A 3.1. Then for every t ∈ N, M ∈ N∗, ξ ∈ M1(X0:t), k0 ∈ N∗, k > k0 and +N ∈ N∗ such that N > Nt, +��Eξ[Π(k0,k),N(ht)] − η0:tht +�� ≤ σbias +(3.15) +Eξ +�� +Π(k0,k),N(ht) − η0:tht +�2� +≤ σ2 +mse, +where +σbias := +cbias +t +κk0 +t,N +�t−1 +m=0 ∥˜hm∥∞ +(k − k0)(1 − κt,N)N +, +σ2 +mse := (�t−1 +m=0 ∥˜hm∥∞)2 +N(k − k0) +� +cmse +t ++ +2ccov +t +N 1/2(1 − κt,N) +� +and cbias +t +, cmse +t +and ccov +t +are constants that do not depend on N and Eξ denotes the expectation under +the law of the Markov chain formed by the PPG when initialized according to ξ. +The proof is provided in the supplementary material. Importantly, (3.15) provides a bound on the +bias of the roll-out estimator that decreases exponentially with the burn-in period k0 and is inversely +proportional to the number N of particles. This means that we can improve the bias of the PARIS +estimator with a better allocation of the computational resources. +7 + +4. +Parameter learning with PPG +We now turn to parameter learning using PPG and gradient-based methods. +We set the focus on +learning the parameter θ of a function V (θ) whose gradient is the smoothed expectation of an additive +functional s0:t,θ in the form (2.6). Algorithm 4 defines a stochastic approximation (SA) scheme where +the noise forms a parameter dependent Markov chain with associated invariant measure πθ. +We +follow the approach of [Karimi et al., 2019] to establish a non-asymptotic bound over the mean field +h(θ) := πθs0:t,θ. Such a setting encompasses for instance the following estimation procedures. +(1) Score ascent. +In the case of fully dominated HMMs, we are often interested in optimizing the +log-likelihood of the observations given by V (θ) = log +� +γ0:t,θ(dx0:t). By applying Fisher’s identity, +we may express its gradient as a smoothed expectation of an additive functional according to +∇θV (θ) = +� +∇θ log γ0:t(x0:t) η0:t,θ(dx0:t), += +� +t−1 +� +ℓ=0 +sℓ,θ(xℓ, xℓ+1) η0:t,θ(dx0:t), +where sℓ,θ : Xℓ:ℓ+1 ∋ (x, x′) �→ ∇θ log{gℓ,θ(x)mℓ,θ(x, x′)} and s0:t,θ := �t−1 +ℓ=0 sℓ,θ. +(2) Inclusive KL surrogates. +Inspired by [Naesseth et al., 2020], we may consider the problem of +learning a surrogate model for η0:t,θ in the form qφ(x0:t) = qφ(x0) �t−1 +ℓ=0 qφ(xℓ+1, xℓ) by minimizing +V (φ) = KL(η0:t,θ, qφ). +Algorithm 3 Gradient estimation with roll-out PPG (GdEst) +Input: θ, ζ0:t[0], {sℓ,θ}t−1 +ℓ=0, number k of PPG iterations, burn-in k0. +Result: β1:N +t +[k0 : k], ζ0:t[k] +10 for ℓ ← 0 to k − 1 do +11 +run (˜β1:N +t +[ℓ + 1], ζ0:t[ℓ + 1]) ← PPG(θ; ζ0:t[ℓ], {sℓ,θ}t−1 +ℓ=0) +12 +if ℓ ≥ k0 − 1 then +13 +set β1:N +t +[ℓ + 1] = ˜β1:N +t +[ℓ + 1] +Algorithm 4 Score ascent with PPG. +Input: θ0, ζ0:t[0], number k of PPG iterations, burn-in k0, number of SA iterations n, learning-rate +sequence {γℓ}ℓ∈N. +Result: θn +14 for i ← 0 to n − 1 do +15 +run (β1:N +t +[k0 : k], ζ0:t[i + 1]) ← GdEst(θi, ζ0:t[i], {sℓ,θi}t−1 +ℓ=0, k, k0) +16 +set Π(k0,k),N(s0:t,θi) = (N(k − k0))−1 �k−1 +ℓ=k0 +�N +j=1 βj +t [ℓ] +17 +set θi+1 ← θi + γi+1Π(k0,k),N(s0:t,θi) +Note that Algorithm 3 defines a (collapsed) Markov kernel Pθ,t defining for each path ζ0:t a measure +Pθ,t(ζ0:t, d(˜ζ0:t, ˜β1:N +t +[k0 : k])) over the extended space of paths and sufficient statistics. Note that by +evaluating the function b1:N +t +[k0 : k] �→ [N(k − k0)]−1 �k +ℓ=k0+1 +�N +j=1 bj +t[ℓ] at a realisation of this kernel +gives the roll-out estimator whose properties are analysed in Theorem 1. The Markov kernel Pθ,t is +detailed in (B.72). +The following assumptions, are vital when analysing the convergence of Algorithm 4. +A 4.1. +(i) The function θ �→ V (θ) is LV -smooth. +8 + +(ii) The function θ �→ η0:t,θ is Lη-Lipschitz in total variation distance. +(iii) For each path ζ0:t ∈ X0:t, the function +θ �→ Kθ,t(ζ0:t, d˜ζ0:t) +(4.16) +is LP +1 -Lipschitz in total variation distance, where Kθ,t is path-marginalized Markov transition +kernel associated with the PPG algorithm when the model is parameterized by θ, see (A.41). +(iv) For each path ζ0:t ∈ X0:t, the function +θ �→ Pθ,tΠk0−1,k,N(s0:t,θ)(ζ0:t) +(4.17) +is LP +2 -Lipschitz in total variation distance. +In the case of score ascent we check, in Appendix B, that these assumptions hold if the strong +mixing assumption A 3.1 is satisfied uniformly in θ, and with additional assumptions on the model. +We are now ready to state a bound on the mean field h(θ) for Algorithm 4. +Theorem 2. Assume A 3.1 uniformly in θ and A 4.1 and suppose that the stepsizes {γℓ+1}ℓ∈�0,n−1� +satisfy γℓ+1 ≤ γℓ, γℓ < aγℓ+1, γℓ − γℓ+1 < a′γ2 +ℓ and γ1 ≤ 0.5(LV + Ch) for some a > 0, a′ > 0 and all +n ∈ N. Then, +E +� +∥h(θϖ)∥2� +≤ 2V0,n + C0,n + C0,γ +�n +k=0 γ2 +k+1 +�n +k=0 γk+1 +, +(4.18) +where V0,n = E [V (θ) − V (θn)] and +C0,n := γ1h(θ0)C0 + σbias(γ1 − γn+1 + 1)δ−1 +k,N,t , +(4.19) +C0,γ := σ2 +mseLV + σmseC1 + σmseσbias +� +LV + +C2 +1 − κN,t +� +δ−1 +k,N,t + σbiasLV δ−1 +k,N,t , +(4.20) +Ch := +� +C1 + σbias +C2 +(1 − κN,t)δk,N,t +� +[(a + 1)/2 + aσmse] + (LV + a′ + 1)σbiasδ−1 +k,N,t , +(4.21) +C1 = LP +2 +� +1 + κk +N,tδ−1 +k,N,t +� ++ LV +(4.22) +C2 = LP +1 δ−1 +k,N,t + Lηκk +N,t . +(4.23) +where C0 is independent of σbias, σmse, N and where δk,N,t = 1 − κk +N,t. +Theorem 2 establishes not only the convergence of Algorithm 4, but also illustrates the impact of +the bias and the variance of the PPG on the convergence rate. +Remark 1. Under additional assumptions on the model (cf Appendix B), if we consider γ1 ≤ +0.5(LV + Ch), γℓ = γ1ℓ−1/2 for all ℓ ∈ �1, n�, then �n +k=0 γ2 +k+1/ �n +k=0 γk+1 ∼ log n/√n, showing +that E +� +∥h(θϖ)∥2� +is O(log n/√n), where the leading constant depends on σbias and σmse. +Remark 1 establishes the rate of convergence of Algorithm 4. In principle we could try to optimize +the parameters k, k0 and N of the algorithm using these bounds, but one of the main challenges with +this approach is the determination of the mixing rate, which is underestimated by κN,t. Still, our +bound provides interesting information of the role of both bias and MSE. +5. +Numerics +In this section, we focus on the numerical analysis of the two main results of the paper, namely the bias +and MSE bounds of the roll-out estimator established in Theorem 1 and the efficiency of using PPG for +learning in the framework developed in Section 4. For the latter, we will restrict ourselves to the case +9 + +of parameter learning via score ascent. In this setting, the competing method that corresponds most +closely to the one presented here consists of using, as presented in Algorithm 5, a standard particle +Gibbs sampler Πθ instead of the PPG. One of the most common such samplers is the particle Gibbs with +ancestor sampling (PGAS) presented in [Lindsten et al., 2014b]. In [Lindholm and Lindsten, 2018], the +PGAS is used for parameter learning in HMMs via the Expectation Maximization (EM) algorithm. +Algorithm 5 Score ascent with particle Gibbs kernel. +Data: ζ0:t[0], θ0, number k of paths per trajectory, burn-in k0, number n of SA iterations, learning-rate +sequence {γℓ}ℓ∈N, Πθ(ζ0:t, d˜ζ0:t) a Markov kernel targeting η0:t. +Result: θn +18 for i ← 0 to n − 1 do +19 +for j ← 0 to k − 1 do +20 +sample ˜ζ0:t[j + 1] ∼ Πθ(˜ζ0:t[j], ·) +21 +set θi+1 ← θi + γi+1 +k−k0 +�k +ℓ=k0+1 s0:t,θi(˜ζ0:t[ℓ]) +22 +set ζ0:t[i + 1] = ˜ζ0:t[k] +5.1 +PPG +Linear Gaussian state-space model (LGSSM). +We first consider a linear Gaussian HMM +Xm+1 = AXm + Qϵm+1, +Ym = BXm + Rζm, +m ∈ N, +(5.24) +where {ϵm}m∈N∗ and {ζm}m∈N are sequences of independent standard normally distributed random +variables, independent of X0. The coefficients A, Q, B, and R are assumed to be known and equal +to 0.97, 0.60, 0.54, and 0.33, respectively. Using this parameterisation, we generate, by simulation, a +record of t = 999 observations. +In this setting, we aim at computing smoothed expectations of the state one-lag covariance ht(x0:t) := +�t−1 +m=0 xmxm+1. In the linear Gaussian case, the disturbance smoother (see [Capp´e et al., 2005, Algo- +rithm 5.2.15]) provides the exact values of the smoothed sufficient statistics, which allows us to study +the bias of the estimator for a given computational budget C. Figure 1 displays, for three different +total budgets C, the distribution of estimates of η0:nhn using the PARIS as well as three different +configurations of the PPG corresponding to k ∈ {2, 4, 10} (and N = C/k) with k0 = k/2 and k0 = k/4. +The reference value is shown as a red-dashed line and the mean value of each distribution is shown as a +black-dashed line. Each boxplot is based on 1000 independent replicates of the corresponding estima- +tor. We observe that in this example, all configurations of the PPG are less biased than the equivalent +PARIS estimator. The illustration of the bounds from Theorem 1 is postponed to Appendix D.1. +5.2 +Score ascent +LGSSM. +We consider the LGSSM with state and observation spaces being R5. We assume that the +parameters R and Q are known and consider the inference of θ = (A, B) on the basis of a simulated +sequence of n = 999 observations. In this setting, the M-step of the EM algorithm can be solved +exactly with the disturbance smoother [Capp´e et al., 2005, Chapter 11]. The parameter obtained by +this procedure (denoted θmle) is the reference value for any likelihood maximization algorithm. Table 1 +shows the L2 distance between the singular values of θmle and those of the parameters obtained by +Algorithm 4 and Algorithm 5. The CLT confidence intervals were obtained on the basis of 25 replicates. +The configurations respect a given particle budget kN = C = 1024. The choice of keeping k0 = k/2 is +a heuristic rule to achieve a good bias–variance trade-off, but other combinations of k0 and k may lead +to better performance for different problems. We analyse this for the LGSMM in Appendix D.2.All +settings are the same for both algorithms and are described in Appendix D.2. +The PPG achieves +10 + +PaRIS + N = 500 +N=10 +N=25 +N=50 +N=100 +5641 +5642 +5643 +5644 +5645 +5646 +PaRIS + N = 500 +N=10 +N=25 +N=50 +N=100 +5641 +5642 +5643 +5644 +5645 +5646 +Figure 1: PARIS and PPG outputs for the LGSSM for C = 500, yellow boxes correspond to PPG outputs +produced using k ∈ {50, 20, 10, 5} iterations and N ∈ {C/50, C/20, C/10, C/5} particles. The image +on the left corresponds to taking k0 = k/2 and the one on the right to k0 = k/4. +consistently a smaller distance to θmle. Figure 2 displays, for each estimator and configuration, the +evolution of the distance to the MLE estimator as a function of the iteration index. +100 +101 +102 +103 +104 +100 +3 × 10 +1 +4 × 10 +1 +6 × 10 +1 +2 × 100 +PGAS(N=32, k=64) +PGAS(N=64, k=32) +PGAS(N=128, k=16) +PGAS(N=256, k=8) +PPG(N=32, k=64) +PPG(N=64, k=32) +PPG(N=128, k=16) +PPG(N=256, k=8) +Figure 2: Distance to the MLE estimator as a function of the iteration step for PGAS and PPG with +different parameters while keeping the particle budget fixed for LGSSM for 25 different seeds. +CRNN. +We consider now the problem of inference in a non-linear HMM and in particular the chaotic +recurrent neural network introduced by [Zhao et al., 2021]. We use the same setting as in the original +paper. The state and observation equations are +Xm+1 = Xm + τ −1∆ (−Xm + γW tanh(Xm)) + ϵm+1, +Ym = BXm + ζm, +m ∈ N, +where {ϵm}m∈N∗ is a sequence of 20-dimensional independent multivariate Gaussian random variables +with zero mean and covariance 0.01I and {ζm}m∈N is a sequence of independent random variables +11 + +Table 1: Distance to θMLE for each configuration in the LGSSM case. +Algorithm +N +k0 +k +Dmle +PGAS +32 +32 +64 +0.793 ± 0.048 +PGAS +64 +16 +32 +0.751 ± 0.052 +PGAS +128 +8 +16 +0.633 ± 0.054 +PGAS +256 +4 +8 +0.580 ± 0.049 +PPG +32 +32 +64 +0.358 ± 0.038 +PPG +64 +16 +32 +0.373 ± 0.031 +PPG +128 +8 +16 +0.355 ± 0.043 +PPG +256 +4 +8 +0.351 ± 0.042 +Table 2: Per configuration negative loglikelihood for the CRNN model. +Algorithm +N +k0 +k +NLL +PGAS +32 +16 +32 +31364.932 ± 173.708 +PGAS +64 +8 +16 +31083.408 ± 380.527 +PGAS +128 +4 +8 +30264.836 ± 265.880 +PPG +32 +16 +32 +22291.971 ± 47.683 +PPG +64 +8 +16 +22314.537 ± 25.028 +PPG +128 +4 +8 +22353.416 ± 39.443 +where each component is distributed independently according to a Student’s t-distribution with scale +0.1 and 2 degrees of freedom. +In this case, the natural metric used to evaluate the different estimators is the negative log likelihood +(NLL). We use the unbiased estimator of the likelihood given by the mean of the log weights produced +by a particle filter [Douc et al., 2014, Section 12.1] using N = 104 particles. Table 2 shows the results +obtained for 25 different replications for several different configurations of PPG and PGAS, while keeping +total budget of particles fixed. Further numerical details are given in Appendix D.2. We observe that +PPG achieves the a considerably lower NLL than PGAS in all configurations. +6. +Conclusion +We have presented a new algorithm, referred to as PPG as well as bounds on its bias and MSE in +Theorem 1. We then propose a way of using PPG in a learning framework and derive a non-asymptotic +bound over the gradient of the updates when doing score ascent with the PPG with explicit dependence +on the bias and MSE of the estimator. We provide numerical simulations to support our claims, and we +show that our algorithm outperforms the current competitors in the two different examples analysed. +12 + +Contents +1 +Introduction +1 +2 +Background +3 +2.1 +Hidden Markov models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +2.2 +Particle filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +2.3 +Backward smoothing and the PARIS algorithm +. . . . . . . . . . . . . . . . . . . . . . . +4 +3 +PARIS particle Gibbs +5 +3.1 +Particle Gibbs methods +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +3.2 +The PPG algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +4 +Parameter learning with PPG +8 +5 +Numerics +9 +5.1 +PPG +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +5.2 +Score ascent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +6 +Conclusion +12 +A PPG +14 +A.1 Many-body Feynman–Kac models +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +A.2 Backward interpretation of Feynman–Kac path flows . . . . . . . . . . . . . . . . . . . . +15 +A.3 Conditional dual processes and particle Gibbs . . . . . . . . . . . . . . . . . . . . . . . . +16 +A.4 The PARIS algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +A.5 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +A.6 Proofs of intermediate results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +A.6.1 +Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +A.6.2 +Proof of Theorem 3 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +A.6.3 +Proof of Theorem 5 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +A.6.4 +Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +A.6.5 +Proof of Theorem 7 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +31 +A.6.6 +Proof of Proposition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +34 +A.6.7 +Proof of Proposition 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +35 +B Learning with PPG +36 +B.1 +Non-asymptotic bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +B.2 +Application to Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +38 +B.2.1 +Verification of the assumptions of Theorem 8 . . . . . . . . . . . . . . . . . . . . +38 +B.2.2 +Proof of Theorem 2 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +42 +B.3 +Conditions on the model to verify A 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . +42 +C Lipschitz properties +44 +C.1 Lipschitz continuity of Pθ, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +44 +C.1.1 +θ �→ Cm,θ is Lipschitz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +46 +C.1.2 +θ �→ Bt,θ(x0:t, ·) is Lipschitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +48 +C.1.3 +θ �→ +� +St,θ(x0:t, dbt)µ(bt)(id) is Lipschitz . . . . . . . . . . . . . . . . . . . . . . +49 +C.2 Lipschitz properties of Markov Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +D Additional numerical results +53 +D.1 PPG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +53 +D.2 Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +53 +13 + +A. +PPG +In this section, we develop the theoretical framework necessary to establish Theorem 1. We recall +the notions of Feynman–Kac models, many-body Feynman–Kac models, backward interpretations, and +conditional dual processes. Our presentation follows closely [Del Moral et al., 2016] but with a different +and hopefully more transparent definition of the many-body extensions. We restate (in Theorem 3 +below) a duality formula of [Del Moral et al., 2016] relating these concepts. This formula provides a +foundation for the particle Gibbs sampler described in Algorithm 2. +Notations. +Let (Z, Z) be a measurable space and L another possibly unnormalised transition +kernel on Y × Z. Define, with K as above, +KL : X × Z ∋ (x, A) �→ +� +L(y, A) K(x, dy) +and +K � L : X × (Y � Z) ∋ (x, A) �→ +�� +1A(y, z) K(x, dy) L(y, dz), +whenever these are well defined. This also defines the � products of a kernel K on X×Y and a measure +ν on X as well as of a kernel L on Y × X and a measure µ on Y as the measures +ν � K : X � Y ∋ A �→ +�� +1A(x, y) K(x, dy) ν(dx), +L � µ : X � Y ∋ A �→ +�� +1A(x, y) L(y, dx) µ(dy). +A.1 +Many-body Feynman–Kac models +In the following, we assume that all random variables are defined on a common probability space +(Ω, F, P). The distribution flow {ηm}m∈N defined in eq. (2.4) is intractable in general, but can be +approximated by random samples ξm = {ξi +m}N +i=1, m ∈ N, referred to as particles, where N ∈ N∗ is a +fixed Monte Carlo sample size and each particle ξi +m is an Xm-valued random variable. Such particle +approximation is based on the recursion ηm+1 = Φm(ηm), m ∈ N, where Φm denotes the mapping +Φm : M1(Xm) ∋ η �→ ηQm +ηgm +(A.25) +taking on values in M1(Xm+1). In order to describe recursively the evolution of the particle popu- +lation, let m ∈ N and assume that the particles ξm form a consistent approximation of ηm in the +sense that µ(ξm)h, where µ(ξm) := N −1 �N +i=1 δξim, with δx denotes the Dirac measure located at +x, is the occupation measure formed by ξm, which serves as a proxy for ηmh for all ηm-integrable +test functions h. Under general conditions, µ(ξm)h converges in probability to ηm with N → ∞; +see [Del Moral, 2004, Chopin and Papaspiliopoulos, 2020] and references therein. Then, in order to +generate an updated particle sample approximating ηm+1, new particles ξm+1 = {ξi +m+1}N +i=1 are drawn +conditionally independently given ξm according to +ξi +m+1 ∼ Φm(µ(ξm)) = +N +� +ℓ=1 +gm(ξℓ +m) +�N +ℓ′=1 gm(ξℓ′ +m) +Mm(ξℓ +m, ·), +i ∈ �1, N�. +Since this process of particle updating involves sampling from the mixture distribution Φm(µ(ξm)), +it can be naturally decomposed into two substeps: selection and mutation. The selection step con- +sists of randomly choosing the ℓ-th mixture stratum with probability gm(ξℓ +m)/ �N +ℓ′=1 gm(ξℓ′ +m) and the +mutation step consists of drawing a new particle ξi +m+1 from the selected stratum Mm(ξℓ +m, ·). +In +14 + +[Del Moral et al., 2016], the term many-body Feynman–Kac models is related to the law of process +{ξm}m∈N. For all m ∈ N, let Xm := XN +m and X m := X �N +m +; then {ξm}m∈N is an inhomogeneous +Markov chain on {Xm}m∈N with transition kernels +M m : Xm × X m+1 ∋ (xm, A) �→ Φm(µ(xm))�N(A) +and initial distribution η0 = η�N +0 +. Now, denote X0:n := �n +m=0 Xm and X 0:n := �n +m=0 X m. In the +following, we use a bold symbol to stress that a quantity is related to the many-body process. The +many-body Feynman–Kac path model refers to the flows {γm}m∈N and {ηm}m∈N of the unnormalised +and normalised, respectively, probability distributions on {X 0:m}m∈N generated by (2.4) and (2.3) for +the Markov kernels {M m}m∈N, the initial distribution η0, the potential functions +gm : Xm ∋ xm �→ µ(xm)gm = 1 +N +N +� +i=1 +gm(xi +m), +m ∈ N, +and the corresponding unnormalised transition kernels +Qm : Xm × X m+1 ∋ (xm, A) �→ gm(xm)M m(xm, A), +m ∈ N. +A.2 +Backward interpretation of Feynman–Kac path flows +Suppose that each kernel Qn, n ∈ N, defined in (2.2), has a transition density qn with respect to some +dominating measure λn+1 ∈ M(Xn+1). Then for n ∈ N and η ∈ M1(Xn) we may define the backward +kernel +←− +Q n,η : Xn+1 × Xn ∋ (xn+1, A) �→ +� +1A(xn)qn(xn, xn+1) η(dxn) +� +qn(x′n, xn+1) η(dx′n) +. +(A.26) +Now, denoting, for n ∈ N∗, +Bn : Xn × X0:n−1 ∋ (xn, A) �→ +� +· · · +� +1A(x0:n−1) +n−1 +� +m=0 +←− +Q m,ηm(xm+1, dxm), +(A.27) +we may state the following—now classical—backward decomposition of the Feynman–Kac path +measures, a result that plays a pivotal role in this paper. +Proposition 1. For every n ∈ N∗ it holds that γ0:n = γn � Bn and η0:n = ηn � Bn. +Although the decomposition in Proposition 1 is well known (see, e.g., [Del Moral et al., 2010, +Del Moral et al., 2016]), we provide a proof in Appendix A.6.1 for completeness. +Using the back- +ward decomposition, a particle approximation of a given Feynman–Kac path measure η0:n is obtained +by first sampling, in an initial forward pass, particle clouds {ξm}n +m=0 from η0 � M 0 � · · · � M n−1 +and then sampling, in a subsequent backward pass, for instance N conditionally independent paths +{˜ξi +0:n}N +i=1 from Bn(ξ0, . . . , ξn, ·), where +Bn : X0:n × X0:n ∋ (x0:n, A) �→ +� +· · · +� +1A(x0:n) +�n−1 +� +m=0 +←− +Q m,µ(xm)(xm+1, dxm) +� +µ(xn)(dxn) +(A.28) +is a Markov kernel describing the time-reversed dynamics induced by the particle approximations +generated in the forward pass. Here and in the following we use blackboard notation to denote kernels +related to many-body path spaces. Finally, µ({˜ξi +0:n}N +i=1)h is returned as an estimator of η0:nh for +any η0:n-integrable test function h. This algorithm is in the literature referred to as the forward– +filtering backward–simulation (FFBSi) algorithm and was introduced in [Godsill et al., 2004]; see also +[Capp´e et al., 2007, Douc et al., 2011]. More precisely, given the forward particles {ξm}n +m=0, each path +15 + +˜ξi +0:n is generated by first drawing ˜ξi +n uniformly among the particles ξn in the last generation and then +drawing, recursively, +˜ξi +m ∼ ←− +Q m,µ(ξm)(˜ξi +m+1, ·) = +N +� +j=1 +qm(ξj +m, ˜ξi +m+1) +�N +ℓ=1 qm(ξℓm, ˜ξi +m+1) +δξj +m(·), +(A.29) +i.e., given ˜ξi +m+1, ˜ξi +m is picked at random among the ξm according to weights proportional to {qm(ξj +m, ˜ξi +m+1)}N +j=1. +Note that in this basic formulation of the FFBSi algorithm, each backward-sampling operation (A.29) +requires the computation of the normalising constant �N +ℓ=1 qm(ξℓ +m, ˜ξi +m+1), which implies an overall +quadratic complexity of the algorithm. Still, this heavy computational burden can eased by means of +an effective accept–reject technique discussed in Appendix A.4. +A.3 +Conditional dual processes and particle Gibbs +The dual process associated with a given Feynman–Kac model (2.4–2.3) and a given trajectory {zn}n∈N, +where zn ∈ Xn for every n ∈ N, is defined as the canonical Markov chain with kernels +M n⟨zn+1⟩ : Xn × X n+1 ∋ (xn, A) �→ 1 +N +N−1 +� +i=0 +� +Φn(µ(xn))�i � δzn+1 � Φn(µ(xn))�(N−i−1)� +(A), +(A.30) +for n ∈ N, and initial distribution +η0⟨z0⟩ := 1 +N +N−1 +� +i=0 +� +η�i +0 +� δz0 � η�(N−i−1) +0 +� +. +(A.31) +As clear from (A.30–A.31), given {zn}n∈N, a realisation {ξn}n∈N of the dual process is generated as +follows. At time zero, the process is initialised by inserting z0 at a randomly selected position in the +vector ξ0 while drawing independently the remaining components from η0. Then, given ξn at step n, +zn+1 is inserted at a randomly selected position in ξn+1 while drawing independently the remaining +components from Φn(µ(ξn)). +In order to describe compactly the law of the conditional dual process, we define the Markov kernel +Cn : X0:n × X 0:n ∋ (z0:n, A) �→ η0⟨z0⟩ � M 0⟨z1⟩ � · · · � M n−1⟨zn⟩(A). +The following result elegantly combines the underlying model (2.4–2.3), the many-body Feynman–Kac +model, the backward decomposition, and the conditional dual process. +Theorem 3 ([Del Moral et al., 2016]). For all n ∈ N, +Bn � γ0:n = γ0:n � Cn. +(A.32) +In [Del Moral et al., 2016], each state ξn of the many-body process maps an outcome ω of the +sample space Ω into an unordered set of N elements in Xn. However, we have chosen to let each +ξn take on values in the standard product space XN +n for two reasons: +first, the construction of +[Del Moral et al., 2016] requires sophisticated measure-theoretic arguments to endow such unordered +sets with suitable σ-fields and appropriate measures; second, we see no need to ignore the index order +of the particles as long as the Markovian dynamics (A.30–A.31) of the conditional dual process is sym- +metrised over the particle cloud. Therefore, in Appendix A.6.2, we include our own proof of duality +(A.32) for completeness. Note that the measure (A.32) on X0:n � X 0:n is unnormalised, but since the +kernels Bn and Cn are both Markovian, normalising the identity with γ0:n(X0:n) = γ0:n(X0:n) yields +immediately +Bn � η0:n = η0:n � Cn. +(A.33) +16 + +Since the two sides of (A.33) provide the full conditionals, it is natural to choose a data-augmentation +approach and sample the target (A.33) using a two-stage deterministic-scan Gibbs sampler [Andrieu et al., 2010b, +Chopin and Singh, 2015a]. More specifically, assume that we have generated a state (ξ0:n[ℓ], ζ0:n[ℓ]) +comprising a dual process with associated path on the basis of ℓ ∈ N iterations of the sampler; +then the next state (ξ0:n[ℓ + 1], ζ0:n[ℓ + 1]) is generated in a Markovian fashion by sampling first +ξ0:n[ℓ + 1] ∼ Cn(ζ0:n[ℓ], ·) and then sampling ζ0:n[ℓ + 1] ∼ Bn(ξ0:n[ℓ + 1], ·). After arbitrary initiali- +sation (and the discard of possible burn-in iterations), this procedure produces a Markov trajectory +{(ξ0:n[ℓ], ζ0:n[ℓ])}ℓ∈N, and under weak additional technical conditions this Markov chain admits (A.33) +as its unique invariant distribution. In such a case, the Markov chain is ergodic [Douc et al., 2018, +Chapter 5], and the marginal distribution of the conditioning path ζ0:n[ℓ] converges to the target +distribution η0:n. Therefore, for every h ∈ F(X0:n), +lim +L→∞ +1 +L +L +� +ℓ=1 +h(ζ0:n[ℓ]) = η0:nh, +P-a.s. +A.4 +The PARIS algorithm +In the following, we assume that we are given a sequence {hn}n∈N of additive state functionals as in +(2.6). This problem is particularly relevant in the context of maximum-likelihood-based parameter +estimation in general state-space models, e.g., when computing the score-function, i.e. the gradient +of the log-likelihood function, via the Fisher identity or when computing the intermediate quantity +of the Expectation Maximization (EM) algorithm, in which case η0:n and hn correspond to the joint +state posterior and an element of some sufficient statistic, respectively; see [Capp´e and Moulines, 2005, +Douc et al., 2011, Del Moral et al., 2010, Poyiadjis et al., 2011, Olsson and Westerborn, 2017] and the +references therein. Interestingly, as noted in [Capp´e, 2011, Del Moral et al., 2010], the backward de- +composition allows, when applied to additive state functionals, a forward recursion for the expecta- +tions {η0:nhn}n∈N. +More specifically, using the forward decomposition hn+1(x0:n+1) = hn(x0:n) + +˜hn(xn, xn+1) and the backward kernel Bn+1 defined in (A.27), we may write, for xn+1 ∈ Xn+1, +Bn+1hn+1(xn+1) = +� ←− +Q n,ηn(xn+1, dxn) +� � +hn(x0:n) + ˜hn(xn, xn+1) +� +Bn(xn, dx0:n−1) += ←− +Q n,ηn(Bnhn + ˜hn)(xn+1), +(A.34) +which by Proposition 1 implies that +η0:n+1hn+1 = ηn+1 +←− +Q n,ηn(Bnhn + ˜hn). +(A.35) +Since the marginal flow {ηn}n∈N can be expressed recursively via the mappings {Φn}n∈N, (A.35) +provides, in principle, a basis for online computation of {η0:nhn}n∈N. To handle the fact that the +marginals are generally intractable we may, following [Del Moral et al., 2010], plug particle approx- +imations µ(ξn+1) and ←− +Q n,µ(ξn) (see (A.29)) of ηn+1 and ←− +Q n,µ(ηn), respectively, into the recursion +(A.35). More precisely, we proceed recursively and assume that at time n we have at hand a sample +{(ξi +n, βi +n)}N +i=1 of particles with associated statistics, where each statistic βi +n serves as an approxima- +tion of Bnhn(ξi +n); then evolving the particle cloud according to ξn+1 ∼ M n(ξn, ·) and updating the +statistics using (A.34), with ←− +Q n,ηn replaced by ←− +Q n,µ(ξn), yields the particle-wise recursion +βi +n+1 = +N +� +ℓ=1 +qn(ξℓ +n, ξi +n+1) +�N +ℓ′=1 qn(ξℓ′ +n , ξi +n+1) +� +βℓ +n + ˜hn(ξℓ +n, ξi +n+1) +� +, +i ∈ �1, N�, +(A.36) +and, finally, the estimator +µ(βn)(id) = 1 +N +N +� +i=1 +βi +n +(A.37) +17 + +of η0:nhn, where βn := (β1 +n, . . . , βN +n ), i ∈ �1, N�. The procedure is initialised by simply letting βi +0 = 0 +for all i ∈ �1, N�. Note that (A.37) provides a particle interpretation of the backward decomposition in +Proposition 1. This algorithm is a special case of the forward–filtering backward–smoothing (FFBSm) +algorithm (see [Andrieu and Doucet, 2003, Godsill et al., 2004, Douc et al., 2011, S¨arkk¨a, 2013]) for +additive functionals satisfying (2.6). It allows for online processing of the sequence {η0:nhn}n∈N, but +has also the appealing property that only the current particles ξn and statistics βn need to be stored. +However, since each update (A.36) requires the summation of N terms, the scheme has an overall +quadratic complexity in the number of particles, leading to a computational bottleneck in applications +to complex models that require large particle sample sizes N. +In order to detour the computational burden of this forward-only implementation of FFBSm, the +PARIS algorithm [Olsson and Westerborn, 2017] updates the statistics βn by replacing each sum (A.36) +by a Monte Carlo estimate +βi +n+1 = 1 +M +M +� +j=1 +� +˜βi,j +n + ˜hn(˜ξi,j +n , ξi +n+1) +� +, +i ∈ �1, N�, +(A.38) +where {(˜ξi,j +n , ˜βi,j +n )}M +j=1 are drawn randomly among {(ξi +n, βi +n)}N +i=1 with replacement, by assigning (˜ξi,j +n , ˜βi,j +n ) the +value of (ξℓ +n, βℓ +n) with probability qn(ξℓ +n, ξi +n+1)/ �N +ℓ′=1 qn(ξℓ′ +n , ξi +n+1), and the Monte Carlo sample size +M ∈ N∗ is supposed to be much smaller than N (say, less than 5). Formally, +{(˜ξi,j +n , ˜βi,j +n )}M +j=1 ∼ +� N +� +ℓ=1 +qn(ξℓ +n, ξi +n+1) +�N +ℓ′=1 qn(ξℓ′ +n , ξi +n+1) +δ(ξℓn,βℓn) +��M +, +i ∈ �1, N�. +The resulting procedure, summarised in Algorithm 1, allows for online processing with constant mem- +ory requirements, since it only needs to store the current particle cloud and the estimated auxiliary +statistics at each iteration. Moreover, in the case where the Markov transition densities of the model +can be uniformly bounded, i.e. when there exists, for every n ∈ N, an upper bound ¯σn > 0 such +that for all (xn, xn+1) ∈ Xn × Xn+1, mn(xn, xn+1) ≤ ¯σn (a weak assumption satisfied for most +models of interest), a sample (˜ξi,j +n , βi,j +n ) can be generated by drawing, with replacement and un- +til acceptance, candidates (˜ξi,∗ +n , ˜βi,∗ +n ) from {(ξi +n, βi +n)}N +i=1 according to the normalised particle weights +{gn(ξℓ +n)/ � +ℓ′ gn(ξℓ′ +n )}N +ℓ=1, obtained as a by-product in the generation of ξn+1, and accepting the same +with probability mn(˜ξi,∗ +n , ξi +n+1)/¯σn. As this sampling procedure bypasses completely the calculation +of the normalising constant �N +ℓ′=1 qn(ξℓ′ +n , ξi +n+1) of the targeted categorical distribution, it yields an +overall O(MN) complexity of the algorithm as a whole; see [Douc et al., 2011] for details. +Increasing M improves the accuracy of the algorithm at the cost of additional computational +complexity. As shown in [Olsson and Westerborn, 2017], there is a qualitative difference between the +cases M = 1 and M ≥ 2, and it turns out that the latter is required to keep PARIS numerically stable. +More precisely, in the latter case, it can be shown that the PARIS estimator µ(βn) satisfies, as N +tends to infinity while M is held fixed, a central limit theorem (CLT) at the rate +√ +N and with an +n-normalised asymptotic variance of order O(1 − 1/(M − 1)). As clear from this bound, using a large +M only yields a waste of computational work, and setting M to 2 or 3 typically works well in practice. +We now introduce the Parisian particle Gibbs (PPG) algorithm. For all t ∈ N∗, let Yt := X0:t×R and +Yt := X0:t � B(R). Moreover, let Y0 := X0 × {0} and Y0 := X0 � {{0}, ∅}. An element of Yt will always +be denoted by yt = (x0:t|t, bt). The Parisian particle Gibbs sampler comprises, as a key ingredient, a +conditional PARIS step, which updates recursively a set of Yt-valued random variables υi +t := (ξi +0:t|t, βi +t), +i ∈ �1, N�. Let (υt)t∈N denote the corresponding many-body process, each υt := {(ξi +0:t|t, βi +t)}N +i=1 taking +on values in the space Yt := YN +t , which we furnish with a σ-field Yt := Y�N +t +. The space Y0 and the +corresponding σ-field Y0 are defined accordingly. For every t ∈ N, we write ξ0:t|t for the collection +{ξi +0:t|t}N +i=1 of paths in υt, and ξt|t for the collection {ξi +t|t}N +i=1 of end points of the same. +In the following, we let t ∈ N be a fixed time horizon, and describe in detail how the PPG approx- +imates η0:tht iteratively. In short, at each iteration ℓ, the PPG produces, given an input conditional +18 + +path ζ0:t[ℓ], a many-body system υt[ℓ + 1] by means of a series of conditional PARIS operations; then, +an updated path ζ0:t[ℓ + 1], serving as input at the next iteration, is generated by picking one of the +paths ξ0:t|t[ℓ + 1] in υt[ℓ + 1] at random. At each iteration, the produced statistics βt in υt provides +an approximation of η0:tht according to (A.37). +More precisely, given the path ζ0:t[ℓ], the conditional PARIS operations are executed as follows. In +the initial step, ξ0|0[ℓ + 1] are drawn from η0⟨ζ0[ℓ]⟩ defined in (A.31) and υi +0[ℓ + 1] ← (ξi +0|0[ℓ + 1], 0) +for all i ∈ �1, N�; then, recursively for m ∈ �0, t�, assuming access to υm[ℓ + 1], +(1) we generate an updated particle cloud ξm+1[ℓ + 1] ∼ M m⟨ζm+1[ℓ]⟩(ξm|m[ℓ + 1], ·), +(2) we pick at random, for each i ∈ �1, N�, an ancestor path with associated statistics (˜ξi,1 +0:m[ℓ + +1], ˜βi,1 +m [ℓ + 1]) among υm[ℓ + 1] by drawing +(˜ξi,1 +0:m[ℓ + 1], ˜βi,1 +m [ℓ + 1]) ∼ +N +� +s=1 +qm(ξs +m|m[ℓ + 1], ξi +m+1[ℓ + 1]) +�N +s′=1 qm(ξs′ +m|m[ℓ + 1], ξi +m+1[ℓ + 1]) +δυsm[ℓ+1], +i ∈ �1, N�, +(3) we draw, with replacement, M −1 ancestor particles and associated statistics {(˜ξi,j +m [ℓ+1], ˜βi,j +m [ℓ+ +1])}M +j=2 at random from {(ξs +m|m[ℓ + 1], βs +m)[ℓ + 1]}N +s=1 according to +{(˜ξi,j +m [ℓ+1], ˜βi,j +m [ℓ+1])}M +j=2 ∼ +� N +� +s=1 +qm(ξs +m|m[ℓ + 1], ξi +m+1[ℓ + 1]) +�N +s′=1 qm(ξs′ +m|m[ℓ + 1], ξi +m+1[ℓ + 1]) +δ(ξs +m|m[ℓ+1],βsm[ℓ+1]) +��(M−1) +, +(4) we set, for all i ∈ �1, N�, ξi +0:m+1|m+1[ℓ + 1] ← (˜ξi,1 +0:m[ℓ + 1], ξi +m+1[ℓ + 1]) and υi +m+1[ℓ + 1] ← +(ξi +0:m+1|m+1[ℓ + 1], βi +m+1[ℓ + 1]), where +βi +m+1[ℓ + 1] ← M −1 +M +� +j=1 +� +˜βi,j +m [ℓ + 1] + ˜hm(˜ξi,j +m [ℓ + 1], ξi +m+1[ℓ + 1]) +� +. +This conditional PARIS procedure is summarised in Algorithm 1. +Once the set of trajectories and associated statistics υt[ℓ + 1] is formed by means of n recursive +conditional PARIS updates, an updated path ζ0:t[ℓ + 1] is drawn from µ(ξ0:t|t[ℓ + 1]). A full sweep of +the PPG is summarised in Algorithm 2. +The following Markov kernels will play an instrumental role in the following. For a given path +{zm}m∈N, the conditional PARIS update in Algorithm 1 defines an inhomogeneous Markov chain on +the spaces {(Ym, Ym)}m∈N with kernels +Ym × Ym+1 ∋ (ym, A) �→ +� +M m⟨zm+1⟩(xm|m, dxm+1) Sm(ym, xm+1, A), +m ∈ N, +where +Sm : Ym × Xm+1 × Ym+1 ∋ (ym, xm+1, A) +(A.39) +�→ +� +· · · +� +1A +� +� +�� +(˜xi,1 +0:m, xi +m+1), 1 +M +M +� +j=1 +� +˜bi,j +m + ˜hm(˜xi,j +m , xi +m+1) +� ��N +i=1 +� +� +× +N +� +i=1 +� N +� +ℓ=1 +qm(xℓ +m|m, xi +m+1) +�N +ℓ′=1 qm(xℓ′ +m|m, xi +m+1) +δyℓm(d(˜xi,1 +0:m,˜bi,1 +m )) +× +� N +� +ℓ=1 +qm(xℓ +m|m, xi +m+1) +�N +ℓ′=1 qm(xℓ′ +m|m, xi +m+1) +δ(xℓ +m|m,bℓm) +��(M−1) +(d(˜xi,2 +m ,˜bi,2 +m , . . . , ˜xi,M +m ,˜bi,M +m )) +� +� . +19 + +In addition, we introduce the joint law +St : X0:t × Yt ∋ (x0:t, A) �→ +� +· · · +� +1A(yt) S0(Jx0, x1, dy1) +t−1 +� +m=1 +Sm(ym, xm+1, dym+1), +(A.40) +where we have defined J := IdN �(0, 1)⊺. +The kernel St can be viewed as a superincumbent sampling kernel describing the distribution of +the output υt generated by a sequence of PARIS iterates when the many-body process {ξm}t +m=0 +associated with the underlying SMC algorithm is given. This allows us to describe alternatively the +PPG as follows: given ζ0:t[ℓ], draw ξ0:t[ℓ + 1] ∼ Ct(ζ0:t[ℓ], ·); then, draw υt[ℓ + 1] ∼ St(ξ0:t[ℓ + 1], ·) +and pick a trajectory ζ0:t[ℓ + 1] from ξ0:t|t[ℓ + 1] at random. The following proposition, which will be +instrumental in the coming developments, establishes that the conditional distribution of ζ0:t[ℓ + 1] +given ξ0:t[ℓ + 1] coincides, as expected, with the particle-induced backward dynamics Bt. +Proposition 2. For all t ∈ N∗, N ∈ N∗, x0:t ∈ X0:t, and h ∈ F(X0:t), +� +St(x0:t, dyt) µ(x0:t|t)h = Bth(x0:t). +Finally, we define the Markov kernel induced by the PPG as well as the extended probability distri- +bution targeted by the same. For this purpose, we introduce the extended measurable space (Et, Et) +with +Et := Yt × X0:t, +Et := Yt � X0:t. +The PPG described in Algorithm 2 defines a Markov chain on (Et, Et) with Markov transition kernel +Kt : Et × Et ∋ (yt, z0:t, A) �→ +��� +1A(˜yt, ˜z0:t) Ct(z0:t, d˜x0:t) St(˜x0:t, d˜yt) µ(˜x0:t|t)(d˜z0:t). +(A.41) +Note that the values of Kt defined above do not depend on yt, but only on (z0:t, A). For any given +initial distribution ξ ∈ M1(X0:t), let Pξ be the distribution of the canonical Markov chain induced by +the kernel Kt and the initial distribution ξ. In the special case where ξ = δz0:t for some given path +z0:t ∈ X0:t, we use the short-hand notation Pδz0:t = Pz0:t. In addition, denote by +Kt : X0:t × X0:t ∋ (z0:t, A) �→ +��� +1A(˜z0:t) Ct(z0:t, d˜x0:t) St(˜x0:t, d˜yt) µ(˜x0:t|t)(d˜z0:t) +(A.42) +the path-marginalised version of Kt. By Proposition 2 it holds that Kt = CtBt, which shows that Kt +coincides with the Markov transition kernel of the backward-sampling-based particle Gibbs sampler +discussed in Appendix A.3. It is also possible to specify the invariant distribution of Kt. +Proposition 3. For all t ∈ N∗, it holds that +η0:tCtStKt = η0:tCtSt . +(A.43) +Proof. Let f ∈ M(E�(k−k0) +t +). +� +f(˜yt, ˜z0:t)η0:t(dz0:t)CtSt(z0:t, d(yt, z′ +0:t))Kt(z′ +0:t, yt, d(˜yt, ˜z0:t)) += +� +f(˜yt, ˜z0:t)η0:t(dz0:t)CtSt(z0:t, d(yt, z′ +0:t))CtSt(z′ +0:t, d(˜yt, ˜z0:t)) += +� +f(˜yt, ˜z0:t)η0:t(dz0:t)Kt(z0:t, dz′ +0:t)CtSt(z′ +0:t, d(˜yt, ˜z0:t)) += +� +f(˜yt, ˜z0:t)η0:t(dz′ +0:t)CtSt(z′ +0:t, d(˜yt, ˜z0:t)) . +20 + +Finally, in order prepare for the statement of our theoretical results on the PPG we need to introduce +the following Feynman–Kac path model with a frozen path. More precisely, for a given path z0:t ∈ X0:t, +define, for every m ∈ �0, t − 1�, the unnormalised kernel +Qm⟨zm+1⟩ : Xm × Xm+1 ∋ (xm, A) �→ +� +1 − 1 +N +� +Qm(xm, A) + 1 +N gm(xm) δzm+1(A) +and the initial distribution η0⟨z0⟩ : X0 ∋ A �→ (1 − 1/N)η0(A) + δz0(A)/N. Given these quantities, +define, for m ∈ �0, t�, γm⟨z0:m⟩ := η0⟨z0⟩Q0⟨z1⟩ · · · Qm−1⟨zm⟩ along with the normalised counterpart +ηm⟨z0:m⟩ := γm⟨z0:m⟩/γm⟨z0:m⟩1X0:m. Finally, we introduce, for m ∈ �0, t�, the kernels +Bm⟨z0:m−1⟩ : Xm × X0:m−1 ∋ (xm, A) �→ +� +· · · +� +1A(x0:m−1) +t−1 +� +m=0 +←− +Q m,ηm⟨z0:m⟩(xm+1, dxm), +as well as the path model η0:m⟨z0:m⟩ := Bm⟨z0:m−1⟩ � ηm⟨z0:m⟩. +A.5 +Proof of Theorem 1 +We start by establishing bias, MSE and covariance bounds for a fixed iteration of the PPG estimator. +Theorem 4. Assume A 3.1. Then for every t ∈ N there exist cbias +t +, cmse +t +, and ccov +t +in R∗ ++ such that +for every M ∈ N∗, ξ ∈ M1(X0:t), ℓ ∈ N∗, s ∈ N∗, and N ∈ N∗ such that N > Nt, +|Eξ [µ(βt[ℓ])(id)] − η0:tht| ≤ cbias +t +� t−1 +� +m=0 +∥˜hm∥∞ +� +N −1κℓ +N,t, +(A.44) +Eξ +� +(µ(βt[ℓ])(id) − η0:tht)2� +≤ cmse +t +� t−1 +� +m=0 +∥˜hm∥∞ +�2 +N −1, +(A.45) +|Eξ [(µ(βt[ℓ])(id) − η0:tht) (µ(βt[ℓ + s])(id) − η0:tht)]| ≤ ccov +t +� t−1 +� +m=0 +∥˜hm∥∞ +�2 +N −3/2κs +N,t. +(A.46) +The constants cbias +t +, cmse +t +, and ccov +t +are explicitly given in the proof. Since the focus of this paper is on +the dependence on N and the index ℓ, we have made no attempt to optimise the dependence of these +constants on t in our proofs; still, we believe that it is possible to prove, under the stated assumptions, +that this dependence is linear. The proof of the bound in Theorem 4 is based on four key ingredients. +The first is the following unbiasedness property of the PARIS under the many-body Feynman–Kac path +model. +Theorem 5. For every t ∈ N, N ∈ N∗, and ℓ ∈ N∗, +Eη0:t [µ(βt[ℓ])(id)] = +� +η0:tCtSt(dbt) µ(bt)(id) = +� +η0:tSt(dbt) µ(bt)(id) = η0:tht. +The proof of Theorem 5 is postponed to Appendix A.6.3. The second ingredient of the proof of +Theorem 4 is the uniform geometric ergodicity of the particle Gibbs with backward sampling established +in [Del Moral and Jasra, 2018]. +Theorem 6. Assume A 3.1. Then, for every t ∈ N, (µ, ν) ∈ M1(X0:t)2, ℓ ∈ N∗, and N ∈ N∗ such +that N > 1 + 5ρ2 +tt/2, ∥µKℓ +t − νKℓ +t ∥TV ≤ κℓ +N,t, where κN,t is defined in (3.14). +As a third ingredient, we require the following uniform exponential concentration inequality of the +conditional PARIS with respect to the frozen-path Feynman–Kac model defined in the previous section. +21 + +Theorem 7. For every t ∈ N there exist ct > 0 and dt > 0 such that for every M ∈ N∗, z0:t ∈ X0:t, +N ∈ N∗, and ε > 0, +� +CtSt(z0:t, dbt)1 {|µ(bt)(id) − η0:t⟨z0:t⟩ht| ≥ ε} ≤ ct exp +� +− +dtNε2 +2(�t−1 +m=0 ∥˜hm∥∞)2 +� +. +Theorem 7, whose proof is postponed to Appendix A.6.5, implies, in turn, the following conditional +variance bound. +Proposition 4. For every t ∈ N, M ∈ N∗, z0:t ∈ X0:t, and N ∈ N∗, +� +CtSt(z0:t, dbt) |µ(bt)(id) − η0:t⟨z0:t⟩ht|2 ≤ ct +dt +� t−1 +� +m=0 +∥˜hm∥∞ +�2 +N −1. +Using Proposition 4, we deduce, in turn, the following bias bound, whose proof is postponed to +Appendix A.6.7. +Proposition 5. For every t ∈ N there exists ¯cbias +t +> 0 such that for every M ∈ N∗, z0:t ∈ X0:t, and +N ∈ N∗, +���� +� +CtSt(z0:t, dbt) µ(bt)(id) − η0:t⟨z0:t⟩ht +���� ≤ ¯cbias +t +N −1 +� t−1 +� +m=0 +∥˜hm∥∞ +� +. +A fourth and last ingredient in the proof of Theorem 4 is the following bound on the discrepancy +between additive expectations under the original and frozen-path Feynman–Kac models. This bound +is established using novel results in [Gloaguen et al., 2022]. More precisely, since for every m ∈ N, +(x, z) ∈ X2 +m, N ∈ N∗, and h ∈ F(Xm+1), using A 3.1, +|Qm⟨z⟩h(x) − Qmh(x)| ≤ 1 +N ∥gm∥∞∥h∥∞ ≤ 1 +N ¯τm∥h∥∞, +applying [Gloaguen et al., 2022, Theorem 4.3] yields the following. +Proposition 6. Assume A 3.1. Then there exists c > 0 such that for every t ∈ N, N ∈ N, and +z0:t ∈ X0:t, +|η0:t⟨z0:t⟩ht − η0:tht| ≤ cN −1 +t−1 +� +m=0 +∥˜hm∥∞. +Note that assuming, in addition, that supt∈N ∥˜ht∥∞ < ∞ yields an O(n/N) bound in Proposition 6. +Finally, by combining these ingredients we are now ready to present a proof of Theorem 4. +Proof of Theorem 4. Write, using the tower property, +Eξ [µ(βt [ℓ])(id)] = Eξ +� +Eζ0:t[ℓ] [µ(βt [0])(id)] +� += +� +ξKℓ +t CtSt(dbt) µ(bt)(id). +Thus, by the unbiasedness property in Theorem 5, +|Eξ [µ(βt [ℓ])(id)] − η0:tht| = +���� +� +ξKℓ +t CtSt(dbt) µ(bt)(id) − +� +η0:tCtSt(dbt) µ(bt)(id) +���� +≤ +��ξKℓ +t − η0:t +�� +TV osc +�� +CtSt(·, dbt) µ(bt)(id) +� +, +22 + +where, by Theorem 6, ∥ξKℓ +t − η0:t∥TV ≤ κℓ +N,t. Moreover, to derive an upper bound on the oscillation, +we consider the decomposition +osc +�� +CtSt(·, dbt) µ(bt)(id) +� +≤ 2 +����� +� +CtSt(·, dbt) µ(bt)(id) − η0:t⟨·⟩ht +���� +∞ ++ ∥η0:t⟨·⟩ht − η0:tht∥∞ +� +, +where the two terms on the right-hand side can be bounded using Proposition 6 and Proposition 5, +respectively. This completes the proof of (A.44). We now consider the proof of (A.45). Writing +Eξ +� +(µ(βt[ℓ])(id) − η0:tht)2� += +� +ξKℓ +t (dz0:t)CtSt(z0:t, dbt) (µ(bt)(id) − η0:tht)2 , +we may establish (A.45) using Proposition 4 and Proposition 6. We finally consider (A.46). Using the +Markov property we obtain +Eξ [(µ(βt[ℓ])(id) − η0:tht) (µ(βt[ℓ + s])(id) − η0:tht)] += Eξ +� +(µ(βt[ℓ])(id) − η0:tht) +� +Eζ0:t[ℓ][µ(βt[s])(id)] − η0:tht +�� +, +from which (A.46) follows by (A.44) and (A.45). +We are finally equipped to prove Theorem 1. +Proof of Theorem 1. We first consider the bias, which can be bounded according to +��Eξ[Π(k0,k),N(f)] − η0:tht +�� ≤ (k − k0)−1 +k +� +ℓ=k0+1 +|Eξµ(βt[ℓ])(id) − η0:tht| +≤ (k − k0)−1N −1cbias +t +� t−1 +� +m=0 +∥˜hm∥∞ +� +k +� +ℓ=k0+1 +κℓ +N,t, +from which the bound (3.15) follows immediately. +We turn to the MSE. Using the decomposition +Eξ[(Π(k0,k),N(f) − η0:tht)2] ≤ (k − k0)−2 +� +k +� +ℓ=k0+1 +Eξ[(µ(βt[ℓ])(id) − η0:tht)2] ++ 2 +k +� +ℓ=k0+1 +k +� +j=ℓ+1 +Eξ[(µ(βt[ℓ])(id) − η0:tht)(µ(βt[j])(id) − η0:tht)] +� +� +� , +the MSE bound in Theorem 4 implies that +k +� +ℓ=k0+1 +Eξ[(µ(βt[ℓ])(id) − η0:tht)2] ≤ cmse +t +� t−1 +� +m=0 +∥˜hm∥∞ +�2 +N −1(k − k0). +Moreover, using the covariance bound in Theorem 4, we deduce that +k +� +ℓ=k0+1 +k +� +j=ℓ+1 +Eξ[(µ(βt[ℓ])(id)−η0:tht)(µ(βt[j])(id)−η0:tht)] ≤ ccov +t +� t−1 +� +m=0 +∥˜hm∥∞ +�2 +N −3/2 +� +� +k +� +ℓ=k0+1 +k +� +j=ℓ+1 +κ(j−ℓ) +N,t +� +� . +Thus, the proof is concluded by noting that �k +ℓ=k0+1 +�k +j=ℓ+1 κ(j−ℓ) +N,t +≤ (k − k0)/(1 − κN,t). +23 + +A.6 +Proofs of intermediate results +A.6.1 +Proof of Proposition 1 +Using the identity +η0Q0 · · · Qt−11Xt = +t−1 +� +m=0 +ηmQm1Xm+1 +and the fact that each kernel Qm has a transition density, write, for h ∈ F(X0:t), +η0:th = +� +· · · +� +h(x0:t) η0(dx0) +t−1 +� +m=0 +�ηm[qm(·, xm+1)] λm+1(dxm+1) +ηmQm1Xm+1 +� � qm(xm, xm+1) +ηm[qm(·, xm+1)] +� += +� +· · · +� +h(x0:t) ηt(dxt) +t−1 +� +m=0 +ηm(dxm) qm(xm, xm+1) +ηm[qm(·, xm+1)] +(A.47) += +�←− +Q 0,η0 � · · · � ←− +Q n−1,ηt−1 � ηt +� +h, +which was to be established. +A.6.2 +Proof of Theorem 3 +Lemma 1. For all t ∈ N, xt ∈ Xt, and h ∈ F(X t+1 � Xt+1), +�� +h(xt+1, zt+1) Qt(xt, dxt+1) µ(xt+1)(dzt+1) = +�� +h(xt+1, zt+1) µ(xt)Qt(dzt+1) M t⟨zt+1⟩(xt, dxt+1). +(A.48) +In addition, for all h ∈ F(X 0 � X0), +�� +h(x0, z0) η0(dx0) µ(x0)(dz0) = +�� +h(x0, z0) η0⟨z0⟩(dx0) η0(dz0). +(A.49) +Proof. Since µ(xt) Qt(dzt+1) = gt(xt) Φt(µ(xt))(dzt+1), we may rewrite the right-hand side of (A.48) +according to +�� +h(xt+1, zt+1) µ(xt)Qt(dzt+1) M t⟨zt+1⟩(xt, dxt+1) += gt(xt) 1 +N +N−1 +� +i=0 +�� +h(xt+1, zt+1) Φt(µ(xt))(dzt+1) +× +� +Φt(µ(xt))�i � δzt+1 � Φt(µ(xt))�(N−i−1)� +(dxt+1) += gt(xt) 1 +N +N +� +i=1 +� +· · · +� +h((x1 +t+1, . . . , xi−1 +t+1, zt+1, xi+1 +t+1, . . . , xN +t+1), zt+1) +× Φt(µ(xt))(dzt+1) +� +ℓ̸=i +Φt(µ(xt))(dxℓ +t+1) += gt(xt) 1 +N +N +� +i=1 +� +h(xt+1, xi +t+1) M t(xt, dxt+1). +On the other hand, note that the left-hand side of (A.48) can be expressed as +�� +h(xt+1, zt+1) Qt(xt, dxt+1) µ(xt+1)(dzt+1) = gt(xt) 1 +N +N +� +i=1 +� +h(xt+1, xi +t+1) M t(xt, dxt+1), +which establishes the identity. The identity (A.49) is established along similar lines. +24 + +We establish Theorem 3 by induction; thus, assume that the claim holds true for n and show that +for all h ∈ F(X 0:t+1 � X0:t+1), +�� +h(x0:t+1, z0:t+1) γ0:t+1(dx0:t+1) Bt+1(x0:t+1, dz0:t+1) += +�� +h(x0:t+1, z0:t+1) γ0:t+1(dz0:t+1) Ct+1(z0:t+1, dx0:t+1). +(A.50) +To prove this, we process, using definition (C.85), the left-hand side of (A.50) according to +�� +h(x0:t+1, z0:t+1) γ0:t+1(dx0:t+1) Bt+1(x0:t+1, dz0:t+1) += +�� +γ0:t(dx0:t) Bt(x0:t, dz0:t) +× +�� +¯h(x0:t+1, z0:t+1) Qt(xt, dxt+1) µ(xt+1)(dzt+1), +(A.51) +where we have defined the function +¯h(x0:t+1, z0:t+1) := qt(zt, zt+1)h(x0:t+1, z0:t+1) +µ(xt)[qt(·, zt+1)] +. +Now, applying Lemma 1 to the inner integral and using that +µ(xt)Qt(dzt+1) = µ(xt)[qt(·, zt+1)] λt+1(dzt+1) +yields, for every x0:t and z0:t, +�� +¯h(x0:t+1, z0:t+1) Qt(xt, dxt+1) µ(xt+1)(dzt+1) += +�� +¯h(x0:t+1, z0:t+1) µ(xt)Qt(dzt+1) M t⟨zt+1⟩(xt, dxt+1) += +�� +h(x0:t+1, z0:t+1) Qt(zt, dzt+1) M t⟨zt+1⟩(xt, dxt+1). +Inserting the previous identity into (A.51) and using the induction hypothesis provides +�� +h(x0:t+1, z0:t+1) γ0:t+1(dx0:t+1) Bt+1(x0:t+1, dz0:t+1) += +�� +γ0:t(dz0:t) Ct(z0:t, dx0:t) +× +�� +h(x0:t+1, z0:t+1) Qt(zt, dzt+1) M t⟨zt+1⟩(xt, dxt+1) += +�� +h(x0:t+1, z0:t+1) γ0:t+1(dz0:t+1) Ct+1(z0:t+1, dx0:t+1), +which establishes (A.50). +A.6.3 +Proof of Theorem 5 +First, define, for m ∈ N, +P m : Ym × Ym+1 ∋ (ym, A) �→ +� +M m(xm|m, dxm+1) Sm(ym, xm+1, A). +(A.52) +25 + +For any given initial distribution ψ0 ∈ M1(Y0), let PP +ψ0 be the distribution of the canonical Markov +chain induced by the Markov kernels {P m}m∈N and the initial distribution ψ0. By abuse of notation +we write, for η0 ∈ M1(X 0), PP +η0 instead of PP +ψ0[η0], where we have defined the extension ψ0[η0](A) = +� +1A(Jx0) η0(dx0), A ∈ Y0. We preface the proof of Theorem 5 by some technical lemmas and a +proposition. +Lemma 2. For all t ∈ N and (ft+1, ˜ft+1) ∈ F(Xt+1)2, +γt+1(ft+1Bt+1ht+1 + ˜ft+1) = γt{Qtft+1Btht + Qt(˜htft+1 + ˜ft+1)}. +Proof. Pick arbitrarily ϕ ∈ F(Xt:t+1) and write, using definition (A.27) and the fact that Qt has a +transition density, +�� +ϕ(xt:t+1) γt(dxt) Qt(xt, dxt+1) += +�� +ϕ(xt:t+1)γt[qt(·, xt+1)] λt+1(dxt+1) γt(dxt)qt(xt, xt+1) +γt[qt(·, xt+1)] += +�� +ϕ(xt:t+1) γt+1(dxt+1) ←− +Q n,ηt(xt+1, dxt). +(A.53) +Now, by (A.34) it holds that +Bt+1ht+1(xt+1) = +� ←− +Q n,ηt(xt+1, dxt) +� +˜ht(xt:t+1) + +� +ht(x0:t) Bt(xt, dx0:t−1) +� +; +therefore, by applying (A.53) with +ϕ(xt:t+1) := ft+1(xt+1) +� +˜ht(xt:t+1) + +� +ht(x0:t) Bt(xt, dx0:t−1) +� +we obtain that +γt+1(ft+1Bt+1ht+1) = +�� +ϕ(xt:t+1) γt+1(dxt+1) ←− +Q n,ηt(xt+1, dxt) += +�� +ϕ(xt:t+1) γt(dxt) Qt(xt, dxt+1) += γt(Qtft+1Btht + Qt˜htft+1). +Now the proof is concluded by noting that since γt+1 = γtQt, γt+1 ˜ft+1 = γtQt ˜ft+1. +Lemma 3. For every t ∈ N∗, ht ∈ F(Yt), and η0 ∈ M1(X 0) it holds that +EP +η0[ht(υt) | ξ0|0, . . . , ξt|t] = Stht(ξ0|0, . . . , ξt|t), +PP +η0-a.s. +Proof. Pick arbitrarily vt ∈ F(X0:t). We show that +EP +η0[vt(ξ0|0, . . . , ξt|t)ht(υt)] = EP +η0[vt(ξ0|0, . . . , ξt|t)Stht(ξ0|0, . . . , ξt|t)], +(A.54) +from which the claim follows. Using the definition (A.52), the left-hand side of the previous identity +26 + +may be rewritten as +� +· · · +� +ψ0[η0](dy0) +t−1 +� +m=0 +P m(ym, dym+1) ht(yt)vt(x0|0, . . . , xt|t) += +� +· · · +� +η0(dx0|0) +t−1 +� +m=0 +M m(xm|m, dxm+1) S0(Jx0|0, x1, dy1) +× +t−1 +� +m=0 +Sm(ym, xm+1, dym+1) ht(yt)vt(x0|0, . . . , xt|t) += +� +· · · +� +η0(dx0) +t−1 +� +m=0 +M m(xm, dxm+1) S0(Jx0, x1, dy1) +× +t−1 +� +m=0 +Sm(ym, xm+1, dym+1) ht(yt)vt(x0, . . . , xt). +Thus, we may conclude the proof by using the definition (A.40) of St together with Fubini’s theorem. +Lemma 4. For every t ∈ N∗ and ht ∈ F(Yt), +Eη0 +�� t−1 +� +m=0 +gm(ξm|m) +� +ht(υt) +� += +� +γ0:tSt(dyt) ht(yt). +Proof. The claim of the lemma is a direct implication of Lemma 3; indeed, by applying the tower +property and the latter we obtain +EP +η0 +�� t−1 +� +m=0 +gm(ξm|m) +� +ht(υt) +� += EP +η0 +�� t−1 +� +m=0 +gm(ξm|m) +� +Stht(ξ0|0, . . . , ξt|t) +� += +� +· · · +� +η0(dx0) +t−1 +� +m=0 +gm(xm) M m(xm, dxm+1) Stht(x0:t) += +� +γ0:tSt(dyt) ht(yt). +Proposition 7. For all t ∈ N∗, (N, M) ∈ (N∗)2, and (ft, ˜ft) ∈ F(Xt)2, +� +γ0:tSt(dyt) +� +1 +N +N +� +i=1 +{bi +tft(xi +t|t) + ˜ft(xi +t|t)} +� += γt(ftBtht + ˜ft). +Proof. Applying Lemma 4 yields +� +γ0:tSt(dyt) +� +1 +N +N +� +i=1 +{bi +tft(xi +t|t) + ˜ft(xi +t|t)} +� += EP +η0 +�� t−1 +� +m=0 +gm(ξm|m) +� +1 +N +N +� +i=1 +{βi +tft(ξi +t|t) + ˜ft(ξi +t|t)} +� +. +(A.55) +27 + +In the following we will use repeatedly the following filtrations. Let ˜Ft := σ({υm}t +m=0) be the σ-field +generated by the output of the PARIS (Algorithm 1) during the first t iterations. In addition, let +Ft := ˜Ft−1 ∨ σ(ξt|t). +We proceed by induction. Thus, assume that the statement of the proposition holds true for a +given t ∈ N∗ and consider, for arbitrarily chosen (ft+1, ˜ft+1) ∈ F(Xt+1)2, +EP +η0 +�� +t� +m=0 +gm(ξm|m) +� +1 +N +N +� +i=1 +{βi +t+1ft+1(ξi +t+1|t+1) + ˜ft+1(ξi +t+1|t+1)} | ˜Ft +� += +� +t� +m=0 +gm(ξm|m) +� +EP +η0[β1 +t+1ft+1(ξ1 +t+1|t+1) + ˜ft+1(ξ1 +t+1|t+1) | ˜Ft] , +where we used that the variables {βi +t+1ft+1(ξi +t+1|t+1) + ˜ft+1(ξi +t+1|t+1)}N +i=1 are conditionally i.i.d. given +˜Ft. Note that, by symmetry, +EP +η0 +� +β1 +t+1 | Ft+1 +� += +� +St(υt, ξt+1|t+1, dyt+1) b1 +t+1 += +� +· · · +� � +� +M +� +j=1 +N +� +ℓ=1 +qt(ξℓ +t|t, ξ1 +t+1|t+1) +�N +ℓ′=1 qt(ξℓ′ +t|t, ξ1 +t+1|t+1) +δ(ξℓ +t|t,βℓ +t)(d˜x1,j +t , d˜b1,j +t ) +� +� +× 1 +M +M +� +j=1 +� +˜b1,j +t ++ ˜ht(˜x1,j +t , ξ1 +t+1|t+1) +� += +N +� +ℓ=1 +qt(ξℓ +t|t, ξ1 +t+1|t+1) +�N +ℓ′=1 qt(ξℓ′ +t|t, ξ1 +t+1|t+1) +� +βℓ +t + ˜ht(ξℓ +t|t, ξ1 +t+1|t+1) +� +. +(A.56) +Thus, using the tower property, +EP +η0 +� +β1 +t+1ft+1(ξ1 +t+1|t+1) | ˜Ft +� += +� +Φt(µ(ξt|t))(dxt+1) ft+1(xt+1) +N +� +ℓ=1 +qt(ξℓ +t|t, xt+1) +�N +ℓ′=1 qt(ξℓ′ +t|t, xt+1) +� +βℓ +t + ˜ht(ξℓ +t|t, xt+1) +� +, +and consequently, using definition (A.25), +� +t� +m=0 +gm(ξm|m) +� +EP +η0 +� +β1 +t+1ft+1(ξ1 +t+1|t+1) | ˜Ft +� += +� t−1 +� +m=0 +gm(ξm|m) +� � +1 +N +N +� +i=1 +qt(ξi +t|t, xt+1) +× ft+1(xt+1) +N +� +ℓ=1 +qt(ξℓ +t|t, xt+1) +�N +ℓ′=1 qt(ξℓ′ +t|t, xt+1) +� +βℓ +t + ˜ht(ξℓ +t|t, xt+1) +� +λt+1(dxt+1) += +� t−1 +� +m=0 +gm(ξm|m) +� +1 +N +N +� +ℓ=1 +� +βℓ +tQtft+1(ξℓ +t|t) + Qt(˜htft+1)(ξℓ +t|t) +� +. +28 + +Thus, applying the induction hypothesis, +EP +η0 +�� +t� +m=0 +gm(ξm|m) +� +1 +N +N +� +i=1 +βi +t+1ft+1(ξi +t+1|t+1) +� += EP +η0 +�� t−1 +� +m=0 +gm(ξm|m) +� +1 +N +N +� +ℓ=1 +� +βℓ +tQtft+1(ξℓ +t|t) + Qt(˜htft+1)(ξℓ +t|t) +�� += γt +� +Qtft+1Btht + Qt(˜htft+1) +� +. +(A.57) +In the same manner, it can be shown that +EP +η0 +�� +t� +m=0 +gm(ξm|m) +� +1 +N +N +� +i=1 +˜ft+1(ξi +t+1|t+1) +� += γtQt ˜ft+1. +(A.58) +Now, by (A.57–A.58) and Lemma 2, +EP +η0 +�� +t� +m=0 +gm(ξm|m) +� +1 +N +N +� +i=1 +{βi +t+1ft+1(ξi +t+1|t+1) + ˜ft+1(ξi +t+1|t+1)} +� += γt +� +Qtft+1Btht + Qt(˜htft+1 + Qt ˜ft+1) +� += γt+1(ft+1Bt+1ht+1 + ˜ft+1), +which shows that the claim of the proposition holds at time n + 1. +It remains to check the base case n = 0, which holds trivially true as β0 = 0, B0h0 = 0 by +convention, and the initial particles ξ0|0 are drawn from η0. This completes the proof. +Proof of Theorem 5. The identity +� +η0:t(dx0:t) St(x0:t, dbt) µ(bt)(id) = η0:tht follows immediately by +letting ft ≡ 1 and ˜ft ≡ 0 in Proposition 7 and using that γ0:t(X0:t) = γ0:t(X0:t). Moreover, applying +Theorem 3 yields +� +η0:tCtSt(dbt) µ(bt)(id) = +�� +η0:t(dz0:t) Ct(z0:t, dx0:t) +� +St(x0:t, dbt) µ(bt)(id) += +�� +η0:t(dx0:t) Bt(x0:t, dz0:t) +� +St(x0:t, dbt) µ(bt)(id) += +� +η0:tSt(dbt) µ(bt)(id). +Finally, the first identity holds true since Kt leaves η0:t invariant. +A.6.4 +Proof of Proposition 2 +First, note that, by definitions (A.39) and (A.40), +Ht(x0:t) := +� +St(x0:t, dyt) µ(x[0 : n|n])h += +� +· · · +� � +� 1 +N +N +� +jt=1 +h(xjt +0:t−1|t, xjt +t ) +� +� +× +t−1 +� +m=0 +N +� +im+1=1 +� +N +� +jm=1 +qm(xjm +m , xim+1 +m+1) +�N +j′m=1 qm(xj′ +m +m , xim+1 +m+1) +δxjm +0:m|m(dxim+1 +0:m|m+1), +29 + +where xi +0:−1|0 = ∅ for all i ∈ �1, N� by convention. We will show that for every k ∈ �0, t�, Hk,t ≡ Ht, +where +Hk,n(x0:t) := 1 +N +N +� +jt=1 +· · · +N +� +jk=1 +t−1 +� +ℓ=k +qℓ(xjℓ +ℓ , xjℓ+1 +ℓ+1 ) +�N +j′ +ℓ=1 qℓ(x +j′ +ℓ +ℓ , xjℓ+1 +ℓ+1 ) +ak,n(x0, . . . , xk−1, xjk +k , . . . , xjt +t ) +with +ak,n(x0, . . . , xk−1, xjk +k , . . . , xjt +t ) += +� +k−1 +� +m=0 +N +� +im+1=1 +N +� +jm=1 +qm(xjm +m , xim+1 +m+1) +�N +j′m=1 qm(xj′m +m , xim+1 +m+1) +δxjm +0:m|m(dxim+1 +0:m|m+1)h(xjk +0:k−1|k, xjk +k , . . . , xjt +t ). +Since, by convention, �t−1 +ℓ=n . . . = 1, Hn,n(x0:t) = N −1 �N +jt=1 an,n(x0, . . . , x[n − 1], xjt +t ), and we note +that Ht ≡ Hn,n. We now show that Hk,n ≡ Hk−1,n for every k ∈ �1, t�; for this purpose, note that +ak,n(x0, . . . , xk−1, xjk +k , . . . , xjt +t ) += +� +k−2 +� +m=0 +N +� +im+1=1 +N +� +jm=1 +qm(xjm +m , xim+1 +m+1) +�N +j′m=1 qm(xj′m +m , xim+1 +m+1) +δxjm +0:m|m(dxim+1 +0:m|m+1) +× +� +N +� +ik=1 +N +� +jk−1=1 +qk−1(xjk−1 +k−1 , xik +k ) +�N +j′ +k−1=1 qk−1(x +j′ +k−1 +k−1 , xik +k ) +δx +jk−1 +0:k−1|k−1(dxik +0:k−1|k) h(xjk +0:k−1|k, xjk +k , . . . , xjt +t ), +and since xjk−1 +0:k−1|k−1 = (xjk−1 +0:k−2|k−1, xjk−1 +k−1 ), it holds that +� +N +� +ik=1 +N +� +jk−1=1 +qk−1(xjk−1 +k−1 , xik +k ) +�N +j′ +k−1=1 qk−1(x +j′ +k−1 +k−1 , xik +k ) +δx +jk−1 +0:k−1|k−1(dxik +0:k−1|k) h(xjk +0:k−1|k, xjk +k , . . . , xjt +t ) += +N +� +jk−1=1 +qk−1(xjk−1 +k−1 , xjk +k ) +�N +j′ +k−1=1 qk−1(x +j′ +k−1 +k−1 , xjk +k ) +h(xjk−1 +0:k−2|k−1, xjk−1 +k−1 , xjk +k , . . . , xjt +t ). +Therefore, we obtain +ak,n(x0, . . . , xk−1, xjk +k , . . . , xjt +t ) += +� +k−2 +� +m=0 +N +� +im+1=1 +N +� +jm=1 +qm(xjm +m , xim+1 +m+1) +�N +j′m=1 qm(xj′ +m +m , xim+1 +m+1) +δxjm +0:m|m(dxim+1 +0:m|m+1) +× +N +� +jk−1=1 +qk−1(xjk−1 +k−1 , xjk +k ) +�N +j′ +k−1=1 qk−1(x +j′ +k−1 +k−1 , xjk +k ) +h(xjk−1 +0:k−2|k−1, xjk−1 +k−1 , xjk +k , . . . , xjt +t ). +Now, changing the order of summation with respect to jk−1 and integration on the right hand side of +the previous display yields +ak,n(x0, . . . , xk−1, xjk +k , . . . , xjt +t ) += +N +� +jk−1=1 +qk−1(xjk−1 +k−1 , xjk +k ) +�N +j′ +k−1=1 qk−1(x +j′ +k−1 +k−1 , xjk +k ) +ak−1,n(x0, . . . , xk−2, xjk−1 +k−1 , . . . , xjt +t ). +30 + +Thus, +Hk,n(x0:t) += 1 +N +N +� +jt=1 +· · · +N +� +jk=1 +t−1 +� +ℓ=k +qℓ(xjℓ +ℓ , xjℓ+1 +ℓ+1 ) +�N +j′ +ℓ=1 qℓ(x +j′ +ℓ +ℓ , xjℓ+1 +ℓ+1 ) +× +N +� +jk−1=1 +qk−1(xjk−1 +k−1 , xjk +k ) +�N +j′ +k−1=1 qk−1(x +j′ +k−1 +k−1 , xjk +k ) +ak−1,n(x0, . . . , xk−2, xjk−1 +k−1 , . . . , xjt +t ) += 1 +N +N +� +jt=1 +· · · +N +� +jk−1=1 +t−1 +� +ℓ=k−1 +qℓ(xjℓ +ℓ , xjℓ+1 +ℓ+1 ) +�N +j′ +ℓ=1 qℓ(x +j′ +ℓ +ℓ , xjℓ+1 +ℓ+1 ) +ak−1,n(x0, . . . , xk−2, xjk−1 +k−1 , . . . , xjt +t ) += Hk−1,n(x0:t), +which establishes the recursion. Therefore, Ht ≡ H0,n and we may now conclude the proof by noting +that Bth ≡ H0,n. +A.6.5 +Proof of Theorem 7 +In order to establish Theorem 7 we will prove the following more general result, of which Theorem 7 +is a direct consequence. +Proposition 8. For every t ∈ N and M ∈ N∗ there exist ct > 0 and dt > 0 such that for every +N ∈ N∗, z0:t ∈ X0:t, (ft, ˜ft) ∈ F(Xt)2, and ε > 0, +� +CtSt(z0:t, dbt)1 +������ +1 +N +N +� +i=1 +{bi +tft(xi +t|t) + ˜ft(xi +t|t)} − ηt⟨z0:t⟩(ftBt⟨z0:t−1⟩ht + ˜ft) +����� ≥ ε +� +≤ ct exp +� +−dtNε2 +2κ2 +t +� +, +where +κt := ∥ft∥∞ +t−1 +� +m=0 +∥˜hm∥∞ + ∥ ˜ft∥∞. +(A.59) +To prove Proposition 8 we need the following technical lemma. +Lemma 5. For every t ∈ N, (ft+1, ˜ft+1) ∈ F(Xt+1)2, z0:t+1 ∈ X0:t+1, and N ∈ N∗, +γt+1⟨z0:t+1⟩(ft+1Bt+1⟨z0:t⟩ht+1 + ˜ft+1) += +� +1 − 1 +N +� +γt⟨z0:t⟩{Qtft+1Bt⟨z0:t−1⟩ht + Qt(˜htft+1 + ˜ft+1)} ++ 1 +N γt⟨z0:t⟩gt +� +ft+1(zt+1)Bt+1⟨z0:t⟩ht+1(zt+1) + ˜ft+1(zt+1) +� +. +Proof. Since Lemma 2 holds also for the Feynman–Kac model with a frozen path, we obtain +γt+1⟨z0:t+1⟩(ft+1Bt+1⟨z0:t⟩ht+1 + ˜ft+1) = γt⟨z0:t⟩{Qt⟨zt+1⟩ft+1Bt⟨z0:t⟩ht + Qt⟨zt+1⟩(˜htft+1 + ˜ft+1)}. +Thus, the proof is concluded by noting that for every xt ∈ Xt and h ∈ F(Xt:t+1), +Qt⟨zt+1⟩h(xt) = +� +1 − 1 +N +� +Qth(xt) + 1 +N g(xt)h(xt, zt+1). +31 + +Finally, before proceeding to the proof of Proposition 8, we introduce the law of the PARIS evolving +conditionally on a frozen path z = {zm}m∈N. Define, for m ∈ N and zm+1 ∈ Xm+1, +P m⟨zm+1⟩ : Ym × Ym+1 ∋ (ym, A) �→ +� +M m⟨zm+1⟩(xm|m, dxm+1) Sm(ym, xm+1, A). +For any given initial distribution ψ0 ∈ M1(Y0), let PP ,z +ψ0 be the distribution of the canonical Markov +chain induced by the Markov kernels {P m⟨zm+1⟩}m∈N and the initial distribution ψ0. By abuse of +notation we write PP ,z +η0 +instead of PP ,z +ψ0[η0⟨z0⟩], where the extension ψ0[η0] is defined in Appendix A.6.3. +Proof of Proposition 8. We proceed by forward induction over t. Let the σ-fields ˜Ft and Ft be defined +as in the proof of Theorem 5, but for the conditional PARIS dual process. Then, under the law PP ,z +η0 , +reusing (A.56), +EP ,z +η0 +� +β1 +t ft(ξ1 +t ) + ˜ft(ξ1 +t ) | ˜Ft−1 +� += EP ,z +η0 +� +EP ,z +η0 +� +β1 +t | Ft +� +ft(ξ1 +t ) + ˜ft(ξ1 +t ) | ˜Ft−1 +� += EP ,z +η0 +� +ft(ξ1 +t ) +N +� +ℓ=1 +qt−1(ξℓ +t−1, ξ1 +t ) +�N +ℓ′=1 qt−1(ξℓ′ +t−1, ξ1 +t ) +� +βℓ +t−1 + ˜ht−1(ξℓ +t−1, ξ1 +t ) +� ++ ˜ft(ξ1 +t ) | ˜Ft−1 +� +. +Using (A.30), we get +EP ,z +η0 +� +β1 +t ft(ξ1 +t ) + ˜ft(ξ1 +t ) | ˜Ft−1 +� += +� +1 − 1 +N +� �N +ℓ=1{βℓ +t−1Qt−1ft(ξℓ +t−1) + Qt−1(˜ht−1ft + ˜ft)(ξℓ +t−1)} +�N +ℓ′=1 gt−1(ξℓ′ +t−1) ++ 1 +N +� +ft(zt) +N +� +ℓ=1 +qt−1(ξℓ +t−1, zt) +�N +ℓ′=1 qt−1(ξℓ′ +t−1, zt) +� +βℓ +t−1 + ˜ht(ξℓ +t−1, zt) +� ++ ˜ft(zt) +� +. +(A.60) +In order to apply the induction hypothesis to each term on the right-hand side of the previous identity, +note that +Bt⟨z0:t−1⟩ht(zt) = ηt−1⟨z0:t−1⟩[qt−1(·, zt){Bt−1⟨z0:t−2⟩ht−1(·) + ˜ht−1(·, zt)}] +ηt−1⟨z0:t−1⟩[qt−1(·, zt)] +. +Therefore, using Lemma 5 and noting that γt⟨z0:t⟩1Xt/γt−1⟨z0:t⟩1Xt−1 = ηt−1⟨z0:t−1⟩gt−1 yields +ηt⟨z0:t⟩(ftBt⟨z0:t−1⟩ht + ˜ft) = 1 +N +� +ft(zt)Bt⟨z0:t−1⟩ht(zt) + ˜ft(zt) +� ++ +� +1 − 1 +N +� ηt−1⟨z0:t−1⟩{Qt−1ftBt−1⟨z0:t−2⟩ht + Qt−1(˜ht−1ft + ˜ft)} +ηt−1⟨z0:t−1⟩gt−1 +. +(A.61) +By combining (A.60) with (A.61), we decompose the error according to +1 +N +N +� +i=1 +{βi +tft(ξi +t|t) + ˜ft(ξi +t|t)} − ηt⟨z0:t⟩(ftBt⟨z0:t−1⟩ht + ˜ft) += 1 +N +N +� +i=1 +{βi +tft(ξi +t|t) + ˜ft(ξi +t|t)} − EP ,z +η0 +� +β1 +t ft(ξ1 +t ) + ˜ft(ξ1 +t ) | ˜Ft−1 +� ++ EP ,z +η0 +� +β1 +t ft(ξ1 +t ) + ˜ft(ξ1 +t ) | ˜Ft−1 +� +− ηt⟨z0:t⟩(ftBt⟨z0:t−1⟩ht + ˜ft) += I(1) +N + +� +1 − 1 +N +� +I(2) +N + 1 +N I(3) +N , +(A.62) +32 + +where +I(1) +N := 1 +N +N +� +i=1 +{βi +tft(ξi +t) + ˜ft(ξi +t)} − EP ,z +η0 +� +β1 +t ft(ξ1 +t ) + ˜ft(ξ1 +t ) | ˜Ft−1 +� +, +I(2) +N := +�N +ℓ=1{βℓ +t−1Qt−1ft(ξℓ +t−1) + Qt−1(˜ht−1ft + ˜ft)(ξℓ +t−1)} +�N +ℓ′=1 gt−1(ξℓ′ +t−1) +− ηt−1⟨z0:t−1⟩{Qt−1ftBt⟨z0:t−1⟩ht + Qt−1(˜ht−1ft + ˜ft)} +ηt−1⟨z0:t−1⟩gt−1 +, +(A.63) +and +I(3) +N := ft(zt) +N +� +ℓ=1 +qt−1(ξℓ +t−1, zt) +�N +ℓ′=1 qt−1(ξℓ′ +t−1, zt) +� +βℓ +t−1 + ˜ht−1(ξℓ +t−1, zt) +� +− ft(zt)ηt−1⟨z0:t−1⟩[qt−1(·, zt){Bt−1⟨z0:t−2⟩ht−1(·) + ˜ht−1(·, zt)}] +ηt−1⟨z0:t−1⟩[qt−1(·, zt)] +. +(A.64) +The proof is now completed by treating the terms I(1) +N , I(2) +N , and I(3) +N separately, using Hoeffding’s +inequality and its generalisation in [Douc et al., 2011, Lemma 4]. Choose ε > 0; then, by Hoeffding’s +inequality, +PP ,z +η0 +� +| I(1) +N | ≥ ε +� +≤ 2 exp +� +−1 +2 +ε2 +κ2 +t +N +� +. +(A.65) +To treat I(2) +N , we apply the induction hypothesis to the numerator and denominator, each normalised +by 1/N, yielding, since ∥Qt−1h∥∞ ≤ ¯τt−1∥h∥∞ for all h ∈ F(Xt−1 � Xt), +PP ,z +η0 +������ +1 +N +N +� +ℓ=1 +{βℓ +t−1Qt−1ft(ξℓ +t−1) + Qt−1(˜ht−1ft + ˜ft)(ξℓ +t−1)} +−ηt−1⟨z0:t−1⟩{Qt−1ftBt⟨z0:t−1⟩ht + Qt−1(˜ht−1ft + ˜ft)} +����� ≥ ε +� +≤ ct−1 exp +� +−dt−1 +ε2 +¯τ 2 +t−1κ2 +t +N +� +and +PP ,z +η0 +������ +1 +N +N +� +ℓ=1 +gt−1(ξℓ +t−1) − ηt−1⟨z0:t−1⟩gt−1 +����� ≥ ε +� +≤ ct−1 exp +� +−dt−1 +ε2 +¯τ 2 +t−1 +N +� +. +Combining the previous two bounds with the generalised Hoeffding inequality in [Douc et al., 2011, +Lemma 4] yields, using also the bounds +�N +ℓ=1{βℓ +t−1Qt−1ft(ξℓ +t−1) + Qt−1(˜ht−1ft + ˜ft)(ξℓ +t−1)} +�N +ℓ′=1 gt−1(ξℓ′ +t−1) +≤ κt +and ηt−1⟨z0:t−1⟩gt−1 ≥ ¯τt−1, the inequality +PP ,z +η0 +� +| I(2) +N | ≥ ε +� +≤ ct−1 exp +� +−dt−1¯τ 2 +t−1ε2 +¯τ 2 +t−1κ2 +t +N +� +. +(A.66) +33 + +The last term I(3) +N is treated along similar lines; indeed, by the induction hypothesis, since ∥qt−1∥∞ ≤ +¯τt−1¯σt−1, +PP ,z +η0 +������ +1 +N +N +� +ℓ=1 +qt−1(ξℓ +t−1, zt) +� +βℓ +t−1 + ˜ht−1(ξℓ +t−1, zt) +� +− ηt−1⟨z0:t−1⟩[qt−1(·, zt){Bt−1⟨z0:t−1⟩ht−1(·) + ˜ht−1(·, zt)}] +����� ≥ ε +� +≤ ct−1 exp +� +�−dt−1 +� +ε +¯τt−1¯σt−1 +�t−1 +m=0 ∥˜hm∥∞ +�2 +N +� +� +and +PP ,z +η0 +������ +1 +N +N +� +ℓ=1 +qt−1(ξℓ +t−1, zt) − ηt−1⟨z0:t−1⟩[qt−1(·, zt)] +����� ≥ ε +� +≤ ct−1 exp +� +−dt−1 +� +ε +¯τt−1¯σt−1 +�2 +N +� +. +Thus, since +N +� +ℓ=1 +qt−1(ξℓ +t−1, zt) +�N +ℓ′=1 qt−1(ξℓ′ +t−1, zt) +� +βℓ +t−1 + ˜ht−1(ξℓ +t−1, zt) +� +≤ +t−1 +� +m=0 +∥˜hm∥∞ +and ηt−1⟨z0:t−1⟩[qt−1(·, zt)] ≥ ¯τt−1, the generalised Hoeffding inequality provides +PP ,z +η0 +� +| I(3) +N | ≥ ε +� +≤ ct−1 exp +� +�−dt−1 +� +¯τt−1ε +2¯τt−1¯σt−1∥ft∥∞ +�t−1 +m=0 ∥˜hm∥∞ +�2 +N +� +� . +(A.67) +Finally, combining the bounds (A.65–A.67) completes the proof. +A.6.6 +Proof of Proposition 4 +The statement of Proposition 4 is implied by the following more general result, which we will prove +below. +Proposition 9. For every t ∈ N, M ∈ N∗, N ∈ N∗, z0:t ∈ X0:t, (ft, ˜ft) ∈ F(Xt)2, and p ≥ 2, it holds +that +� +CtSt(z0:t, dbt) +����� +1 +N +N +� +i=1 +{bi +tft(xi +t|t) + ˜ft(xi +t|t)} − ηt⟨z0:t⟩(ftBt⟨z0:t−1⟩ht + ˜ft) +����� +p +≤ ct(p/dt)p/2N −p/2κp +t , +where ct > 0, dt > 0 and κt are defined in Proposition 8 and (A.59), respectively. +Before proving Proposition 9, we establish the following result. +Lemma 6. Let X be an Rd-valued random variable, defined on some probability space (Ω, F, P), +satisfying P(|X| ≥ t) ≤ c exp(−t2/(2σ2)) for every t ≥ 0 and some c > 0 and σ > 0. Then for every +p ≥ 2 it holds that E[|X|p] ≤ cpp/2σp. +Proof. Using Fubini’s theorem and the change of variable formula, +E [|X|p] = +� ∞ +0 +ptp−1P(|X| ≥ t) dt = cp2p/2−1σpΓ(p/2), +where Γ is the Gamma function. It remains to apply the bound Γ(p/2) ≤ (p/2)p/2−1 (see [Anderson and Qiu, 1997]), +which holds for p ≥ 2 by [2, Theorem 1.5]. +34 + +Proof of Proposition 9. By combining Proposition 8 and Lemma 6 we obtain +N +� +CtSt(z0:t, dbt) +���� +1 +N +�N +i=1{bi +tft(xi +t|t) + ˜ft(xi +t|t)} − ηt⟨z0:t⟩(ftBt⟨z0:t−1⟩ht + ˜ft) +���� +2 +≤ ct(p/dt)p/2N −p/2 +� +∥ft∥∞ +t−1 +� +m=0 +∥˜hm∥∞ + ∥ ˜ft∥∞ +�p +, +which was to be established. +A.6.7 +Proof of Proposition 5 +Like previously, we establish Proposition 5 via a more general result, namely the following. +Proposition 10. For every t ∈ N, the exists ¯cbias +t +< ∞ such that for every M ∈ N∗, N ∈ N∗, +z0:t ∈ X0:t, and (ft, ˜ft) ∈ F(Xt)2, +����� +� +CtSt(z0:t, dbt) 1 +N +N +� +i=1 +{bi +tft(xi +t|t) + ˜ft(xi +t|t)} − ηt⟨z0:t⟩(ftBt⟨z0:t−1⟩ht + ˜ft) +����� ≤ ¯cbias +t +κtN −1, +where κt is defined in (A.59). +We preface the proof of Proposition 10 by a technical lemma providing a bound on the bias of +ratios of random variables. +Lemma 7. Let α and β be (possibly dependent) random variables defined on some probability space +(Ω, F, P) and such that E[α2] < ∞ and E[β2] < ∞. Moreover, assume that there exist c > 0 and d > 0 +such that |α/β| ≤ c, P-a.s., |a/b| ≤ c, E[(α − a)2] ≤ c2d2, and E[(β − b)2] ≤ d2. Then +|E[α/β] − a/b| ≤ 2c(d/b)2 + c|E[β − b]|/|b| + |E[α − a]|/|b|. +(A.68) +Proof. Using the identity +E[α/β] − a/b = E[(α/β)(b − β)2]/b2 + E[(α − a)(b − β)]/b2 + aE[b − β]/b2 + E[α − a]/b, +the claim is established by applying the Cauchy–Schwarz inequality and the assumptions of the lemma +according to +|E[α/β] − a/b| +≤ cE[(β − b)2]/b2 + {E[(α − a)2]E[(β − b)2]}1/2/b2 + |a||E[β − b]|/b2 + |E[α − a]|/b2 +≤ 2c(d/b)2 + c|E[β − b]|/|b| + |E[α − a]|/|b|. +Proof of Proposition 5. We proceed by induction and assume that the claim holds true for n − 1. +Reusing the error decomposition (A.62), it is enough to bound the expectations of the terms I(2) +N +and I(3) +N +given in (A.63) and (A.64), respectively (since EP ,z +η0 [I(1) +N ] = 0). This will be done using the +induction hypothesis, Lemma 7, and Proposition 9. More precisely, to bound the expectation of I(2) +N , +we use Lemma 7 with α ← αt, β ← βt, a ← at, and b ← bt, where +αt := 1 +N +N +� +ℓ=1 +{βℓ +t−1Qt−1ft(ξℓ +t−1) + Qt−1(˜ht−1ft + ˜ft)(ξℓ +t−1)}, +βt := 1 +N +N +� +ℓ=1 +gt−1(ξℓ +t−1), +at := ηt−1⟨z0:t−1⟩{Qt−1ftBt⟨z0:t−1⟩ht + Qt−1(˜ht−1ft + ˜ft)}, +bt := ηt−1⟨z0:t−1⟩gt−1. +35 + +For this purpose, note that |αt/βt| ≤ κt and |at/bt| ≤ κt, where κt is defined in (A.59). On the other +hand, using Proposition 9 (applied with p = 2), we obtain +EP ,z +η0 [(αt − at)2] ≤ d2 +tκ2 +t +and +EP ,z +η0 [(βt − bt)2] ≤ d2 +t, +where d2 +t := ct¯τ 2 +t−1/(dtN). Using the induction assumption, we get +|EP ,z +η0 [αt] − at| ≤ ¯cbias +t−1N −1¯τt−1κt +and +|EP ,z +η0 [βt] − bt| ≤ ¯cbias +t−1N −1¯τt−1. +Hence, the conditions of Lemma 7 are satisfied and we deduce that +|EP ,z +η0 [I(2) +N ]| = |EP ,z +η0 [αt/βt] − at/bt| ≤ 2κt +ct +dtN +¯τ 2 +t−1 +¯τ 2 +t−1 ++ 2¯cbias +t−1κt +¯τt−1 +¯τt−1N . +The bound on |EP ,z +η0 [I(2) +N ]| is obtained along the same lines. +B. +Learning with PPG +This section is divided into three subsections. Appendix B.1 establishes, following closely [Karimi et al., 2019], +a non-asymptotic bound for stochastic approximation schemes under general assumptions. Appendix B.2 +shows how assumptions A 4.1 and A 3.1 imply the assumptions provided in Appendix B.1 and there- +fore allow to establish Theorem 2. Finally, Appendix B.3 provides sufficient assumptions on the model +ensuring that A 4.1 holds. +B.1 +Non-asymptotic bound +We follow closely [Karimi et al., 2019]. Consider the recursion +θn+1 = θn − γn+1Hθn(Xn+1), +n ∈ N, +where θn ∈ Θ ⊂ Rd for some d ∈ N∗ and {Xn}n∈N is a state-dependent Markov chain on some +measurable space (X, X) in the sense that Xn+1 ∼ Pθn(Xn, ·) with Pθ being some Markov kernel on +(X, X). Let h(θ) = +� +Hθ(x) πθ(dx), where πθ is the invariant measure of Pθ and en+1 := Hθn(Xn+1)− +h(θn). As all norms are equivalent in finite dimensional vector spaces, we use ∥ · ∥ to denote a generic +norm. We denote by {Fn}n∈N the natural filtration of the Markov chain {Xn}n∈N. +A B.1. There exists a Borel measurable function V : Θ → R such that for every θ ∈ Θ, ∇V (θ) = h(θ). +A B.2. There exists LV ∈ R≥0 such that for every (θ, θ′) ∈ Θ2, +∥∇V (θ) − ∇V (θ′)∥ ≤ LV ∥θ − θ′∥. +A B.3. There exists a Borel measurable function �H : Θ×X → Θ such that for every θ ∈ Θ and x ∈ X, +�Hθ(x) − Pθ �Hθ(x) = Hθ(x) − h(θ) . +A B.4. There exists LP � +H ∈ R≥0 such that for every (θ0, θ1) ∈ Θ2, +sup +x∈X +∥Pθ0 �Hθ0(x) − Pθ0 �Hθ1(x)∥ ≤ LP � +H∥θ0 − θ1∥ . +A B.5. There exists LP � +H +0 +∈ R≥0 such that +sup +θ∈Θ +∥Pθ �Hθ∥ ≤ LP � +H +0 +. +36 + +A B.6. There exists σmse ∈ R≥0 such that for every x ∈ X and θ ∈ Θ, +� +∥Hθ(x′) − h(θ)∥2 Pθ(x, dx′) ≤ σ2 +mse . +A B.7. There exists L � +H ∈ R≥0 such that for every x ∈ X, +sup +θ∈Θ +� +∥ �Hθ∥ Pθ(x, dx′) ≤ L +� +H . +Theorem 8. Assume that A B.1–A B.7 hold. In addition, assume that there exist a > 0 and a′ > 0 +such that for all n ∈ N, +γn+1 ≤ γn ≤ aγn+1 , +γn − γn+1 ≤ a′γ2 +n , +γ1 ≤ (LV + Ch)−1/2 . +Moreover, for any n ∈ N∗, let ϖ be a �0, n�-valued random variable, independent of {Fℓ}ℓ≥0 and such +that P(ϖ = k) = γk+1/ �n +ℓ=0 γℓ+1 for k ∈ �0, n�. Then, +E +� +∥h(θϖ)∥2� +≤ 2V0,n + C0,n + (σ2 +mseLV + Cγ) �n +k=0 γ2 +k+1 +�n +k=0 γk+1 +, +where V0,n := E [V (θ) − V (θn)] and +C0,n := γ1h(θ0)L +� +H + LP � +H +0 (γ1 − γn+1 + 1) , +(B.69) +Cγ := σmseLP � +H + (1 + σmse)LV LP � +H +0 +, +(B.70) +Ch := LP � +H ((a + 1)/2 + aσmse) + (LV + a′ + 1)LP � +H +0 +. +(B.71) +Proof. We follow closely the proof of [Karimi et al., 2019, Theorem 2] and adapt it to our setting. +First, note that by A B.1, assumptions A1 and A2 of [Karimi et al., 2019, Theorem 2] hold with +c0 = d0 = 0 and c1 = d1 = 1. In addition, the claim in [Karimi et al., 2019, Lemma 1] holds true since +by AB.2, A3 holds. Moreover, [Karimi et al., 2019, Equation 17] can also be established under AB.6, +as we may rewrite it as +n +� +ℓ=0 +γ2 +ℓ+1E +� +∥eℓ+1∥2� += +n +� +ℓ=0 +γ2 +ℓ+1E +� +E +� +∥eℓ+1∥2 | Fℓ +�� +≤ σ2 +mse +n +� +ℓ=0 +γ2 +ℓ+1 . +Following the proof of [Karimi et al., 2019, Lemma 2], consider the decomposition +E +� +− +n +� +ℓ=0 +γℓ+1 ⟨∇V (θℓ), eℓ+1⟩ +� += E [A1 + A2 + A3 + A4 + A5] , +where +A1 := − +n +� +ℓ=1 +γℓ+1 +� +∇V (θℓ), �Hθℓ(Xℓ+1) − Pθℓ �Hθℓ(Xℓ) +� +, +A2 := − +n +� +ℓ=1 +γℓ+1 +� +∇V (θℓ), Pθℓ �Hθℓ(Xℓ) − Pθℓ−1 �Hθℓ−1(Xℓ) +� +, +A3 := − +n +� +ℓ=1 +γℓ+1 +� +∇V (θℓ) − ∇V (θℓ−1), Pθℓ−1 �Hθℓ−1(Xℓ) +� +, +A4 := − +n +� +ℓ=1 +(γℓ+1 − γℓ) +� +∇V (θℓ−1), Pθℓ−1 �Hθℓ−1(Xℓ) +� +, +A5 := −γ1 +� +∇V (θ0), �Hθ0(X1) +� ++ γn+1 +� +∇V (θn), Pθn �Hθn(Xn+1) +� +. +37 + +As �Hθℓ(Xℓ+1) − Pθℓ �Hθℓ(Xℓ) is a martingale difference, it holds that E [A1] = 0. The upper bounds on +the expectations of A2, A3 and A4 are obtained similarly as in [Karimi et al., 2019]. Using A B.4, +A2 ≤ LP � +H +� +σmse +n +� +k=1 +γ2 +k + 1 +2 (1 + 2aσmse + a) +n +� +k=0 +γ2 +k+1∥h(θk)∥2 +� +. +By A B.2 and B.5, +A3 ≤ LV LP � +H +0 +� +(1 + σmse) +n +� +k=1 +γ2 +k + +n +� +k=1 +γ2 +k∥h(θk)∥2) +� +. +On the other hand, +A4 ≤ LP � +H +0 +� +γ1 − γn+1 + a′ +n +� +k=1 +γ2 +k∥h(θk−1)∥2 +� +. +We now focus on A5. As in the proof of [Karimi et al., 2019, Lemma 2], the expectation of the first +term can be straightforwardly bounded by γ1∥h(θ0)∥L � +H using the Cauchy–Schwarz inequality and +A B.7. The second term can, using A B.5 and γn+1∥h(θn)∥ ≤ 1 + γ2 +n+1∥h(θn)∥2, be bounded in the +same way according to +γn+1 +� +∇V (θn), Pθn �Hθn(Xn+1) +� +≤ LP � +H +0 γn+1∥h(θn)∥ ≤ LP � +H +0 +� +1 + γ2 +n+1∥h(θn)∥2� +≤ LP � +H +0 +� +1 + +n +� +ℓ=0 +γ2 +ℓ+1∥h(θℓ)∥2 +� +. +The rest of the proof follows that of [Karimi et al., 2019, Theorem 2]. +B.2 +Application to Theorem 2 +The goal of this section is to establish that the assumptions of Theorem 2 ensure all the assumptions +in appendix B.1, which in turn allows Theorem 8 to be applied. First, we start by explicitly defining +the kernel Pθ and the function h in terms of the kernels presented in appendix A. We write Pθ,t instead +of Pθ to explicit the dependence of the kernel on the fixed number of observations t. +B.2.1 +Verification of the assumptions of Theorem 8 +For (k0, k) ∈ (N∗)2 such that k0 < k, define +Pθ,t : Ek−k0 +t +× E�(k−k0) +t +∋ (yt[k0 : k], z0:t[k0 : k], A) �→ Kk0 +θ,t � K�(k−k0) +θ,t +(z0:t[k], A), +(B.72) +where Kθ,t is the PPG kernel defined in (A.41). Note that Pθ,t depends only on the last frozen path, +namely z0:t[k]. Note also that, since Kθ,t depends only on the paths, there is no dependence between +yt,ℓ[k0 : k] and yt,ℓ+1[k0 : k]. The score ascent algorithm (Algorithm 4) can be formulated as follows. +1. Sample (z0:t,ℓ[k0 : k], yt,ℓ[k0 : k]) ∼ Pθℓ,t +� +(z0:t,ℓ−1[k0 : k], yt,ℓ−1[k0 : k]), · +� +. +2. Update the parameter according to ηℓ+1 = ηℓ + γℓ+1H(z0:t,ℓ[k0 : k], yt,ℓ[k0 : k]), where +H(z0:t,ℓ[k0 : k], yt,ℓ[k0 : k]) = +1 +k − k0 + 1 +k +� +i=k0 +µ(βt,ℓ[i])(id) = Π(k0−1,k),N(ht), +where Π(k0−1,k),N(ht) is defined in (3.12). We denote by πθ,t the invariant distribution of Pθ,t, which, +by Proposition 3, is given by πθ,t = (η0:t � CtSt)�(k−k0). +We also require the strong mixing assumption to hold uniformly in θ. +38 + +A B.8 (Strong mixing uniformly in θ). For every s ∈ N there exist ¯τs, ¯τs, ¯σs, and ¯σs in R∗ ++ such that +for all θ ∈ Θ, +(i) ¯τs ≤ gs,θ(xs) ≤ ¯τs for every xs ∈ Xs, +(ii) ¯σs ≤ ms,θ(xs, xs+1) ≤ ¯σs for every (xs, xs+1) ∈ Xs:s+1. +Note that the assumption above implies that κN,t is also uniform in θ. +Proof that A B.1 holds. +Proposition 11. For all θ ∈ Θ, h(θ) = ∇V (θ), where V (θ) = log γ0:t,θ(X0:t) is the log-likelihood +function. +Proof. By Theorem 5, +h(θ) = +� +H(˜yt[k0 : k], ˜x0:t[k0 : k]) πθ,t(d(˜yt[k0 : k], ˜x0:t[k0 : k])) += +1 +k − k0 + 1 +k +� +i=k0 +� +[η0:t,θ � Ct,θSt,θ] (d(˜yt[i], ˜x0:t[i]))µ(˜βt,ℓ[i])(id) += η0:t,θ (s0:t,θ) = ∇V (θ). +Proof that A B.2 holds. +A B.2 is trivially implied by A 4.1(i). +Proof that A B.3 and B.5 hold. +Let �Hθ be given by +�Hθ : Ek−k0 +t +∋ (yt[k0 : k], z0:t[k0 : k]) �→ +∞ +� +r=0 +{Pr +θ,tH(yt[k0 : k], z0:t[k0 : k]) − h(θ)}. +(B.73) +Then the following holds true. +Lemma 8. Assume A B.8. Then for all θ ∈ Θ and t ∈ N∗, +∥Pθ,t �Hθ∥∞ ≤ σbias(1 − κk +N,t)−1 . +Proof. By Theorem 1, we have for any r > 0 +��Pr +θ,tH(yt[k0 : k], z0:t[k0 : k]) − h(θ) +�� ≤ σbiasκ(r−1)k +N,t +and thus +∥Pθ,t �Hθ∥∞ ≤ +∞ +� +r=1 +��Pr +θ,tH − h(θ) +�� +∞ ≤ σbias +∞ +� +r=0 +κrk +N,t ≤ σbias(1 − κk +N,t)−1 , +where κN,t ∈ (0, 1). +Lemma 8 proves A B.3 and B.5 with LP � +H +0 +:= σbias(1 − κk +N,t)−1. +39 + +Proof that A B.4 holds. +Theorem 9. Assume A B.8 and A 4.1. +Then for every t ∈ N, θ ∈ Θ and N ∈ N∗ such that +N > 1 + 5ρ2 +tt/2, +���Pθ1,t �Hθ1 − Pθ2,t �Hθ2 +��� +∞ ≤ LP � +H∥θ1 − θ2∥ , +where +LP � +H := ∥LP +2 ∥∞ +� +1 + κk +N,t(1 − κk +N,t) +� ++ LV + +σbias(1 − κN,t)−1(1 − κk +N,t)−1 � +∥LP +1 ∥∞(1 − κk +N,t)−1 + Lηκk +N,t +� +. +(B.74) +Proof. We establish the claim by adapting the proof of [Karimi et al., 2019, Lemma 7]. First, recall +that the kernel Kθ,t defined in (A.42) is the path marginalized version of Kθ,t given in (A.41). Note +that for every x ∈ Ek−k0 +t +, +Pθ1,t �Hθ1(x) = +∞ +� +n=0 +δxPθ1,t +� +Pn +θ1,tH − h(θ1) +� += +∞ +� +n=0 +δxKkn +θ1,t {Pθ1,tH − η0:t,θ1Pθ1,tH} , +where we have used (i) the fact that the backward statistics output by Pθ,t are independent of the +input backward statistics and (ii) the penultimate line in the computation of h(θ) above. We follow +the proof of [Fort et al., 2011, Lemma 4.2] and consider the following decomposition: for n ∈ N∗, +δxKkn +θ1,t (Pθ1,tH − η0:t,θ1Pθ1,tH) − δxKkn +θ2,t (Pθ2,tH − η0:t,θ2Pθ2,tH) +(B.75) += +n−1 +� +j=0 +� +δxKkj +θ1,t − η0:t,θ1 +� � +Kkj +θ1,t − Kkj +θ2,t +� � +Kk(n−j−1) +θ2,t +Pθ1,tH − η0:t,θ2Pθ1,tH +� +− +� +δxKkn +θ2,tPθ2,tH − η0:t,θ2Pθ2,tH +� ++ +� +δxKkn +θ2,tPθ1,tH − η0:t,θ2Pθ1,tH +� +− η0:t,θ1 +� +Kkn +θ2,tPθ1,tH − η0:t,θ2Pθ1,tH +� +. +Applying Theorem 6 with µ = δx and ν = η0:t,θ and using the fact that η0:t,θKℓ +θ,t = η0:t,θ for all ℓ ∈ N, +we obtain that for all ℓ ∈ N and all θ ∈ Θ, +���δxKℓ +θ,t − η0:t,θ +��� +TV ≤ κℓ +N,t. Note that by A4.1(iii), Kθ,t is +Lipschitz; therefore, for all r ∈ N∗, by Lemma 18, Kr +θ,t is Lipschitz with constant ∥LP +1 ∥∞(1 − κN,t)−1. +Combining all this together, we obtain +��� +� +δxKkj +θ1,t − η0:t,θ1 +� � +Kkj +θ1,t − Kkj +θ2,t +� � +Kk(n−j−1) +θ2,t +Pθ1,tH − η0:t,θ2Pθ1,tH +���� += +��� +� +δxKkj +θ1,t − η0:t,θ1 +� � +Kkj +θ1,t − Kkj +θ2,t +� � +Kk(n−j−1) +θ2,t +[Pθ1,tH − h(θ1)] − η0:t,θ2 [Pθ1,tH − h(θ1)] +���� +≤ ∥LP +1 ∥∞(1 − κN,t)−1κkj +N,tκk(n−j−1) +N,t +∥Pθ1,tH − h(θ1)∥∞∥θ1 − θ2∥ +≤ σbias∥LP +1 ∥∞(1 − κN,t)−1κk(n−1) +N,t +∥θ1 − θ2∥ , +where the last inequality is due to Theorem 1. Therefore, the first term of the right side of (B.75) +is upper bounded by σbias∥LP +1 ∥∞(1 − κN,t)−1nκk(n−1) +N,t +∥θ1 − θ2∥. The second term of (B.75) can be +written +− +� +δxKkn +θ2,tPθ2,tH − η0:t,θ2Pθ2,tH +� ++ +� +δxKkn +θ2,tPθ1,tH − η0:t,θ2Pθ1,tH +� += +� +δxKkn +θ2,t − η0:t,θ2 +� +(Pθ1,tH − Pθ2,tH) , +and using again the ergodicity of Kθ,t and the fact that θ �→ Pθ,tH is uniformly Lipschitz by A4.1(iv), +we may conclude that it is upper bounded by ∥LP +2 ∥∞κkn +N,t∥θ1 − θ2∥. Finally, for the last term, using +40 + +the facts that Kk +θ,t is η0:t,θ-invariant and geometrically ergodic and that θ �→ η0:t,θ is Lipschitz by +A 4.1(iv) yields +��η0:t,θ1 +� +Kkn +θ2,tPθ1,tH − η0:t,θ2Pθ1,tH +��� += +��(η0:t,θ1 − η0:t,θ2) +� +Kkn +θ2,t [Pθ1,tH − h(θ1)] − η0:t,θ2 [Pθ1,tH − h(θ1)] +��� +≤ Lηκkn +N,t∥Pθ1,tH − h(θ1)∥∞∥θ1 − θ2∥ +≤ Lησbias(1 − κN,t)−1κkn +N,t∥θ1 − θ2∥ . +Therefore, we have that +δxKkn +θ1,t (Pθ1,tH − η0:t,θ1Pθ1,tH) − δxKkn +θ2,t (Pθ2,tH − η0:t,θ2Pθ2,tH) +≤ +� +σbias∥LP +1 ∥∞(1 − κN,t)−1nκk(n−1) +N,t ++ +� +∥LP +2 ∥∞ + Lησbias(1 − κN,t)−1� +κkn +N,t +� +∥θ1 − θ2∥ . +Therefore, we obtain +���Pθ1,t �Hθ1(x) − Pθ2,t �Hθ2(x) +��� +≤ |δxPθ1,tH − δxPθ2,tH| + |η0:t,θ1Pθ1,tH − η0:t,θ2Pθ2,tH| ++ +����� +∞ +� +n=1 +δxKkn +θ1,t (Pθ1,tH − η0:t,θ1Pθ1,tH) − δxKkn +θ2,t (Pθ2,tH − η0:t,θ2Pθ2,tH) +����� +≤ |δxPθ1,tH − δxPθ2,tH| + |η0:t,θ1Pθ1,tH − η0:t,θ2Pθ2,tH| ++ +� +σbias∥LP +1 ∥∞(1 − κN,t)−1(1 − κk +N,t)−2 ++ +� +∥LP +2 ∥∞ + Lησbias(1 − κN,t)−1� +κk +N,t(1 − κk +N,t)−1 +� +∥θ1 − θ2∥ . +To conclude, note that by A 4.1(iv), ∥δxPθ1,tH − δxPθ2,tH∥ ≤ ∥LP +2 ∥∞∥θ1 − θ2∥. Furthermore, note +that by Theorem 5 we obtain that for all θ ∈ Θ, η0:t,θPθ,tH = η0:t,θs0:t,θ = ∇V (θ). Therefore, by +A 4.1(i) we obtain that ∥η0:t,θ1Pθ1,tH − η0:t,θ2Pθ2,tH∥ ≤ LV ∥θ1 − θ2∥, concluding the proof. +Proof that AB.6 holds. +AB.6 is simply a bound on the MSE of the roll-out PPG estimator, given +by Theorem 1. +Proof that A B.7 holds. +Proposition 12. For all θ ∈ Θ and all ℓ ∈ �1, t − 1� +E +� +∥ �Hθ∥ | Fℓ +� +≤ 2∥s0:t,θ∥∞ + σbias(1 − κk +N,t)−1 . +Proof. Note that for all x ∈ Ek−k0 +t +and all θ ∈ Θ, +�Hθ(x) = H(x) − h(θ) + Pθ,t �Hθ(x) . +(B.76) +Lemma 8 shows that ∥Pθ,t �Hθ∥∞ ≤ σbias(1 − κk +N,t)−1. Note that h(θ) ≤ ∥s0:t,θ∥∞ We write +E [∥H∥ | Fℓ] ≤ +1 +(k − k0 + 1)N +k +� +i=k0 +N +� +j=1 +E +� +∥βj +t,ℓ[i]∥ | Fℓ +� +. +By Proposition 14, E +� +∥βj +t,ℓ[i]∥ | Fℓ +� +≤ ∥s0:t,θ∥∞, concluding the proof. +A B.7 follows directly by Proposition 12 and by considering supθ∈Θ ∥s0:t,θ∥∞. +41 + +B.2.2 +Proof of Theorem 2 +We have shown in Appendix B.2.1 that under A 4.1 and B.8, it is possible to apply Theorem 8. To +conclude the proof of Theorem 2 we just have to rearrange the constants. We start by rewriting the +constant in Theorem 9 +LP � +H = C1 + σbias(1 − κN,t)−1(1 − κk +N,t)−1C2, +with +C1 = +��LP +2 +�� +∞ +� +1 + κk +N,t(1 − κk +N,t)−1� ++ LV +C2 = +��LP +1 +�� +∞ (1 − κk +N,t)−1 + Lηκk +N,t . +By (B.70) and Lemma 8, +Cγ = σmseLP � +H + (1 + σmse)LV LP � +H +0 += σmse +� +C1 + σbias(1 − κN,t)−1(1 − κk +N,t)−1C2 +� ++ (1 + σmse)LV σbias(1 − κk +N,t)−1 += σmseC1 + σmseσbias(1 − κk +N,t)−1 � +LV + (1 − κN,t)−1C2 +� ++ σbiasLV (1 − κk +N,t)−1 . +Therefore, +C0,γ := σ2 +mseLV + Cγ += σ2 +mseLV + σmseC1 + σmseσbias(1 − κk +N,t)−1 � +LV + (1 − κN,t)−1C2 +� ++ σbiasLV (1 − κk +N,t)−1 . +In the same way, we can rewrite (B.71) as +Ch = LP � +H [(a + 1)/2 + aσmse] + (LV + a′ + 1)LP � +H +0 += +� +C1 + σbias(1 − κN,t)−1(1 − κk +N,t)−1C2 +� +[(a + 1)/2 + aσmse] + (LV + a′ + 1)σbias(1 − κk +N,t)−1 . +The constant C0 from Theorem 2 is L � +H = 2 supθ∈Θ ∥s0:t,θ∥∞ + σbias(1 − κk +N,t)−1 which completes the +proof. +B.3 +Conditions on the model to verify A 4.1 +In our specific application to score ascent, we work with the following assumptions. +A B.9 (Lipschitz). +(i) For all t ∈ N, there exists Ls +t ∈ M(Xt:t+1) such that for all (xt, xt+1) ∈ Xt:t+1, +the function θ �→ st,θ(xt, xt+1) is Ls +t(xt, xt+1)-Lipschitz and Xt:t+1 ∋ (xt, xt+1) �→ st,θ(xt, xt+1) +is bounded by ∥st(θ)∥∞ for all θ ∈ Θ. Furthermore, ∥Ls +k∥∞ < ∞. +(ii) For all t ∈ N, there exists Lq +t ∈ Xt:t+1 such that ∥Lq +t∥∞ < ∞ and that for all (xt, xt+1) ∈ Xt:t+1, +θ �→ qt,θ(xt, xt+1) is Lq +t(xt, xt+1)-Lipschitz. +Lemma 9 (A B.2(i) holds). Assume A B.8 and A 4.1. There exists a constant LV such that the +Lyapunov function V satisfies, for all (θ1, θ2) ∈ Θ2, +∥∇V (θ1) − ∇V (θ2)∥ ≤ LV ∥θ1 − θ2∥. +Proof. For all θ1, θ2, +∥∇V (θ1) − ∇V (θ2)∥ = ∥η0:t,θ1(s0:t,θ1) − η0:t,θ2(s0:t,θ2)∥ +≤ ∥η0:t,θ1(s0:t,θ1) − η0:t,θ1(s0:t,θ2)∥ + ∥η0:t,θ1(s0:t,θ2) − η0:t,θ2(s0:t,θ2)∥ . +By (3.1) and by [Gloaguen et al., 2022, Theorem 4.10] there exists a constant c such that +∥η0:t,θ1(s0:t,θ2) − η0:t,θ2(s0:t,θ2)∥ ≤ ct∥θ1 − θ2∥ supθ supk ∥sk(θ)∥∞ , +42 + +Using A 3.1 and A 4.1[i], we can write: +∥η0:t,θ1(s0:t,θ1) − η0:t,θ1(s0:t,θ2)∥ ≤ +t−1 +� +u=0 +η0:t,θ1 [∥su,θ1(xu:u+1) − su,θ2(xu:u+1)∥], +≤ +t−1 +� +u=0 +η0:t,θ1 [Ls +u(xu:u+1)] ∥θ1 − θ2∥, +≤ σ+ +σ− +supu∈�0,t−1� [Ls +u] ∥θ1 − θ2∥t. +Theorem 10 (Lipschitz continuity of Particle Gibbs with Backward Sampling). Assume A B.9. For +every t ∈ N, θ ∈ Θ and N ∈ N∗ +sup +x0:t∈X0:t +∥Kθ1,t(x0:t, .) − Kθ2,t(x0:t, .)∥TV ≤ LK +t,N∥θ1 − θ2∥ , +where +LK +t,N := +t−1 +� +ℓ=0 +¯τ −1 +ℓ +� +¯σ−1 +ℓ ++ (N − 1) +� +∥Lq +ℓ∥∞ . +(B.77) +Proof. We know that Kθ,t = Cm,θBt,θ. Therefore, by Lemmas 14, 16 and 19, we have that Kθ,t is +Lipschitz with constant equals LC +t + supθ Ct,θLB +t . +Corollary 1 (A 4.1(iii) holds.). Assume A B.9. For every t ∈ N, θ ∈ Θ, r ∈ N∗ and N ∈ N∗ such +that N > 1 + 5ρ2 +tt/2 +sup +x0:t∈X0:t +��Kr +θ1,t(x0:t, .) − Kr +θ2,t(x0:t, .) +�� +TV ≤ LP +t,N∥θ1 − θ2∥ +where +LP +t,N := (1 − κt,N)−1∥LK +t,N∥∞ +(B.78) +where LK +t,N is defined in (B.77). +Proof. Under B.8, the Particle Gibbs with backward sampling is geometrically ergodic with contraction +rate κt,N and thus LK +t,N is bounded and the result follows from Lemma 18 +Corollary 2 (A 4.1(i)). Assume A B.8 and A B.9. For all t ∈ N∗, (θ0, θ1) ∈ Θ2, +∥η0:t,θ0 − η0:t,θ1∥TV ≤ Lη∥θ0 − θ1∥, +where +Lη := LP +t,N ∗ , +(B.79) +and LP +t,N is defined in (B.78) and N ∗ = ⌈1 + 5ρ2 +t/2⌉. +Proof. Consider the following decomposition, valid for all k ∈ N∗ and N ≥ 1+5ρ2 +t/2, and all x0:t ∈ X0:t, +∥η0:t,θ1 − η0:t,θ2∥TV ≤ +��η0:t,θ1 − Kk +θ1,t(x0:t, ·) +�� +TV + +��η0:t,θ2 − Kk +θ2,t(x0:t, ·) +�� +TV + +��Kk +θ1,t(x0:t, ·) − Kk +θ2,t(x0:t, ·) +�� +TV +≤ +��η0:t,θ1 − Kk +θ1,t(x0:t, ·) +�� +TV + +��η0:t,θ2 − Kk +θ2,t(x0:t, ·) +�� +TV + LP +t,N∥θ1 − θ2∥ , +where we applied Corollary 1. Since the Lipschitz constant of Kθ,t is independent of k, and Kθ,t is +geometrically ergodic for all θ, we obtain by taking the limit when k goes to infinity with N fixed, +∥η0:t,θ1 − η0:t,θ2∥TV ≤ +∥LK +t,N∥∞ +1 − κt,N +∥θ1 − θ2 ∥ , +for all N ≥ 1 + 5ρ2 +t/2, where the dependence in N is hidden in LP +t,N. The result follows by choosing +N = ⌈1 + 5ρ2 +t/2⌉. +43 + +Remark 2. As noted by [Lindholm and Lindsten, 2018], the Lipschitz constant appearing in Corol- +lary 1 possesses an unexpected dependence on N − 1. One would expect it not to be true, in that we +know that Kθ,t converges geometrically fast and uniformly to η0:t and this is faster as N gets bigger. +Therefore, for large N the Lipschitz constant is expected to converge to that of η0:t whose Lipschitz +constant is independent of N. +Proposition 13 (Lipschitz continuity of θ �→ Kθ,tµ(βt)(id)). Assume A B.9. For every t ∈ N, θ ∈ Θ +and N ∈ N∗, +∥Kθ1,tµ(βt)(id) − Kθ2,tµ(βt)(id)∥∞ ≤ LK +t ∥θ1 − θ2∥ , +where +LK +t := (N − 1) +t−1 +� +ℓ=0 +¯τℓ∥Lq +ℓ∥∞ + +m +� +j=1 +∥L +← +− +Q +j ∥∞ +�m−1 +� +ℓ=0 +s∞ +ℓ +� ++ +m +� +j=1 +∥Ls +j∥∞ . +(B.80) +Proof. Consider e = (x0:t, y0:t) ∈ Et and fθ(e) := +� +Sm,θ(x0:t, d˜yt)µ(bt)(id). Then Kθ,tµ(bt)(id) = +Cm,θfθ(x0:t) is a composition of a Markov kernel and a Lipschitz function, therefore Lipschitz. +Corollary 3 (A 4.1(iv) holds.). Assume A B.9. For every t ∈ N, θ ∈ Θ and N ∈ N∗ +sup +x0:t∈X0:t +∥Pθ1,tH − Pθ2,tH∥ ≤ LP +2 ∥θ1 − θ2∥ , +where +LP +2 = LP +t,N + LK +t , +(B.81) +with LP and LK +t are defined in (B.80) and (B.78). +Proof. Let ˜f : Ek−k0 ∋ (x0:t[k0 : k], x0:t|t[k0 : k], bt[k0 : k]) �→ (k − k0)−1 �k +ℓ=k0+1 µ(bt[ℓ])(id). As Kθ,t +depends only on the path, with a slight abuse of notation, we can define fθ(x0:t) := K�k−k0 +θ,t +( ˜f)(x0:t). +By proposition 13, we have that fθ is Lipschitz with Lf = LK +t . Note that Pθ,tH(x0:t, yt) = Kk0 +θ,tfθ(x0:t), +therefore, by lemma 19 Lipschitz with constant LP + LK +t . +C. +Lipschitz properties +C.1 +Lipschitz continuity of Pθ, +In this section we prove the following items: +• Cm,θ(z0:m, ·) is Lipschitz, see Appendix C.1.1 +• Bm,θ(x0:m, ·) is Lipschitz, see Appendix C.1.2 +• +� +Sm,θ(x0:m, dbm)µ(bm)(Id) is Lipschitz, see Appendix C.1.3 +The following technical lemma will be useful. +Lemma 10. Let α ∈]0, 1], x ∈ R≥0 and ℓ ∈ N. +Then for all λi ∈ R≥0, i ∈ �0, ℓ�, such that +α ≥ �ℓ +i=0(1 − λix) it holds that α ≥ 1 − x �ℓ +i=0 λi. +Proof. Consider first the case where xλi ≤ 1 for all i ∈ �0, ℓ�. We prove the result by induction. The +case ℓ = 0 is straightforward. Assume now that the result holds for some r ∈ �0, ℓ − 1�. Then, +r+1 +� +i=0 +(1 − λix) = (1 − λr+1x) +r +� +i=0 +(1 − λix) ≥ (1 − λr+1x)(1 − x +r +� +i=0 +λi) += 1 − x +r+1 +� +i=0 +λi + x2 +r +� +i=0 +λiλr+1 ≥ 1 − x +r+1 +� +i=0 +λi . +44 + +Consider now the case where there is a index j ∈ �0, ℓ� such that xλj ≥ 1. +Then α ≥ 0 ≥ 1 − +(�ℓ +i=0 λi)x. +We begin with some important definitions. Let P and Q be probability distributions on some +common measurable space (X, X), and assume that these distributions admit densities p and q w.r.t +some common reference measure λ. Let M [P, Q] denote a maximal coupling between P and Q. As +in [Lindholm and Lindsten, 2018, Theorem 2], it is possible to explicitly construct one such maximal +coupling by +M [P, Q] (d(x, y)) := min{p(x), g(x)}λ(dx)δx(dy)+ +� +P(dx) − min{p(x), g(x)}λ(dx) +�� +Q(dy) − min{p(y), g(y)}λ(dy) +� +1 − λ +� +min{p, q} +� +. +(C.82) +From this definition it follows that for continuous and discrete dominating measures λ, +� +1{x=y}M [P, Q] d(x, y) = +� +min{p(x), g(x)}λ(dx) . +Moreover, for two Markov transition kernels K1 and K2 on (X, X), which are assumed to admit +transition densities with respect to some common dominating measure, we let, for (x1, x2) ∈ X2, +M [K1, K2] ((x1, x2), ·) denote the maximal coupling between the measures K1(x1, ·) and K2(x2, ·). +Defined in this way, M [K1, K2] defines a Markov transition kernel on the product space (X2, X �2) +The following Lemma will be crucial in what follows. +Lemma 11. +(i) Let (µ1, µ2) be two probability measures admitting a density with respect to a com- +mon dominating measure and let (K1, K2) two Markov transition kernels also admitting transition +densities with respect to some dominating measure. Then the probability measure +M [µ1, µ2] M [K1, K2] (d(x1, x2)) = +� +M [µ1, µ2] (d(z1, z2))M [K1, K2] ((z1, z2), d(x1, x2)), +is a coupling of (µ1K1, µ2K2), and it holds that +� +1x1=x2M [µ1K1, µ2K2] (d(x1, x2)) +≥ +� � +1z1=z21x1=x2M [µ1, µ2] (d(z1, z2))M [K1, K2] ((z1, z2), d(x1, x2)). +(ii) Let (µ1, · · · , µn) and (ν1, · · · , νn) be probability measures such that for all i ∈ �1, n�, µi and νi +admit densities with respect to the same dominating measure. Then �n +i=1 M [µi, νi] is a coupling +of �n +i=1 µi and �n +i=1 νi, and thus +� +n +� +i=1 +1xi=yiM +� n +� +i=1 +µi, +n +� +i=1 +νi +� +(d(x1, . . . , xn, y1, . . . , yn)) +≥ +� +n +� +i=1 +1xi=yi +n +� +i=1 +M [µi, νi] (d(x1, . . . , xn, y1, . . . , yn)). +Proof. It is enough to show that M [µ1, µ2] M [K1, K2] admits µ1K1 and µ2K2 as marginal distribu- +tions. This follows immediately from the fact that M [µ1, µ1] and M [K1, K2] admit the right marginal +45 + +distributions; indeed, +M [µ1, µ2]M [K1, K2] (X × A) += +� +M [µ1, µ2] (dz1, d2) M [K1, K2] (z1, z2, d(x1, x2)) 1X×A(x1, x2)1X2(z1, z2) += +� +M [µ1, µ2] (dz1, d2)K2(z2, A) += +� +µ2(dz2)K2(z2, A) += µ2K2(A). +The derivation for the first marginal distribution follows similarly. For the second point, since M [µ1, µ2] M [K1, K2] +is a coupling of (µ1K1, µ2K2) and M [µ1K1, µ2K2] is the maximal coupling, we have that +� +1x1=x2M [µ1K1, µ2K2] (d(x1, x2)) +≥ +�� +1x1=x2M [µ1, µ2] (d(z1, z2)) M [K1, K2] (z1, z2; d(x1, x2)) +≥ +�� +1x1=x21z1=z2M [µ1, µ2] (d(z1, z2)) M [K1, K2] (z1, z2; d(x1, x2)). +The proof of the second item follows similarly. +C.1.1 +θ �→ Cm,θ is Lipschitz. +We proceed by a coupling method that is inspired by [Lindholm and Lindsten, 2018, Theorem 2]. The +coupling we consider is that where the selection and mutation steps of the particle filter are respectively +coupled maximally. +Algorithm 6 Coupling Cm,θ +Data: θ1, θ2, ζ0:m +Result: x0:m,1, x0:m,1 +23 draw x0,1, x0,2 ∼ M [η0⟨ζ0⟩, η0⟨ζ0⟩] +24 for s ← 1 to t do +25 +draw (xs,1, xs,2) ∼ M [M s−1,θ1⟨ζs⟩(xs−1,1, ·), M s−1,θ2⟨ζs⟩(xs−1,2, ·)] +First, let us prove that the one step selection–mutation kernel is Lipschitz. +Lemma 12. For all t ∈ N, xt−1 ∈ Xt−1 and (θ1, θ2) ∈ Θ2, +� +1{x1=x2}M [Φt−1,θ1(µ(xt−1)), Φt−1,θ2(µ(xt−1))] (d(x1, x2)) ≥ 1 − +�N +i=1 λt +� +Lq +t−1(xi +t−1, ·) +� +N ¯τn +∥θ1 − θ2∥. +(C.83) +46 + +Proof. By A 3.1(i) and A 4.1(iii), +� +1{x1=x2}M [Φt−1,θ1(µ(xt−1)), Φt−1,θ2(µ(xt−1))] (d(x1, x2)) += +� +min +� N +� +i=1 +qt−1,θ1(xi +t−1, x) +�N +j=1 gt−1,θ1(xj +t−1) +, +N +� +i=1 +qt−1,θ2(xi +t−1, x) +�N +j=1 gt−1,θ2(xj +t−1) +� +λt(dx) +≥ +N +� +j=1 +� +min +� +qt−1,θ1(xi +t−1, x) +�N +j=1 gt−1,θ1(xj +t−1) +, +qt−1,θ2(xi +t−1, x) +�N +j=1 gt−1,θ2(xj +t−1) +� +λt(dx) +≥ +1 +�N +j=1 max +� +gt−1,θ1(xj +t−1), gt−1,θ2(xj +t−1) +� +N +� +j=1 +� +min +� +qt−1,θ1(xj +t−1, x), qt−1,θ2(xj +t−1, x) +� +λt(dx) +≥ +�N +j=1 max +� +gt−1,θ1(xj +t−1), gt−1,θ2(xj +t−1) +� +− �N +i=1 λt +� +Lq +t−1(xi +t−1, ·) +� +∥θ1 − θ2∥ +�N +j=1 max +� +gt−1,θ1(xj +t−1), gt−1,θ2(xj +t−1) +� +≥ 1 − +�N +i=1 λt +� +Lq +t−1(xi +t−1, ·) +� +N ¯τn +∥θ1 − θ2∥, +where we have used that +� +max(qt−1,θ1(xi +t−1, x), qt−1,θ2(xi +t−1, x))λt(dx) ≥ max +�� +qt−1,θ1(xi +t−1, x)λt(dx), +� +qt−1,θ2(xi +t−1, x)λt(dx) +� +≥ max(gt−1,θ1(xi +t−1), gt−1,θ2(xi +t−1)). +Lemma 13. For all t ∈ N, xt−1 ∈ Xt−1, z ∈ Xt and (θ1, θ2) ∈ Θ2, +∥M t−1,θ1⟨z⟩(xt−1, ·) − M t−1,θ2⟨z⟩(xt−1, ·)∥TV ≤ LM +t−1(xt−1)∥θ1 − θ2∥ +where LM +t−1(xt−1) = (1 − N −1)¯τ −1 +t−1 +�N +i=1 λt +� +Lq +t−1(xi +t−1, ·) +� +. +Proof. Let us denote by U�1, n� the uniform distribution on �1, n�. +By definition of the kernel +M t−1,θ⟨z⟩, we have that +M t−1,θ⟨z⟩(xt−1, dxt) = +� +U�1, n�(dj) +� +Φt−1(µ(xt−1))�j � δz � Φt−1(µ(xt−1))�(N−j−1)� +(dxt) +and thus, applying the two items of Lemma 11 combined with the fact that M [µ, µ] +� +d(x1, x2) +� += +47 + +µ(dx1)δx1(dx2) for any probability measure µ, we get that +� +1{xt,1=xt,2}M [M t−1,θ1⟨z⟩(xt−1, ·), M t−1,θ2⟨z⟩(xt−1, ·)] d(xt,1, xt,2) +≥ +� +1xt,1=xt,2,i1=i2M [U�1, n�, U�1, n�] +� +d(i1, i2) +� +× M [Φt−1,θ1(µ(xt−1)), Φt−1,θ2(µ(xt−1))]⊗i1 ⊗ M [δz, δz] +⊗ M [Φt−1,θ1(µ(xt−1)), Φt−1,θ2(µ(xt−1))]⊗N−i1−1 d(xt,1, xt,2) += 1 +N +N +� +i=1 +� +n +� +k=1,k̸=i +1xi +t,1=xi +t,2M [Φt−1,θ1(µ(xt−1)), Φt−1,θ2(µ(xt−1))] +� +d(xi +t,1, xi +t,2) +� +≥ +� +1 − +�N +i=1 λt +� +Lq +t−1(xi +t−1, ·) +� +N ¯τt−1 +∥θ1 − θ2∥ +�N−1 +≥ 1 − N − 1 +¯τt−1N +N +� +i=1 +λt +� +Lq +t−1(xi +t−1, ·) +� +∥θ1 − θ2∥ . +where we have applied Lemma 12 in the penultimate line and Lemma 10 in the last one. +Lemma 14. For every t ∈ N∗, there exists LC +t ∈ M(X0:t) such that +∥Ct,θ1(z0:t) − Ct,θ2(z0:t)∥TV ≤ LC +t (z0:t)∥θ1 − θ2∥ , +(C.84) +where LC +t (z0:t) = supθ Ct,θ +��t−1 +i=0 LM +i +� +(z0:t). Under AB.9(i), we obtain that ∥LC +t ∥∞ ≤ (N−1) �t−1 +ℓ=0 ¯τℓ∥Lq +ℓ∥∞. +Proof. This is a direct application of lemma 20. +C.1.2 +θ �→ Bt,θ(x0:t, ·) is Lipschitz +We start by recalling the definition of Bm +Bt,θ : X0:t × X0:t ∋ (x0:t, A) �→ +� +· · · +� +1A(x0:t) +�t−1 +� +s=0 +←− +Q s,µ(xs)(xs+1, dxs) +� +µ(xt)(dxt) . +(C.85) +Lemma 15. For all s ∈ �0, t�, xt+1 ∈ Xt+1, xt ∈ Xt and (θ1, θ2) ∈ Θ2 +���←− +Q s,µ(xs),θ1(xs+1, ·) − ←− +Q s,µ(xs),θ2(xs+1, ·) +��� +TV ≤ L +← +− +Q +s (xs+1, xs)∥θ1 − θ2∥ . +(C.86) +with L +← +− +Q +s (xs+1, xs) = (N ¯τt¯σs)−1 �N +i=1 Lq +s(xi +s, xs+1). Under AB.9(i), we have ∥L +← +− +Q +m∥∞ = (¯τm¯σm)−1∥Lq +m∥∞. +Proof. Note that ←− +Q t,µ(xt)(xt+1, ·) = �N +ℓ=1 +qt(xℓ +t,xt+1) +�N +ℓ′=1 qt(xℓ′ +t ,xt+1)δxℓ +t. Therefore, similarly to the proof of +Lemma 12, +� +1{xt,1=xt,2}M +�←− +Q t,µ(xt),θ1(xt+1, ·), ←− +Q t,µ(xt),θ2(xt+1, ·) +� +d(xt,1, xt,2) +≥ +�N +ℓ=1 max(qt,θ1(xℓ +t, xt+1), qt,θ2(xℓ +t, xt+1)) − Lq +t(xℓ +t, xt+1)∥θ1 − θ2∥ +�N +ℓ=1 max(qt,θ1(xℓ +t, xt+1), qt,θ2(xℓ +t, xt+1)) +≥ 1 − +�N +ℓ=1 Lq +t(xℓ +t, xt+1) +N ¯τt¯σt +∥θ1 − θ2∥ . +48 + +Lemma 16. For all t ∈ N, x0:t ∈ X0:t and (θ1, θ2) ∈ Θ2 +∥Bt,θ1(x0:t, ·) − Bt,θ2(x0:t, ·)∥TV ≤ LB +t (x0:t)∥θ1 − θ2∥ +(C.87) +where LB +t (x0:t) = supθ Bt +��t−1 +i=0 L +← +− +Q +i +� +(x0:t). Under AB.9(i), we have that ∥LB +t ∥∞ = �t−1 +i=0(¯τi¯σi)−1∥Lq +i ∥∞. +Proof. Apply lemma 19 and lemma 15. +C.1.3 +θ �→ +� +St,θ(x0:t, dbt)µ(bt)(id) is Lipschitz +Define the backward ancestors kernel +Bθ,t : Xt+1 × Xt × σ(�1, N�) �→ +� +1A(˜j) +� N +� +ℓ=1 +qt(xℓ +t, xt+1) +�N +ℓ′=1 qt(xℓ′ +t , xt+1) +δℓ(d˜j) +� +. +Lemma 17. (Bθ,t is Lipschitz) For every m ∈ �0, t�, there exists LBK +m +∈ M(X m:m+1) such that +∥Bθ1,m(xm+1, xm) − Bθ2,m(xm+1, xm)∥TV ≤ L +← +− +Q +m(xm+1, xm)∥θ1 − θ2∥ , +(C.88) +where L +← +− +Q +s +is defined in Lemma 15 +Proof. Bθ,s is the index version of the kernel (C.85) and thus it is Lipschitz with the same constant. +Proposition 14. For every m ∈ �0, t�, we have that +�� +� +CmSm,θ(z0:m, dbm)µ(bm)(Id) +�� ≤ +m−1 +� +ℓ=0 +s∞ +ℓ +(C.89) +and +���� +� +Sm,θ1(x0:m, dbm)µ(bm)(Id) − +� +Sm,θ2(x0:m, dbm)µ(bm)(Id) +���� ≤ LSµ +m (x0:m)∥θ1 − θ2∥ . +(C.90) +where LSµ +m (x0:m) = N −1 �N +i=1 LB +m(xk +m, x0:m) and LB +m is defined recursively as +LB +m+1(xk +m+1, x0:m) = L +← +− +Q +m(xk +m+1, xm) +m +� +ℓ=0 +s∞ +ℓ + +� +Bθ,m(xk +m+1, xm, dJ) +� +Ls +m(xJ +m, xk +m+1) + LB +m(xJ +m, x0:m−1) +� +. +(C.91) +In particular, under A B.9, we have that LB +m ≤ �m +j=1 ∥L +← +− +Q +j ∥∞ +��m−1 +ℓ=0 s∞ +ℓ +� ++ �m +j=1 ∥Ls +j∥∞. +Proof. Consider the following kernels, +�Sm,θ(x0:m+1, d(Ji,j +0 , . . . , Ji,j +m )N,M +i=1,j=1) := +m +� +ℓ=0 +N +� +k=1 +�Sℓ,θ(xk +ℓ+1, xℓ, d +� +Jk,j +ℓ +�M +j=1) , +(C.92) +�Sℓ,θ(xk +ℓ+1, xℓ, d(Jk,j +ℓ )M +j=1) := +M +� +j=1 +Bθ,ℓ(xk +ℓ+1, xℓ, dJk,j +ℓ ) . +(C.93) +Define for all k ∈ [1 : N], m ∈ N>0, +Bm+1,k : θ �→ +� +�Sm,θ(x0:m+1, d +� +Ji,j +0 , . . . , Ji,j +m +�N,M +i=1,j=1)bk +m+1 +� +x0:m+1, +� +Ji,j +0 , . . . , Ji,j +m +�N,M +i=1,j=1 +� +, +49 + +where bk +m+1 +� +x0:m+1, +� +Ji,j +0 , . . . , Ji,j +m +�N,M +i=1,j=1 +� +is defined recursively as +bk +m+1 +� +x0:m+1, +� +Ji,j +0 , . . . , Ji,j +m +�N,M +i=1,j=1 +� += M −1 +M +� +ℓ=1 +bJk,ℓ +m +m +� +x0:m, +� +Ji,j +0 , . . . , Ji,j +m−1 +�N,M +i=1,j=1 +� ++sm,θ(xJk,ℓ +m +m , xk +m+1). +For notational convenience, we henceforth drop the arguments and simply write bk +m+1. +We herebelow show that Bm+1,k is Lipschitz with constant LB +m(xk +m+1, xm) and bounded by �m−1 +ℓ=0 s∞ +ℓ . +For m > 2 and k ∈ [1 : N], +Bm+1,k(θ) = +� +�Sm,θ(x0:m+1, d(Ji,j +0 , . . . , Ji,j +m )N,M +i=1,j=1)bk +m+1 += +� +· · · +� +�Sm−1,θ(x0:m, d(Ji,j +0 , . . . , Ji,j +m−1)N,M +i=1,j=1)�Sm,θ(xk +m+1, xm, d(Jk,j +m )M +j=1) +× +� +M −1 +M +� +ℓ=1 +bJk,ℓ +m +m ++ sm,θ(xJk,ℓ +m +m , xk +m+1) +� += +� +· · · +� +�Sm,θ(xk +m+1, xm, d{Jk,j +m }M +j=1) +� +M −1 +M +� +ℓ=1 +� +sm,θ(xJk,ℓ +m +m , xk +m+1) ++ +� +�Sm−1,θ(x0:m, d(Ji,j +0 , . . . , Ji,j +m−1)N,M +i=1,j=1)bJk,ℓ +m +m +�� += +� +· · · +� +�Sm,θ(xk +m+1, xm, d(Jk,j +m )M +j=1) +� +M −1 +M +� +ℓ=1 +� +sm,θ(xJk,ℓ +m +m , xk +m+1) + Bm,Jk,ℓ +m (θ) +�� += +� +Bθ,m(xk +m+1, xm, dJ) +� +sm,θ(xJ +m, xk +m+1) + Bm,J(θ) +� +Applying the induction hypothesis conditionally on Jk,ℓ +m , Bm,Jk,ℓ +m is Lipschitz with constant LB +m(xJk,ℓ +m +m , x0:m−1) +and thus the Lipschitz constant of Bm+1,k is +LB +m+1(xk +m+1, x0:m) = L +← +− +Q +m(xk +m+1, xm) +m +� +ℓ=0 +s∞ +ℓ + +� +Bθ,m(xk +m+1, xm, dJ) +� +Ls +m(xJ +m, xk +m+1) + LB +m(xJ +m, x0:m−1) +� +. +(C.94) +where we have used the fact that Bθ,m and sm,θ are also Lipschitz. Again by induction Bm+1,k is +bounded uniformly by �m +ℓ=0 s∞ +ℓ . The induction is concluded by noting that for the base case m = 0, +βk +m = 0 for all k ∈ N and thus the result holds. +It now remains to check that for all θ ∈ Θ, m ∈ �0, t� and k ∈ [1 : N], +Bm,k(θ) = +� +Sm(x0:m, dbm)bk +m . +50 + +Again, we proceed by induction. +� +Sm(x0:m, dbm)bk +m += +� +· · · +� +Sm−1(x0:m−1, dbm−1)Sm(bm−1, xm−1:m, dbm)bk +m += +� +· · · +� +Sm−1(x0:m−1, dbm−1) +× +M +� +j=1 +� N +� +p=1 +qm−1(xp +m−1, xk +m) +�N +ℓ=1 qm−1(xℓ +m−1, xkm) +δxp +m−1,bp +m−1 +� +d(˜xk,j +m−1,˜bk,j +m−1) +� +� +× +� +M −1 +M +� +n=1 +� +˜bk,n +m−1 + sm,θ(˜xk,n +m−1, xk +m) +�� += +� +· · · +� +Sm−1(x0:m−1, dbm−1) +× +M +� +j=1 +� N +� +p=1 +qm−1(xp +m−1, xk +m) +�N +ℓ=1 qm−1(xℓ +m−1, xkm) +δp(dJk,j +m−1) +� � +M −1 +M +� +n=1 +� +b +Jk,n +m−1 +m−1 + sm,θ(x +Jk,n +m−1 +m−1 , xk +m) +�� += +� +· · · +� +�Sm,θ(xk +m−1, xℓ−1, d(Jk,j +ℓ−1)M +j=1) +× +� +M −1 +M +� +ℓ=1 +� +sm,θ(x +Jk,ℓ +m−1 +m−1 , xk +m) + Sm−1(x0:m−1, dbm−1)b +Jk,ℓ +m−1 +m−1 +�� += +� +· · · +� +�Sm,θ(xk +m−1, xℓ−1, d(Jk,j +ℓ−1)M +j=1) +× +� +M −1 +M +� +ℓ=1 +� +sm,θ(x +Jk,ℓ +m−1 +m−1 , xk +m) + +� +Sm−1(x0:m−1, dbm−1)b +Jk,ℓ +m−1 +m−1 +�� += +� +· · · +� +�Sm,θ(xk +m−1, xℓ−1, d(Jk,j +ℓ−1)M +j=1) +� +M −1 +M +� +ℓ=1 +� +sm,θ(x +Jk,ℓ +m−1 +m−1 , xk +m) + Bm−1,Jk,ℓ +m−1(θ) +�� += Bm,k(θ) +The proof is finalized by noting that +� +Sm(x0:m, dbm)µ(bm)(Id) = N −1 +N +� +k=1 +Bm,k(θ) +and thus it is Lipschitz with constant LSµ +m (x0:m) = N −1 �N +i=1 LB +m(xk +m, xm−1). +C.2 +Lipschitz properties of Markov Kernels +Lemma 18 (Composition of ergodic Lipschitz kernels is lipschitz). Let Pθ be a Markov kernel over +X × Y that is uniformly π-geometrically ergodic for any θ with contraction constant ρ independent of +θ and such that there exists Lp > 0 such that for every x ∈ X +∥Pθ0(x, ·) − Pθ1(x, ·)∥TV ≤ LP ∥θ0 − θ1∥. +Then, for all k > 0 +��P k +θ0(x, ·) − P k +θ1(x, ·) +�� +TV ≤ +LP +1 − ρ∥θ0 − θ1∥. +51 + +Proof. We use the following decomposition borrowed from [Fort et al., 2011]. For any k ≥ 1, +P k +θ0f − P k +θ1f = +k−1 +� +j=0 +P j +θ0(Pθ0 − Pθ1) +� +P k−j−1 +θ1 +f − πf +� +. +Then, for any f s.t. ∥f∥∞ ≤ 1 and x ∈ X, +|P k +θ0f(x) − P k +θ1f(x)| ≤ +k−1 +� +j=0 +���� +� +P j +θ0(x, dy) sup +z∈X +|P k−j−1 +θ1 +f(z) − πf| +���� LP ∥θ0 − θ1∥ +≤ LP +� k−1 +� +j=0 +ρk−j−1 +� +∥θ0 − θ1∥ +≤ +LP +1 − ρ∥θ0 − θ1∥. +Lemma 19 (Composition of Lipschitz kernels is lipschitz). Let Pθ, Qθ be two kernels defined over +X × Y and Y × Z such that for ever x ∈ X, y ∈ Y there are Lp ∈ M(X), Lq ∈ M(Y ) that satisfy +∥Pθ0(x, ·) − Pθ1(x, ·)∥TV ≤ Lp(x)∥θ0 − θ1∥ +and +∥Qθ0(y, ·) − Qθ1(y, ·)∥TV ≤ Lq(y)∥θ0 − θ1∥ . +Then +∥Pθ0Qθ0(x, ·) − Pθ1Qθ1(x, ·)∥TV ≤ Lpq∥θ0 − θ1∥ , +where Lpq = (supθ PθLq(x) + Lp(x) supy supθ Qθ(y, Z)). +Proof. Let f ∈ M such that ∥f∥∞ ≤ 1. +∥Pθ1Qθ1f − Pθ2Qθ2f∥ ≤ ∥Pθ1 [Qθ1f − Qθ2f] ∥ + ∥(Pθ1 − Pθ2)Qθ2f∥ +≤ (Pθ1Lq(x) + Lp(x)∥Qθ2f∥∞)∥θ1 − θ2∥ . +Corollary 4. Let Pθ, Qθ be two Markov kernels defined over X × Y and Y × Z such that for ever +x ∈ X, y ∈ Y there are Lp ∈ M(X), Lq ∈ M(Y ) that satisfy +∥Pθ0(x, ·) − Pθ1(x, ·)∥TV ≤ Lp(x)∥θ0 − θ1∥ +and +∥Qθ0(y, ·) − Qθ1(y, ·)∥TV ≤ Lq(y)∥θ0 − θ1∥ . +Then +∥Pθ0Qθ0(x, ·) − Pθ1Qθ1(x, ·)∥TV ≤ Lpq∥θ0 − θ1∥ , +where Lpq = (supθ PθLq(x) + Lp(x)). +Lemma 20 (Product of Lipschitz kernels is lipschitz). Let Pθ, Qθ be two markov kernels that are +uniformly Lipschitz with constants LP , LQ. Then Pθ�Qθ is uniformly Lipschitz with constant LP +LQ. +Proof. Let hθ : y �→ +� +Qθ(y, dz)f(y, z). Then (Pθi ⊗ Qθi)(f) = Pθi(hθi) and the proof is similar to +that of the previous Lemma since hθ is Lipschitz with constant LQ and ∥hθ∥∞ ≤ 1. +52 + +0 +10 +20 +30 +40 +50 +K +10 +1 +100 +101 +102 +N = 10 +N = 25 +N = 50 +N = 100 +PaRIS +0 +10 +20 +30 +40 +50 +K +10 +1 +100 +101 +102 +N = 10 +N = 25 +N = 50 +N = 100 +PaRIS +Figure 3: Output of the PPG roll-out estimator for the LGSSM. The curves describe the evolution of +the bias with increasing k for different particle sample sizes N. The left and right panels correspond +to k0 = k − 1 and k0 = ⌊k/2⌋, respectively. +D. +Additional numerical results +D.1 +PPG +D.2 +Learning +For both experiments, all the parameters were initialized by sampling from a centered multivari- +ate gaussian distribution with covariance matrix of 0.01I. +We have used the ADAM optimizer +[Kingma and Ba, 2014] with a learning rate decay of 1/ +√ +ℓ where ℓ is the iteration index, with a +starting learning rate of 0.2. We rescale the gradients by T. +LGSSM +For LGSSM we evaluated for fixed number of particles (N = 64) and number of gibbs +iterations (k = 8) the influence of the burn-in phase (k0) over the final distance obtained to the +MLE estimator. Table 3 indicates that configurations with smaller k0 perform better. 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Particle approximations +of the score and observed information matrix in state space models with application to parameter +estimation. Biometrika, 98(1):65–80. +[S¨arkk¨a, 2013] S¨arkk¨a, S. (2013). Bayesian Filtering and Smoothing. Cambridge University Press. +[Singh et al., 2017] Singh, S. S., Lindsten, F., and Moulines, E. (2017). Blocking strategies and stability +of particle gibbs samplers. Biometrika, 104(4):953–969. +[Whiteley, 2010] Whiteley, N. (2010). Discussion on particle markov chain monte carlo methods. pages +306–307. +[Zhao et al., 2021] Zhao, Y., Nassar, J., Jordan, I., Bugallo, M., and Park, I. M. (2021). Streaming +variational monte carlo. +56 + diff --git a/idAyT4oBgHgl3EQf-_pB/content/tmp_files/load_file.txt b/idAyT4oBgHgl3EQf-_pB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b666dedaf77cd2c2f35a64bdcecebc695673e39c --- /dev/null +++ b/idAyT4oBgHgl3EQf-_pB/content/tmp_files/load_file.txt @@ -0,0 +1,3216 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf,len=3215 +page_content='State and parameter learning with PARIS particle Gibbs Gabriel Cardoso†, Yazid Janati El Idrissi‡, Sylvain Le Corff⋆, ´Eric Moulines†, and Jimmy Olsson⊤ †CMAP, ´Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ‡Samovar, T´el´ecom SudParis, d´epartement CITI, TIPIC, Institut Polytechnique de Paris, Palaiseau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ⋆LPSM, Sorbonne Universit´e, UMR CNRS 8001, 4 Place Jussieu, 75005 Paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ⊤Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Abstract Non-linear state-space models, also known as general hidden Markov models, are ubiquitous in statistical machine learning, being the most classical generative models for serial data and sequences in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The particle-based, rapid incremental smoother (PARIS) is a sequential Monte Carlo (SMC) technique allowing for efficient online approximation of expectations of additive functionals under the smoothing distribution in these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Such expectations appear naturally in several learning contexts, such as likelihood estimation (MLE) and Markov score climbing (MSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' PARIS has linear computational complexity, limited memory requirements and comes with non-asymptotic bounds, convergence results and stability guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Still, being based on self- normalised importance sampling, the PARIS estimator is biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Our first contribution is to design a novel additive smoothing algorithm, the Parisian particle Gibbs (PPG) sampler, which can be viewed as a PARIS algorithm driven by conditional SMC moves, resulting in bias-reduced estimates of the targeted quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We substantiate the PPG algorithm with theoretical results, including new bounds on bias and variance as well as deviation inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Our second contribution is to apply PPG in a learning framework, covering MLE and MSC as special examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In this context, we establish, under standard assumptions, non-asymptotic bounds highlighting the value of bias reduction and the implicit Rao–Blackwellization of PPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' These are the first non-asymptotic results of this kind in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We illustrate our theoretical results with numerical experiments supporting our claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Introduction Sequential Monte Carlo (SMC) methods, or particle filters, are simulation-based approaches used for the online approximation of posterior distributions in the context of Bayesian inference in state space models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In nonlinear hidden Markov models (HMM), they have been successfully applied for ap- proximating online the typically intractable posterior distributions of sequences of unobserved states (Xs1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , Xs2) given observations (Yt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , Yt2) for 0 ≤ s1 ≤ s2 and 0 ≤ t1 ≤ t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Standard SMC methods use Monte Carlo samples generated recursively by means of sequential importance sampling and resampling steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A particle filter approximates the flow of marginal posteriors by a sequence of occupation measures associated with a sequence {ξi t}N i=1, t ∈ N, of Monte Carlo samples, each parti- cle ξi t being a random draw in the state space of the hidden process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Particle filters revolve around two operations: a selection step duplicating/discarding particles with large/small importance weights, respectively, and a mutation step evolving randomly the selected particles in the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Ap- plying alternatingly and iteratively selection and mutation results in swarms of particles being both temporally and spatially dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The joint state posteriors of an HMM can also be interpreted as laws associated with a certain kind of Markovian backward dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' this interpretation is useful, 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='00900v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='ME] 2 Jan 2023 for instance, when designing backward-sampling-based particle algorithms for nonlinear smoothing [Douc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2011, Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Throughout the years, several convergence results as the number N of particles tends to infinity have been established;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', [Del Moral, 2004, Douc and Moulines, 2008, Capp´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2005] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In addition, a number of non-asymptotic results have been established, including time-uniform bounds on the SMC Lp error and bias as well as bounds describing the propagation of chaos among the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Extensions to the backward-sampling-based particle algorithms can also be found for instance in [Douc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2011, Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2010, Dubarry and Le Corff, 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In this paper, we focus on the problem of recursively computing smoothed expectations η0:tht = E[ht(X0:t) | Y0:t] for additive functionals ht in the form ht(x0:t) := t−1 � s=0 ˜hs(xs:s+1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1) where X0:n and Y0:n denote vectors of states and observations (see below for precise definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Such expectations appear frequently in the context of maximum-likelihood parameter estimation in nonlin- ear HMMs, for instance, when computing the score function (the gradient of the log-likelihood function) or the Expectation Maximization intermediate quantity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' see [Capp´e, 2001, Andrieu and Doucet, 2003, Poyiadjis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2005, Capp´e, 2011, Poyiadjis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The particle-based, rapid incremental smoother (PARIS) proposed in [Olsson and Westerborn, 2017] is tailored for solving online this additive smooth- ing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' When the transition density of the latent states is lower and upper bounded, this algorithm can be shown to have a linear computational complexity in the number N of particles and limited memory requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' An interesting feature of the PARIS, which samples on-the-fly from the backward dynamics induced by the particle filter, is that it requires two or more backward draws per particle to cope with the degeneracy of the sampled trajectories and remain numerically stable in the long run, with an asymptotic variance that grows only linearly with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In this paper, we introduce a method to reduce the bias of the PARIS estimator of η0:tht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The idea is to mix—by introducing a conditional PARIS algorithm—the PARIS algorithm with a backward- sampling-based version of the particle Gibbs sampler [Andrieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2010b, Lindsten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2014a, Chopin and Singh, 2015a, Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2016, Del Moral and Jasra, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This leads to a batch mode PARIS particle Gibbs (PPG) sampler, which we furnish with an upper bound of the bias that decreases inversely proportionally to the number N of particles and exponentially fast with the particle Gibbs iteration index (under the assumption that the particle Gibbs sampler is uniformly ergodic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' As an application we consider the problem of likelihood maximization with stochastic gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In this specific context, where the smoothing estimator is employed repeatedly to produce mean-field estimates, controlling the bias becomes critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Thus, it is natural to aim at minimizing the bias for a fixed computational budget, provided that the variance does not explode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For this reason, bias reduction in stochastic simulation has been the subject of extensive research during the last decades [Jacob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2020, Glynn and Rhee, 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The present paper contributes to this line of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In particular, we show that stochastic approximation (SA) with PPG achieves a O(log(n)/√n) rate, where n is the number of SA steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This improves on a previous result of [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ], which establishes the almost sure convergence (to a stationary point of the likelihood) of an SA Expectation Maximization (EM) algorithm based on particle Gibbs with ancestor sampling (PGAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In Section 2, we recall the hidden Markov model framework, the particle filter and the PARIS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In Section 3, we lay out the PPG algorithm and present the first central result of this paper, an upper bound on the bias of our estimator as a function of the number of particles and the iteration index of the Gibbs algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In addition, we provide an upper bound on the mean-squared error (MSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In Section 4, we undertake the learning problem and present the second result of this paper, a O(log(n)/√n) non-asymptotic bound on the expectation of the squared gradient norm taken at a random index K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1, we illustrate our results through numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' All the proofs are collected in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 2 Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For a given measurable space (X, X), where X is a countably generated σ-algebra, we denote by F(X) the set of bounded X/B(R)-measurable functions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For any h ∈ F(X), we let ∥h∥∞ := supx∈X |h(x)| and osc(h) := sup(x,x′)∈X2 |h(x) − h(x′)| denote the supremum and oscillator norms of h, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let M(X) be the set of σ-finite measures on (X, X) and M1(X) ⊂ M(X) the probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For any h ∈ F(X) and µ ∈ M(X) we write µ(f) = � h(x)µ(dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For a Markov kernel K from (X, X) to another measurable space (Y, Y), we define the measurable function Kh : X ∋ x �→ � h(y)K(x, dy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The composition µK is a probability measure on (Y, Y) such that µK : X ∋ A �→ � µ(dx)K(x, dy)1A(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all sequences {au}u∈Z and {bu}u∈Z, and all s ≤ t we write as:t = {as, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , at} and bs:t = {bs, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , bt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 Hidden Markov models Hidden Markov models consist of an unobserved state process {Xt}t∈N and observations {Yt}t∈N, where, at each time t ∈ N, the unobserved state Xt and the observation Yt are assumed to take values in some general measurable spaces (Xt, Xt) and (Yt, Yt), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' It is assumed that {Xt}t∈N is a Markov chain with transition kernels {Mt+1}t∈N and initial distribution η0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Given the states {Xt}t∈N, the observations {Yt}t∈N are assumed to be independent and such that for all t ∈ N, the conditional distribution of the observation Yt depends only on the current state Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This distribution is assumed to admit a density gt(Xt, ·) with respect to some reference measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In the following we assume that we are given a fixed sequence {yt}t∈N of observations and define, abusing notations, gt(·) = gt(·, yt) for each t ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We denote, for 0 ≤ s ≤ t, Xs:t := �t u=s Xu and Xs:t := �t u=s Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Consider the unnormalized transition kernel Qs : Xs × Xs+1 ∋ (x, A) �→ gs(x)Ms(x, A) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2) and let γ0:t : X0:t ∋ A �→ � 1A(x0:t) η0(dx0) t−1 � s=0 Qs(xs, dxs+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3) Using these quantities, we may define the joint-smoothing and predictor distributions at time t ∈ N as η0:t : X0:t ∋ A �→ γ0:t(A) γ0:t(X0:t), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='4) ηt : Xt ∋ A �→ η0:t(X0:t−1 × A), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='5) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' It can be shown (see [Capp´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2005, Section 3]) that η0:t and ηt are the condi- tional distributions of X0:t and Xt given Y0:t−1 respectively, evaluated at y0:t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Unfortunately, these distributions, which are vital in Bayesian smoothing and filtering as they enable the estimation of hidden states through the observed data stream, are available in a closed form only in the cases of linear Gaussian models or models with finite state spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' see [Capp´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2009] for a comprehensive coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2 Particle filters For most models of interest in practice, the joint smoothing and predictor distributions are intractable, and so are also any expectation associated with these distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Still, such expectations can typically be efficiently estimated using particle methods, which are based on the predictor recursion ηt+1 = ηtQt/ηtgt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' At time t, if we assume that we have at hand a consistent particle approximation of ηt, formed by N random draws {ξi t}N i=1, so-called particles, in Xt and given by ηN t = N −1 �N i=1 δξi t, plugging ηN t into the recursion tying ηt+1 and ηt yields the mixture ηN t Qt, from which a sample of N new particles can be drawn in order to construct ηN t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' To do so, we sample, for all 1 ≤ i ≤ N, ancestor 3 indices αi t ∼ Categorical({gt(ξℓ t)}N ℓ=1) and then propagate ξi t+1 ∼ Mt(ξαi t t , ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This procedure, which is initialized by sampling the initial particles {ξi 0}N i=1 independently from η0, describes the particle filter with multinomial resampling and produces consistent estimators such that for every h ∈ F(Xt), ηN t (h) converges almost surely to ηt(h) as the number N of particles tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This procedure can also be extended to produce particle approximations of the joint-smoothing distributions {η0:t}t∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Note that the successive ancestor selection steps described previously generates an ancestor line for each terminal particle ξi t, which we denote by ξi 0:t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' It can then be easily shown that ηN 0:t = N −1 �N i=1 δξi 0:t forms a particle approximation of the joint-smoothing distribution η0:t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' However, it is well known that the same selection operation also depletes the ancestor lines, since, at each step, two different particles are likely to originate from the same parent in the previous generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Thus, eventually, all the particles end up having a large portion of their initial ancestry in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This means that in practice, this naive approach, which we refer to as the poor man’s smoother, suffers generally from high variance when used for estimating joint-smoothing expectations of objective functionals depending on the whole state trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3 Backward smoothing and the PARIS algorithm We now discuss how to avoid the problem of particle degeneracy relative to the smoothing problem by means of so-called backward sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' While this line of research has broader applicability, we restrict ourselves for the sake of simplicity to the case of additive state functionals in the form ht(x0:t) := t−1 � s=0 ˜hs(xs:s+1), x0:t ∈ X0:t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6) Appealingly, using the poor man’s smoother described in the previous section, smoothing of additive functionals can be performed online alongside the particle filter by letting, for each s, ηN 0:shs := N −1 N � i=1 βi s, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='7) where the statistics {βi s}N i=1 satisfy the recursion βi s+1 = βαi s s + ˜hs(ξαi s s , ξi s+1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='8) where αi s is, as described, the ancestor at time s of particle ξi s+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' As mentioned above, the previous estimator suffers from high variance when s is relatively large with respect to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' However, assume now that the model is fully dominated in the sense that each state process kernel Ms has a transition density ms with respect to some reference measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' then, interestingly, it is easily seen that the conditional probability that αi s = j given the offspring ξi s+1 and the ancestors {ξℓ s}N ℓ=1 is given by Λs(i, j) := ωj sms(ξj s, ξi s+1) �N ℓ=1 ωℓsms(ξℓs, ξi s+1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='9) Here Λs forms a backward Markov transition kernel on �1, N� × �1, N�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Using this observation, we may avoid completely the particle-path degeneracy of the poor man’s smoother by simply replacing the naive update (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='8) by the Rao–Blackwellized counterpart βi s+1 = N � j=1 Λs(i, j){βj s + ˜hs(ξj s, ξi s+1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='10) This approach, proposed in [Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2010], avoids elegantly the path degeneracy as is elimi- nates the ancestral connection between the particles by means of averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Furthermore, it is entirely 4 online since at step s only the particle populations ξ1:N s and ξ1:N s+1 are needed to perform the update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Still, a significant drawback is the overall O(N 2) complexity for the computation of β1:N t , since the calculation of each βi s+1 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='10) involves the computation of N 2 terms, which can be prohibitive when the number N of particles is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Thus, in [Olsson and Westerborn, 2017], the authors propose to sample M ≪ N conditionally independent indices {Ji,j s }M j=1 from the distribution Λs(i, ·) and to update the statistics according to βi s+1 = M −1 M � j=1 � βJi,j s s + ˜hs(ξJi,j s s , ξi s+1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='11) If the transition density ms is uniformly bounded from above and below, an accept-reject approach allows the sampling-based update (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='11) to be performed for i ∈ �1, N� at an O(N(M + 1)) overall complexity if a pre-initialized multinomial sampler is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A key aspect of this approach is that the number M of sampled indices at each step can be very small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' indeed, for any fixed M ≥ 2, the algorithm, which is referred to as the PARIS, can be shown to be stochastically stable with an O(t) variance (see [Olsson and Westerborn, 2017, Section 1] for details), and setting M to 2 or 3 yields typically fully satisfying results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The PARIS estimator can be viewed as an alternative to the FFBSm, rather than the FFBSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Even if the PARIS and FFBSi are both randomised versions of the FFBSm estimator, the PARIS is of a fundamentally different nature than the FFBSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The PARIS approximates the forward-only FFBSm online in the context of additive functionals by approximating each updating step by additional Monte Carlo sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The sample size M is an accuracy parameter that determines the precision of this approximation, and by increasing M the statistical properties of the PARIS approaches those of the forward-only FFBSm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' On the other hand, as shown in [Douc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2011, Corollary 9], the asymptotic variance of FFBSi is always larger than that of the FFBSm, with a gap given by the variance of the state functional under the joint-smoothing distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Thus, we expect, especially in the case of a low signal-to-noise ratio, the PARIS to be more accurate than the FFBSi for a given computational budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Another important reason to focus on the PARIS estimator rather than the FFBSi is the appealing online properties of the latter, whose interplay with and relevance to the particle MCMC methodology is to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Our results can be naturally extended to the FFBSi and PGAS but since the PARIS has a practical edge, we chose to center our contribution around it although the main idea behind our paper is more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' PARIS particle Gibbs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 Particle Gibbs methods The conditional particle filter (CPF) introduced in [Andrieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2010a] serves the basis of a particle- based MCMC algorithm targeting the joint-smoothing distribution η0:t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let ℓ ∈ N∗ be an iteration index and ζ0:t[ℓ] a conditional path used at iteration ℓ of the CPF to construct a particle approximation of η0:t as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' At step s ∈ �1, t� of the CPF, a randomly selected particle, with uniform probability 1/N, is set to ζs[ℓ], whereas the remaining N − 1 particles are all drawn from the mixture ηN s−1Qs−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' At the final step, a new particle path ζ0:t[ℓ + 1] is drawn either: by selecting randomly, again with uniform probability 1/N, a genealogical trace from the an- cestral tree of the particles {ξ1:N s }t s=0 produced by the CPF, as in the vanilla particle Gibbs sampler;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' or by generating the path by means of backward sampling, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', by drawing indices J0:t backwards in time according to Jt ∼ Categorical({1/N}N i=1) and, conditionally to Js+1, Js ∼ Λs(Js+1, ·), s ∈ �0, t−1�, and letting ζ0:t[ℓ+1] = (ξJ0 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ξJt t ) (where the transition kernels {Λs}t s=0, defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='9), are induced by the particles produced by the CPF), as proposed in [Whiteley, 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 5 The theoretical properties of the different versions of the particle Gibbs sampler are well studied [Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2017, Chopin and Singh, 2015b, Andrieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In short, the produced conditional paths (ζ0:t[ℓ])ℓ∈N form a Markov chain whose marginal law converges geometrically fast in total vari- ation to the target distribution η0:t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' As it is the case for smoothing algorithms, the vanilla particle Gibbs sampler suffers from bad mixing due to particle path degeneracy while its backward-sampling counterpart exhibits superior performance as t increases [Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2 The PPG algorithm Remarkably, in order for the standard particle Gibbs samplers to output a single conditional path, a whole particle filter is run and then discarded, resulting in significant waste of computational work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Thus, we now introduce a variant of the PARIS algorithm, coined the PARIS particle Gibbs (PPG), in which the conditional path of particle Gibbs with backward sampling is merged with the intermediate particles, ensuring less computational waste and reduced bias with respect to the vanilla PARIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In the following we let t ∈ N be a fixed time horizon, and describe in detail how the PPG ap- proximates iteratively η0:tht, where ht is an additive functional in the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Using a given conditional path ζ0:t[ℓ − 1] as input, the ℓ-th iteration of the PPG outputs a many-body system υt[ℓ] = ((ξ1 0:t, β1 t ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , (ξN 0:t, βN t )) comprising N backward particle paths {ξi 0:t}N i=1 with associated PARIS statistics {βi t}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This is the so-called conditional PARIS update detailed in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' After this, an updated conditional path is selected with probability 1/N among the N particle paths {ξi 0:t}N i=1 and used as input in the next conditional PARIS operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' At each iteration, the produced statistics {βi t}N i=1 provide an approximation of η0:tht according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The overall algorithm is summarized in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The function CPFs describes one step of the conditional particle filter and is given in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In addition, the PPG algorithm defines a Markov chain with Markov transition kernel denoted by Kt and detailed in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Algorithm 1 One conditional PARIS update (CondPaRIS) Input: {(ξi 0:s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' βi s)}N i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ζs+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ˜hs−1 Result: {(ξi 0:s+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' βi s+1)}N i=1 1 draw ξ1:N s+1 ∼ CPFs(ζs+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ξ1:N s ) for i ← 1 to N do 2 draw {Ji,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='ℓ s }M ℓ=1 ∼ Λ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ·)�M 3 set βi s+1 ← M −1 �M ℓ=1 � βi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='Ji,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='ℓ s s + ˜hs(ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='Ji,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='ℓ s s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ξi s+1) � 4 set ξi 0:s+1 ← (ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='Ji,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 s 0:s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ξi s+1) Algorithm 2 One iteration of PPG Input: Initial path ζ0:t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' {˜hs}t−1 s=0 Result: {βi t}N i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ζ′ 0:t 5 draw ξ1:N 0 ∼ CPF0(ζ0) 6 set βi 0 ← 0 for i ∈ �1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' N� 7 for s ← 0 to t − 1 do 8 set {(ξi 0:s+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' βi s+1)}N i=1 ← CondPaRIS({(ξi 0:s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' βi s)}N i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ζs+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ˜hs) 9 draw ζ′ 0:t ∼ N −1 �N i=1 δξi 0:t As performing k steps of the PPG results in k many-body systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' it is natural to consider the 6 following roll-out estimator which combines the backward statistics from step k0 < k to k: Π(k0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='N(ht) = [N(k − k0)]−1 k � ℓ=k0+1 N � j=1 βj t [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='12) The total number of particles used in this estimator is C = (N − 1)k per time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We denote by υ = (k−k0)/k the ratio of the number of particles used in the estimator to the total number of sampled particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We now state the first main results of the present paper, in the form of theoretical bounds on the bias and mean-squared error (MSE) of the roll-out estimator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' These results are ob- tained under the following strong mixing assumptions, which are now standard in the literature (see [Del Moral, 2004, Douc and Moulines, 2008, Del Moral, 2013, Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2016]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' It is crucial for obtaining quantitative bounds for particle smoothing algorithms, see [Olsson and Westerborn, 2017] or [Gloaguen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2022] but also for the coupled conditional backward sampling particle filter [Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 (strong mixing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every s ∈ N there exist ¯τs, ¯τs, ¯σs, and ¯σs in R∗ + such that (i) ¯τs ≤ gs(xs) ≤ ¯τs for every xs ∈ Xs, (ii) ¯σs ≤ ms(xs, xs+1) ≤ ¯σs for every (xs, xs+1) ∈ Xs:s+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Under A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1, define, for every s ∈ N, ρs := max m∈�0,s� ¯τm¯σm ¯τm¯σm (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='13) and, for every N ∈ N∗ and t ∈ N such that N > Nt := (1 + 5ρ2 t/2) ∨ 2t(1 + ρ2 t), κN,t := 1 − 1 − (1 + 5tρ2 t/2)/N 1 + 4t(1 + 2ρ2 t)/N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='14) Note that κN,t ∈ (0, 1) for all N and t as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Assume A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then for every t ∈ N, M ∈ N∗, ξ ∈ M1(X0:t), k0 ∈ N∗, k > k0 and N ∈ N∗ such that N > Nt, ��Eξ[Π(k0,k),N(ht)] − η0:tht �� ≤ σbias (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='15) Eξ �� Π(k0,k),N(ht) − η0:tht �2� ≤ σ2 mse, where σbias := cbias t κk0 t,N �t−1 m=0 ∥˜hm∥∞ (k − k0)(1 − κt,N)N , σ2 mse := (�t−1 m=0 ∥˜hm∥∞)2 N(k − k0) � cmse t + 2ccov t N 1/2(1 − κt,N) � and cbias t , cmse t and ccov t are constants that do not depend on N and Eξ denotes the expectation under the law of the Markov chain formed by the PPG when initialized according to ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The proof is provided in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Importantly, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='15) provides a bound on the bias of the roll-out estimator that decreases exponentially with the burn-in period k0 and is inversely proportional to the number N of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This means that we can improve the bias of the PARIS estimator with a better allocation of the computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Parameter learning with PPG We now turn to parameter learning using PPG and gradient-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We set the focus on learning the parameter θ of a function V (θ) whose gradient is the smoothed expectation of an additive functional s0:t,θ in the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Algorithm 4 defines a stochastic approximation (SA) scheme where the noise forms a parameter dependent Markov chain with associated invariant measure πθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We follow the approach of [Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2019] to establish a non-asymptotic bound over the mean field h(θ) := πθs0:t,θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Such a setting encompasses for instance the following estimation procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (1) Score ascent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In the case of fully dominated HMMs, we are often interested in optimizing the log-likelihood of the observations given by V (θ) = log � γ0:t,θ(dx0:t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' By applying Fisher’s identity, we may express its gradient as a smoothed expectation of an additive functional according to ∇θV (θ) = � ∇θ log γ0:t(x0:t) η0:t,θ(dx0:t), = � t−1 � ℓ=0 sℓ,θ(xℓ, xℓ+1) η0:t,θ(dx0:t), where sℓ,θ : Xℓ:ℓ+1 ∋ (x, x′) �→ ∇θ log{gℓ,θ(x)mℓ,θ(x, x′)} and s0:t,θ := �t−1 ℓ=0 sℓ,θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (2) Inclusive KL surrogates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Inspired by [Naesseth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2020], we may consider the problem of learning a surrogate model for η0:t,θ in the form qφ(x0:t) = qφ(x0) �t−1 ℓ=0 qφ(xℓ+1, xℓ) by minimizing V (φ) = KL(η0:t,θ, qφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Algorithm 3 Gradient estimation with roll-out PPG (GdEst) Input: θ, ζ0:t[0], {sℓ,θ}t−1 ℓ=0, number k of PPG iterations, burn-in k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Result: β1:N t [k0 : k], ζ0:t[k] 10 for ℓ ← 0 to k − 1 do 11 run (˜β1:N t [ℓ + 1], ζ0:t[ℓ + 1]) ← PPG(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ζ0:t[ℓ], {sℓ,θ}t−1 ℓ=0) 12 if ℓ ≥ k0 − 1 then 13 set β1:N t [ℓ + 1] = ˜β1:N t [ℓ + 1] Algorithm 4 Score ascent with PPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Input: θ0, ζ0:t[0], number k of PPG iterations, burn-in k0, number of SA iterations n, learning-rate sequence {γℓ}ℓ∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Result: θn 14 for i ← 0 to n − 1 do 15 run (β1:N t [k0 : k], ζ0:t[i + 1]) ← GdEst(θi, ζ0:t[i], {sℓ,θi}t−1 ℓ=0, k, k0) 16 set Π(k0,k),N(s0:t,θi) = (N(k − k0))−1 �k−1 ℓ=k0 �N j=1 βj t [ℓ] 17 set θi+1 ← θi + γi+1Π(k0,k),N(s0:t,θi) Note that Algorithm 3 defines a (collapsed) Markov kernel Pθ,t defining for each path ζ0:t a measure Pθ,t(ζ0:t, d(˜ζ0:t, ˜β1:N t [k0 : k])) over the extended space of paths and sufficient statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Note that by evaluating the function b1:N t [k0 : k] �→ [N(k − k0)]−1 �k ℓ=k0+1 �N j=1 bj t[ℓ] at a realisation of this kernel gives the roll-out estimator whose properties are analysed in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The Markov kernel Pθ,t is detailed in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The following assumptions, are vital when analysing the convergence of Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (i) The function θ �→ V (θ) is LV -smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 8 (ii) The function θ �→ η0:t,θ is Lη-Lipschitz in total variation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (iii) For each path ζ0:t ∈ X0:t, the function θ �→ Kθ,t(ζ0:t, d˜ζ0:t) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='16) is LP 1 -Lipschitz in total variation distance, where Kθ,t is path-marginalized Markov transition kernel associated with the PPG algorithm when the model is parameterized by θ, see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (iv) For each path ζ0:t ∈ X0:t, the function θ �→ Pθ,tΠk0−1,k,N(s0:t,θ)(ζ0:t) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='17) is LP 2 -Lipschitz in total variation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In the case of score ascent we check, in Appendix B, that these assumptions hold if the strong mixing assumption A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 is satisfied uniformly in θ, and with additional assumptions on the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We are now ready to state a bound on the mean field h(θ) for Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Assume A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 uniformly in θ and A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 and suppose that the stepsizes {γℓ+1}ℓ∈�0,n−1� satisfy γℓ+1 ≤ γℓ, γℓ < aγℓ+1, γℓ − γℓ+1 < a′γ2 ℓ and γ1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='5(LV + Ch) for some a > 0, a′ > 0 and all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then, E � ∥h(θϖ)∥2� ≤ 2V0,n + C0,n + C0,γ �n k=0 γ2 k+1 �n k=0 γk+1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='18) where V0,n = E [V (θ) − V (θn)] and C0,n := γ1h(θ0)C0 + σbias(γ1 − γn+1 + 1)δ−1 k,N,t , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='19) C0,γ := σ2 mseLV + σmseC1 + σmseσbias � LV + C2 1 − κN,t � δ−1 k,N,t + σbiasLV δ−1 k,N,t , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='20) Ch := � C1 + σbias C2 (1 − κN,t)δk,N,t � [(a + 1)/2 + aσmse] + (LV + a′ + 1)σbiasδ−1 k,N,t , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='21) C1 = LP 2 � 1 + κk N,tδ−1 k,N,t � + LV (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='22) C2 = LP 1 δ−1 k,N,t + Lηκk N,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='23) where C0 is independent of σbias, σmse, N and where δk,N,t = 1 − κk N,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Theorem 2 establishes not only the convergence of Algorithm 4, but also illustrates the impact of the bias and the variance of the PPG on the convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Under additional assumptions on the model (cf Appendix B), if we consider γ1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='5(LV + Ch), γℓ = γ1ℓ−1/2 for all ℓ ∈ �1, n�, then �n k=0 γ2 k+1/ �n k=0 γk+1 ∼ log n/√n, showing that E � ∥h(θϖ)∥2� is O(log n/√n), where the leading constant depends on σbias and σmse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Remark 1 establishes the rate of convergence of Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In principle we could try to optimize the parameters k, k0 and N of the algorithm using these bounds, but one of the main challenges with this approach is the determination of the mixing rate, which is underestimated by κN,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Still, our bound provides interesting information of the role of both bias and MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Numerics In this section, we focus on the numerical analysis of the two main results of the paper, namely the bias and MSE bounds of the roll-out estimator established in Theorem 1 and the efficiency of using PPG for learning in the framework developed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For the latter, we will restrict ourselves to the case 9 of parameter learning via score ascent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In this setting, the competing method that corresponds most closely to the one presented here consists of using, as presented in Algorithm 5, a standard particle Gibbs sampler Πθ instead of the PPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' One of the most common such samplers is the particle Gibbs with ancestor sampling (PGAS) presented in [Lindsten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2014b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In [Lindholm and Lindsten, 2018], the PGAS is used for parameter learning in HMMs via the Expectation Maximization (EM) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Algorithm 5 Score ascent with particle Gibbs kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Data: ζ0:t[0], θ0, number k of paths per trajectory, burn-in k0, number n of SA iterations, learning-rate sequence {γℓ}ℓ∈N, Πθ(ζ0:t, d˜ζ0:t) a Markov kernel targeting η0:t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Result: θn 18 for i ← 0 to n − 1 do 19 for j ← 0 to k − 1 do 20 sample ˜ζ0:t[j + 1] ∼ Πθ(˜ζ0:t[j], ·) 21 set θi+1 ← θi + γi+1 k−k0 �k ℓ=k0+1 s0:t,θi(˜ζ0:t[ℓ]) 22 set ζ0:t[i + 1] = ˜ζ0:t[k] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 PPG Linear Gaussian state-space model (LGSSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We first consider a linear Gaussian HMM Xm+1 = AXm + Qϵm+1, Ym = BXm + Rζm, m ∈ N, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='24) where {ϵm}m∈N∗ and {ζm}m∈N are sequences of independent standard normally distributed random variables, independent of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The coefficients A, Q, B, and R are assumed to be known and equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='97, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='54, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='33, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Using this parameterisation, we generate, by simulation, a record of t = 999 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In this setting, we aim at computing smoothed expectations of the state one-lag covariance ht(x0:t) := �t−1 m=0 xmxm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In the linear Gaussian case, the disturbance smoother (see [Capp´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2005, Algo- rithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='15]) provides the exact values of the smoothed sufficient statistics, which allows us to study the bias of the estimator for a given computational budget C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Figure 1 displays, for three different total budgets C, the distribution of estimates of η0:nhn using the PARIS as well as three different configurations of the PPG corresponding to k ∈ {2, 4, 10} (and N = C/k) with k0 = k/2 and k0 = k/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The reference value is shown as a red-dashed line and the mean value of each distribution is shown as a black-dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Each boxplot is based on 1000 independent replicates of the corresponding estima- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We observe that in this example, all configurations of the PPG are less biased than the equivalent PARIS estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The illustration of the bounds from Theorem 1 is postponed to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2 Score ascent LGSSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We consider the LGSSM with state and observation spaces being R5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We assume that the parameters R and Q are known and consider the inference of θ = (A, B) on the basis of a simulated sequence of n = 999 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In this setting, the M-step of the EM algorithm can be solved exactly with the disturbance smoother [Capp´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2005, Chapter 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The parameter obtained by this procedure (denoted θmle) is the reference value for any likelihood maximization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Table 1 shows the L2 distance between the singular values of θmle and those of the parameters obtained by Algorithm 4 and Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The CLT confidence intervals were obtained on the basis of 25 replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The configurations respect a given particle budget kN = C = 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The choice of keeping k0 = k/2 is a heuristic rule to achieve a good bias–variance trade-off, but other combinations of k0 and k may lead to better performance for different problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We analyse this for the LGSMM in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='All settings are the same for both algorithms and are described in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The PPG achieves 10 PaRIS N = 500 N=10 N=25 N=50 N=100 5641 5642 5643 5644 5645 5646 PaRIS N = 500 N=10 N=25 N=50 N=100 5641 5642 5643 5644 5645 5646 Figure 1: PARIS and PPG outputs for the LGSSM for C = 500, yellow boxes correspond to PPG outputs produced using k ∈ {50, 20, 10, 5} iterations and N ∈ {C/50, C/20, C/10, C/5} particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The image on the left corresponds to taking k0 = k/2 and the one on the right to k0 = k/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' consistently a smaller distance to θmle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Figure 2 displays, for each estimator and configuration, the evolution of the distance to the MLE estimator as a function of the iteration index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 100 101 102 103 104 100 3 × 10 1 4 × 10 1 6 × 10 1 2 × 100 PGAS(N=32, k=64) PGAS(N=64, k=32) PGAS(N=128, k=16) PGAS(N=256, k=8) PPG(N=32, k=64) PPG(N=64, k=32) PPG(N=128, k=16) PPG(N=256, k=8) Figure 2: Distance to the MLE estimator as a function of the iteration step for PGAS and PPG with different parameters while keeping the particle budget fixed for LGSSM for 25 different seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' CRNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We consider now the problem of inference in a non-linear HMM and in particular the chaotic recurrent neural network introduced by [Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We use the same setting as in the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The state and observation equations are Xm+1 = Xm + τ −1∆ (−Xm + γW tanh(Xm)) + ϵm+1, Ym = BXm + ζm, m ∈ N, where {ϵm}m∈N∗ is a sequence of 20-dimensional independent multivariate Gaussian random variables with zero mean and covariance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='01I and {ζm}m∈N is a sequence of independent random variables 11 Table 1: Distance to θMLE for each configuration in the LGSSM case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Algorithm N k0 k Dmle PGAS 32 32 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='793 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='048 PGAS 64 16 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='751 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='052 PGAS 128 8 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='633 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='054 PGAS 256 4 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='580 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='049 PPG 32 32 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='358 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='038 PPG 64 16 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='373 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='031 PPG 128 8 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='355 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='043 PPG 256 4 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='351 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='042 Table 2: Per configuration negative loglikelihood for the CRNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Algorithm N k0 k NLL PGAS 32 16 32 31364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='932 ± 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='708 PGAS 64 8 16 31083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='408 ± 380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='527 PGAS 128 4 8 30264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='836 ± 265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='880 PPG 32 16 32 22291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='971 ± 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='683 PPG 64 8 16 22314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='537 ± 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='028 PPG 128 4 8 22353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='416 ± 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='443 where each component is distributed independently according to a Student’s t-distribution with scale 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 and 2 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In this case, the natural metric used to evaluate the different estimators is the negative log likelihood (NLL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We use the unbiased estimator of the likelihood given by the mean of the log weights produced by a particle filter [Douc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2014, Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1] using N = 104 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Table 2 shows the results obtained for 25 different replications for several different configurations of PPG and PGAS, while keeping total budget of particles fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Further numerical details are given in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We observe that PPG achieves the a considerably lower NLL than PGAS in all configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Conclusion We have presented a new algorithm, referred to as PPG as well as bounds on its bias and MSE in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We then propose a way of using PPG in a learning framework and derive a non-asymptotic bound over the gradient of the updates when doing score ascent with the PPG with explicit dependence on the bias and MSE of the estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We provide numerical simulations to support our claims, and we show that our algorithm outperforms the current competitors in the two different examples analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 12 Contents 1 Introduction 1 2 Background 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 Hidden Markov models .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 24 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 24 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3 Proof of Theorem 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 25 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='4 Proof of Proposition 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 29 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='5 Proof of Theorem 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 31 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6 Proof of Proposition 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 42 C Lipschitz properties 44 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 Lipschitz continuity of Pθ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 44 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 θ �→ Cm,θ is Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 46 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2 θ �→ Bt,θ(x0:t, ·) is Lipschitz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 48 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3 θ �→ � St,θ(x0:t, dbt)µ(bt)(id) is Lipschitz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 53 13 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' PPG In this section, we develop the theoretical framework necessary to establish Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We recall the notions of Feynman–Kac models, many-body Feynman–Kac models, backward interpretations, and conditional dual processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Our presentation follows closely [Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2016] but with a different and hopefully more transparent definition of the many-body extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We restate (in Theorem 3 below) a duality formula of [Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2016] relating these concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This formula provides a foundation for the particle Gibbs sampler described in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let (Z, Z) be a measurable space and L another possibly unnormalised transition kernel on Y × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Define, with K as above, KL : X × Z ∋ (x, A) �→ � L(y, A) K(x, dy) and K � L : X × (Y � Z) ∋ (x, A) �→ �� 1A(y, z) K(x, dy) L(y, dz), whenever these are well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This also defines the � products of a kernel K on X×Y and a measure ν on X as well as of a kernel L on Y × X and a measure µ on Y as the measures ν � K : X � Y ∋ A �→ �� 1A(x, y) K(x, dy) ν(dx), L � µ : X � Y ∋ A �→ �� 1A(x, y) L(y, dx) µ(dy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 Many-body Feynman–Kac models In the following, we assume that all random variables are defined on a common probability space (Ω, F, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The distribution flow {ηm}m∈N defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='4) is intractable in general, but can be approximated by random samples ξm = {ξi m}N i=1, m ∈ N, referred to as particles, where N ∈ N∗ is a fixed Monte Carlo sample size and each particle ξi m is an Xm-valued random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Such particle approximation is based on the recursion ηm+1 = Φm(ηm), m ∈ N, where Φm denotes the mapping Φm : M1(Xm) ∋ η �→ ηQm ηgm (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='25) taking on values in M1(Xm+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In order to describe recursively the evolution of the particle popu- lation, let m ∈ N and assume that the particles ξm form a consistent approximation of ηm in the sense that µ(ξm)h, where µ(ξm) := N −1 �N i=1 δξim, with δx denotes the Dirac measure located at x, is the occupation measure formed by ξm, which serves as a proxy for ηmh for all ηm-integrable test functions h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Under general conditions, µ(ξm)h converges in probability to ηm with N → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' see [Del Moral, 2004, Chopin and Papaspiliopoulos, 2020] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then, in order to generate an updated particle sample approximating ηm+1, new particles ξm+1 = {ξi m+1}N i=1 are drawn conditionally independently given ξm according to ξi m+1 ∼ Φm(µ(ξm)) = N � ℓ=1 gm(ξℓ m) �N ℓ′=1 gm(ξℓ′ m) Mm(ξℓ m, ·), i ∈ �1, N�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Since this process of particle updating involves sampling from the mixture distribution Φm(µ(ξm)), it can be naturally decomposed into two substeps: selection and mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The selection step con- sists of randomly choosing the ℓ-th mixture stratum with probability gm(ξℓ m)/ �N ℓ′=1 gm(ξℓ′ m) and the mutation step consists of drawing a new particle ξi m+1 from the selected stratum Mm(ξℓ m, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In 14 [Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2016], the term many-body Feynman–Kac models is related to the law of process {ξm}m∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all m ∈ N, let Xm := XN m and X m := X �N m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' then {ξm}m∈N is an inhomogeneous Markov chain on {Xm}m∈N with transition kernels M m : Xm × X m+1 ∋ (xm, A) �→ Φm(µ(xm))�N(A) and initial distribution η0 = η�N 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Now, denote X0:n := �n m=0 Xm and X 0:n := �n m=0 X m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In the following, we use a bold symbol to stress that a quantity is related to the many-body process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The many-body Feynman–Kac path model refers to the flows {γm}m∈N and {ηm}m∈N of the unnormalised and normalised, respectively, probability distributions on {X 0:m}m∈N generated by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3) for the Markov kernels {M m}m∈N, the initial distribution η0, the potential functions gm : Xm ∋ xm �→ µ(xm)gm = 1 N N � i=1 gm(xi m), m ∈ N, and the corresponding unnormalised transition kernels Qm : Xm × X m+1 ∋ (xm, A) �→ gm(xm)M m(xm, A), m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2 Backward interpretation of Feynman–Kac path flows Suppose that each kernel Qn, n ∈ N, defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2), has a transition density qn with respect to some dominating measure λn+1 ∈ M(Xn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then for n ∈ N and η ∈ M1(Xn) we may define the backward kernel ←− Q n,η : Xn+1 × Xn ∋ (xn+1, A) �→ � 1A(xn)qn(xn, xn+1) η(dxn) � qn(x′n, xn+1) η(dx′n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='26) Now, denoting, for n ∈ N∗, Bn : Xn × X0:n−1 ∋ (xn, A) �→ � · · � 1A(x0:n−1) n−1 � m=0 ←− Q m,ηm(xm+1, dxm), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='27) we may state the following—now classical—backward decomposition of the Feynman–Kac path measures, a result that plays a pivotal role in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every n ∈ N∗ it holds that γ0:n = γn � Bn and η0:n = ηn � Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Although the decomposition in Proposition 1 is well known (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', [Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2010, Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2016]), we provide a proof in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Using the back- ward decomposition, a particle approximation of a given Feynman–Kac path measure η0:n is obtained by first sampling, in an initial forward pass, particle clouds {ξm}n m=0 from η0 � M 0 � · · · � M n−1 and then sampling, in a subsequent backward pass, for instance N conditionally independent paths {˜ξi 0:n}N i=1 from Bn(ξ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , ξn, ·), where Bn : X0:n × X0:n ∋ (x0:n, A) �→ � · · � 1A(x0:n) �n−1 � m=0 ←− Q m,µ(xm)(xm+1, dxm) � µ(xn)(dxn) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='28) is a Markov kernel describing the time-reversed dynamics induced by the particle approximations generated in the forward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Here and in the following we use blackboard notation to denote kernels related to many-body path spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Finally, µ({˜ξi 0:n}N i=1)h is returned as an estimator of η0:nh for any η0:n-integrable test function h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This algorithm is in the literature referred to as the forward– filtering backward–simulation (FFBSi) algorithm and was introduced in [Godsill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2004];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' see also [Capp´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2007, Douc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' More precisely, given the forward particles {ξm}n m=0, each path 15 ˜ξi 0:n is generated by first drawing ˜ξi n uniformly among the particles ξn in the last generation and then drawing, recursively, ˜ξi m ∼ ←− Q m,µ(ξm)(˜ξi m+1, ·) = N � j=1 qm(ξj m, ˜ξi m+1) �N ℓ=1 qm(ξℓm, ˜ξi m+1) δξj m(·), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='29) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', given ˜ξi m+1, ˜ξi m is picked at random among the ξm according to weights proportional to {qm(ξj m, ˜ξi m+1)}N j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Note that in this basic formulation of the FFBSi algorithm, each backward-sampling operation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='29) requires the computation of the normalising constant �N ℓ=1 qm(ξℓ m, ˜ξi m+1), which implies an overall quadratic complexity of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Still, this heavy computational burden can eased by means of an effective accept–reject technique discussed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3 Conditional dual processes and particle Gibbs The dual process associated with a given Feynman–Kac model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='4–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3) and a given trajectory {zn}n∈N, where zn ∈ Xn for every n ∈ N, is defined as the canonical Markov chain with kernels M n⟨zn+1⟩ : Xn × X n+1 ∋ (xn, A) �→ 1 N N−1 � i=0 � Φn(µ(xn))�i � δzn+1 � Φn(µ(xn))�(N−i−1)� (A), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='30) for n ∈ N, and initial distribution η0⟨z0⟩ := 1 N N−1 � i=0 � η�i 0 � δz0 � η�(N−i−1) 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='31) As clear from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='30–A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='31), given {zn}n∈N, a realisation {ξn}n∈N of the dual process is generated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' At time zero, the process is initialised by inserting z0 at a randomly selected position in the vector ξ0 while drawing independently the remaining components from η0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then, given ξn at step n, zn+1 is inserted at a randomly selected position in ξn+1 while drawing independently the remaining components from Φn(µ(ξn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In order to describe compactly the law of the conditional dual process, we define the Markov kernel Cn : X0:n × X 0:n ∋ (z0:n, A) �→ η0⟨z0⟩ � M 0⟨z1⟩ � · · · � M n−1⟨zn⟩(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The following result elegantly combines the underlying model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='4–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3), the many-body Feynman–Kac model, the backward decomposition, and the conditional dual process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Theorem 3 ([Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2016]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all n ∈ N, Bn � γ0:n = γ0:n � Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='32) In [Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2016], each state ξn of the many-body process maps an outcome ω of the sample space Ω into an unordered set of N elements in Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' However, we have chosen to let each ξn take on values in the standard product space XN n for two reasons: first, the construction of [Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2016] requires sophisticated measure-theoretic arguments to endow such unordered sets with suitable σ-fields and appropriate measures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' second, we see no need to ignore the index order of the particles as long as the Markovian dynamics (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='30–A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='31) of the conditional dual process is sym- metrised over the particle cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Therefore, in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2, we include our own proof of duality (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='32) for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Note that the measure (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='32) on X0:n � X 0:n is unnormalised, but since the kernels Bn and Cn are both Markovian, normalising the identity with γ0:n(X0:n) = γ0:n(X0:n) yields immediately Bn � η0:n = η0:n � Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='33) 16 Since the two sides of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='33) provide the full conditionals, it is natural to choose a data-augmentation approach and sample the target (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='33) using a two-stage deterministic-scan Gibbs sampler [Andrieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2010b, Chopin and Singh, 2015a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' More specifically, assume that we have generated a state (ξ0:n[ℓ], ζ0:n[ℓ]) comprising a dual process with associated path on the basis of ℓ ∈ N iterations of the sampler;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' then the next state (ξ0:n[ℓ + 1], ζ0:n[ℓ + 1]) is generated in a Markovian fashion by sampling first ξ0:n[ℓ + 1] ∼ Cn(ζ0:n[ℓ], ·) and then sampling ζ0:n[ℓ + 1] ∼ Bn(ξ0:n[ℓ + 1], ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' After arbitrary initiali- sation (and the discard of possible burn-in iterations), this procedure produces a Markov trajectory {(ξ0:n[ℓ], ζ0:n[ℓ])}ℓ∈N, and under weak additional technical conditions this Markov chain admits (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='33) as its unique invariant distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In such a case, the Markov chain is ergodic [Douc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2018, Chapter 5], and the marginal distribution of the conditioning path ζ0:n[ℓ] converges to the target distribution η0:n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Therefore, for every h ∈ F(X0:n), lim L→∞ 1 L L � ℓ=1 h(ζ0:n[ℓ]) = η0:nh, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='4 The PARIS algorithm In the following, we assume that we are given a sequence {hn}n∈N of additive state functionals as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This problem is particularly relevant in the context of maximum-likelihood-based parameter estimation in general state-space models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', when computing the score-function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' the gradient of the log-likelihood function, via the Fisher identity or when computing the intermediate quantity of the Expectation Maximization (EM) algorithm, in which case η0:n and hn correspond to the joint state posterior and an element of some sufficient statistic, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' see [Capp´e and Moulines, 2005, Douc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2011, Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2010, Poyiadjis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2011, Olsson and Westerborn, 2017] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Interestingly, as noted in [Capp´e, 2011, Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2010], the backward de- composition allows, when applied to additive state functionals, a forward recursion for the expecta- tions {η0:nhn}n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' More specifically, using the forward decomposition hn+1(x0:n+1) = hn(x0:n) + ˜hn(xn, xn+1) and the backward kernel Bn+1 defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='27), we may write, for xn+1 ∈ Xn+1, Bn+1hn+1(xn+1) = � ←− Q n,ηn(xn+1, dxn) � � hn(x0:n) + ˜hn(xn, xn+1) � Bn(xn, dx0:n−1) = ←− Q n,ηn(Bnhn + ˜hn)(xn+1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='34) which by Proposition 1 implies that η0:n+1hn+1 = ηn+1 ←− Q n,ηn(Bnhn + ˜hn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='35) Since the marginal flow {ηn}n∈N can be expressed recursively via the mappings {Φn}n∈N, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='35) provides, in principle, a basis for online computation of {η0:nhn}n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' To handle the fact that the marginals are generally intractable we may, following [Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2010], plug particle approx- imations µ(ξn+1) and ←− Q n,µ(ξn) (see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='29)) of ηn+1 and ←− Q n,µ(ηn), respectively, into the recursion (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' More precisely, we proceed recursively and assume that at time n we have at hand a sample {(ξi n, βi n)}N i=1 of particles with associated statistics, where each statistic βi n serves as an approxima- tion of Bnhn(ξi n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' then evolving the particle cloud according to ξn+1 ∼ M n(ξn, ·) and updating the statistics using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='34), with ←− Q n,ηn replaced by ←− Q n,µ(ξn), yields the particle-wise recursion βi n+1 = N � ℓ=1 qn(ξℓ n, ξi n+1) �N ℓ′=1 qn(ξℓ′ n , ξi n+1) � βℓ n + ˜hn(ξℓ n, ξi n+1) � , i ∈ �1, N�, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='36) and, finally, the estimator µ(βn)(id) = 1 N N � i=1 βi n (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='37) 17 of η0:nhn, where βn := (β1 n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , βN n ), i ∈ �1, N�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The procedure is initialised by simply letting βi 0 = 0 for all i ∈ �1, N�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Note that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='37) provides a particle interpretation of the backward decomposition in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This algorithm is a special case of the forward–filtering backward–smoothing (FFBSm) algorithm (see [Andrieu and Doucet, 2003, Godsill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2004, Douc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2011, S¨arkk¨a, 2013]) for additive functionals satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' It allows for online processing of the sequence {η0:nhn}n∈N, but has also the appealing property that only the current particles ξn and statistics βn need to be stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' However, since each update (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='36) requires the summation of N terms, the scheme has an overall quadratic complexity in the number of particles, leading to a computational bottleneck in applications to complex models that require large particle sample sizes N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In order to detour the computational burden of this forward-only implementation of FFBSm, the PARIS algorithm [Olsson and Westerborn, 2017] updates the statistics βn by replacing each sum (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='36) by a Monte Carlo estimate βi n+1 = 1 M M � j=1 � ˜βi,j n + ˜hn(˜ξi,j n , ξi n+1) � , i ∈ �1, N�, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='38) where {(˜ξi,j n , ˜βi,j n )}M j=1 are drawn randomly among {(ξi n, βi n)}N i=1 with replacement, by assigning (˜ξi,j n , ˜βi,j n ) the value of (ξℓ n, βℓ n) with probability qn(ξℓ n, ξi n+1)/ �N ℓ′=1 qn(ξℓ′ n , ξi n+1), and the Monte Carlo sample size M ∈ N∗ is supposed to be much smaller than N (say, less than 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Formally, {(˜ξi,j n , ˜βi,j n )}M j=1 ∼ � N � ℓ=1 qn(ξℓ n, ξi n+1) �N ℓ′=1 qn(ξℓ′ n , ξi n+1) δ(ξℓn,βℓn) ��M , i ∈ �1, N�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The resulting procedure, summarised in Algorithm 1, allows for online processing with constant mem- ory requirements, since it only needs to store the current particle cloud and the estimated auxiliary statistics at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Moreover, in the case where the Markov transition densities of the model can be uniformly bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' when there exists,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' for every n ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' an upper bound ¯σn > 0 such that for all (xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xn+1) ∈ Xn × Xn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' mn(xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xn+1) ≤ ¯σn (a weak assumption satisfied for most models of interest),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' a sample (˜ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' βi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j n ) can be generated by drawing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' with replacement and un- til acceptance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' candidates (˜ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='∗ n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ˜βi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='∗ n ) from {(ξi n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' βi n)}N i=1 according to the normalised particle weights {gn(ξℓ n)/ � ℓ′ gn(ξℓ′ n )}N ℓ=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' obtained as a by-product in the generation of ξn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' and accepting the same with probability mn(˜ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='∗ n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ξi n+1)/¯σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' As this sampling procedure bypasses completely the calculation of the normalising constant �N ℓ′=1 qn(ξℓ′ n , ξi n+1) of the targeted categorical distribution, it yields an overall O(MN) complexity of the algorithm as a whole;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' see [Douc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2011] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Increasing M improves the accuracy of the algorithm at the cost of additional computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' As shown in [Olsson and Westerborn, 2017], there is a qualitative difference between the cases M = 1 and M ≥ 2, and it turns out that the latter is required to keep PARIS numerically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' More precisely, in the latter case, it can be shown that the PARIS estimator µ(βn) satisfies, as N tends to infinity while M is held fixed, a central limit theorem (CLT) at the rate √ N and with an n-normalised asymptotic variance of order O(1 − 1/(M − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' As clear from this bound, using a large M only yields a waste of computational work, and setting M to 2 or 3 typically works well in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We now introduce the Parisian particle Gibbs (PPG) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all t ∈ N∗, let Yt := X0:t×R and Yt := X0:t � B(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Moreover, let Y0 := X0 × {0} and Y0 := X0 � {{0}, ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' An element of Yt will always be denoted by yt = (x0:t|t, bt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The Parisian particle Gibbs sampler comprises, as a key ingredient, a conditional PARIS step, which updates recursively a set of Yt-valued random variables υi t := (ξi 0:t|t, βi t), i ∈ �1, N�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let (υt)t∈N denote the corresponding many-body process, each υt := {(ξi 0:t|t, βi t)}N i=1 taking on values in the space Yt := YN t , which we furnish with a σ-field Yt := Y�N t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The space Y0 and the corresponding σ-field Y0 are defined accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every t ∈ N, we write ξ0:t|t for the collection {ξi 0:t|t}N i=1 of paths in υt, and ξt|t for the collection {ξi t|t}N i=1 of end points of the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In the following, we let t ∈ N be a fixed time horizon, and describe in detail how the PPG approx- imates η0:tht iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In short, at each iteration ℓ, the PPG produces, given an input conditional 18 path ζ0:t[ℓ], a many-body system υt[ℓ + 1] by means of a series of conditional PARIS operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' then, an updated path ζ0:t[ℓ + 1], serving as input at the next iteration, is generated by picking one of the paths ξ0:t|t[ℓ + 1] in υt[ℓ + 1] at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' At each iteration, the produced statistics βt in υt provides an approximation of η0:tht according to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' More precisely, given the path ζ0:t[ℓ], the conditional PARIS operations are executed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In the initial step, ξ0|0[ℓ + 1] are drawn from η0⟨ζ0[ℓ]⟩ defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='31) and υi 0[ℓ + 1] ← (ξi 0|0[ℓ + 1], 0) for all i ∈ �1, N�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' recursively for m ∈ �0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' t�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' assuming access to υm[ℓ + 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (1) we generate an updated particle cloud ξm+1[ℓ + 1] ∼ M m⟨ζm+1[ℓ]⟩(ξm|m[ℓ + 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (2) we pick at random,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' for each i ∈ �1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' N�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' an ancestor path with associated statistics (˜ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 0:m[ℓ + 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ˜βi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 m [ℓ + 1]) among υm[ℓ + 1] by drawing (˜ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 0:m[ℓ + 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ˜βi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 m [ℓ + 1]) ∼ N � s=1 qm(ξs m|m[ℓ + 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ξi m+1[ℓ + 1]) �N s′=1 qm(ξs′ m|m[ℓ + 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ξi m+1[ℓ + 1]) δυsm[ℓ+1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' i ∈ �1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' N�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (3) we draw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' with replacement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' M −1 ancestor particles and associated statistics {(˜ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j m [ℓ+1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ˜βi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j m [ℓ+ 1])}M j=2 at random from {(ξs m|m[ℓ + 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' βs m)[ℓ + 1]}N s=1 according to {(˜ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j m [ℓ+1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ˜βi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j m [ℓ+1])}M j=2 ∼ � N � s=1 qm(ξs m|m[ℓ + 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ξi m+1[ℓ + 1]) �N s′=1 qm(ξs′ m|m[ℓ + 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ξi m+1[ℓ + 1]) δ(ξs m|m[ℓ+1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='βsm[ℓ+1]) ��(M−1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (4) we set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' for all i ∈ �1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' N�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ξi 0:m+1|m+1[ℓ + 1] ← (˜ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 0:m[ℓ + 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ξi m+1[ℓ + 1]) and υi m+1[ℓ + 1] ← (ξi 0:m+1|m+1[ℓ + 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' βi m+1[ℓ + 1]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' where βi m+1[ℓ + 1] ← M −1 M � j=1 � ˜βi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j m [ℓ + 1] + ˜hm(˜ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j m [ℓ + 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ξi m+1[ℓ + 1]) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This conditional PARIS procedure is summarised in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Once the set of trajectories and associated statistics υt[ℓ + 1] is formed by means of n recursive conditional PARIS updates, an updated path ζ0:t[ℓ + 1] is drawn from µ(ξ0:t|t[ℓ + 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A full sweep of the PPG is summarised in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The following Markov kernels will play an instrumental role in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For a given path {zm}m∈N, the conditional PARIS update in Algorithm 1 defines an inhomogeneous Markov chain on the spaces {(Ym, Ym)}m∈N with kernels Ym × Ym+1 ∋ (ym, A) �→ � M m⟨zm+1⟩(xm|m, dxm+1) Sm(ym, xm+1, A), m ∈ N, where Sm : Ym × Xm+1 × Ym+1 ∋ (ym, xm+1, A) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='39) �→ � · · � 1A � � �� (˜xi,1 0:m, xi m+1), 1 M M � j=1 � ˜bi,j m + ˜hm(˜xi,j m , xi m+1) � ��N i=1 � � × N � i=1 � N � ℓ=1 qm(xℓ m|m, xi m+1) �N ℓ′=1 qm(xℓ′ m|m, xi m+1) δyℓm(d(˜xi,1 0:m,˜bi,1 m )) × � N � ℓ=1 qm(xℓ m|m, xi m+1) �N ℓ′=1 qm(xℓ′ m|m, xi m+1) δ(xℓ m|m,bℓm) ��(M−1) (d(˜xi,2 m ,˜bi,2 m , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , ˜xi,M m ,˜bi,M m )) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 19 In addition, we introduce the joint law St : X0:t × Yt ∋ (x0:t, A) �→ � · · � 1A(yt) S0(Jx0, x1, dy1) t−1 � m=1 Sm(ym, xm+1, dym+1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='40) where we have defined J := IdN �(0, 1)⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The kernel St can be viewed as a superincumbent sampling kernel describing the distribution of the output υt generated by a sequence of PARIS iterates when the many-body process {ξm}t m=0 associated with the underlying SMC algorithm is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This allows us to describe alternatively the PPG as follows: given ζ0:t[ℓ], draw ξ0:t[ℓ + 1] ∼ Ct(ζ0:t[ℓ], ·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' then, draw υt[ℓ + 1] ∼ St(ξ0:t[ℓ + 1], ·) and pick a trajectory ζ0:t[ℓ + 1] from ξ0:t|t[ℓ + 1] at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The following proposition, which will be instrumental in the coming developments, establishes that the conditional distribution of ζ0:t[ℓ + 1] given ξ0:t[ℓ + 1] coincides, as expected, with the particle-induced backward dynamics Bt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all t ∈ N∗, N ∈ N∗, x0:t ∈ X0:t, and h ∈ F(X0:t), � St(x0:t, dyt) µ(x0:t|t)h = Bth(x0:t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Finally, we define the Markov kernel induced by the PPG as well as the extended probability distri- bution targeted by the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For this purpose, we introduce the extended measurable space (Et, Et) with Et := Yt × X0:t, Et := Yt � X0:t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The PPG described in Algorithm 2 defines a Markov chain on (Et, Et) with Markov transition kernel Kt : Et × Et ∋ (yt, z0:t, A) �→ ��� 1A(˜yt, ˜z0:t) Ct(z0:t, d˜x0:t) St(˜x0:t, d˜yt) µ(˜x0:t|t)(d˜z0:t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='41) Note that the values of Kt defined above do not depend on yt, but only on (z0:t, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For any given initial distribution ξ ∈ M1(X0:t), let Pξ be the distribution of the canonical Markov chain induced by the kernel Kt and the initial distribution ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In the special case where ξ = δz0:t for some given path z0:t ∈ X0:t, we use the short-hand notation Pδz0:t = Pz0:t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In addition, denote by Kt : X0:t × X0:t ∋ (z0:t, A) �→ ��� 1A(˜z0:t) Ct(z0:t, d˜x0:t) St(˜x0:t, d˜yt) µ(˜x0:t|t)(d˜z0:t) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='42) the path-marginalised version of Kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' By Proposition 2 it holds that Kt = CtBt, which shows that Kt coincides with the Markov transition kernel of the backward-sampling-based particle Gibbs sampler discussed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' It is also possible to specify the invariant distribution of Kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all t ∈ N∗, it holds that η0:tCtStKt = η0:tCtSt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='43) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let f ∈ M(E�(k−k0) t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' � f(˜yt, ˜z0:t)η0:t(dz0:t)CtSt(z0:t, d(yt, z′ 0:t))Kt(z′ 0:t, yt, d(˜yt, ˜z0:t)) = � f(˜yt, ˜z0:t)η0:t(dz0:t)CtSt(z0:t, d(yt, z′ 0:t))CtSt(z′ 0:t, d(˜yt, ˜z0:t)) = � f(˜yt, ˜z0:t)η0:t(dz0:t)Kt(z0:t, dz′ 0:t)CtSt(z′ 0:t, d(˜yt, ˜z0:t)) = � f(˜yt, ˜z0:t)η0:t(dz′ 0:t)CtSt(z′ 0:t, d(˜yt, ˜z0:t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 20 Finally, in order prepare for the statement of our theoretical results on the PPG we need to introduce the following Feynman–Kac path model with a frozen path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' More precisely, for a given path z0:t ∈ X0:t, define, for every m ∈ �0, t − 1�, the unnormalised kernel Qm⟨zm+1⟩ : Xm × Xm+1 ∋ (xm, A) �→ � 1 − 1 N � Qm(xm, A) + 1 N gm(xm) δzm+1(A) and the initial distribution η0⟨z0⟩ : X0 ∋ A �→ (1 − 1/N)η0(A) + δz0(A)/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Given these quantities, define, for m ∈ �0, t�, γm⟨z0:m⟩ := η0⟨z0⟩Q0⟨z1⟩ · · · Qm−1⟨zm⟩ along with the normalised counterpart ηm⟨z0:m⟩ := γm⟨z0:m⟩/γm⟨z0:m⟩1X0:m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Finally, we introduce, for m ∈ �0, t�, the kernels Bm⟨z0:m−1⟩ : Xm × X0:m−1 ∋ (xm, A) �→ � · · � 1A(x0:m−1) t−1 � m=0 ←− Q m,ηm⟨z0:m⟩(xm+1, dxm), as well as the path model η0:m⟨z0:m⟩ := Bm⟨z0:m−1⟩ � ηm⟨z0:m⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='5 Proof of Theorem 1 We start by establishing bias, MSE and covariance bounds for a fixed iteration of the PPG estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Assume A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then for every t ∈ N there exist cbias t , cmse t , and ccov t in R∗ + such that for every M ∈ N∗, ξ ∈ M1(X0:t), ℓ ∈ N∗, s ∈ N∗, and N ∈ N∗ such that N > Nt, |Eξ [µ(βt[ℓ])(id)] − η0:tht| ≤ cbias t � t−1 � m=0 ∥˜hm∥∞ � N −1κℓ N,t, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='44) Eξ � (µ(βt[ℓ])(id) − η0:tht)2� ≤ cmse t � t−1 � m=0 ∥˜hm∥∞ �2 N −1, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='45) |Eξ [(µ(βt[ℓ])(id) − η0:tht) (µ(βt[ℓ + s])(id) − η0:tht)]| ≤ ccov t � t−1 � m=0 ∥˜hm∥∞ �2 N −3/2κs N,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='46) The constants cbias t , cmse t , and ccov t are explicitly given in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Since the focus of this paper is on the dependence on N and the index ℓ, we have made no attempt to optimise the dependence of these constants on t in our proofs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' still, we believe that it is possible to prove, under the stated assumptions, that this dependence is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The proof of the bound in Theorem 4 is based on four key ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The first is the following unbiasedness property of the PARIS under the many-body Feynman–Kac path model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every t ∈ N, N ∈ N∗, and ℓ ∈ N∗, Eη0:t [µ(βt[ℓ])(id)] = � η0:tCtSt(dbt) µ(bt)(id) = � η0:tSt(dbt) µ(bt)(id) = η0:tht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The proof of Theorem 5 is postponed to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The second ingredient of the proof of Theorem 4 is the uniform geometric ergodicity of the particle Gibbs with backward sampling established in [Del Moral and Jasra, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Assume A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then, for every t ∈ N, (µ, ν) ∈ M1(X0:t)2, ℓ ∈ N∗, and N ∈ N∗ such that N > 1 + 5ρ2 tt/2, ∥µKℓ t − νKℓ t ∥TV ≤ κℓ N,t, where κN,t is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' As a third ingredient, we require the following uniform exponential concentration inequality of the conditional PARIS with respect to the frozen-path Feynman–Kac model defined in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 21 Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every t ∈ N there exist ct > 0 and dt > 0 such that for every M ∈ N∗, z0:t ∈ X0:t, N ∈ N∗, and ε > 0, � CtSt(z0:t, dbt)1 {|µ(bt)(id) − η0:t⟨z0:t⟩ht| ≥ ε} ≤ ct exp � − dtNε2 2(�t−1 m=0 ∥˜hm∥∞)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Theorem 7, whose proof is postponed to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='5, implies, in turn, the following conditional variance bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every t ∈ N, M ∈ N∗, z0:t ∈ X0:t, and N ∈ N∗, � CtSt(z0:t, dbt) |µ(bt)(id) − η0:t⟨z0:t⟩ht|2 ≤ ct dt � t−1 � m=0 ∥˜hm∥∞ �2 N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Using Proposition 4, we deduce, in turn, the following bias bound, whose proof is postponed to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every t ∈ N there exists ¯cbias t > 0 such that for every M ∈ N∗, z0:t ∈ X0:t, and N ∈ N∗, ���� � CtSt(z0:t, dbt) µ(bt)(id) − η0:t⟨z0:t⟩ht ���� ≤ ¯cbias t N −1 � t−1 � m=0 ∥˜hm∥∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A fourth and last ingredient in the proof of Theorem 4 is the following bound on the discrepancy between additive expectations under the original and frozen-path Feynman–Kac models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This bound is established using novel results in [Gloaguen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' More precisely, since for every m ∈ N, (x, z) ∈ X2 m, N ∈ N∗, and h ∈ F(Xm+1), using A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1, |Qm⟨z⟩h(x) − Qmh(x)| ≤ 1 N ∥gm∥∞∥h∥∞ ≤ 1 N ¯τm∥h∥∞, applying [Gloaguen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2022, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3] yields the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Assume A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then there exists c > 0 such that for every t ∈ N, N ∈ N, and z0:t ∈ X0:t, |η0:t⟨z0:t⟩ht − η0:tht| ≤ cN −1 t−1 � m=0 ∥˜hm∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Note that assuming, in addition, that supt∈N ∥˜ht∥∞ < ∞ yields an O(n/N) bound in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Finally, by combining these ingredients we are now ready to present a proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Write, using the tower property, Eξ [µ(βt [ℓ])(id)] = Eξ � Eζ0:t[ℓ] [µ(βt [0])(id)] � = � ξKℓ t CtSt(dbt) µ(bt)(id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Thus, by the unbiasedness property in Theorem 5, |Eξ [µ(βt [ℓ])(id)] − η0:tht| = ���� � ξKℓ t CtSt(dbt) µ(bt)(id) − � η0:tCtSt(dbt) µ(bt)(id) ���� ≤ ��ξKℓ t − η0:t �� TV osc �� CtSt(·, dbt) µ(bt)(id) � , 22 where, by Theorem 6, ∥ξKℓ t − η0:t∥TV ≤ κℓ N,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Moreover, to derive an upper bound on the oscillation, we consider the decomposition osc �� CtSt(·, dbt) µ(bt)(id) � ≤ 2 ����� � CtSt(·, dbt) µ(bt)(id) − η0:t⟨·⟩ht ���� ∞ + ∥η0:t⟨·⟩ht − η0:tht∥∞ � , where the two terms on the right-hand side can be bounded using Proposition 6 and Proposition 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This completes the proof of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We now consider the proof of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Writing Eξ � (µ(βt[ℓ])(id) − η0:tht)2� = � ξKℓ t (dz0:t)CtSt(z0:t, dbt) (µ(bt)(id) − η0:tht)2 , we may establish (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='45) using Proposition 4 and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We finally consider (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Using the Markov property we obtain Eξ [(µ(βt[ℓ])(id) − η0:tht) (µ(βt[ℓ + s])(id) − η0:tht)] = Eξ � (µ(βt[ℓ])(id) − η0:tht) � Eζ0:t[ℓ][µ(βt[s])(id)] − η0:tht �� , from which (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='46) follows by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='44) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We are finally equipped to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We first consider the bias, which can be bounded according to ��Eξ[Π(k0,k),N(f)] − η0:tht �� ≤ (k − k0)−1 k � ℓ=k0+1 |Eξµ(βt[ℓ])(id) − η0:tht| ≤ (k − k0)−1N −1cbias t � t−1 � m=0 ∥˜hm∥∞ � k � ℓ=k0+1 κℓ N,t, from which the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='15) follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We turn to the MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Using the decomposition Eξ[(Π(k0,k),N(f) − η0:tht)2] ≤ (k − k0)−2 � k � ℓ=k0+1 Eξ[(µ(βt[ℓ])(id) − η0:tht)2] + 2 k � ℓ=k0+1 k � j=ℓ+1 Eξ[(µ(βt[ℓ])(id) − η0:tht)(µ(βt[j])(id) − η0:tht)] � � � , the MSE bound in Theorem 4 implies that k � ℓ=k0+1 Eξ[(µ(βt[ℓ])(id) − η0:tht)2] ≤ cmse t � t−1 � m=0 ∥˜hm∥∞ �2 N −1(k − k0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Moreover, using the covariance bound in Theorem 4, we deduce that k � ℓ=k0+1 k � j=ℓ+1 Eξ[(µ(βt[ℓ])(id)−η0:tht)(µ(βt[j])(id)−η0:tht)] ≤ ccov t � t−1 � m=0 ∥˜hm∥∞ �2 N −3/2 � � k � ℓ=k0+1 k � j=ℓ+1 κ(j−ℓ) N,t � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Thus, the proof is concluded by noting that �k ℓ=k0+1 �k j=ℓ+1 κ(j−ℓ) N,t ≤ (k − k0)/(1 − κN,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 23 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6 Proofs of intermediate results A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 Proof of Proposition 1 Using the identity η0Q0 · · · Qt−11Xt = t−1 � m=0 ηmQm1Xm+1 and the fact that each kernel Qm has a transition density, write, for h ∈ F(X0:t), η0:th = � · · � h(x0:t) η0(dx0) t−1 � m=0 �ηm[qm(·, xm+1)] λm+1(dxm+1) ηmQm1Xm+1 � � qm(xm, xm+1) ηm[qm(·, xm+1)] � = � · · � h(x0:t) ηt(dxt) t−1 � m=0 ηm(dxm) qm(xm, xm+1) ηm[qm(·, xm+1)] (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='47) = �←− Q 0,η0 � · · · � ←− Q n−1,ηt−1 � ηt � h, which was to be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2 Proof of Theorem 3 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all t ∈ N, xt ∈ Xt, and h ∈ F(X t+1 � Xt+1), �� h(xt+1, zt+1) Qt(xt, dxt+1) µ(xt+1)(dzt+1) = �� h(xt+1, zt+1) µ(xt)Qt(dzt+1) M t⟨zt+1⟩(xt, dxt+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='48) In addition, for all h ∈ F(X 0 � X0), �� h(x0, z0) η0(dx0) µ(x0)(dz0) = �� h(x0, z0) η0⟨z0⟩(dx0) η0(dz0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='49) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Since µ(xt) Qt(dzt+1) = gt(xt) Φt(µ(xt))(dzt+1), we may rewrite the right-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='48) according to �� h(xt+1, zt+1) µ(xt)Qt(dzt+1) M t⟨zt+1⟩(xt, dxt+1) = gt(xt) 1 N N−1 � i=0 �� h(xt+1, zt+1) Φt(µ(xt))(dzt+1) × � Φt(µ(xt))�i � δzt+1 � Φt(µ(xt))�(N−i−1)� (dxt+1) = gt(xt) 1 N N � i=1 � · · � h((x1 t+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xi−1 t+1, zt+1, xi+1 t+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xN t+1), zt+1) × Φt(µ(xt))(dzt+1) � ℓ̸=i Φt(µ(xt))(dxℓ t+1) = gt(xt) 1 N N � i=1 � h(xt+1, xi t+1) M t(xt, dxt+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' On the other hand, note that the left-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='48) can be expressed as �� h(xt+1, zt+1) Qt(xt, dxt+1) µ(xt+1)(dzt+1) = gt(xt) 1 N N � i=1 � h(xt+1, xi t+1) M t(xt, dxt+1), which establishes the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The identity (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='49) is established along similar lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 24 We establish Theorem 3 by induction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' thus, assume that the claim holds true for n and show that for all h ∈ F(X 0:t+1 � X0:t+1), �� h(x0:t+1, z0:t+1) γ0:t+1(dx0:t+1) Bt+1(x0:t+1, dz0:t+1) = �� h(x0:t+1, z0:t+1) γ0:t+1(dz0:t+1) Ct+1(z0:t+1, dx0:t+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='50) To prove this, we process, using definition (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='85), the left-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='50) according to �� h(x0:t+1, z0:t+1) γ0:t+1(dx0:t+1) Bt+1(x0:t+1, dz0:t+1) = �� γ0:t(dx0:t) Bt(x0:t, dz0:t) × �� ¯h(x0:t+1, z0:t+1) Qt(xt, dxt+1) µ(xt+1)(dzt+1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='51) where we have defined the function ¯h(x0:t+1, z0:t+1) := qt(zt, zt+1)h(x0:t+1, z0:t+1) µ(xt)[qt(·, zt+1)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Now, applying Lemma 1 to the inner integral and using that µ(xt)Qt(dzt+1) = µ(xt)[qt(·, zt+1)] λt+1(dzt+1) yields, for every x0:t and z0:t, �� ¯h(x0:t+1, z0:t+1) Qt(xt, dxt+1) µ(xt+1)(dzt+1) = �� ¯h(x0:t+1, z0:t+1) µ(xt)Qt(dzt+1) M t⟨zt+1⟩(xt, dxt+1) = �� h(x0:t+1, z0:t+1) Qt(zt, dzt+1) M t⟨zt+1⟩(xt, dxt+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Inserting the previous identity into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='51) and using the induction hypothesis provides �� h(x0:t+1, z0:t+1) γ0:t+1(dx0:t+1) Bt+1(x0:t+1, dz0:t+1) = �� γ0:t(dz0:t) Ct(z0:t, dx0:t) × �� h(x0:t+1, z0:t+1) Qt(zt, dzt+1) M t⟨zt+1⟩(xt, dxt+1) = �� h(x0:t+1, z0:t+1) γ0:t+1(dz0:t+1) Ct+1(z0:t+1, dx0:t+1), which establishes (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3 Proof of Theorem 5 First, define, for m ∈ N, P m : Ym × Ym+1 ∋ (ym, A) �→ � M m(xm|m, dxm+1) Sm(ym, xm+1, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='52) 25 For any given initial distribution ψ0 ∈ M1(Y0), let PP ψ0 be the distribution of the canonical Markov chain induced by the Markov kernels {P m}m∈N and the initial distribution ψ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' By abuse of notation we write, for η0 ∈ M1(X 0), PP η0 instead of PP ψ0[η0], where we have defined the extension ψ0[η0](A) = � 1A(Jx0) η0(dx0), A ∈ Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We preface the proof of Theorem 5 by some technical lemmas and a proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all t ∈ N and (ft+1, ˜ft+1) ∈ F(Xt+1)2, γt+1(ft+1Bt+1ht+1 + ˜ft+1) = γt{Qtft+1Btht + Qt(˜htft+1 + ˜ft+1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Pick arbitrarily ϕ ∈ F(Xt:t+1) and write, using definition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='27) and the fact that Qt has a transition density, �� ϕ(xt:t+1) γt(dxt) Qt(xt, dxt+1) = �� ϕ(xt:t+1)γt[qt(·, xt+1)] λt+1(dxt+1) γt(dxt)qt(xt, xt+1) γt[qt(·, xt+1)] = �� ϕ(xt:t+1) γt+1(dxt+1) ←− Q n,ηt(xt+1, dxt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='53) Now, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='34) it holds that Bt+1ht+1(xt+1) = � ←− Q n,ηt(xt+1, dxt) � ˜ht(xt:t+1) + � ht(x0:t) Bt(xt, dx0:t−1) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' therefore, by applying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='53) with ϕ(xt:t+1) := ft+1(xt+1) � ˜ht(xt:t+1) + � ht(x0:t) Bt(xt, dx0:t−1) � we obtain that γt+1(ft+1Bt+1ht+1) = �� ϕ(xt:t+1) γt+1(dxt+1) ←− Q n,ηt(xt+1, dxt) = �� ϕ(xt:t+1) γt(dxt) Qt(xt, dxt+1) = γt(Qtft+1Btht + Qt˜htft+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Now the proof is concluded by noting that since γt+1 = γtQt, γt+1 ˜ft+1 = γtQt ˜ft+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every t ∈ N∗, ht ∈ F(Yt), and η0 ∈ M1(X 0) it holds that EP η0[ht(υt) | ξ0|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , ξt|t] = Stht(ξ0|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , ξt|t), PP η0-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Pick arbitrarily vt ∈ F(X0:t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We show that EP η0[vt(ξ0|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , ξt|t)ht(υt)] = EP η0[vt(ξ0|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , ξt|t)Stht(ξ0|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , ξt|t)], (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='54) from which the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Using the definition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='52), the left-hand side of the previous identity 26 may be rewritten as � · · � ψ0[η0](dy0) t−1 � m=0 P m(ym, dym+1) ht(yt)vt(x0|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xt|t) = � · · � η0(dx0|0) t−1 � m=0 M m(xm|m, dxm+1) S0(Jx0|0, x1, dy1) × t−1 � m=0 Sm(ym, xm+1, dym+1) ht(yt)vt(x0|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xt|t) = � · · � η0(dx0) t−1 � m=0 M m(xm, dxm+1) S0(Jx0, x1, dy1) × t−1 � m=0 Sm(ym, xm+1, dym+1) ht(yt)vt(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Thus, we may conclude the proof by using the definition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='40) of St together with Fubini’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every t ∈ N∗ and ht ∈ F(Yt), Eη0 �� t−1 � m=0 gm(ξm|m) � ht(υt) � = � γ0:tSt(dyt) ht(yt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The claim of the lemma is a direct implication of Lemma 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' indeed, by applying the tower property and the latter we obtain EP η0 �� t−1 � m=0 gm(ξm|m) � ht(υt) � = EP η0 �� t−1 � m=0 gm(ξm|m) � Stht(ξ0|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , ξt|t) � = � · · � η0(dx0) t−1 � m=0 gm(xm) M m(xm, dxm+1) Stht(x0:t) = � γ0:tSt(dyt) ht(yt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all t ∈ N∗, (N, M) ∈ (N∗)2, and (ft, ˜ft) ∈ F(Xt)2, � γ0:tSt(dyt) � 1 N N � i=1 {bi tft(xi t|t) + ˜ft(xi t|t)} � = γt(ftBtht + ˜ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Applying Lemma 4 yields � γ0:tSt(dyt) � 1 N N � i=1 {bi tft(xi t|t) + ˜ft(xi t|t)} � = EP η0 �� t−1 � m=0 gm(ξm|m) � 1 N N � i=1 {βi tft(ξi t|t) + ˜ft(ξi t|t)} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='55) 27 In the following we will use repeatedly the following filtrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let ˜Ft := σ({υm}t m=0) be the σ-field generated by the output of the PARIS (Algorithm 1) during the first t iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In addition, let Ft := ˜Ft−1 ∨ σ(ξt|t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We proceed by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Thus, assume that the statement of the proposition holds true for a given t ∈ N∗ and consider, for arbitrarily chosen (ft+1, ˜ft+1) ∈ F(Xt+1)2, EP η0 �� t� m=0 gm(ξm|m) � 1 N N � i=1 {βi t+1ft+1(ξi t+1|t+1) + ˜ft+1(ξi t+1|t+1)} | ˜Ft � = � t� m=0 gm(ξm|m) � EP η0[β1 t+1ft+1(ξ1 t+1|t+1) + ˜ft+1(ξ1 t+1|t+1) | ˜Ft] , where we used that the variables {βi t+1ft+1(ξi t+1|t+1) + ˜ft+1(ξi t+1|t+1)}N i=1 are conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' given ˜Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Note that, by symmetry, EP η0 � β1 t+1 | Ft+1 � = � St(υt, ξt+1|t+1, dyt+1) b1 t+1 = � · · � � � M � j=1 N � ℓ=1 qt(ξℓ t|t, ξ1 t+1|t+1) �N ℓ′=1 qt(ξℓ′ t|t, ξ1 t+1|t+1) δ(ξℓ t|t,βℓ t)(d˜x1,j t , d˜b1,j t ) � � × 1 M M � j=1 � ˜b1,j t + ˜ht(˜x1,j t , ξ1 t+1|t+1) � = N � ℓ=1 qt(ξℓ t|t, ξ1 t+1|t+1) �N ℓ′=1 qt(ξℓ′ t|t, ξ1 t+1|t+1) � βℓ t + ˜ht(ξℓ t|t, ξ1 t+1|t+1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='56) Thus, using the tower property, EP η0 � β1 t+1ft+1(ξ1 t+1|t+1) | ˜Ft � = � Φt(µ(ξt|t))(dxt+1) ft+1(xt+1) N � ℓ=1 qt(ξℓ t|t, xt+1) �N ℓ′=1 qt(ξℓ′ t|t, xt+1) � βℓ t + ˜ht(ξℓ t|t, xt+1) � , and consequently, using definition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='25), � t� m=0 gm(ξm|m) � EP η0 � β1 t+1ft+1(ξ1 t+1|t+1) | ˜Ft � = � t−1 � m=0 gm(ξm|m) � � 1 N N � i=1 qt(ξi t|t, xt+1) × ft+1(xt+1) N � ℓ=1 qt(ξℓ t|t, xt+1) �N ℓ′=1 qt(ξℓ′ t|t, xt+1) � βℓ t + ˜ht(ξℓ t|t, xt+1) � λt+1(dxt+1) = � t−1 � m=0 gm(ξm|m) � 1 N N � ℓ=1 � βℓ tQtft+1(ξℓ t|t) + Qt(˜htft+1)(ξℓ t|t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 28 Thus, applying the induction hypothesis, EP η0 �� t� m=0 gm(ξm|m) � 1 N N � i=1 βi t+1ft+1(ξi t+1|t+1) � = EP η0 �� t−1 � m=0 gm(ξm|m) � 1 N N � ℓ=1 � βℓ tQtft+1(ξℓ t|t) + Qt(˜htft+1)(ξℓ t|t) �� = γt � Qtft+1Btht + Qt(˜htft+1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='57) In the same manner, it can be shown that EP η0 �� t� m=0 gm(ξm|m) � 1 N N � i=1 ˜ft+1(ξi t+1|t+1) � = γtQt ˜ft+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='58) Now, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='57–A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='58) and Lemma 2, EP η0 �� t� m=0 gm(ξm|m) � 1 N N � i=1 {βi t+1ft+1(ξi t+1|t+1) + ˜ft+1(ξi t+1|t+1)} � = γt � Qtft+1Btht + Qt(˜htft+1 + Qt ˜ft+1) � = γt+1(ft+1Bt+1ht+1 + ˜ft+1), which shows that the claim of the proposition holds at time n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' It remains to check the base case n = 0, which holds trivially true as β0 = 0, B0h0 = 0 by convention, and the initial particles ξ0|0 are drawn from η0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The identity � η0:t(dx0:t) St(x0:t, dbt) µ(bt)(id) = η0:tht follows immediately by letting ft ≡ 1 and ˜ft ≡ 0 in Proposition 7 and using that γ0:t(X0:t) = γ0:t(X0:t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Moreover, applying Theorem 3 yields � η0:tCtSt(dbt) µ(bt)(id) = �� η0:t(dz0:t) Ct(z0:t, dx0:t) � St(x0:t, dbt) µ(bt)(id) = �� η0:t(dx0:t) Bt(x0:t, dz0:t) � St(x0:t, dbt) µ(bt)(id) = � η0:tSt(dbt) µ(bt)(id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Finally, the first identity holds true since Kt leaves η0:t invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='4 Proof of Proposition 2 First, note that, by definitions (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='39) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='40), Ht(x0:t) := � St(x0:t, dyt) µ(x[0 : n|n])h = � · · � � � 1 N N � jt=1 h(xjt 0:t−1|t, xjt t ) � � × t−1 � m=0 N � im+1=1 � N � jm=1 qm(xjm m , xim+1 m+1) �N j′m=1 qm(xj′ m m , xim+1 m+1) δxjm 0:m|m(dxim+1 0:m|m+1), 29 where xi 0:−1|0 = ∅ for all i ∈ �1, N� by convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We will show that for every k ∈ �0, t�, Hk,t ≡ Ht, where Hk,n(x0:t) := 1 N N � jt=1 · · N � jk=1 t−1 � ℓ=k qℓ(xjℓ ℓ , xjℓ+1 ℓ+1 ) �N j′ ℓ=1 qℓ(x j′ ℓ ℓ , xjℓ+1 ℓ+1 ) ak,n(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xk−1, xjk k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xjt t ) with ak,n(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xk−1, xjk k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xjt t ) = � k−1 � m=0 N � im+1=1 N � jm=1 qm(xjm m , xim+1 m+1) �N j′m=1 qm(xj′m m , xim+1 m+1) δxjm 0:m|m(dxim+1 0:m|m+1)h(xjk 0:k−1|k, xjk k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xjt t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Since, by convention, �t−1 ℓ=n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' = 1, Hn,n(x0:t) = N −1 �N jt=1 an,n(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , x[n − 1], xjt t ), and we note that Ht ≡ Hn,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We now show that Hk,n ≡ Hk−1,n for every k ∈ �1, t�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' for this purpose, note that ak,n(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xk−1, xjk k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xjt t ) = � k−2 � m=0 N � im+1=1 N � jm=1 qm(xjm m , xim+1 m+1) �N j′m=1 qm(xj′m m , xim+1 m+1) δxjm 0:m|m(dxim+1 0:m|m+1) × � N � ik=1 N � jk−1=1 qk−1(xjk−1 k−1 , xik k ) �N j′ k−1=1 qk−1(x j′ k−1 k−1 , xik k ) δx jk−1 0:k−1|k−1(dxik 0:k−1|k) h(xjk 0:k−1|k, xjk k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xjt t ), and since xjk−1 0:k−1|k−1 = (xjk−1 0:k−2|k−1, xjk−1 k−1 ), it holds that � N � ik=1 N � jk−1=1 qk−1(xjk−1 k−1 , xik k ) �N j′ k−1=1 qk−1(x j′ k−1 k−1 , xik k ) δx jk−1 0:k−1|k−1(dxik 0:k−1|k) h(xjk 0:k−1|k, xjk k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xjt t ) = N � jk−1=1 qk−1(xjk−1 k−1 , xjk k ) �N j′ k−1=1 qk−1(x j′ k−1 k−1 , xjk k ) h(xjk−1 0:k−2|k−1, xjk−1 k−1 , xjk k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xjt t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Therefore, we obtain ak,n(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xk−1, xjk k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xjt t ) = � k−2 � m=0 N � im+1=1 N � jm=1 qm(xjm m , xim+1 m+1) �N j′m=1 qm(xj′ m m , xim+1 m+1) δxjm 0:m|m(dxim+1 0:m|m+1) × N � jk−1=1 qk−1(xjk−1 k−1 , xjk k ) �N j′ k−1=1 qk−1(x j′ k−1 k−1 , xjk k ) h(xjk−1 0:k−2|k−1, xjk−1 k−1 , xjk k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xjt t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Now, changing the order of summation with respect to jk−1 and integration on the right hand side of the previous display yields ak,n(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xk−1, xjk k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xjt t ) = N � jk−1=1 qk−1(xjk−1 k−1 , xjk k ) �N j′ k−1=1 qk−1(x j′ k−1 k−1 , xjk k ) ak−1,n(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xk−2, xjk−1 k−1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xjt t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 30 Thus, Hk,n(x0:t) = 1 N N � jt=1 · · N � jk=1 t−1 � ℓ=k qℓ(xjℓ ℓ , xjℓ+1 ℓ+1 ) �N j′ ℓ=1 qℓ(x j′ ℓ ℓ , xjℓ+1 ℓ+1 ) × N � jk−1=1 qk−1(xjk−1 k−1 , xjk k ) �N j′ k−1=1 qk−1(x j′ k−1 k−1 , xjk k ) ak−1,n(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xk−2, xjk−1 k−1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xjt t ) = 1 N N � jt=1 · · N � jk−1=1 t−1 � ℓ=k−1 qℓ(xjℓ ℓ , xjℓ+1 ℓ+1 ) �N j′ ℓ=1 qℓ(x j′ ℓ ℓ , xjℓ+1 ℓ+1 ) ak−1,n(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xk−2, xjk−1 k−1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xjt t ) = Hk−1,n(x0:t), which establishes the recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Therefore, Ht ≡ H0,n and we may now conclude the proof by noting that Bth ≡ H0,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='5 Proof of Theorem 7 In order to establish Theorem 7 we will prove the following more general result, of which Theorem 7 is a direct consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every t ∈ N and M ∈ N∗ there exist ct > 0 and dt > 0 such that for every N ∈ N∗, z0:t ∈ X0:t, (ft, ˜ft) ∈ F(Xt)2, and ε > 0, � CtSt(z0:t, dbt)1 ������ 1 N N � i=1 {bi tft(xi t|t) + ˜ft(xi t|t)} − ηt⟨z0:t⟩(ftBt⟨z0:t−1⟩ht + ˜ft) ����� ≥ ε � ≤ ct exp � −dtNε2 2κ2 t � , where κt := ∥ft∥∞ t−1 � m=0 ∥˜hm∥∞ + ∥ ˜ft∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='59) To prove Proposition 8 we need the following technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every t ∈ N, (ft+1, ˜ft+1) ∈ F(Xt+1)2, z0:t+1 ∈ X0:t+1, and N ∈ N∗, γt+1⟨z0:t+1⟩(ft+1Bt+1⟨z0:t⟩ht+1 + ˜ft+1) = � 1 − 1 N � γt⟨z0:t⟩{Qtft+1Bt⟨z0:t−1⟩ht + Qt(˜htft+1 + ˜ft+1)} + 1 N γt⟨z0:t⟩gt � ft+1(zt+1)Bt+1⟨z0:t⟩ht+1(zt+1) + ˜ft+1(zt+1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Since Lemma 2 holds also for the Feynman–Kac model with a frozen path, we obtain γt+1⟨z0:t+1⟩(ft+1Bt+1⟨z0:t⟩ht+1 + ˜ft+1) = γt⟨z0:t⟩{Qt⟨zt+1⟩ft+1Bt⟨z0:t⟩ht + Qt⟨zt+1⟩(˜htft+1 + ˜ft+1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Thus, the proof is concluded by noting that for every xt ∈ Xt and h ∈ F(Xt:t+1), Qt⟨zt+1⟩h(xt) = � 1 − 1 N � Qth(xt) + 1 N g(xt)h(xt, zt+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 31 Finally, before proceeding to the proof of Proposition 8, we introduce the law of the PARIS evolving conditionally on a frozen path z = {zm}m∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Define, for m ∈ N and zm+1 ∈ Xm+1, P m⟨zm+1⟩ : Ym × Ym+1 ∋ (ym, A) �→ � M m⟨zm+1⟩(xm|m, dxm+1) Sm(ym, xm+1, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For any given initial distribution ψ0 ∈ M1(Y0), let PP ,z ψ0 be the distribution of the canonical Markov chain induced by the Markov kernels {P m⟨zm+1⟩}m∈N and the initial distribution ψ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' By abuse of notation we write PP ,z η0 instead of PP ,z ψ0[η0⟨z0⟩], where the extension ψ0[η0] is defined in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We proceed by forward induction over t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let the σ-fields ˜Ft and Ft be defined as in the proof of Theorem 5, but for the conditional PARIS dual process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then, under the law PP ,z η0 , reusing (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='56), EP ,z η0 � β1 t ft(ξ1 t ) + ˜ft(ξ1 t ) | ˜Ft−1 � = EP ,z η0 � EP ,z η0 � β1 t | Ft � ft(ξ1 t ) + ˜ft(ξ1 t ) | ˜Ft−1 � = EP ,z η0 � ft(ξ1 t ) N � ℓ=1 qt−1(ξℓ t−1, ξ1 t ) �N ℓ′=1 qt−1(ξℓ′ t−1, ξ1 t ) � βℓ t−1 + ˜ht−1(ξℓ t−1, ξ1 t ) � + ˜ft(ξ1 t ) | ˜Ft−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='30), we get EP ,z η0 � β1 t ft(ξ1 t ) + ˜ft(ξ1 t ) | ˜Ft−1 � = � 1 − 1 N � �N ℓ=1{βℓ t−1Qt−1ft(ξℓ t−1) + Qt−1(˜ht−1ft + ˜ft)(ξℓ t−1)} �N ℓ′=1 gt−1(ξℓ′ t−1) + 1 N � ft(zt) N � ℓ=1 qt−1(ξℓ t−1, zt) �N ℓ′=1 qt−1(ξℓ′ t−1, zt) � βℓ t−1 + ˜ht(ξℓ t−1, zt) � + ˜ft(zt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='60) In order to apply the induction hypothesis to each term on the right-hand side of the previous identity, note that Bt⟨z0:t−1⟩ht(zt) = ηt−1⟨z0:t−1⟩[qt−1(·, zt){Bt−1⟨z0:t−2⟩ht−1(·) + ˜ht−1(·, zt)}] ηt−1⟨z0:t−1⟩[qt−1(·, zt)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Therefore, using Lemma 5 and noting that γt⟨z0:t⟩1Xt/γt−1⟨z0:t⟩1Xt−1 = ηt−1⟨z0:t−1⟩gt−1 yields ηt⟨z0:t⟩(ftBt⟨z0:t−1⟩ht + ˜ft) = 1 N � ft(zt)Bt⟨z0:t−1⟩ht(zt) + ˜ft(zt) � + � 1 − 1 N � ηt−1⟨z0:t−1⟩{Qt−1ftBt−1⟨z0:t−2⟩ht + Qt−1(˜ht−1ft + ˜ft)} ηt−1⟨z0:t−1⟩gt−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='61) By combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='60) with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='61), we decompose the error according to 1 N N � i=1 {βi tft(ξi t|t) + ˜ft(ξi t|t)} − ηt⟨z0:t⟩(ftBt⟨z0:t−1⟩ht + ˜ft) = 1 N N � i=1 {βi tft(ξi t|t) + ˜ft(ξi t|t)} − EP ,z η0 � β1 t ft(ξ1 t ) + ˜ft(ξ1 t ) | ˜Ft−1 � + EP ,z η0 � β1 t ft(ξ1 t ) + ˜ft(ξ1 t ) | ˜Ft−1 � − ηt⟨z0:t⟩(ftBt⟨z0:t−1⟩ht + ˜ft) = I(1) N + � 1 − 1 N � I(2) N + 1 N I(3) N , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='62) 32 where I(1) N := 1 N N � i=1 {βi tft(ξi t) + ˜ft(ξi t)} − EP ,z η0 � β1 t ft(ξ1 t ) + ˜ft(ξ1 t ) | ˜Ft−1 � , I(2) N := �N ℓ=1{βℓ t−1Qt−1ft(ξℓ t−1) + Qt−1(˜ht−1ft + ˜ft)(ξℓ t−1)} �N ℓ′=1 gt−1(ξℓ′ t−1) − ηt−1⟨z0:t−1⟩{Qt−1ftBt⟨z0:t−1⟩ht + Qt−1(˜ht−1ft + ˜ft)} ηt−1⟨z0:t−1⟩gt−1 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='63) and I(3) N := ft(zt) N � ℓ=1 qt−1(ξℓ t−1, zt) �N ℓ′=1 qt−1(ξℓ′ t−1, zt) � βℓ t−1 + ˜ht−1(ξℓ t−1, zt) � − ft(zt)ηt−1⟨z0:t−1⟩[qt−1(·, zt){Bt−1⟨z0:t−2⟩ht−1(·) + ˜ht−1(·, zt)}] ηt−1⟨z0:t−1⟩[qt−1(·, zt)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='64) The proof is now completed by treating the terms I(1) N , I(2) N , and I(3) N separately, using Hoeffding’s inequality and its generalisation in [Douc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2011, Lemma 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Choose ε > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' then, by Hoeffding’s inequality, PP ,z η0 � | I(1) N | ≥ ε � ≤ 2 exp � −1 2 ε2 κ2 t N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='65) To treat I(2) N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' we apply the induction hypothesis to the numerator and denominator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' each normalised by 1/N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' yielding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' since ∥Qt−1h∥∞ ≤ ¯τt−1∥h∥∞ for all h ∈ F(Xt−1 � Xt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' PP ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='z η0 ������ 1 N N � ℓ=1 {βℓ t−1Qt−1ft(ξℓ t−1) + Qt−1(˜ht−1ft + ˜ft)(ξℓ t−1)} −ηt−1⟨z0:t−1⟩{Qt−1ftBt⟨z0:t−1⟩ht + Qt−1(˜ht−1ft + ˜ft)} ����� ≥ ε � ≤ ct−1 exp � −dt−1 ε2 ¯τ 2 t−1κ2 t N � and PP ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='z η0 ������ 1 N N � ℓ=1 gt−1(ξℓ t−1) − ηt−1⟨z0:t−1⟩gt−1 ����� ≥ ε � ≤ ct−1 exp � −dt−1 ε2 ¯τ 2 t−1 N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Combining the previous two bounds with the generalised Hoeffding inequality in [Douc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2011, Lemma 4] yields, using also the bounds �N ℓ=1{βℓ t−1Qt−1ft(ξℓ t−1) + Qt−1(˜ht−1ft + ˜ft)(ξℓ t−1)} �N ℓ′=1 gt−1(ξℓ′ t−1) ≤ κt and ηt−1⟨z0:t−1⟩gt−1 ≥ ¯τt−1, the inequality PP ,z η0 � | I(2) N | ≥ ε � ≤ ct−1 exp � −dt−1¯τ 2 t−1ε2 ¯τ 2 t−1κ2 t N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='66) 33 The last term I(3) N is treated along similar lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' indeed, by the induction hypothesis, since ∥qt−1∥∞ ≤ ¯τt−1¯σt−1, PP ,z η0 ������ 1 N N � ℓ=1 qt−1(ξℓ t−1, zt) � βℓ t−1 + ˜ht−1(ξℓ t−1, zt) � − ηt−1⟨z0:t−1⟩[qt−1(·, zt){Bt−1⟨z0:t−1⟩ht−1(·) + ˜ht−1(·, zt)}] ����� ≥ ε � ≤ ct−1 exp � �−dt−1 � ε ¯τt−1¯σt−1 �t−1 m=0 ∥˜hm∥∞ �2 N � � and PP ,z η0 ������ 1 N N � ℓ=1 qt−1(ξℓ t−1, zt) − ηt−1⟨z0:t−1⟩[qt−1(·, zt)] ����� ≥ ε � ≤ ct−1 exp � −dt−1 � ε ¯τt−1¯σt−1 �2 N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Thus, since N � ℓ=1 qt−1(ξℓ t−1, zt) �N ℓ′=1 qt−1(ξℓ′ t−1, zt) � βℓ t−1 + ˜ht−1(ξℓ t−1, zt) � ≤ t−1 � m=0 ∥˜hm∥∞ and ηt−1⟨z0:t−1⟩[qt−1(·, zt)] ≥ ¯τt−1, the generalised Hoeffding inequality provides PP ,z η0 � | I(3) N | ≥ ε � ≤ ct−1 exp � �−dt−1 � ¯τt−1ε 2¯τt−1¯σt−1∥ft∥∞ �t−1 m=0 ∥˜hm∥∞ �2 N � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='67) Finally, combining the bounds (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='65–A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='67) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6 Proof of Proposition 4 The statement of Proposition 4 is implied by the following more general result, which we will prove below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every t ∈ N, M ∈ N∗, N ∈ N∗, z0:t ∈ X0:t, (ft, ˜ft) ∈ F(Xt)2, and p ≥ 2, it holds that � CtSt(z0:t, dbt) ����� 1 N N � i=1 {bi tft(xi t|t) + ˜ft(xi t|t)} − ηt⟨z0:t⟩(ftBt⟨z0:t−1⟩ht + ˜ft) ����� p ≤ ct(p/dt)p/2N −p/2κp t , where ct > 0, dt > 0 and κt are defined in Proposition 8 and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='59), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Before proving Proposition 9, we establish the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let X be an Rd-valued random variable, defined on some probability space (Ω, F, P), satisfying P(|X| ≥ t) ≤ c exp(−t2/(2σ2)) for every t ≥ 0 and some c > 0 and σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then for every p ≥ 2 it holds that E[|X|p] ≤ cpp/2σp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Using Fubini’s theorem and the change of variable formula, E [|X|p] = � ∞ 0 ptp−1P(|X| ≥ t) dt = cp2p/2−1σpΓ(p/2), where Γ is the Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' It remains to apply the bound Γ(p/2) ≤ (p/2)p/2−1 (see [Anderson and Qiu, 1997]), which holds for p ≥ 2 by [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 34 Proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' By combining Proposition 8 and Lemma 6 we obtain N � CtSt(z0:t, dbt) ���� 1 N �N i=1{bi tft(xi t|t) + ˜ft(xi t|t)} − ηt⟨z0:t⟩(ftBt⟨z0:t−1⟩ht + ˜ft) ���� 2 ≤ ct(p/dt)p/2N −p/2 � ∥ft∥∞ t−1 � m=0 ∥˜hm∥∞ + ∥ ˜ft∥∞ �p , which was to be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='7 Proof of Proposition 5 Like previously, we establish Proposition 5 via a more general result, namely the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every t ∈ N, the exists ¯cbias t < ∞ such that for every M ∈ N∗, N ∈ N∗, z0:t ∈ X0:t, and (ft, ˜ft) ∈ F(Xt)2, ����� � CtSt(z0:t, dbt) 1 N N � i=1 {bi tft(xi t|t) + ˜ft(xi t|t)} − ηt⟨z0:t⟩(ftBt⟨z0:t−1⟩ht + ˜ft) ����� ≤ ¯cbias t κtN −1, where κt is defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We preface the proof of Proposition 10 by a technical lemma providing a bound on the bias of ratios of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let α and β be (possibly dependent) random variables defined on some probability space (Ω, F, P) and such that E[α2] < ∞ and E[β2] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Moreover, assume that there exist c > 0 and d > 0 such that |α/β| ≤ c, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', |a/b| ≤ c, E[(α − a)2] ≤ c2d2, and E[(β − b)2] ≤ d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then |E[α/β] − a/b| ≤ 2c(d/b)2 + c|E[β − b]|/|b| + |E[α − a]|/|b|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='68) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Using the identity E[α/β] − a/b = E[(α/β)(b − β)2]/b2 + E[(α − a)(b − β)]/b2 + aE[b − β]/b2 + E[α − a]/b, the claim is established by applying the Cauchy–Schwarz inequality and the assumptions of the lemma according to |E[α/β] − a/b| ≤ cE[(β − b)2]/b2 + {E[(α − a)2]E[(β − b)2]}1/2/b2 + |a||E[β − b]|/b2 + |E[α − a]|/b2 ≤ 2c(d/b)2 + c|E[β − b]|/|b| + |E[α − a]|/|b|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We proceed by induction and assume that the claim holds true for n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Reusing the error decomposition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='62), it is enough to bound the expectations of the terms I(2) N and I(3) N given in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='63) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='64), respectively (since EP ,z η0 [I(1) N ] = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This will be done using the induction hypothesis, Lemma 7, and Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' More precisely, to bound the expectation of I(2) N , we use Lemma 7 with α ← αt, β ← βt, a ← at, and b ← bt, where αt := 1 N N � ℓ=1 {βℓ t−1Qt−1ft(ξℓ t−1) + Qt−1(˜ht−1ft + ˜ft)(ξℓ t−1)}, βt := 1 N N � ℓ=1 gt−1(ξℓ t−1), at := ηt−1⟨z0:t−1⟩{Qt−1ftBt⟨z0:t−1⟩ht + Qt−1(˜ht−1ft + ˜ft)}, bt := ηt−1⟨z0:t−1⟩gt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 35 For this purpose, note that |αt/βt| ≤ κt and |at/bt| ≤ κt, where κt is defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' On the other hand, using Proposition 9 (applied with p = 2), we obtain EP ,z η0 [(αt − at)2] ≤ d2 tκ2 t and EP ,z η0 [(βt − bt)2] ≤ d2 t, where d2 t := ct¯τ 2 t−1/(dtN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Using the induction assumption, we get |EP ,z η0 [αt] − at| ≤ ¯cbias t−1N −1¯τt−1κt and |EP ,z η0 [βt] − bt| ≤ ¯cbias t−1N −1¯τt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Hence, the conditions of Lemma 7 are satisfied and we deduce that |EP ,z η0 [I(2) N ]| = |EP ,z η0 [αt/βt] − at/bt| ≤ 2κt ct dtN ¯τ 2 t−1 ¯τ 2 t−1 + 2¯cbias t−1κt ¯τt−1 ¯τt−1N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The bound on |EP ,z η0 [I(2) N ]| is obtained along the same lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Learning with PPG This section is divided into three subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 establishes, following closely [Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2019], a non-asymptotic bound for stochastic approximation schemes under general assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2 shows how assumptions A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 and A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 imply the assumptions provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 and there- fore allow to establish Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Finally, Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3 provides sufficient assumptions on the model ensuring that A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 Non-asymptotic bound We follow closely [Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Consider the recursion θn+1 = θn − γn+1Hθn(Xn+1), n ∈ N, where θn ∈ Θ ⊂ Rd for some d ∈ N∗ and {Xn}n∈N is a state-dependent Markov chain on some measurable space (X, X) in the sense that Xn+1 ∼ Pθn(Xn, ·) with Pθ being some Markov kernel on (X, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let h(θ) = � Hθ(x) πθ(dx), where πθ is the invariant measure of Pθ and en+1 := Hθn(Xn+1)− h(θn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' As all norms are equivalent in finite dimensional vector spaces, we use ∥ · ∥ to denote a generic norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We denote by {Fn}n∈N the natural filtration of the Markov chain {Xn}n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' There exists a Borel measurable function V : Θ → R such that for every θ ∈ Θ, ∇V (θ) = h(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' There exists LV ∈ R≥0 such that for every (θ, θ′) ∈ Θ2, ∥∇V (θ) − ∇V (θ′)∥ ≤ LV ∥θ − θ′∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' There exists a Borel measurable function �H : Θ×X → Θ such that for every θ ∈ Θ and x ∈ X, �Hθ(x) − Pθ �Hθ(x) = Hθ(x) − h(θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' There exists LP � H ∈ R≥0 such that for every (θ0, θ1) ∈ Θ2, sup x∈X ∥Pθ0 �Hθ0(x) − Pθ0 �Hθ1(x)∥ ≤ LP � H∥θ0 − θ1∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' There exists LP � H 0 ∈ R≥0 such that sup θ∈Θ ∥Pθ �Hθ∥ ≤ LP � H 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 36 A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' There exists σmse ∈ R≥0 such that for every x ∈ X and θ ∈ Θ, � ∥Hθ(x′) − h(θ)∥2 Pθ(x, dx′) ≤ σ2 mse .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' There exists L � H ∈ R≥0 such that for every x ∈ X, sup θ∈Θ � ∥ �Hθ∥ Pθ(x, dx′) ≤ L � H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Assume that A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1–A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='7 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In addition, assume that there exist a > 0 and a′ > 0 such that for all n ∈ N, γn+1 ≤ γn ≤ aγn+1 , γn − γn+1 ≤ a′γ2 n , γ1 ≤ (LV + Ch)−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Moreover, for any n ∈ N∗, let ϖ be a �0, n�-valued random variable, independent of {Fℓ}ℓ≥0 and such that P(ϖ = k) = γk+1/ �n ℓ=0 γℓ+1 for k ∈ �0, n�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then, E � ∥h(θϖ)∥2� ≤ 2V0,n + C0,n + (σ2 mseLV + Cγ) �n k=0 γ2 k+1 �n k=0 γk+1 , where V0,n := E [V (θ) − V (θn)] and C0,n := γ1h(θ0)L � H + LP � H 0 (γ1 − γn+1 + 1) , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='69) Cγ := σmseLP � H + (1 + σmse)LV LP � H 0 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='70) Ch := LP � H ((a + 1)/2 + aσmse) + (LV + a′ + 1)LP � H 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='71) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We follow closely the proof of [Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2019, Theorem 2] and adapt it to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' First, note that by A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1, assumptions A1 and A2 of [Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2019, Theorem 2] hold with c0 = d0 = 0 and c1 = d1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In addition, the claim in [Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2019, Lemma 1] holds true since by AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2, A3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Moreover, [Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2019, Equation 17] can also be established under AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6, as we may rewrite it as n � ℓ=0 γ2 ℓ+1E � ∥eℓ+1∥2� = n � ℓ=0 γ2 ℓ+1E � E � ∥eℓ+1∥2 | Fℓ �� ≤ σ2 mse n � ℓ=0 γ2 ℓ+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Following the proof of [Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2019, Lemma 2], consider the decomposition E � − n � ℓ=0 γℓ+1 ⟨∇V (θℓ), eℓ+1⟩ � = E [A1 + A2 + A3 + A4 + A5] , where A1 := − n � ℓ=1 γℓ+1 � ∇V (θℓ), �Hθℓ(Xℓ+1) − Pθℓ �Hθℓ(Xℓ) � , A2 := − n � ℓ=1 γℓ+1 � ∇V (θℓ), Pθℓ �Hθℓ(Xℓ) − Pθℓ−1 �Hθℓ−1(Xℓ) � , A3 := − n � ℓ=1 γℓ+1 � ∇V (θℓ) − ∇V (θℓ−1), Pθℓ−1 �Hθℓ−1(Xℓ) � , A4 := − n � ℓ=1 (γℓ+1 − γℓ) � ∇V (θℓ−1), Pθℓ−1 �Hθℓ−1(Xℓ) � , A5 := −γ1 � ∇V (θ0), �Hθ0(X1) � + γn+1 � ∇V (θn), Pθn �Hθn(Xn+1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 37 As �Hθℓ(Xℓ+1) − Pθℓ �Hθℓ(Xℓ) is a martingale difference, it holds that E [A1] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The upper bounds on the expectations of A2, A3 and A4 are obtained similarly as in [Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Using A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='4, A2 ≤ LP � H � σmse n � k=1 γ2 k + 1 2 (1 + 2aσmse + a) n � k=0 γ2 k+1∥h(θk)∥2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' By A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='5, A3 ≤ LV LP � H 0 � (1 + σmse) n � k=1 γ2 k + n � k=1 γ2 k∥h(θk)∥2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' On the other hand, A4 ≤ LP � H 0 � γ1 − γn+1 + a′ n � k=1 γ2 k∥h(θk−1)∥2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We now focus on A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' As in the proof of [Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2019, Lemma 2], the expectation of the first term can be straightforwardly bounded by γ1∥h(θ0)∥L � H using the Cauchy–Schwarz inequality and A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The second term can, using A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='5 and γn+1∥h(θn)∥ ≤ 1 + γ2 n+1∥h(θn)∥2, be bounded in the same way according to γn+1 � ∇V (θn), Pθn �Hθn(Xn+1) � ≤ LP � H 0 γn+1∥h(θn)∥ ≤ LP � H 0 � 1 + γ2 n+1∥h(θn)∥2� ≤ LP � H 0 � 1 + n � ℓ=0 γ2 ℓ+1∥h(θℓ)∥2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The rest of the proof follows that of [Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2019, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2 Application to Theorem 2 The goal of this section is to establish that the assumptions of Theorem 2 ensure all the assumptions in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1, which in turn allows Theorem 8 to be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' First, we start by explicitly defining the kernel Pθ and the function h in terms of the kernels presented in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We write Pθ,t instead of Pθ to explicit the dependence of the kernel on the fixed number of observations t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 Verification of the assumptions of Theorem 8 For (k0, k) ∈ (N∗)2 such that k0 < k, define Pθ,t : Ek−k0 t × E�(k−k0) t ∋ (yt[k0 : k], z0:t[k0 : k], A) �→ Kk0 θ,t � K�(k−k0) θ,t (z0:t[k], A), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='72) where Kθ,t is the PPG kernel defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Note that Pθ,t depends only on the last frozen path, namely z0:t[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Note also that, since Kθ,t depends only on the paths, there is no dependence between yt,ℓ[k0 : k] and yt,ℓ+1[k0 : k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The score ascent algorithm (Algorithm 4) can be formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Sample (z0:t,ℓ[k0 : k], yt,ℓ[k0 : k]) ∼ Pθℓ,t � (z0:t,ℓ−1[k0 : k], yt,ℓ−1[k0 : k]), · � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Update the parameter according to ηℓ+1 = ηℓ + γℓ+1H(z0:t,ℓ[k0 : k], yt,ℓ[k0 : k]), where H(z0:t,ℓ[k0 : k], yt,ℓ[k0 : k]) = 1 k − k0 + 1 k � i=k0 µ(βt,ℓ[i])(id) = Π(k0−1,k),N(ht), where Π(k0−1,k),N(ht) is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We denote by πθ,t the invariant distribution of Pθ,t, which, by Proposition 3, is given by πθ,t = (η0:t � CtSt)�(k−k0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We also require the strong mixing assumption to hold uniformly in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 38 A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='8 (Strong mixing uniformly in θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every s ∈ N there exist ¯τs, ¯τs, ¯σs, and ¯σs in R∗ + such that for all θ ∈ Θ, (i) ¯τs ≤ gs,θ(xs) ≤ ¯τs for every xs ∈ Xs, (ii) ¯σs ≤ ms,θ(xs, xs+1) ≤ ¯σs for every (xs, xs+1) ∈ Xs:s+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Note that the assumption above implies that κN,t is also uniform in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof that A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all θ ∈ Θ, h(θ) = ∇V (θ), where V (θ) = log γ0:t,θ(X0:t) is the log-likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' By Theorem 5, h(θ) = � H(˜yt[k0 : k], ˜x0:t[k0 : k]) πθ,t(d(˜yt[k0 : k], ˜x0:t[k0 : k])) = 1 k − k0 + 1 k � i=k0 � [η0:t,θ � Ct,θSt,θ] (d(˜yt[i], ˜x0:t[i]))µ(˜βt,ℓ[i])(id) = η0:t,θ (s0:t,θ) = ∇V (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof that A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2 is trivially implied by A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof that A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='5 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let �Hθ be given by �Hθ : Ek−k0 t ∋ (yt[k0 : k], z0:t[k0 : k]) �→ ∞ � r=0 {Pr θ,tH(yt[k0 : k], z0:t[k0 : k]) − h(θ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='73) Then the following holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Assume A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then for all θ ∈ Θ and t ∈ N∗, ∥Pθ,t �Hθ∥∞ ≤ σbias(1 − κk N,t)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' By Theorem 1, we have for any r > 0 ��Pr θ,tH(yt[k0 : k], z0:t[k0 : k]) − h(θ) �� ≤ σbiasκ(r−1)k N,t and thus ∥Pθ,t �Hθ∥∞ ≤ ∞ � r=1 ��Pr θ,tH − h(θ) �� ∞ ≤ σbias ∞ � r=0 κrk N,t ≤ σbias(1 − κk N,t)−1 , where κN,t ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 8 proves A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='5 with LP � H 0 := σbias(1 − κk N,t)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 39 Proof that A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='4 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Assume A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='8 and A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then for every t ∈ N, θ ∈ Θ and N ∈ N∗ such that N > 1 + 5ρ2 tt/2, ���Pθ1,t �Hθ1 − Pθ2,t �Hθ2 ��� ∞ ≤ LP � H∥θ1 − θ2∥ , where LP � H := ∥LP 2 ∥∞ � 1 + κk N,t(1 − κk N,t) � + LV + σbias(1 − κN,t)−1(1 − κk N,t)−1 � ∥LP 1 ∥∞(1 − κk N,t)−1 + Lηκk N,t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='74) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We establish the claim by adapting the proof of [Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2019, Lemma 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' First, recall that the kernel Kθ,t defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='42) is the path marginalized version of Kθ,t given in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Note that for every x ∈ Ek−k0 t , Pθ1,t �Hθ1(x) = ∞ � n=0 δxPθ1,t � Pn θ1,tH − h(θ1) � = ∞ � n=0 δxKkn θ1,t {Pθ1,tH − η0:t,θ1Pθ1,tH} , where we have used (i) the fact that the backward statistics output by Pθ,t are independent of the input backward statistics and (ii) the penultimate line in the computation of h(θ) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We follow the proof of [Fort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2011, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2] and consider the following decomposition: for n ∈ N∗, δxKkn θ1,t (Pθ1,tH − η0:t,θ1Pθ1,tH) − δxKkn θ2,t (Pθ2,tH − η0:t,θ2Pθ2,tH) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='75) = n−1 � j=0 � δxKkj θ1,t − η0:t,θ1 � � Kkj θ1,t − Kkj θ2,t � � Kk(n−j−1) θ2,t Pθ1,tH − η0:t,θ2Pθ1,tH � − � δxKkn θ2,tPθ2,tH − η0:t,θ2Pθ2,tH � + � δxKkn θ2,tPθ1,tH − η0:t,θ2Pθ1,tH � − η0:t,θ1 � Kkn θ2,tPθ1,tH − η0:t,θ2Pθ1,tH � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Applying Theorem 6 with µ = δx and ν = η0:t,θ and using the fact that η0:t,θKℓ θ,t = η0:t,θ for all ℓ ∈ N, we obtain that for all ℓ ∈ N and all θ ∈ Θ, ���δxKℓ θ,t − η0:t,θ ��� TV ≤ κℓ N,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Note that by A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1(iii), Kθ,t is Lipschitz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' therefore, for all r ∈ N∗, by Lemma 18, Kr θ,t is Lipschitz with constant ∥LP 1 ∥∞(1 − κN,t)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Combining all this together, we obtain ��� � δxKkj θ1,t − η0:t,θ1 � � Kkj θ1,t − Kkj θ2,t � � Kk(n−j−1) θ2,t Pθ1,tH − η0:t,θ2Pθ1,tH ���� = ��� � δxKkj θ1,t − η0:t,θ1 � � Kkj θ1,t − Kkj θ2,t � � Kk(n−j−1) θ2,t [Pθ1,tH − h(θ1)] − η0:t,θ2 [Pθ1,tH − h(θ1)] ���� ≤ ∥LP 1 ∥∞(1 − κN,t)−1κkj N,tκk(n−j−1) N,t ∥Pθ1,tH − h(θ1)∥∞∥θ1 − θ2∥ ≤ σbias∥LP 1 ∥∞(1 − κN,t)−1κk(n−1) N,t ∥θ1 − θ2∥ , where the last inequality is due to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Therefore, the first term of the right side of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='75) is upper bounded by σbias∥LP 1 ∥∞(1 − κN,t)−1nκk(n−1) N,t ∥θ1 − θ2∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The second term of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='75) can be written − � δxKkn θ2,tPθ2,tH − η0:t,θ2Pθ2,tH � + � δxKkn θ2,tPθ1,tH − η0:t,θ2Pθ1,tH � = � δxKkn θ2,t − η0:t,θ2 � (Pθ1,tH − Pθ2,tH) , and using again the ergodicity of Kθ,t and the fact that θ �→ Pθ,tH is uniformly Lipschitz by A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1(iv), we may conclude that it is upper bounded by ∥LP 2 ∥∞κkn N,t∥θ1 − θ2∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Finally, for the last term, using 40 the facts that Kk θ,t is η0:t,θ-invariant and geometrically ergodic and that θ �→ η0:t,θ is Lipschitz by A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1(iv) yields ��η0:t,θ1 � Kkn θ2,tPθ1,tH − η0:t,θ2Pθ1,tH ��� = ��(η0:t,θ1 − η0:t,θ2) � Kkn θ2,t [Pθ1,tH − h(θ1)] − η0:t,θ2 [Pθ1,tH − h(θ1)] ��� ≤ Lηκkn N,t∥Pθ1,tH − h(θ1)∥∞∥θ1 − θ2∥ ≤ Lησbias(1 − κN,t)−1κkn N,t∥θ1 − θ2∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Therefore, we have that δxKkn θ1,t (Pθ1,tH − η0:t,θ1Pθ1,tH) − δxKkn θ2,t (Pθ2,tH − η0:t,θ2Pθ2,tH) ≤ � σbias∥LP 1 ∥∞(1 − κN,t)−1nκk(n−1) N,t + � ∥LP 2 ∥∞ + Lησbias(1 − κN,t)−1� κkn N,t � ∥θ1 − θ2∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Therefore, we obtain ���Pθ1,t �Hθ1(x) − Pθ2,t �Hθ2(x) ��� ≤ |δxPθ1,tH − δxPθ2,tH| + |η0:t,θ1Pθ1,tH − η0:t,θ2Pθ2,tH| + ����� ∞ � n=1 δxKkn θ1,t (Pθ1,tH − η0:t,θ1Pθ1,tH) − δxKkn θ2,t (Pθ2,tH − η0:t,θ2Pθ2,tH) ����� ≤ |δxPθ1,tH − δxPθ2,tH| + |η0:t,θ1Pθ1,tH − η0:t,θ2Pθ2,tH| + � σbias∥LP 1 ∥∞(1 − κN,t)−1(1 − κk N,t)−2 + � ∥LP 2 ∥∞ + Lησbias(1 − κN,t)−1� κk N,t(1 − κk N,t)−1 � ∥θ1 − θ2∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' To conclude, note that by A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1(iv), ∥δxPθ1,tH − δxPθ2,tH∥ ≤ ∥LP 2 ∥∞∥θ1 − θ2∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Furthermore, note that by Theorem 5 we obtain that for all θ ∈ Θ, η0:t,θPθ,tH = η0:t,θs0:t,θ = ∇V (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Therefore, by A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1(i) we obtain that ∥η0:t,θ1Pθ1,tH − η0:t,θ2Pθ2,tH∥ ≤ LV ∥θ1 − θ2∥, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof that AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='6 is simply a bound on the MSE of the roll-out PPG estimator, given by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof that A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='7 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all θ ∈ Θ and all ℓ ∈ �1, t − 1� E � ∥ �Hθ∥ | Fℓ � ≤ 2∥s0:t,θ∥∞ + σbias(1 − κk N,t)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Note that for all x ∈ Ek−k0 t and all θ ∈ Θ, �Hθ(x) = H(x) − h(θ) + Pθ,t �Hθ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='76) Lemma 8 shows that ∥Pθ,t �Hθ∥∞ ≤ σbias(1 − κk N,t)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Note that h(θ) ≤ ∥s0:t,θ∥∞ We write E [∥H∥ | Fℓ] ≤ 1 (k − k0 + 1)N k � i=k0 N � j=1 E � ∥βj t,ℓ[i]∥ | Fℓ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' By Proposition 14, E � ∥βj t,ℓ[i]∥ | Fℓ � ≤ ∥s0:t,θ∥∞, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='7 follows directly by Proposition 12 and by considering supθ∈Θ ∥s0:t,θ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 41 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2 Proof of Theorem 2 We have shown in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 that under A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='8, it is possible to apply Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' To conclude the proof of Theorem 2 we just have to rearrange the constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We start by rewriting the constant in Theorem 9 LP � H = C1 + σbias(1 − κN,t)−1(1 − κk N,t)−1C2, with C1 = ��LP 2 �� ∞ � 1 + κk N,t(1 − κk N,t)−1� + LV C2 = ��LP 1 �� ∞ (1 − κk N,t)−1 + Lηκk N,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' By (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='70) and Lemma 8, Cγ = σmseLP � H + (1 + σmse)LV LP � H 0 = σmse � C1 + σbias(1 − κN,t)−1(1 − κk N,t)−1C2 � + (1 + σmse)LV σbias(1 − κk N,t)−1 = σmseC1 + σmseσbias(1 − κk N,t)−1 � LV + (1 − κN,t)−1C2 � + σbiasLV (1 − κk N,t)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Therefore, C0,γ := σ2 mseLV + Cγ = σ2 mseLV + σmseC1 + σmseσbias(1 − κk N,t)−1 � LV + (1 − κN,t)−1C2 � + σbiasLV (1 − κk N,t)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In the same way, we can rewrite (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='71) as Ch = LP � H [(a + 1)/2 + aσmse] + (LV + a′ + 1)LP � H 0 = � C1 + σbias(1 − κN,t)−1(1 − κk N,t)−1C2 � [(a + 1)/2 + aσmse] + (LV + a′ + 1)σbias(1 − κk N,t)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The constant C0 from Theorem 2 is L � H = 2 supθ∈Θ ∥s0:t,θ∥∞ + σbias(1 − κk N,t)−1 which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3 Conditions on the model to verify A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 In our specific application to score ascent, we work with the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='9 (Lipschitz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (i) For all t ∈ N, there exists Ls t ∈ M(Xt:t+1) such that for all (xt, xt+1) ∈ Xt:t+1, the function θ �→ st,θ(xt, xt+1) is Ls t(xt, xt+1)-Lipschitz and Xt:t+1 ∋ (xt, xt+1) �→ st,θ(xt, xt+1) is bounded by ∥st(θ)∥∞ for all θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Furthermore, ∥Ls k∥∞ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (ii) For all t ∈ N, there exists Lq t ∈ Xt:t+1 such that ∥Lq t∥∞ < ∞ and that for all (xt, xt+1) ∈ Xt:t+1, θ �→ qt,θ(xt, xt+1) is Lq t(xt, xt+1)-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 9 (A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2(i) holds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Assume A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='8 and A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' There exists a constant LV such that the Lyapunov function V satisfies, for all (θ1, θ2) ∈ Θ2, ∥∇V (θ1) − ∇V (θ2)∥ ≤ LV ∥θ1 − θ2∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all θ1, θ2, ∥∇V (θ1) − ∇V (θ2)∥ = ∥η0:t,θ1(s0:t,θ1) − η0:t,θ2(s0:t,θ2)∥ ≤ ∥η0:t,θ1(s0:t,θ1) − η0:t,θ1(s0:t,θ2)∥ + ∥η0:t,θ1(s0:t,θ2) − η0:t,θ2(s0:t,θ2)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1) and by [Gloaguen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2022, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='10] there exists a constant c such that ∥η0:t,θ1(s0:t,θ2) − η0:t,θ2(s0:t,θ2)∥ ≤ ct∥θ1 − θ2∥ supθ supk ∥sk(θ)∥∞ , 42 Using A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 and A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1[i], we can write: ∥η0:t,θ1(s0:t,θ1) − η0:t,θ1(s0:t,θ2)∥ ≤ t−1 � u=0 η0:t,θ1 [∥su,θ1(xu:u+1) − su,θ2(xu:u+1)∥], ≤ t−1 � u=0 η0:t,θ1 [Ls u(xu:u+1)] ∥θ1 − θ2∥, ≤ σ+ σ− supu∈�0,t−1� [Ls u] ∥θ1 − θ2∥t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Theorem 10 (Lipschitz continuity of Particle Gibbs with Backward Sampling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Assume A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every t ∈ N, θ ∈ Θ and N ∈ N∗ sup x0:t∈X0:t ∥Kθ1,t(x0:t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=') − Kθ2,t(x0:t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' )∥TV ≤ LK t,N∥θ1 − θ2∥ , where LK t,N := t−1 � ℓ=0 ¯τ −1 ℓ � ¯σ−1 ℓ + (N − 1) � ∥Lq ℓ∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='77) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We know that Kθ,t = Cm,θBt,θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Therefore, by Lemmas 14, 16 and 19, we have that Kθ,t is Lipschitz with constant equals LC t + supθ Ct,θLB t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Corollary 1 (A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1(iii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Assume A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every t ∈ N, θ ∈ Θ, r ∈ N∗ and N ∈ N∗ such that N > 1 + 5ρ2 tt/2 sup x0:t∈X0:t ��Kr θ1,t(x0:t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=') − Kr θ2,t(x0:t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=') �� TV ≤ LP t,N∥θ1 − θ2∥ where LP t,N := (1 − κt,N)−1∥LK t,N∥∞ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='78) where LK t,N is defined in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Under B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='8, the Particle Gibbs with backward sampling is geometrically ergodic with contraction rate κt,N and thus LK t,N is bounded and the result follows from Lemma 18 Corollary 2 (A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Assume A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='8 and A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all t ∈ N∗, (θ0, θ1) ∈ Θ2, ∥η0:t,θ0 − η0:t,θ1∥TV ≤ Lη∥θ0 − θ1∥, where Lη := LP t,N ∗ , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='79) and LP t,N is defined in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='78) and N ∗ = ⌈1 + 5ρ2 t/2⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Consider the following decomposition, valid for all k ∈ N∗ and N ≥ 1+5ρ2 t/2, and all x0:t ∈ X0:t, ∥η0:t,θ1 − η0:t,θ2∥TV ≤ ��η0:t,θ1 − Kk θ1,t(x0:t, ·) �� TV + ��η0:t,θ2 − Kk θ2,t(x0:t, ·) �� TV + ��Kk θ1,t(x0:t, ·) − Kk θ2,t(x0:t, ·) �� TV ≤ ��η0:t,θ1 − Kk θ1,t(x0:t, ·) �� TV + ��η0:t,θ2 − Kk θ2,t(x0:t, ·) �� TV + LP t,N∥θ1 − θ2∥ , where we applied Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Since the Lipschitz constant of Kθ,t is independent of k, and Kθ,t is geometrically ergodic for all θ, we obtain by taking the limit when k goes to infinity with N fixed, ∥η0:t,θ1 − η0:t,θ2∥TV ≤ ∥LK t,N∥∞ 1 − κt,N ∥θ1 − θ2 ∥ , for all N ≥ 1 + 5ρ2 t/2, where the dependence in N is hidden in LP t,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The result follows by choosing N = ⌈1 + 5ρ2 t/2⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 43 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' As noted by [Lindholm and Lindsten, 2018], the Lipschitz constant appearing in Corol- lary 1 possesses an unexpected dependence on N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' One would expect it not to be true, in that we know that Kθ,t converges geometrically fast and uniformly to η0:t and this is faster as N gets bigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Therefore, for large N the Lipschitz constant is expected to converge to that of η0:t whose Lipschitz constant is independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proposition 13 (Lipschitz continuity of θ �→ Kθ,tµ(βt)(id)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Assume A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every t ∈ N, θ ∈ Θ and N ∈ N∗, ∥Kθ1,tµ(βt)(id) − Kθ2,tµ(βt)(id)∥∞ ≤ LK t ∥θ1 − θ2∥ , where LK t := (N − 1) t−1 � ℓ=0 ¯τℓ∥Lq ℓ∥∞ + m � j=1 ∥L ← − Q j ∥∞ �m−1 � ℓ=0 s∞ ℓ � + m � j=1 ∥Ls j∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='80) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Consider e = (x0:t, y0:t) ∈ Et and fθ(e) := � Sm,θ(x0:t, d˜yt)µ(bt)(id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then Kθ,tµ(bt)(id) = Cm,θfθ(x0:t) is a composition of a Markov kernel and a Lipschitz function, therefore Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Corollary 3 (A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1(iv) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Assume A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every t ∈ N, θ ∈ Θ and N ∈ N∗ sup x0:t∈X0:t ∥Pθ1,tH − Pθ2,tH∥ ≤ LP 2 ∥θ1 − θ2∥ , where LP 2 = LP t,N + LK t , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='81) with LP and LK t are defined in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='80) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let ˜f : Ek−k0 ∋ (x0:t[k0 : k], x0:t|t[k0 : k], bt[k0 : k]) �→ (k − k0)−1 �k ℓ=k0+1 µ(bt[ℓ])(id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' As Kθ,t depends only on the path, with a slight abuse of notation, we can define fθ(x0:t) := K�k−k0 θ,t ( ˜f)(x0:t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' By proposition 13, we have that fθ is Lipschitz with Lf = LK t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Note that Pθ,tH(x0:t, yt) = Kk0 θ,tfθ(x0:t), therefore, by lemma 19 Lipschitz with constant LP + LK t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lipschitz properties C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 Lipschitz continuity of Pθ, In this section we prove the following items: Cm,θ(z0:m, ·) is Lipschitz, see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 Bm,θ(x0:m, ·) is Lipschitz, see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2 � Sm,θ(x0:m, dbm)µ(bm)(Id) is Lipschitz, see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3 The following technical lemma will be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let α ∈]0, 1], x ∈ R≥0 and ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then for all λi ∈ R≥0, i ∈ �0, ℓ�, such that α ≥ �ℓ i=0(1 − λix) it holds that α ≥ 1 − x �ℓ i=0 λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Consider first the case where xλi ≤ 1 for all i ∈ �0, ℓ�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We prove the result by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The case ℓ = 0 is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Assume now that the result holds for some r ∈ �0, ℓ − 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then, r+1 � i=0 (1 − λix) = (1 − λr+1x) r � i=0 (1 − λix) ≥ (1 − λr+1x)(1 − x r � i=0 λi) = 1 − x r+1 � i=0 λi + x2 r � i=0 λiλr+1 ≥ 1 − x r+1 � i=0 λi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 44 Consider now the case where there is a index j ∈ �0, ℓ� such that xλj ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then α ≥ 0 ≥ 1 − (�ℓ i=0 λi)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We begin with some important definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let P and Q be probability distributions on some common measurable space (X, X), and assume that these distributions admit densities p and q w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='t some common reference measure λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let M [P, Q] denote a maximal coupling between P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' As in [Lindholm and Lindsten, 2018, Theorem 2], it is possible to explicitly construct one such maximal coupling by M [P, Q] (d(x, y)) := min{p(x), g(x)}λ(dx)δx(dy)+ � P(dx) − min{p(x), g(x)}λ(dx) �� Q(dy) − min{p(y), g(y)}λ(dy) � 1 − λ � min{p, q} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='82) From this definition it follows that for continuous and discrete dominating measures λ, � 1{x=y}M [P, Q] d(x, y) = � min{p(x), g(x)}λ(dx) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Moreover, for two Markov transition kernels K1 and K2 on (X, X), which are assumed to admit transition densities with respect to some common dominating measure, we let, for (x1, x2) ∈ X2, M [K1, K2] ((x1, x2), ·) denote the maximal coupling between the measures K1(x1, ·) and K2(x2, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Defined in this way, M [K1, K2] defines a Markov transition kernel on the product space (X2, X �2) The following Lemma will be crucial in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (i) Let (µ1, µ2) be two probability measures admitting a density with respect to a com- mon dominating measure and let (K1, K2) two Markov transition kernels also admitting transition densities with respect to some dominating measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then the probability measure M [µ1, µ2] M [K1, K2] (d(x1, x2)) = � M [µ1, µ2] (d(z1, z2))M [K1, K2] ((z1, z2), d(x1, x2)), is a coupling of (µ1K1, µ2K2), and it holds that � 1x1=x2M [µ1K1, µ2K2] (d(x1, x2)) ≥ � � 1z1=z21x1=x2M [µ1, µ2] (d(z1, z2))M [K1, K2] ((z1, z2), d(x1, x2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (ii) Let (µ1, · · · , µn) and (ν1, · · · , νn) be probability measures such that for all i ∈ �1, n�, µi and νi admit densities with respect to the same dominating measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then �n i=1 M [µi, νi] is a coupling of �n i=1 µi and �n i=1 νi, and thus � n � i=1 1xi=yiM � n � i=1 µi, n � i=1 νi � (d(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xn, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , yn)) ≥ � n � i=1 1xi=yi n � i=1 M [µi, νi] (d(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , xn, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , yn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' It is enough to show that M [µ1, µ2] M [K1, K2] admits µ1K1 and µ2K2 as marginal distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This follows immediately from the fact that M [µ1, µ1] and M [K1, K2] admit the right marginal 45 distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' indeed, M [µ1, µ2]M [K1, K2] (X × A) = � M [µ1, µ2] (dz1, d2) M [K1, K2] (z1, z2, d(x1, x2)) 1X×A(x1, x2)1X2(z1, z2) = � M [µ1, µ2] (dz1, d2)K2(z2, A) = � µ2(dz2)K2(z2, A) = µ2K2(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The derivation for the first marginal distribution follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For the second point, since M [µ1, µ2] M [K1, K2] is a coupling of (µ1K1, µ2K2) and M [µ1K1, µ2K2] is the maximal coupling, we have that � 1x1=x2M [µ1K1, µ2K2] (d(x1, x2)) ≥ �� 1x1=x2M [µ1, µ2] (d(z1, z2)) M [K1, K2] (z1, z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' d(x1, x2)) ≥ �� 1x1=x21z1=z2M [µ1, µ2] (d(z1, z2)) M [K1, K2] (z1, z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' d(x1, x2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The proof of the second item follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 θ �→ Cm,θ is Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We proceed by a coupling method that is inspired by [Lindholm and Lindsten, 2018, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The coupling we consider is that where the selection and mutation steps of the particle filter are respectively coupled maximally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Algorithm 6 Coupling Cm,θ Data: θ1, θ2, ζ0:m Result: x0:m,1, x0:m,1 23 draw x0,1, x0,2 ∼ M [η0⟨ζ0⟩, η0⟨ζ0⟩] 24 for s ← 1 to t do 25 draw (xs,1, xs,2) ∼ M [M s−1,θ1⟨ζs⟩(xs−1,1, ·), M s−1,θ2⟨ζs⟩(xs−1,2, ·)] First, let us prove that the one step selection–mutation kernel is Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all t ∈ N, xt−1 ∈ Xt−1 and (θ1, θ2) ∈ Θ2, � 1{x1=x2}M [Φt−1,θ1(µ(xt−1)), Φt−1,θ2(µ(xt−1))] (d(x1, x2)) ≥ 1 − �N i=1 λt � Lq t−1(xi t−1, ·) � N ¯τn ∥θ1 − θ2∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='83) 46 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' By A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1(i) and A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1(iii),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' � 1{x1=x2}M [Φt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ1(µ(xt−1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Φt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ2(µ(xt−1))] (d(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' x2)) = � min � N � i=1 qt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ1(xi t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' x) �N j=1 gt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ1(xj t−1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' N � i=1 qt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ2(xi t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' x) �N j=1 gt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ2(xj t−1) � λt(dx) ≥ N � j=1 � min � qt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ1(xi t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' x) �N j=1 gt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ1(xj t−1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' qt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ2(xi t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' x) �N j=1 gt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ2(xj t−1) � λt(dx) ≥ 1 �N j=1 max � gt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ1(xj t−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' gt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ2(xj t−1) � N � j=1 � min � qt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ1(xj t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' qt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ2(xj t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' x) � λt(dx) ≥ �N j=1 max � gt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ1(xj t−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' gt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ2(xj t−1) � − �N i=1 λt � Lq t−1(xi t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ·) � ∥θ1 − θ2∥ �N j=1 max � gt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ1(xj t−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' gt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ2(xj t−1) � ≥ 1 − �N i=1 λt � Lq t−1(xi t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ·) � N ¯τn ∥θ1 − θ2∥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' where we have used that � max(qt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ1(xi t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' qt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ2(xi t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' x))λt(dx) ≥ max �� qt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ1(xi t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' x)λt(dx),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' � qt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ2(xi t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' x)λt(dx) � ≥ max(gt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ1(xi t−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' gt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ2(xi t−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all t ∈ N, xt−1 ∈ Xt−1, z ∈ Xt and (θ1, θ2) ∈ Θ2, ∥M t−1,θ1⟨z⟩(xt−1, ·) − M t−1,θ2⟨z⟩(xt−1, ·)∥TV ≤ LM t−1(xt−1)∥θ1 − θ2∥ where LM t−1(xt−1) = (1 − N −1)¯τ −1 t−1 �N i=1 λt � Lq t−1(xi t−1, ·) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let us denote by U�1, n� the uniform distribution on �1, n�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' By definition of the kernel M t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ⟨z⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' we have that M t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ⟨z⟩(xt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' dxt) = � U�1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' n�(dj) � Φt−1(µ(xt−1))�j � δz � Φt−1(µ(xt−1))�(N−j−1)� (dxt) and thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' applying the two items of Lemma 11 combined with the fact that M [µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' µ] � d(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' x2) � = 47 µ(dx1)δx1(dx2) for any probability measure µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' we get that � 1{xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1=xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2}M [M t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ1⟨z⟩(xt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' M t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ2⟨z⟩(xt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ·)] d(xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2) ≥ � 1xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1=xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='i1=i2M [U�1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' n�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' U�1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' n�] � d(i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' i2) � × M [Φt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ1(µ(xt−1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Φt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ2(µ(xt−1))]⊗i1 ⊗ M [δz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' δz] ⊗ M [Φt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ1(µ(xt−1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Φt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ2(µ(xt−1))]⊗N−i1−1 d(xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2) = 1 N N � i=1 � n � k=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='k̸=i 1xi t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1=xi t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2M [Φt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ1(µ(xt−1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Φt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ2(µ(xt−1))] � d(xi t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xi t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2) � ≥ � 1 − �N i=1 λt � Lq t−1(xi t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ·) � N ¯τt−1 ∥θ1 − θ2∥ �N−1 ≥ 1 − N − 1 ¯τt−1N N � i=1 λt � Lq t−1(xi t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ·) � ∥θ1 − θ2∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' where we have applied Lemma 12 in the penultimate line and Lemma 10 in the last one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every t ∈ N∗, there exists LC t ∈ M(X0:t) such that ∥Ct,θ1(z0:t) − Ct,θ2(z0:t)∥TV ≤ LC t (z0:t)∥θ1 − θ2∥ , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='84) where LC t (z0:t) = supθ Ct,θ ��t−1 i=0 LM i � (z0:t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Under AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='9(i), we obtain that ∥LC t ∥∞ ≤ (N−1) �t−1 ℓ=0 ¯τℓ∥Lq ℓ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' This is a direct application of lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2 θ �→ Bt,θ(x0:t, ·) is Lipschitz We start by recalling the definition of Bm Bt,θ : X0:t × X0:t ∋ (x0:t, A) �→ � · · � 1A(x0:t) �t−1 � s=0 ←− Q s,µ(xs)(xs+1, dxs) � µ(xt)(dxt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='85) Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all s ∈ �0, t�, xt+1 ∈ Xt+1, xt ∈ Xt and (θ1, θ2) ∈ Θ2 ���←− Q s,µ(xs),θ1(xs+1, ·) − ←− Q s,µ(xs),θ2(xs+1, ·) ��� TV ≤ L ← − Q s (xs+1, xs)∥θ1 − θ2∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='86) with L ← − Q s (xs+1, xs) = (N ¯τt¯σs)−1 �N i=1 Lq s(xi s, xs+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Under AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='9(i), we have ∥L ← − Q m∥∞ = (¯τm¯σm)−1∥Lq m∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Note that ←− Q t,µ(xt)(xt+1, ·) = �N ℓ=1 qt(xℓ t,xt+1) �N ℓ′=1 qt(xℓ′ t ,xt+1)δxℓ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Therefore, similarly to the proof of Lemma 12, � 1{xt,1=xt,2}M �←− Q t,µ(xt),θ1(xt+1, ·), ←− Q t,µ(xt),θ2(xt+1, ·) � d(xt,1, xt,2) ≥ �N ℓ=1 max(qt,θ1(xℓ t, xt+1), qt,θ2(xℓ t, xt+1)) − Lq t(xℓ t, xt+1)∥θ1 − θ2∥ �N ℓ=1 max(qt,θ1(xℓ t, xt+1), qt,θ2(xℓ t, xt+1)) ≥ 1 − �N ℓ=1 Lq t(xℓ t, xt+1) N ¯τt¯σt ∥θ1 − θ2∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 48 Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For all t ∈ N, x0:t ∈ X0:t and (θ1, θ2) ∈ Θ2 ∥Bt,θ1(x0:t, ·) − Bt,θ2(x0:t, ·)∥TV ≤ LB t (x0:t)∥θ1 − θ2∥ (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='87) where LB t (x0:t) = supθ Bt ��t−1 i=0 L ← − Q i � (x0:t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Under AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='9(i), we have that ∥LB t ∥∞ = �t−1 i=0(¯τi¯σi)−1∥Lq i ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Apply lemma 19 and lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='3 θ �→ � St,θ(x0:t, dbt)µ(bt)(id) is Lipschitz Define the backward ancestors kernel Bθ,t : Xt+1 × Xt × σ(�1, N�) �→ � 1A(˜j) � N � ℓ=1 qt(xℓ t, xt+1) �N ℓ′=1 qt(xℓ′ t , xt+1) δℓ(d˜j) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (Bθ,t is Lipschitz) For every m ∈ �0, t�, there exists LBK m ∈ M(X m:m+1) such that ∥Bθ1,m(xm+1, xm) − Bθ2,m(xm+1, xm)∥TV ≤ L ← − Q m(xm+1, xm)∥θ1 − θ2∥ , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='88) where L ← − Q s is defined in Lemma 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Bθ,s is the index version of the kernel (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='85) and thus it is Lipschitz with the same constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For every m ∈ �0, t�, we have that �� � CmSm,θ(z0:m, dbm)µ(bm)(Id) �� ≤ m−1 � ℓ=0 s∞ ℓ (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='89) and ���� � Sm,θ1(x0:m, dbm)µ(bm)(Id) − � Sm,θ2(x0:m, dbm)µ(bm)(Id) ���� ≤ LSµ m (x0:m)∥θ1 − θ2∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='90) where LSµ m (x0:m) = N −1 �N i=1 LB m(xk m, x0:m) and LB m is defined recursively as LB m+1(xk m+1, x0:m) = L ← − Q m(xk m+1, xm) m � ℓ=0 s∞ ℓ + � Bθ,m(xk m+1, xm, dJ) � Ls m(xJ m, xk m+1) + LB m(xJ m, x0:m−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='91) In particular, under A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='9, we have that LB m ≤ �m j=1 ∥L ← − Q j ∥∞ ��m−1 ℓ=0 s∞ ℓ � + �m j=1 ∥Ls j∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Consider the following kernels, �Sm,θ(x0:m+1, d(Ji,j 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , Ji,j m )N,M i=1,j=1) := m � ℓ=0 N � k=1 �Sℓ,θ(xk ℓ+1, xℓ, d � Jk,j ℓ �M j=1) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='92) �Sℓ,θ(xk ℓ+1, xℓ, d(Jk,j ℓ )M j=1) := M � j=1 Bθ,ℓ(xk ℓ+1, xℓ, dJk,j ℓ ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='93) Define for all k ∈ [1 : N], m ∈ N>0, Bm+1,k : θ �→ � �Sm,θ(x0:m+1, d � Ji,j 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , Ji,j m �N,M i=1,j=1)bk m+1 � x0:m+1, � Ji,j 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , Ji,j m �N,M i=1,j=1 � , 49 where bk m+1 � x0:m+1, � Ji,j 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , Ji,j m �N,M i=1,j=1 � is defined recursively as bk m+1 � x0:m+1, � Ji,j 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , Ji,j m �N,M i=1,j=1 � = M −1 M � ℓ=1 bJk,ℓ m m � x0:m, � Ji,j 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , Ji,j m−1 �N,M i=1,j=1 � +sm,θ(xJk,ℓ m m , xk m+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For notational convenience, we henceforth drop the arguments and simply write bk m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We herebelow show that Bm+1,k is Lipschitz with constant LB m(xk m+1, xm) and bounded by �m−1 ℓ=0 s∞ ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For m > 2 and k ∈ [1 : N], Bm+1,k(θ) = � �Sm,θ(x0:m+1, d(Ji,j 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , Ji,j m )N,M i=1,j=1)bk m+1 = � · · � �Sm−1,θ(x0:m, d(Ji,j 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' , Ji,j m−1)N,M i=1,j=1)�Sm,θ(xk m+1, xm, d(Jk,j m )M j=1) × � M −1 M � ℓ=1 bJk,ℓ m m + sm,θ(xJk,ℓ m m , xk m+1) � = � · · � �Sm,θ(xk m+1, xm, d{Jk,j m }M j=1) � M −1 M � ℓ=1 � sm,θ(xJk,ℓ m m , xk m+1) + � �Sm−1,θ(x0:m, d(Ji,j 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Ji,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j m−1)N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='M i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j=1)bJk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='ℓ m m �� = � · · � �Sm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ(xk m+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' d(Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j m )M j=1) � M −1 M � ℓ=1 � sm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ(xJk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='ℓ m m ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xk m+1) + Bm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='ℓ m (θ) �� = � Bθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='m(xk m+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' dJ) � sm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ(xJ m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xk m+1) + Bm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='J(θ) � Applying the induction hypothesis conditionally on Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='ℓ m ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Bm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='ℓ m is Lipschitz with constant LB m(xJk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='ℓ m m ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' x0:m−1) and thus the Lipschitz constant of Bm+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='k is LB m+1(xk m+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' x0:m) = L ← − Q m(xk m+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xm) m � ℓ=0 s∞ ℓ + � Bθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='m(xk m+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' dJ) � Ls m(xJ m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xk m+1) + LB m(xJ m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' x0:m−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='94) where we have used the fact that Bθ,m and sm,θ are also Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Again by induction Bm+1,k is bounded uniformly by �m ℓ=0 s∞ ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The induction is concluded by noting that for the base case m = 0, βk m = 0 for all k ∈ N and thus the result holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' It now remains to check that for all θ ∈ Θ, m ∈ �0, t� and k ∈ [1 : N], Bm,k(θ) = � Sm(x0:m, dbm)bk m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 50 Again, we proceed by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' � Sm(x0:m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' dbm)bk m = � · · � Sm−1(x0:m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' dbm−1)Sm(bm−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xm−1:m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' dbm)bk m = � · · � Sm−1(x0:m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' dbm−1) × M � j=1 � N � p=1 qm−1(xp m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xk m) �N ℓ=1 qm−1(xℓ m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xkm) δxp m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='bp m−1 � d(˜xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='˜bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j m−1) � � × � M −1 M � n=1 � ˜bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='n m−1 + sm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ(˜xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='n m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xk m) �� = � · · � Sm−1(x0:m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' dbm−1) × M � j=1 � N � p=1 qm−1(xp m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xk m) �N ℓ=1 qm−1(xℓ m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xkm) δp(dJk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j m−1) � � M −1 M � n=1 � b Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='n m−1 m−1 + sm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ(x Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='n m−1 m−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xk m) �� = � · · � �Sm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ(xk m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xℓ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' d(Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j ℓ−1)M j=1) × � M −1 M � ℓ=1 � sm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ(x Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='ℓ m−1 m−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xk m) + Sm−1(x0:m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' dbm−1)b Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='ℓ m−1 m−1 �� = � · · � �Sm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ(xk m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xℓ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' d(Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j ℓ−1)M j=1) × � M −1 M � ℓ=1 � sm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ(x Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='ℓ m−1 m−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xk m) + � Sm−1(x0:m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' dbm−1)b Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='ℓ m−1 m−1 �� = � · · � �Sm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ(xk m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xℓ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' d(Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='j ℓ−1)M j=1) � M −1 M � ℓ=1 � sm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='θ(x Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='ℓ m−1 m−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xk m) + Bm−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='ℓ m−1(θ) �� = Bm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='k(θ) The proof is finalized by noting that � Sm(x0:m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' dbm)µ(bm)(Id) = N −1 N � k=1 Bm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='k(θ) and thus it is Lipschitz with constant LSµ m (x0:m) = N −1 �N i=1 LB m(xk m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' xm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2 Lipschitz properties of Markov Kernels Lemma 18 (Composition of ergodic Lipschitz kernels is lipschitz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let Pθ be a Markov kernel over X × Y that is uniformly π-geometrically ergodic for any θ with contraction constant ρ independent of θ and such that there exists Lp > 0 such that for every x ∈ X ∥Pθ0(x, ·) − Pθ1(x, ·)∥TV ≤ LP ∥θ0 − θ1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then, for all k > 0 ��P k θ0(x, ·) − P k θ1(x, ·) �� TV ≤ LP 1 − ρ∥θ0 − θ1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 51 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We use the following decomposition borrowed from [Fort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' For any k ≥ 1, P k θ0f − P k θ1f = k−1 � j=0 P j θ0(Pθ0 − Pθ1) � P k−j−1 θ1 f − πf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then, for any f s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ∥f∥∞ ≤ 1 and x ∈ X, |P k θ0f(x) − P k θ1f(x)| ≤ k−1 � j=0 ���� � P j θ0(x, dy) sup z∈X |P k−j−1 θ1 f(z) − πf| ���� LP ∥θ0 − θ1∥ ≤ LP � k−1 � j=0 ρk−j−1 � ∥θ0 − θ1∥ ≤ LP 1 − ρ∥θ0 − θ1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 19 (Composition of Lipschitz kernels is lipschitz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let Pθ, Qθ be two kernels defined over X × Y and Y × Z such that for ever x ∈ X, y ∈ Y there are Lp ∈ M(X), Lq ∈ M(Y ) that satisfy ∥Pθ0(x, ·) − Pθ1(x, ·)∥TV ≤ Lp(x)∥θ0 − θ1∥ and ∥Qθ0(y, ·) − Qθ1(y, ·)∥TV ≤ Lq(y)∥θ0 − θ1∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then ∥Pθ0Qθ0(x, ·) − Pθ1Qθ1(x, ·)∥TV ≤ Lpq∥θ0 − θ1∥ , where Lpq = (supθ PθLq(x) + Lp(x) supy supθ Qθ(y, Z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let f ∈ M such that ∥f∥∞ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' ∥Pθ1Qθ1f − Pθ2Qθ2f∥ ≤ ∥Pθ1 [Qθ1f − Qθ2f] ∥ + ∥(Pθ1 − Pθ2)Qθ2f∥ ≤ (Pθ1Lq(x) + Lp(x)∥Qθ2f∥∞)∥θ1 − θ2∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let Pθ, Qθ be two Markov kernels defined over X × Y and Y × Z such that for ever x ∈ X, y ∈ Y there are Lp ∈ M(X), Lq ∈ M(Y ) that satisfy ∥Pθ0(x, ·) − Pθ1(x, ·)∥TV ≤ Lp(x)∥θ0 − θ1∥ and ∥Qθ0(y, ·) − Qθ1(y, ·)∥TV ≤ Lq(y)∥θ0 − θ1∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then ∥Pθ0Qθ0(x, ·) − Pθ1Qθ1(x, ·)∥TV ≤ Lpq∥θ0 − θ1∥ , where Lpq = (supθ PθLq(x) + Lp(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Lemma 20 (Product of Lipschitz kernels is lipschitz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let Pθ, Qθ be two markov kernels that are uniformly Lipschitz with constants LP , LQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then Pθ�Qθ is uniformly Lipschitz with constant LP +LQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Let hθ : y �→ � Qθ(y, dz)f(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Then (Pθi ⊗ Qθi)(f) = Pθi(hθi) and the proof is similar to that of the previous Lemma since hθ is Lipschitz with constant LQ and ∥hθ∥∞ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 52 0 10 20 30 40 50 K 10 1 100 101 102 N = 10 N = 25 N = 50 N = 100 PaRIS 0 10 20 30 40 50 K 10 1 100 101 102 N = 10 N = 25 N = 50 N = 100 PaRIS Figure 3: Output of the PPG roll-out estimator for the LGSSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The curves describe the evolution of the bias with increasing k for different particle sample sizes N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' The left and right panels correspond to k0 = k − 1 and k0 = ⌊k/2⌋, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Additional numerical results D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='1 PPG D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2 Learning For both experiments, all the parameters were initialized by sampling from a centered multivari- ate gaussian distribution with covariance matrix of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='01I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We have used the ADAM optimizer [Kingma and Ba, 2014] with a learning rate decay of 1/ √ ℓ where ℓ is the iteration index, with a starting learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' We rescale the gradients by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' LGSSM For LGSSM we evaluated for fixed number of particles (N = 64) and number of gibbs iterations (k = 8) the influence of the burn-in phase (k0) over the final distance obtained to the MLE estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Table 3 indicates that configurations with smaller k0 perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A possible interpretation of this phenomenon is that, since between two gradient ascent iterates the conditioning path is being passed on, this conditioning path from a moment on makes the estimates less biased, so the importance of having k0 high to have less bias vanishes, but the effect of augmenting the variance with k0 is still shown, since the fact of having a conditioning particle from the right marginal does not affect the variance of the estimator, only it’s bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' References [Anderson and Qiu, 1997] Anderson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' and Qiu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' A monotonicity property of the gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 125(11):3355–3362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' [Andrieu and Doucet, 2003] Andrieu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' and Doucet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Online Expectation–Maximization type algorithms for parameter estimation in general state space models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Acoust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', Speech, Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', volume 6, pages 69–72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 53 Table 3: Distance to θMLE for each configuration in the LGSSM case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Algorithm N k0 k Dmle PPG 64 0 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='205 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='013 PPG 64 1 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='213 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='016 PPG 64 2 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='201 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='010 PPG 64 3 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='201 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='010 PPG 64 4 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='207 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='012 PPG 64 5 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='212 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='015 PPG 64 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='210 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='017 PPG 64 7 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='211 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content='018 [Andrieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2010a] Andrieu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', Doucet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', and Holenstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' (2010a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Particle Markov chain Monte Carlo methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B, 72(3):269–342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' [Andrieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', 2010b] Andrieu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', Doucet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=', and Holenstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQf-_pB/content/2301.00900v1.pdf'} +page_content=' 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Dragan +UC Berkeley +Andrea Bajcsy +UC Berkeley +ABSTRACT +Humans have internal models of robots (like their physical capabili- +ties), the world (like what will happen next), and their tasks (like a +preferred goal). However, human internal models are not always +perfect: for example, it is easy to underestimate a robot’s inertia. +Nevertheless, these models change and improve over time as hu- +mans gather more experience. Interestingly, robot actions influence +what this experience is, and therefore influence how people’s inter- +nal models change. In this work we take a step towards enabling +robots to understand the influence they have, leverage it to better +assist people, and help human models more quickly align with real- +ity. Our key idea is to model the human’s learning as a nonlinear +dynamical system which evolves the human’s internal model given +new observations. We formulate a novel optimization problem to +infer the human’s learning dynamics from demonstrations that +naturally exhibit human learning. We then formalize how robots +can influence human learning by embedding the human’s learn- +ing dynamics model into the robot planning problem. Although +our formulations provide concrete problem statements, they are +intractable to solve in full generality. We contribute an approxima- +tion that sacrifices the complexity of the human internal models we +can represent, but enables robots to learn the nonlinear dynamics +of these internal models. We evaluate our inference and planning +methods in a suite of simulated environments and an in-person user +study, where a 7DOF robotic arm teaches participants to be better +teleoperators. While influencing human learning remains an open +problem, our results demonstrate that this influence is possible and +can be helpful in real human-robot interaction. +CCS CONCEPTS +• Computing methodologies → Artificial intelligence. +KEYWORDS +robot influence, human internal model, dynamics of human learning +ACM Reference Format: +Ran Tian, Masayoshi Tomizuka, Anca D. Dragan, and Andrea Bajcsy. 2023. +Towards Modeling and Influencing the Dynamics of Human Learning. In +∗This work supported by ONR YIP, NSF NRI, and WeRide Corp. Author emails: +{rantian, tomizuka, anca, abajcsy}@berkeley.edu. Project website with link +to code: https://sites.google.com/berkeley.edu/midle. +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +HRI ’23, March 13–16, 2023, Stockholm, SE +© 2023 Association for Computing Machinery. +ACM ISBN 978-1-4503-XXXX-X/23/06...$15.00 +https://doi.org/XXXXXXX.XXXXXXX +𝑓𝐿 +Robot influence +𝜃H +𝑡+1 +Dynamics of human learning +𝜋𝑅 𝑥𝑡, 𝜃H +𝑡 +𝑓𝐿 +𝜃H +𝑡 +Figure 1: Human teleoperates a new robot; they update their +internal model by acting and observing outcomes. Planning +with human learning dynamics, the robot influences the hu- +man’s internal model to help them be a better teleoperator. +Proceedings of Human Robot Interaction (HRI ’23). ACM, New York, NY, USA, +12 pages. https://doi.org/XXXXXXX.XXXXXXX +1 +INTRODUCTION +Imagine your first time controlling a robot arm to perform daily +living tasks like throwing away trash or stirring a pot of soup. +Initial interactions with the robot are tough: you aren’t familiar +with the robot’s dynamics so your motions are jerky and imprecise. +In other words, your internal model of the robot is incorrect. And +robot dynamics are not the only thing we, humans, have incorrect +internal models of. We might not fully understand the world’s +dynamics (e.g., result of pouring lemon juice into cream) or our +own preferences (e.g., only liking something after trying it). +However, over time, our internal models evolve with our experi- +ences. As you control the robot, you start to understand how it will +move; as you try different things, you learn what you like. Since the +robot is part of the world, the robot’s actions and their outcomes +become part of these experiences. In other words, robot actions +inevitably influence the change in a human’s internal model. +In this work, we advocate that robots should understand and +use this influence. First, collaborative tasks require understanding +what people are trying to do in order to assist them. Prior work has +shown that inferring a human’s internal model is critical for assis- +tance [38]; in turn, we argue that if this model changes over time, +tracking this change will enhance assistance. Further, purposefully +influencing a change in the human’s internal model opens the door +for teaching: robot actions that are optimized to quickly align the +human’s understanding with reality. For instance, as an operator +controlling the robot (Figure 1), this means you quickly understand +how the robot works and can do the task independently. +A key challenge towards this is modeling how humans learn; +without a proper model of this, the robot cannot plan to change +the human’s internal model. Although we do not know the precise +functional form of how people learn, we observe that a human’s +arXiv:2301.00901v1 [cs.RO] 2 Jan 2023 + +CHRI ’23, March 13–16, 2023, Stockholm, SE +Ran Tian, Masayoshi Tomizuka, Anca D. Dragan, and Andrea Bajcsy +understanding of the robot or the world changes as a function of +what they observe. For example, at first you may mistakenly believe +that the robot doesn’t experience any inertia. However, as soon as +you gesture to move the robot forward, you see the robot lagging +behind. This observation controls the evolution of your internal +robot physics model. The same holds true for your internal model of +the world and personal preferences. In other words, we can model +human learning as a dynamical system where the hu- +man’s internal model is the state, and the observations— +which the robot can influence—evolve the internal model. +Of course, this does not prescribe the functional form of the dy- +namical system. One idea is to draw on computational cognitive +science work to define this function. A predominant lens is that of +probabilistic models [12], which posits that humans perform some +form of approximate Bayesian inference based on the observations +they receive. In reality, people have been shown to have a plethora +of cognitive biases which deviate from perfect Bayesian inference: +they might use gradient information [53], might not process the en- +tire observation due to sensory overload [28], or exhibit systematic +bias like over- or under- estimation [39]. Instead of committing to +a specific model, in our work we treat this as a general dynamics +learning problem, which has roots in controls and robotics [21]. We +leverage demonstrations which naturally exhibit human learning +(e.g., humans teleoperating a robot they have never interacted with +before), to fit a human learning model under the assumption that +observed human actions are approximately optimal given their cur- +rent internal model. This enables flexibility of capturing different +possible learning updates, at the cost of being domain-specific. +Although the most general model learning problem remains +computationally intractable, we introduce a tractable approxima- +tion that is readily solvable via gradient-based optimization, and +is compatible with neural network representations of the human +learning dynamics. Leveraging our approximate dynamics model +of human learning, we formalize robot influence over the human’s +internal model as a Markov Decision Process (MDP) where the hu- +man’s internal model is part of the state and the human’s learning +dynamics are part of the transition function. The solution yields +robot actions that change the human’s internal model by changing +the human’s observations in a way that rewards the robot. +We run experiments with simulated humans to study the fidelity +of the inferred human learning dynamics and investigate robot +teaching and assistance in settings where the human’s understand- +ing of robot physics, motion preferences, or goals can be influenced. +Finally, we conduct a user study with a Kinova Jaco 7DOF robot +arm and find that our method can help teach humans to be better +teleoperators. Overall, while influencing human learning remains +an open problem, we are excited to have taken a step in this domain +via a principled yet tractable learning and planning method. +2 +RELATED WORK +Inferring human preferences and beliefs. A large body of work +has focused on learning human reward functions via inverse rein- +forcement learning (IRL) [19, 22, 31]. This includes inferring hu- +man driving preferences [34, 40], desired exoskeleton gaits [25], +intended goals [17], motion preferences [35], and human under- +standing about physics [38]. A key assumption in these works is +that people have static internal models of preferences or physics. In- +stead, we are interested in learning a dynamic model of how humans +change their preferences, goals, and understanding of physics. +Models of human learning for robot decision-making. Prior +works in robotics model human learning as Bayesian inference +when updating goals or preferences [8, 14, 16], a linear Gaussian +system when updating trust [7], gradient-based IRL when learning +rewards [4], or as a multi-armed bandit algorithm when updating +preferences [6]. Instead of assuming a known model of how people +learn, in this work we seek to learn a model of how humans learn. +Most related to our work is [39] which learns a model of how +people estimate the state of the world. In this work, we propose a +generalization where the human is not estimating world state, but +updating their preferences, goals, and internal physics model. This +induces a significantly harder model learning problem, for which +we propose a tractable approximation. +Cognitive theories of human learning. Models of human infer- +ence have been extensively studied in both computational cognitive +science [2, 13] and psychology [36, 50]. While human cognition +can be broadly modeled at three levels (computational, algorith- +mic, and hardware) [27], most relevant to us are the algorithmic +works. [13] posits that modeling human reasoning as “implement- +ing” an exact Bayesian posterior or a gradient-based point estimate +are both compatible with probabilistic models of human cognition, +and are a potential source of rational process models [45]. Further, +[42] finds evidence that humans may update their forward models +using the models’ prediction error as loss functions. Inspired by +these works, our simulated human experiments leverage exact and +approximate probabilistic inference models, and we study if our +flexible, learning-based method can effectively recover such models. +Robot influencing human behavior. While there are many ways +a robot can influence humans (e.g., through nonverbal cues, ap- +pearance, visuals, or curriculum design [1, 37, 39, 41, 46]), we focus +on robot influence through physical action [30]. A common ap- +proach towards this models human-robot interaction as a game +[15, 23, 32, 40, 44, 48]. While these approaches can capture reactions +from the human, they do not address the internal learning prob- +lem: over repeated interactions, the human may not have learned +anything and is only reacting. Alternatively, model-free methods +learn a latent representation of the human’s policy and then lever- +age the latent dynamics to influence the human [33, 54]. Here the +human’s internal model is implicitly captured by the latent rep- +resentation, and the internal model evolves between interaction +episodes. In contrast, in our work the human’s internal model is an +explicit parameterization (e.g., high-dimensional parameterization +like dynamics) and the human internal model can evolve continu- +ously during an interaction episode. This enables robot behaviors +like teaching the human the correct internal model, which would +otherwise not be possible with implicit, latent representations. +3 +MODELING HOW HUMANS LEARN & ACT +We begin by mathematically modelling the dynamics of human +learning, before diving into how the robot can infer this dynamics +model and use it influence the human’s internal model evolution. + +Towards Modeling and Influencing +the Dynamics of Human Learning +HRI ’23, March 13–16, 2023, Stockholm, SE +Notation. Let 𝑥 ∈ R𝑛 be the state of the world including the robot +(e.g., robot end-effector position, objects, etc.). Both the human and +robot can take actions, 𝑢H ∈ R𝑚 and 𝑢R ∈ R𝑚 respectively, that +affect the next state. Let the deterministic world dynamics be +𝑥𝑡+1 = 𝑓 (𝑥𝑡,𝑢𝑡 +H,𝑢𝑡 +R). +(1) +Human internal model. We model the human as having an in- +ternal parameter vector, 𝜃H, which captures a latent aspect of the +task that the human is uncertain about but continuously learns +about. Going back to our motivating example where the human +teleoperates a robot, 𝜃H can model the human’s current estimate of +the robot’s physical properties, like its inertia. Or, 𝜃H could model +the human’s current preferences for teleoperation: they start off +wanting to move the robot to one goal, but then change their mind +to a new goal after realizing it is easier to reach. Regardless of what +𝜃H represents, it is important to remember that it is time-varying +and that it evolves as a function of what the human observes. +Human policy: acting under the internal model. In our work, +we model the human actions as driven by some reward function, +𝑅H(𝑥,𝑢H;𝜃H), which depends on the current state, the human’s +action, and their internal parameter 𝜃H. Following prior works +[2, 24, 52, 55], we treat the human as a noisily-optimal actor: +P(𝑢H | 𝑥;𝜃H) = 𝑒𝑄H(𝑥,𝑢H;𝜃H) � ∫ +˜𝑢 +𝑒𝑄H(𝑥, ˜𝑢;𝜃H)𝑑 ˜𝑢 +�−1 +, +(2) +where the optimal state-action value is denoted by 𝑄H(𝑥,𝑢H;𝜃H) +and 𝑥 is the current state, 𝑢H is the human action, and 𝜃H the +human’s current parameter estimate. +We make two simplifying assumptions in this model. First, the +human does not explicitly account for the actions 𝑢R the robot +could take. Instead, the human reacts to the current state 𝑥, which +implicitly captures the effect of any robot actions that change the +state. This models scenarios where the human is doing the task on +their own, or where the human is not aware of how the robot is +providing guidance. Second, when the human plans their action, +we assume that they separate the estimation of 𝜃H from policy +generation and they plan with their current estimate. +Dynamics of human learning: updating the internal model. +As the human acts in the environment, they receive new observa- +tions: they may see the next state, including that of the robot’s, or +experience how much they enjoy something (i.e. observe “reward +signal”). This naturally lets the human update their understanding +of the robot, physical aspects of the world, or their preferences. +Leveraging our core idea, we model the human’s learning process +as a nonlinear dynamical system over the human’s internal model +parameter. Let 𝜃0 +H be the human’s initial internal model, and 𝑥0:𝑡 +and 𝑢0:𝑡 +H be the state and action history until timestep 𝑡 and 𝑥𝑡+1 +be the resulting state at the next timestep, possibly including the +influence of robot actions. Given the initial parameter estimate, the +state and action history, and next state data, the human evolves their +internal model to the next estimate, 𝜃𝑡+1 +H . Let the true dynamics of +the human’s learning process be: +𝜃𝑡+1 +H += 𝑓𝐿(𝜃0 +H,𝑥0:𝑡+1,𝑢0:𝑡 +H ). +(3) +Here we are faced with the question “What 𝑓𝐿 models how the +human learns?” Instead of committing to a specific model, here we +take a robotics perspective and view this question as an instance of +a dynamics learning problem. By looking to human data, we aim +to learn an approximate 𝑓𝐿 model that is domain-specific. +4 +INFERRING THE DYNAMICS OF HUMAN +LEARNING +In this section we focus on inferring the dynamics of human learn- +ing by leveraging demonstrations which naturally exhibit human +learning: for example, initial trials of a human teleoperating a robot +they have never interacted with before. We assume these demon- +strations contain only the state and action histories and do not +contain ground-truth human internal model data (since this is not +possible in practice). However, we do assume that the observed +actions are coupled with the human’s internal model, allowing us +to leverage demonstrations to infer the dynamics of the human’s +internal model. Given this dataset, we seek to fit a nonlinear model +to represent the dynamics of human learning, +𝑓 𝜙 +𝐿 ≈ 𝑓𝐿, +(4) +where 𝜙 are the parameters of the approximate model. In the follow- +ing sections, we formalize inferring 𝑓 𝜙 +𝐿 as a maximum likelihood +estimation (MLE) problem and propose a tractable approximation. +4.1 +Formalizing the Inference Problem +Let D𝑑𝑒𝑚𝑜 := {(x, uH)𝑖}𝑁 +𝑖=0 be a collection of 𝑁 demonstrations +containing state and human action trajectories of length 𝑇 time +steps. We want to infer the parameter of the human’s learning +dynamics, 𝜙, and the initial human parameter estimate, 𝜃0 +H, which +maximizes the likelihood of the observed demonstrations. We for- +mulate this inference via the constrained optimization problem: +max +𝜙,𝜃0 +H +∑︁ +(x,uH) ∈D𝑑𝑒𝑚𝑜 +𝑇−1 +∑︁ +𝑡=0 +log +� +P(𝑢𝑡 +H | 𝑥𝑡;𝜃𝑡 +H) +� +, +(5) +s.t. +𝜃𝑡+1 +H += 𝑓 𝜙 +𝐿 (𝜃0 +H,𝑥0:𝑡+1,𝑢0:𝑡 +H ), +(6) +where P(𝑢𝑡 +H | 𝑥𝑡,𝜃𝑡) is the human action likelihood from Equa- +tion (2) and the constraint ensures that the human’s internal param- +eter evolves according to the human’s learning dynamics model. +4.2 +Solving the Inference Problem +Unfortunately, the inference problem in Equation (5) is intractable +to solve directly for two main reasons. First, recall that the human’s +internal model 𝜃H of their preferences, dynamics, or goals, changes +over time. This means that at each timestep the human is gener- +ating data 𝑢H under a possibly different 𝜃H. In other words, the +human acts under a new action policy P(𝑢𝑡 +H | 𝑥𝑡;𝜃𝑡 +H) at each 𝑡, re- +quiring us to solve an entirely new reinforcement learning problem +to obtain the action policy at each time step along the inference +horizon. In the case where 𝜃H is a continuous, high-dimensional +parameter (e.g., physical properties of the robot dynamics), this is +intractable to compute per-timestep. Secondly, even if we could +obtain the human’s policy infinitely fast, our optimization problem +still requires searching over the the high-dimensional space of 𝜙 +and 𝜃H. Gradient-based optimization is a natural choice, but we +need to be able to compute the gradient of the MLE objective and, +therefore, differentiate through 𝑄H with respect to 𝜃H. + +HRI ’23, March 13–16, 2023, Stockholm, SE +Ran Tian, Masayoshi Tomizuka, Anca D. Dragan, and Andrea Bajcsy +In the following subsections, we introduce several approxima- +tions to arrive at a tractable solution to the inference problem. Our +key idea is to use a linear-quadratic (LQ) approximation of the +physical dynamics and the human reward. This enables us to derive +a closed-form expression of the human policy as a function of 𝜃𝑡 +H +at any time and yields a differentiable inference objective. +4.2.1 +Linear-Quadratic approximation. We take inspiration from +infinite-horizon linear-quadratic (LQ) control [20] and assume that +the human’s reward is quadratic and their model of the physical +dynamics is linear. Let the linear physical dynamics be: +𝑥𝑡+1 = 𝑓 (𝑥𝑡,𝑢𝑡 +H,𝑢𝑡 +R ≡ 0) ≈ 𝐴𝑥𝑡 + 𝐵𝑢𝑡 +H +(7) +where 𝐴 ∈ R𝑛×𝑛, 𝐵 ∈ R𝑛×𝑚 are matrices governing the physical +dynamics. Note that in the human’s mind, the robot is not exerting +any control effort, and hence 𝑢R ≡ 0. Let the human’s reward be +approximated by a quadratic function: +𝑟H(𝑥,𝑢H;𝜃H) ≈ −𝑥⊤𝑄𝑥 − 𝑢⊤ +H𝑅𝑢H, +(8) +where the matricies 𝑄 ∈ R𝑛×𝑛 and 𝑅 ∈ R𝑚×𝑚 tradeoff the state +reward (e.g., how much reward the human gets for reaching a state) +and the action reward (e.g., how much effort the human wants to +exert), respectively. Note that 𝜃H enters in different ways depending +on what the human is learning about. For example, if 𝜃H encodes +reward weights (i.e., the human’s preferences about how to do +a task), then 𝜃H := (𝑄, 𝑅). If the parameter encodes a human’s +goal state, then 𝜃H ∈ Θ ⊂ R𝑛 and the human’s reward function +regulates the human towards their desired goal: 𝑟H(𝑥,𝑢H;𝜃H) ≈ +−(𝑥 − 𝜃H)⊤𝑄(𝑥 − 𝜃H) − 𝑢⊤ +H𝑅𝑢H. Finally, if 𝜃H encodes aspects of +the physical dynamics that the human is estimating, then 𝜃H := +(𝐴, 𝐵) from the dynamics in Equation (7), and governs how the +human imagines the physical dynamics evolving. +4.2.2 +Closed-form 𝑄H. Recall that the human plans a policy us- +ing their current estimate 𝜃H; at every step, 𝜃H changes, resulting +in a new policy. In general, obtaining the exact 𝑄H-value via dy- +namic programming in continuous state, action, and 𝜃H-spaces is +computationally demanding. However, under our infinite-horizon +LQ-approximation the human’s 𝑄H-value is: +𝑄H(𝑥,𝑢H;𝜃H) = 𝑟H(𝑥,𝑢H;𝜃H) − (𝑥 ′)⊤𝑃𝜃H (𝑥 ′) +(9) +where the instantaneous reward is quadratic from Equation (8) and +𝑥 ′ is the next physical state as a result of applying 𝑢H from state +𝑥. Note that −(𝑥 ′)⊤𝑃𝜃H (𝑥 ′) is the infinite-horizon optimal value +where 𝑃𝜃H is the well-known positive-definite fixed point of the +discrete-time algebraic Riccati equation (DARE) [3]: +𝑃 = 𝐴⊤𝑃𝐴 − 𝐴⊤𝑃𝐵(𝑅 + 𝐵⊤𝑃𝐵)−1𝐵⊤𝑃𝐴 + 𝑄. +(10) +Obtaining 𝑃𝜃H also yields the optimal human action: 𝑢∗ +H(𝑥;𝜃H) = +−𝐾𝜃H𝑥 where 𝐾𝜃H = (𝑅 + 𝐵⊤𝑃𝜃H𝐵)−1𝐵⊤𝑃𝜃H𝐴. Note that in all of +the equations above, 𝜃H enters differently depending on what the +human’s internal model represents. +4.2.3 +Closed-form human policy. In general, obtaining the human +policy in Equation (2) is computationally intractable in continuous +action spaces due to the integral over 𝑢H. However, plugging in +our closed-form 𝑄H, we see that the exponent is quadratic in 𝑢, +allowing us to take a Gaussian integral [49]. Overall, this yields a +closed-form human policy (see full derivation in Appendix A.1.): +P(𝑢H | 𝑥;𝜃H) = |H|1/2(2𝜋)−𝑚H/2𝑒𝑄H(𝑥,𝑢H;𝜃H)−𝑄H(𝑥,𝑢∗;𝜃H). +(11) +4.2.4 +Representing the dynamics of human learning. Finally, we +are faced with the question of how to functionally represent the +dynamics of human learning; for example, we could take inspiration +from computational cognitive science and model 𝑓 𝜙 +𝐿 as Bayesian +inference [12]. Instead of committing to a specific functional form, +in this work we seek a model that has the potential to capture a +broad range of “learning algorithms” that the human could use +to update their internal parameter. Recently, self-attention based +transformer models [51] have shown success at predicting high- +dimensional sequential tasks [18], at the cost of being domain- +specific. Inspired by this, we represent 𝑓 𝜙 +𝐿 as a transformer encoder +where 𝜙 are the weights of the neural network. At each time step +𝑡, a collection of the state 𝑥𝑡, the human’s action 𝑢𝑡 +H, and the next +state 𝑥𝑡+1 are fed into an encoder to extract embeddings which +are fed into a transformer encoder that predicts the human’s next +internal model. Training details are in Appendix A.3. +4.2.5 +Deriving an efficient, gradient-based solution. To optimize the +transformer-based model of human learning dynamics, we need +the gradient of our inference objective with respect to the neural +network parameters. Here a key challenge lies in the human’s pol- +icy gradient because it requires differentiating through the DARE +function, which is non-obvious. However, we leverage recent work +[10] to obtain the relevant closed-form Jacobians, enabling us to ef- +ficiently infer the parameters of 𝑓 𝜙 +𝐿 via gradient-based optimization. +More details on this approach are in Appendix A.2. +5 +INFLUENCING HUMAN LEARNING WITH +ROBOT ACTIONS +Inferring how humans learn presents an opportunity for human- +robot interaction. For example, when a human teleoperator is mis- +taken about the robot’s inertia, it may take them many interactions +to learn and become better. Instead, could the robot influence the +human so that their understanding improves faster? Here, we math- +ematically formalize this influence by embedding the approximate +dynamics model of human learning into robot planning. +Formalizing the Influence Problem. We formalize the robot in- +fluence problem as a Markov Decision Process (MDP) where the +human’s internal model parameter is part of the state. Our MDP +is a tuple < 𝑆,𝑈R,𝑇,𝑟R > where the state 𝑠 = (𝑥,𝜃H) ∈ 𝑆 is the +joint physical state and human internal model parameter and the +robot’s actions are 𝑢R ∈ 𝑈R. The stochastic state transition function +is defined as 𝑇 (𝑠𝑡+1 | 𝑠𝑡,𝑢𝑡 +R) := � +𝑢H P(𝑢H | 𝑠𝑡) ˜𝑓 (𝑠𝑡,𝑢𝑡 +R,𝑢𝑡 +H,𝑠𝑡+1) +which accounts for the human policy from Equation (2). Impor- +tantly, ˜𝑓 (𝑠𝑡,𝑢𝑡 +R,𝑢𝑡 +H,𝑠𝑡+1) is a deterministic function that evolves 𝑥𝑡 +via the physical dynamics 𝑓 from Equation (1) and the human’s +internal model parameter 𝜃𝑡 +H via the human learning dynamics +𝑓 𝜙 +𝐿 from Equation (6). Finally, the robot optimizes its reward func- +tion 𝑟R(𝑠,𝑢R,𝑢H;𝜃∗) where 𝜃∗ is the robot’s true internal model +parameters (e.g., the robot’s true physical dynamics). Note that + +Towards Modeling and Influencing +the Dynamics of Human Learning +HRI ’23, March 13–16, 2023, Stockholm, SE +because 𝑠 = (𝑥,𝜃H), the robot’s reward depends on the human’s +time-varying internal model, 𝜃H, at each timestep. +The robot seeks an optimal policy 𝜋∗ +R which maximizes it’s re- +ward in expectation over the human’s action sequence, uH: +𝜋∗ +R = arg max +𝜋R EuH +� ∞ +∑︁ +𝑡=0 +𝑟R(𝑠𝑡,𝑢𝑡 +R,𝑢𝑡 +H;𝜃∗) +� +s.t. 𝑇 (𝑠𝑡+1 | 𝑠𝑡,𝑢𝑡 +R), +(12) +Because human’s internal model parameter 𝜃𝑡 +H is part of the state +and the state transition function 𝑇 (𝑠𝑡+1 | 𝑠𝑡,𝑢𝑡 +R) includes the in- +ferred dynamics model of human learning, 𝜋∗ +𝑅 should automatically +influence the human’s internal model if it yields higher reward. +Computing Solutions to the Influence Problem The presence +of the human’s nonlinear learning dynamics 𝑓 𝜙 +𝐿 in the transition +function results in a nonconvex optimization problem. To obtain +the optimal robot policy, we would have to solve the MDP either +exactly with dynamic programming (which suffers from the curse +of dimensionality) [3] or approximately via receding-horizon con- +trol (which requires trading off optimality with computational effi- +ciency) [5]. To achieve both long-horizon reasoning and efficient +runtime performance, we use a Dyna-style algorithm [47] that uses +the samples generated by the transition 𝑇 (𝑠𝑡+1 | 𝑠𝑡,𝑢𝑡 +R) to train 𝜋∗ +𝑅 +using model-free learning (Proximal Policy Optimization [43]). +6 +SIMULATED HUMAN EXPERIMENTS +We want to test two aspects of our approach: our ability to infer +the dynamics of human learning and the effectiveness of our robot +influencing algorithm. To fully validate both, we need access to the +ground-truth human learning dynamics (𝑓𝐿). For this reason, we +first perform a series of simulation experiments with simulated hu- +mans. We explore two shared autonomy contexts: a robot teaching +a human about physics-based robot dynamics (Section 6.1) and a +robot that implicitly influences human objectives, like their goal or +motion preferences (Section 6.2). +Similar to prior work in shared autonomy [9, 17, 26, 29], the robot +combines the human’s commanded action, 𝑢H, with the robot’s +planned guidance, 𝑢R, and executes the action: +𝑢 = 𝛼 · 𝑢R + (1 − 𝛼) · 𝑢H +(13) +where 𝛼 ∈ [0, 1] trades off how much guidance the robot can exert. +In all experiments, we use 𝛼 = 0.5. To generate human demon- +strations and infer the human learning dynamics, we simulate a +suite of human learners (see 6.1.1 and 6.2.1). In each experimental +environment we collect 50 demonstrations for model learning. We +randomize the initial state of the robot for each demonstration, and +randomize the robot actions during each interaction.1 +gradient human +threshold human +Figure 2: Our inference problem lets us learn to predict 𝜃𝑡 +H. +1We randomize 𝑢R to diversely cover how human’s internal model changes. +6.1 +Teaching Physical Dynamics +We focus on shared autonomy settings where the human knows +the task objective (e.g., control a robot arm to follow a path), but +they learn about the true robot dynamics (e.g., inertia). We want to +understand how the human learns about the physical robot dynam- +ics, and if a robot that actively teaches the human about its physics +can help the human quickly improve their task performance. +6.1.1 +Dynamics of human learning. Motivated by computational +cognitive science models [13], we simulated two types of human +learners: gradient-based learners and threshold learners. All hu- +mans update their internal model via Equation (3), but the structure +of 𝑓𝐿 takes various forms. After observing a new state-action pair +(𝑥𝑡,𝑢𝑡), the gradient-based learner updates their parameter𝜃𝑡 +H ac- +cording to a gradient-ascent update rule: 𝑓 grad +𝐿 +:= 𝜃𝑡 +H + 𝜂∇𝜃H𝑃(𝑢𝑡 | +𝑥𝑡;𝜃𝑡 +H) where𝜂 ∈ R+ is the step size. Note that𝑢𝑡 is the observed, to- +tal executed control, possibly combining 𝑢R and 𝑢H. Intuitively, this +learner can be viewed as doing gradient-based maximum likelihood +estimation of their latent parameter, similarly to prior IRL methods +[55]. The threshold learner also uses a gradient-based learning rule, +but only updates their internal parameters if they observe a “large +enough” change: 𝑓 thresh +𝐿 +:= 𝜃𝑡 +H + 𝜂1|∇𝑃 (𝑢𝑡 |𝑥𝑡,𝜃𝑡 +H) |>𝜖 +� +∇𝜃H𝑃(𝑢𝑡 | +𝑥𝑡,𝜃𝑡 +H) +� where 1 is an indicator determining if the magnitude of +the gradient is deemed large enough to induce a learning update +and 𝜖 is a threshold parameter. +6.1.2 +Human internal model. In all experiments, the simulated +humans are learning about the robot’s physical dynamics and thus +𝜃H encodes various aspects from Equation (7). +6.1.3 +Simulated environments. Figure 3 shows our simulated envi- +ronments, all or which have continuous state and action spaces. +(1) Lunar Lander. The human controls the Lunar Lander’s engines +to change its tilt. The human wants to keep the lander upright dur- +ing its descent. Let the state be the tilt angle with respect to the +ground and tilt angular velocity 𝑥 = (𝜓,𝜔) and 𝑢 be the engine +force. The dynamics are 𝑥𝑡+1 = 𝐴𝑥𝑡 + 𝐵𝑢𝑡 where the ground-truth +dynamics are 𝐴∗ = [1, 0.2; 0, 1], 𝐵∗ = [0; 0.5]. Here, the human’s in- +ternal model represents the control matrix 𝜃H := 𝐵, which depends +on the human’s inertia estimate. +(2) Robot Arm Teleoperation. The human controls the end-effector +of a 7DOF robot arm via hand gestures (see Figure 3). They want to +control the robot to reach a series of known goals, 𝑥𝑔. However, one +of the robot motors is slightly defective, causing the robot to con- +sistently lag in one direction. Let the state be the robot end-effector +position 𝑥 = (𝑝𝑥, 𝑝𝑦, 𝑝𝑧) and the control𝑢 be linear velocity. The ro- +bot’s end-effector dynamics can be described by the goal-dependent +system2 : 𝑥𝑡+1 = 𝐴𝑥𝑡 +𝐵 +� +𝑢𝑡 −sign(𝑥𝑡 −𝑥𝑔) ⊙𝑤 +� +where 𝑤 is the bias +induced by the defective robot motor and ⊙ is the Hadamard prod- +uct. Intuitively, this describes a dynamical system that consistently +experiences lag in the 𝑥-direction. The ground-truth dynamics are +𝐴∗ = 𝐼3×3, 𝐵∗ = diag(0.4, 0.4, 0.4), and 𝑤∗ = [−0.15, 0, 0]⊤. The hu- +man’s internal model is 𝜃H := (𝐵,𝑤), which captures their system +responsiveness and bias estimates. +2Although this system is nonlinear, since the robot knows the human’s goal at each +time step, the dynamics can be approximated by a linear system 𝑥𝑡+1 = 𝐴𝑥𝑡 + 𝐵 ˜𝑢𝑡 , +where 𝑥0 is the system state at that time step and ˜𝑢𝑡 := 𝑢𝑡 − sign(𝑥0 − 𝑥𝑔) ⊙ 𝑤. + +Robot Arm Teleoperation +Lunar Lander +0.6 +0.4 +0.4 +-1.2 +0.2 +1.4 +0.2 +-1.4 +tH +0 +-1.6 +0 +0 +10 +20 +30 +40 +50 +0 +10 +20 +30 +40 +50 +Epoch +EpochHRI ’23, March 13–16, 2023, Stockholm, SE +Ran Tian, Masayoshi Tomizuka, Anca D. Dragan, and Andrea Bajcsy +Robot Arm Teleoperation +Lunar Lander +Gradient Human +Threshold Human +passive learn +random +oracle +active teach (ours) +Robot Arm Teleoperation +robot automatically +stops teaching! +higher 𝜃H error; +robot keeps teaching +Gradient Human +Threshold Human +Lunar Lander +robot automatically +stops teaching! +robot eventually +stops teaching! +Figure 3: (left) Visualization of both simulation environments. (right) Mean and standard deviation of human internal model +error, robot effort, and human action optimality for both dynamics teaching environments, and both simulated humans. +6.1.4 +Human objective. We assume the human always knows the +objective, and their reward function is quadratic as in (8). For Lunar +Lander the human was rewarded for keeping the lander upright +and stable (𝜓 = 0, 𝜔 = 0), and for Robot Arm they were rewarded +for reaching all the goals and tracking the path shown in Figure 3. +6.1.5 +Robot objective. The robot objective is to align human’s in- +ternal model with the true robot dynamics model while minimally +intervening. Mathematically, the robot’s reward function is: +𝑟R(𝑠,𝑢R,𝑢H;𝜃∗) = −||𝜃H − 𝜃∗||2 +2 − ||𝑢 − 𝑢H||2 +2, +(14) +where the true dynamics are 𝜃∗ := 𝐵∗ in the Lunar Lander envi- +ronment and 𝜃∗ := (𝐵∗,𝑤∗) in the Robot Arm setting. +6.1.6 +Baselines. We compare our method where the robot actively +teaches by planning with the inferred learning dynamics 𝑓 𝜙 +𝐿 (Active +Teach) to a robot that teaches with the true learning dynamics 𝑓𝐿 +(Oracle), no robot intervention (Passive Learn), and a robot that +randomly perturbs the human actions (Random). +6.1.7 +Hypotheses. H1: We can learn to predict 𝜃𝑡 +H well by maxi- +mizing the MLE objective. H2: Active Teach outperforms Passive +Learn and Random in aligning the human’s internal model. H3: +Robot stops intervening when the human’s internal model is well- +aligned. +6.1.8 +Results. For H1, we study the relationship between the MLE +objective in (5) and our inferred model’s (𝑓 𝜙 +𝐿 ) ability to predict 𝜃H. +Figure 2 shows these curves for both the Robot Arm and Lunar +Lander environments over 50 epochs. We see that across both +gradient and threshold human learners, the log likelihood of the +human’s actions increases (shown in pink) while the 𝜃H prediction +error decreases (shown in blue), supporting H1. +Figure 3 shows the human’s internal model error, the robot’s ef- +fort, and the difference between the human’s action and the optimal +action in the Robot Arm Teleoperation and the Lunar Lander +environment for both types of human learners. We see that across all +environments, our method performs comparably to Oracle model, +and is able to align the human’s internal model of the robot’s dy- +namics with the true dynamics significantly faster than Passive +Learn or Random (supporting H2). Interestingly, in all but one +setting does the robot automatically stop teaching the human since +the human’s internal model is sufficiently correct (supporting H3). +The one exception is in the Robot Arm Teleoperation environ- +ment with the threshold human. Since this human doesn’t learn +when the gradient is too small, the robot must continue to exert +effort to maximize its reward. +6.2 +Implicitly Influencing Human Objectives +We now turn to scenarios where the human has an accurate un- +derstanding of the robot’s dynamics, but their objective (i.e., their +reward function 𝑟H) can be changed by the robot. Specifically, we +study how assistive robots can implicitly influence human motion +preferences and desired goals. Importantly, in this setting influenc- +ing or teaching the human is not explicitly in the robot’s objective: +the robot simply wants to perform the desired task with minimal +assistance. Thus, getting the human to want to reach a goal or +change their preferences should be an emergent behavior of robots +planning with the dynamics of human learning. +6.2.1 +Dynamics of human learning. We simulate3 the gradient hu- +man learner from 6.1.1 and introduce a new human, the Bayesian +learner4 which is inspired by probabilistic models of cognition +[13, 50]. This human’s learning produces a full posterior, 𝑏𝑡+1(𝜃H), +over the model parameters given a state-action observation, and +the dynamics of learning are: 𝑓 Bayes +𝐿 +∝ 𝑃(𝑢𝑡 | 𝑥𝑡,𝜃H)𝑏𝑡 (𝜃H). +6.2.2 +Human internal model. Since the human’s objectives are +influenceable, we model 𝜃H as a reward parameter encoding the +motion preferences 𝜃H := (𝑄, 𝑅) or a desired goal state 𝜃H ∈ Θ. +6.2.3 +Simulated environments. We assume the human knows the +physical robot dynamics (the bias-free RobotArm dynamics from +6.1.3), but can have their reward influenced by new observations. +(1) Goal Influence. The human wants to teleoperate the robot +to put an object in one of the three trays (upper left Figure 4). +However,the human doesn’t notice that only one of the trays is +empty enough. Unlike the human, the robot’s sensors detect that +only one of the trays is empty. We investigate if the robot can +3While we simulate the human as changing their reward, but the human’s reward +could be viewed as static while their subgoals change Nonetheless, it will be common +for a robot to not fully represent this hierarchy. +4Bayesian humans act under their belief: P(𝑢H | 𝑥) = � +𝜃H 𝑏(𝜃H)P(𝑢H | 𝑥,𝜃H). + +Robot Arm Teleoperation +0.4 +error +0.2 +0 +0.2 +0.1 +0 +0.1 +0 +1 +10 +20 +30 +40 +50 +timestepRobot Arm Teleoperation +0.4 +0.2 +0 +0.2 +0.1 +0 +0.1 +0 +1 +10 +20 +30 +40 +50 +timestepLunar Lander +0.6 +0.4 +0.2 +0 +0.2 +0.1 +0 +0.2 +0.1 +0 +1 +10 +20 +timestepLunar Lander +0.6 +0.4 +0.2 +0 +0.2 +0.1 +0 +0.2 +0.1 +0 +1 +10 +20 +timestepTowards Modeling and Influencing +the Dynamics of Human Learning +HRI ’23, March 13–16, 2023, Stockholm, SE +influence the human to change their preferences about which tray +(i.e., goal location) to place their object in. +(2) Preference Influence. The human wants to teleoperate the +robot to pick up a cup on the table. Their initial preference is to +move the robot’s end-effector in a straight line from start to the cup +(lower left Figure 4). However, the robot knows that grasps tend +to fail with this kind of motion. Instead, the robot knows that first +moving directly above the can and then straight down to grasp has +a higher chance of success. We investigate if the robot can influence +the human to change their preferences about how to reach the cup. +6.2.4 +Human objective. In all simulations the human has a qua- +dratic cost function (from (8)). In Goal Influence the simulated +human receives reward for moving the robot end-effector to their +desired tray, and in Preference Influence the human receives +reward according to their current preference matricies, (𝑄, 𝑅). +6.2.5 +Robot objective. We implement an assistive robot that wants +to help the human perform the task while minimally intervening. +However, we assume that the robot knows best: the robot knows +which goal or reward weights lead to success. Let 𝜃∗ capture this +aspect of the robot’s reward. In the Goal setting the robot’s reward +𝑟R(𝑠,𝑢R,𝑢H;𝜃∗) = −(𝑥 −𝜃∗)⊤𝑄(𝑥 −𝜃∗) −𝑢⊤𝑅𝑢 − ||𝑢 −𝑢H||2 +2 and in +Preference the robot’s reward parameter is 𝜃∗ = (𝑄∗, 𝑅∗), yielding +𝑟R(𝑠,𝑢R,𝑢H;𝜃∗) = −𝑥⊤𝑄∗𝑥 − 𝑢⊤𝑅∗𝑢 − ||𝑢 − 𝑢H||2 +2 where 𝑢 is the +combined human and robot action from Equation (13). +6.2.6 +Baselines. We implement our method where the robot as- +sists the human and plans with the inferred dynamics of human +learning (Learning Assist). We compare to a robot assisting with +the ground-truth dynamics of human learning (Oracle), robot as- +sistance that is unaware that humans learn (Static Assist), and a +robot that randomly perturbs the human actions (Random). +6.2.7 +Hypotheses. H4: Learning Assist aligns the human’s mental +model faster. H5: Assistance that accounts for human learning enables +the human-robot team to achieve higher reward under the true 𝜃∗. +6.2.8 +Results. Figure 4 shows the human’s internal model error, ro- +bot effort, and task cost (i.e., just the task-component of 𝑟R, negated) +for both environments. Because the Learning Assist robot knows +that the human’s internal model can be changed, it automatically +exerts higher effort early on to align the human’s internal model +with it’s own, resulting in less long-term assistance and lower task +cost (supporting H4 and H5). In contrast, the Static Assist robot +is not aware that the human can change their mind, and thus does +not exert enough effort to influence the human’s internal model. +After repeatedly incurring task cost because the two agents are at +odds with each other, the Static Assist robot “gives up” and starts +executing the human’s control directly: in other words, 𝑢 = 𝑢H. +7 +USER STUDY: TEACHING TO +TELEOPERATE +So far we conducted experiments with simulated human behavior, +allowing us to analyze the quality of our inferred human learning +dynamics model, and the robot’s ability to influence simulated +humans. Here we investigate if we can infer the dynamics of real +human learning, and enable robots to influence real users. +Goal Influence (Bayesian human) +Preference Influence (gradient human) +static assist +random +oracle +learning assist (ours) +𝑏0(𝜃H) +𝜃∗ +𝜃H +0 +𝜃∗ +Figure 4: (left) Environments for influencing human objec- +tives. (right) Internal model error, robot effort, and task cost. +We focus on scenarios where the robot’s physical dynamics are +different from what the human is used to; for example, perhaps the +human was used to teleoperating a robotic wheelchair, but is now +teleoperating a robotic arm. As they interact with the robotic arm, +they will naturally learn about the new robot dynamics. In our IRB- +approved user study, we investigate if a robot can actively teach +a human the physical dynamics and improve their teleoperation +performance faster than if the human does the task on their own. In +other words, we aim to understand if a robot can align the human’s +internal model with the robot’s. +Experimental Setup. We designed a teleoperation task where the +human controls a 7DOF Jaco robot arm through a webcam-based +gesture interface (Figure 1). The participant uses their index finger +to indicate how the end-effector should move parallel to the tabletop. +The task is to move the end-effector to reach four goals on the table +in a counter-clockwise pattern, tracing out a diamond pattern. All +participants experience a familiarzation task where they perform +the task unassisted, with the default robot dynamics in order to +understand the gesture interface. In software, we then simulate two +“new” robots, each with different physical properties. +Independent Variables. We manipulated the robot strategy with +two levels: no-teaching and active-teaching. The robot either let the +human do the task on their own, or it modified the human’s input +to teach them about the physical robot dynamics via Equation (12). +We also manipulate the robot physical dynamics with two levels: +end-effector dynamics bias in x-direction and bias in y-direction. +Dependent Measures. A challenge in evaluating our experiment +is that we do not have access to the human’s ground-truth internal +model. As a proxy, we measure human action optimality distance: +|| ˆ𝑢𝐻 − 𝑢∗||2 +2. Intuitively, the better the human understands the +robot, the more optimally they should be able to control it to reach +the goals. Since we cannot directly measure a human’s internal +understanding, we instead look at their actions to measure their +deviation from the optimal action under the robot’s true physics. +We also measured subjective measures via a Likert scale survey. +Hypotheses. H6: Participants in the active teaching condition be- +come optimal teleoperators faster than passively learning on their own. + +uR effort +0.2 +0.1 +0uR effort +0.5 +0Task cost +10 +5 +0 +1 +5 +10 +15 +20 +25 +30 +timestepTask cost +40 +20 +0 +1 +5 +10 +15 +20 +25 +30 +timestep1 + error +0.5 +0 +5 +10 +15 +20 +25 +30 +timestep2 +1.5 +)error +(H)9 +1 +0.5 +0 +5 +10 +15 +20 +25 +30 +timestepHRI ’23, March 13–16, 2023, Stockholm, SE +Ran Tian, Masayoshi Tomizuka, Anca D. Dragan, and Andrea Bajcsy +passive learn +active teach (ours) +Human Action Optimality Distance +Participant Trajectories +User Study: Quantitative & Qualitative Results +early on robot teaches +by exaggeration +later humans +more optimal +early & late suboptimality +w/ passively learning +start +end +start +end +Figure 5: (left) Avg. human action optimality distance and +95% confidence interval. (right) Dashed line is desired path. +Participant trajectories reveal that an active teaching robot +initially exaggerates the dynamics bias to teach the human. +H7: Participants feel they learned to teleoperate faster and understood +the robot dynamics better in the active teaching condition. +Participants. We recruited two groups of participants from the +campus community: the first for providing data for inferring the +dynamics of human learning (12 participants; 2 female, 10 male, +age 18-34, all with technical backgrounds), and the second for the +user study (10 participants; 1 female, 8 male, 1 non-binary, age +18-34, all with technical backgrounds). For inferring the human +learning dynamics, all participants learned to teleoperate the robot +unassisted and we counterbalanced the robot physical dynamics. +Procedure. A within-subjects design is challenging, since humans +who experience one condition will learn about the robots and then +carry over that experience to the next condition. To study the effect +of this confound, each participant experienced a combination of +robot strategy and physical dynamics conditions, but in a random +order. For example, one group of participants would interact with +the (active-teaching, bias-x) condition and then (no-teaching, bias-y) +condition. Thus, each participant experiences both robot strategies +and biases. We counterbalance the order in which the participants +experience the combination. All participants experienced a familiar- +ization round at the start and between each experimental condition, +to “reset” their mental model of the robot. Each participant gave 3 +demonstrations per condition, each lasting ∼1 minute. +Quantitative Results. Figure 5 shows how human action opti- +mality distance varies over time with each robot strategy. We con- +ducted an ANOVA with robot strategy and stage (first or second +half of interaction) as factors and robot physical dynamics as ran- +dom effect. We found a significant main effect of the robot strategy +(𝐹 (1, 19) = 12.943, 𝑝 = 0.001) and a marginal interaction effect +between the robot strategy and the interaction stage (𝑝 = 0.098), +so we did not run a post-hoc analysis. However, we hypothesize +that this marginal interaction effect comes from the fact that early- +stage changes in robot behavior (induced by either robot strategy) +influences the human’s later-stage action optimality. Ultimately, +the quantitative results indicate a significant improvement in the +human’s action optimality when the robot actively teaches them +compared to when the human passively learns (supporting H6). +Qualitative & Subjective Results. On the right of Figure 5 we +visualize the executed trajectories from all participants in the active- +teaching (orange) and no-teaching (grey) conditions. The high- +lighted trajectories are two representative examples, the color gra- +dient indicates time along the trajectory, and the dashed line is +the desired path. When participants passively learn on their own, +their trajectories are consistently suboptimal, weaving around the +optimal path. In contrast, in the active teaching condition, the initial +portion of the trajectory exhibits the robots teaching behavior: the +robot intentionally exaggerates the dynamics bias to change the hu- +man’s internal model faster. After this initial exaggerated deviation, +the human trajectory is closer to optimal compared to the passive +learning trajectory at comparable timesteps (see Appendix A.4 for +a detailed visualization of human and robot actions). +We also ran an ANOVA on the Likert survey questions. Survey +questions investigated perceived performance improvements (e.g., +“By the end of the interaction, it was easy to control the robot to +do the task.”) and robot understanding (e.g., “By the end of the +interaction, I understood the robot’s physical properties.”). Across +all questions, we did not find a significant effect of the robot strategy +(rejecting H7). What we found surprising was that even though +participants were quantitatively performing better in the teaching +condition, they did not perceive an improvement in performance +(𝑝 = 0.689) nor in their understanding of the robot physics (𝑝 = +0.299). We hypothesize that this could be because participants only +interacted with each robot strategy for one minute, making the +differences hard to notice. In the future, investigating longer-term +interactions with the robot would shed light on the disconnect. +8 +CONCLUSION +In this work we took a step towards enabling robots to understand +the influence that they have over human internal models. We do +this by modeling human learning as a nonlinear dynamical system +that evolves as a function of new observations that the robot can +influence. We propose a tractable method for inferring approxi- +mate human learning dynamics from demonstrations that naturally +exhibit human learning, and propose how robots can influence hu- +man learning by embedding the approximate dynamics into robot +planning. Our experimental results indicate that robot influence is +possible and can help humans learn better internal models. +Limitations & Future Work. A strength and limitation of our +approach is representing the dynamics of human learning via a +transformer. As a general function approximator, it poses no as- +sumptions on the structure of the human’s learning dynamics; in +fact, we are excited that our results indicate that it is possible to +infer a useful model of human learning from real data, without prior +assumptions. However, since neural networks require abundant +human data, they are not appropriate for low-data settings and +may fail when encountering humans that are out of distribution. A +further limitation is that if the person is not noisily-optimal as in +(2) and has a specific bias (e.g., myopia), then the transformer will +learn parameters that compensate for this; in turn, this could lead +the robot to influence the human in unintended ways. In the future +we are excited to combine the strengths of data-driven models and +cognitive science models of human learning. 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Chicago, IL, +USA, 1433–1438. + +Towards Modeling and Influencing +the Dynamics of Human Learning +HRI ’23, March 13–16, 2023, Stockholm, SE +A +APPENDIX +A.1 +Derivation: Gaussian integral under LQ +approximation +Here we derive the closed-form solution to the denominator from (2) +under the LQ-approximation. First, we recall the Gaussian integral: +Theorem: Gaussian Integral [49]. Let 𝑀 ∈ R𝑛×𝑛 be a symmetric, +positive-definite matrix and 𝑥 ∈ R𝑛. Then: +∫ +exp +� +− 1 +2𝑥⊤𝑀𝑥+𝑏⊤𝑥 +� +𝑑𝑛𝑥 = +√︄ +(2𝜋)𝑛 +det(𝑀) exp +� 1 +2𝑏⊤𝑀−1𝑏 +� +. (15) +Theorem: Infinite-horizon Linear-Quadratic Regulator [3]. Let +the discrete-time dynamics be linear, 𝑥𝑡+1 = 𝐴𝑥𝑡 + 𝐵𝑢𝑡, and the cost +quadratic, 𝑥⊤𝑄𝑥 +𝑢⊤𝑅𝑢. Then the infinite-horizon optimal cost-to-go +𝐽 and optimal control 𝑢∗(𝑥) are: +𝐽 (𝑥) = 𝑥⊤𝑃𝑥 +(16) +𝑢∗(𝑥) = −𝐾𝑥 +(17) +where 𝑃 is the unique, positive-definite fixed point of the infinite- +horizon, discrete-time Ricatti equation (DARE): +𝑃 = 𝐴⊤𝑃𝐴 − 𝐴⊤𝑃𝐵(𝑅 + 𝐵⊤𝑃𝐵)−1𝐵⊤𝑃𝐴 + 𝑄 +(18) +and the feedback matrix 𝐾 = (𝑅 + 𝐵⊤𝑃𝐵)−1𝐵⊤𝑃𝐴. +Derivation. Let that the physical dynamics be linear and the re- +ward is quadratic in state and control. Assume that we have approx- +imated the state-action 𝑄H function as: +𝑄H(𝑥,𝑢) = −𝑥⊤𝑄𝑥 − 𝑢⊤𝑅𝑢 − (𝑥 ′)⊤𝑃(𝑥 ′) +(19) +where 𝑥 ′ = 𝐴𝑥 + 𝐵𝑢 is the next state and 𝑃 is the solution to (18). +Plugging in (19) into the denominator of the human policy, we +obtain: +∫ +exp +� +𝑄H(𝑥,𝑢) +� +𝑑𝑢 +(20) += +∫ +exp +� +− 𝑥⊤𝑄𝑥 − 𝑢⊤𝑅𝑢 − (𝐴𝑥 + 𝐵𝑢)⊤𝑃(𝐴𝑥 + 𝐵𝑢) +� +𝑑𝑢 +(21) += exp +� +− 𝑥⊤𝑄𝑥 − 𝑥⊤𝐴⊤𝑃𝐴𝑥 +� +· +(22) +∫ +exp +� +− 1 +2𝑢⊤(2𝑅 + 2𝐵⊤𝑃𝐵)𝑢 + (−2𝑥⊤𝐴⊤𝑃𝐵)𝑢 +� +𝑑𝑢 +(23) +We see that if we let 𝑀 := 2𝑅 + 2𝐵⊤𝑃𝐵 and 𝑏 := −2𝑥⊤𝐴⊤𝑃𝐵 then +we can directly take the Gaussian integral and obtain: += exp +� +− 𝑥⊤𝑄𝑥 − 𝑥⊤𝐴⊤𝑃𝐴𝑥 +�√︄ +(2𝜋)𝑚 +det(2𝑅 + 2𝐵⊤𝑃𝐵) · +(24) +exp +� +(−2𝑥⊤𝐴⊤𝑃𝐵)⊤[2𝑅 + 2𝐵⊤𝑃𝐵]−1(−2𝑥⊤𝐴⊤𝑃𝐵) +� +(25) += exp +� +𝑥⊤� +𝐴⊤𝑃𝐵(𝑅 + 𝐵⊤𝑃𝐵)−1𝐵⊤𝑃𝐴 − 𝑄 − 𝐴⊤𝑃𝐴 +� +𝑥 +� +· +(26) +√︄ +(2𝜋)𝑚 +det(2𝑅 + 2𝐵⊤𝑃𝐵) . +(27) +Interestingly, we see that the exponent contains the (negated) DARE +equation from (18) within the brackets. Substituting 𝑃 back in, we +obtain: +∫ +exp +� +𝑄H(𝑥,𝑢) +� +𝑑𝑢 = exp +� +− 𝑥⊤𝑃𝑥 +�√︄ +(2𝜋)𝑚 +det(2𝑅 + 2𝐵⊤𝑃𝐵) . (28) +We can further simplify this equation by +𝑥⊤𝑃𝑥 = min +𝑢 +� +𝑥⊤𝑄𝑥 + 𝑢⊤𝑅𝑢 + (𝐴𝑥 + 𝐵𝑢)⊤𝑃(𝐴𝑥 + 𝐵𝑢) +� +(29) += 𝑥⊤𝑄𝑥 + (𝑢∗)⊤𝑅(𝑢∗) + (𝐴𝑥 + 𝐵𝑢∗)⊤𝑃(𝐴𝑥 + 𝐵𝑢∗) +(30) += 𝑄H(𝑥,𝑢∗) +(31) +where 𝑢∗ is the optimal control at state 𝑥. Thus, we can obtain our +final simplified form: +∫ +exp +� +𝑄H(𝑥,𝑢) +� +𝑑𝑢 = exp +� +− 𝑄H(𝑥,𝑢∗) +�√︄ +(2𝜋)𝑚 +det(2𝑅 + 2𝐵⊤𝑃𝐵) . +𝜃H +𝑡+1 +𝜃H +𝑡+2 +𝜃H +𝑡+3 +… +… +𝑥𝑡, 𝑢H +𝑡 , 𝑥𝑡+1 +𝑥𝑡+1, 𝑢H +𝑡+1, 𝑥𝑡+2 +𝑥𝑡+2, 𝑢H +𝑡+2, 𝑥𝑡+3 +𝑓𝐿 +𝜙 +Transformer +encoder +encoder +encoder +Figure 6: Architecture of transformer representing 𝑓 𝜙 +𝐿 . +A.2 +Details on a gradient-based solution to +inferring the dynamics of human learning +To optimize the transformer-based model of human learning dy- +namics, we need to compute the gradient of our inference objective +(in Equation (5) and referred to here as L) with respect to the neural +network parameters, 𝜙: +𝜕L +𝜕𝜙 = 𝜕L +𝜕𝜃H +𝜕𝜃H +𝜕𝜙 . +The second component, 𝜕𝜃H +𝜕𝜙 , is the gradient of the transformer’s in- +ternal model predictions with respect to the neural network weights +and is readily available since the transformer is differentiable. How- +ever, the first component +𝜕L +𝜕𝜃H += +� 𝜕 log P(𝑢𝑡 +H | 𝑥𝑡;𝜃𝑡 +H) +𝜕𝜃H𝑡 +� +, +which is the human’s policy gradient with respect to the human’s +internal model parameter, is a key challenge. This is because the +human’s policy P(𝑢H | 𝑥,𝜃H) depends on 𝑃𝜃H through the 𝑄H- +value. Recall that 𝑃𝜃H is the solution to DARE in Equation (10) +which depends on the matrices 𝐴, 𝐵,𝑄, 𝑅 and 𝑃𝜃H. Regardless of if +the human’s internal model parameter is the physical dynamics +𝜃H := (𝐴, 𝐵) or the reward weights 𝜃H := (𝑄, 𝑅), the human’s +policy gradient requires differentiating through the DARE function, +which is non-obvious. Leveraging recent work [10] that treats the +DARE as an implicit function of (𝐴, 𝐵,𝑄, 𝑅), we obtain closed-form +Jacobians 𝜕𝑃 +𝜕𝐴, 𝜕𝑃 +𝜕𝐵 , 𝜕𝑃 +𝜕𝑄 , and 𝜕𝑃 +𝜕𝑅 . The precise form of these can be + +HRI ’23, March 13–16, 2023, Stockholm, SE +Ran Tian, Masayoshi Tomizuka, Anca D. Dragan, and Andrea Bajcsy +Human input action (𝑢H) +Robot executed action (𝑢) +𝑡 = 0 +𝑡 = 40 𝑠 +later, the human and robot +actions are more aligned +early on the robot +executes exaggerations +of the human’s input +Figure 7: An example participant trajectory from Section 7. +Blue vectors show the executed robot actions (in solid line) +and human input action (in dashed line) at timesteps sam- +pled at 0.5 s. +found in Proposition 2 of [10]. Thus, we can efficiently compute +𝜕L +𝜕𝜙 and infer the human’s learning dynamics via gradient-based +optimization. +A.3 +Training the dynamics model of human +learning +To enhance the reproducibility of inferring 𝑓 𝜙 +𝐿 , we present the ar- +chitecture and optimization details here. The encoder for encoding +(𝑥𝑡,𝑢𝑡 +H,𝑥𝑡+1) is a multilayer perceptron with 3 fully-connected lay- +ers in all settings. We use the Hugging Face’s implementation [11] +of the transformer encoder [51] to represent the human’s learning +dynamics, and use the Adam optimizer to train the neural network. +In both the simulated experiments and in the user study, we use +the same transformer architecture to represent the dynamics of +human learning, with only the output layer size adjusted per each +task to appropriately model the human’s internal model 𝜃H. From +Section 6.1, in Lunar Lander the output size is 2-dimensional, rep- +resenting the 𝐵-vector that the human is estimating and in Robot +Arm Teleoperation the output is 4-dimensional to account for the +diagonal elements of 𝐵 and 𝑤. From Section 6.2, in Goal Influence +the output size is 2-dimensional, representing the probability (i.e. +human belief) over the first and the second tray goals (the prob- +ability over the third goal is implicitly defined as one minus the +probability of the other two goals combined), while in Preference +Influence the output size is 3-dimensional to represent the diago- +nal terms of the 𝑄 ∈ R3×3. Finally, in the user study from Section 7, +the output is 4-dimensional to account for the diagonal elements +of 𝐵 and 𝑤. Note that the human’s initial internal model (𝜃0 +H) is +implicitly estimated at the beginning of the input when predicting +𝜃1 +H. +A.4 +User Study: Human and Robot Action +Alignment +We looked at the user study data and investigated how the human +input actions compared to the executed robot actions under our +teaching method. Recall that the robot executes actions according +to (13): 𝑢 = 𝛼 ·𝑢R + (1−𝛼) ·𝑢H where 𝛼 = 0.5 for the duration of our +user study and 𝑢R is generated according to our influence-aware +planning method. In Fig. 7 we plot a sample participant trajectory +and the robot executed actions (solid blue vector) and human input +actions (dashed blue vector) at 1.5 s time intervals. Qualitatively, +we see that early on the human and robot’s actions are misaligned. +Intuitively, since the robot’s planning objective is to quickly align +the human’s mental model of the physics with the robot’s physics +model, the robot plans to execute an exaggeration of the human’s +input in hopes of quickly changing their mind. Later on, we see that +the human and robot actions become more aligned as the human +learns to be a better teleoperator. + diff --git a/itAyT4oBgHgl3EQf-_qA/content/tmp_files/load_file.txt b/itAyT4oBgHgl3EQf-_qA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..17594a13356e89b25dd2791a62b7145bb9535116 --- /dev/null +++ b/itAyT4oBgHgl3EQf-_qA/content/tmp_files/load_file.txt @@ -0,0 +1,942 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf,len=941 +page_content='Towards Modeling and Influencing the Dynamics of Human Learning Ran Tian∗ UC Berkeley Masayoshi Tomizuka UC Berkeley Anca D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Dragan UC Berkeley Andrea Bajcsy UC Berkeley ABSTRACT Humans have internal models of robots (like their physical capabili- ties), the world (like what will happen next), and their tasks (like a preferred goal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' However, human internal models are not always perfect: for example, it is easy to underestimate a robot’s inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Nevertheless, these models change and improve over time as hu- mans gather more experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Interestingly, robot actions influence what this experience is, and therefore influence how people’s inter- nal models change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In this work we take a step towards enabling robots to understand the influence they have, leverage it to better assist people, and help human models more quickly align with real- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Our key idea is to model the human’s learning as a nonlinear dynamical system which evolves the human’s internal model given new observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We formulate a novel optimization problem to infer the human’s learning dynamics from demonstrations that naturally exhibit human learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We then formalize how robots can influence human learning by embedding the human’s learn- ing dynamics model into the robot planning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Although our formulations provide concrete problem statements, they are intractable to solve in full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We contribute an approxima- tion that sacrifices the complexity of the human internal models we can represent, but enables robots to learn the nonlinear dynamics of these internal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We evaluate our inference and planning methods in a suite of simulated environments and an in-person user study, where a 7DOF robotic arm teaches participants to be better teleoperators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' While influencing human learning remains an open problem, our results demonstrate that this influence is possible and can be helpful in real human-robot interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' CCS CONCEPTS Computing methodologies → Artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' KEYWORDS robot influence, human internal model, dynamics of human learning ACM Reference Format: Ran Tian, Masayoshi Tomizuka, Anca D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Dragan, and Andrea Bajcsy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Towards Modeling and Influencing the Dynamics of Human Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In ∗This work supported by ONR YIP, NSF NRI, and WeRide Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Author emails: {rantian, tomizuka, anca, abajcsy}@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Project website with link to code: https://sites.' metadata={'source': 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owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' HRI ’23, March 13–16, 2023, Stockholm, SE © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' ACM ISBN 978-1-4503-XXXX-X/23/06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='XXXXXXX 𝑓𝐿 Robot influence 𝜃H 𝑡+1 Dynamics of human learning 𝜋𝑅 𝑥𝑡, 𝜃H 𝑡 𝑓𝐿 𝜃H 𝑡 Figure 1: Human teleoperates a new robot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' they update their internal model by acting and observing outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Planning with human learning dynamics, the robot influences the hu- man’s internal model to help them be a better teleoperator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Proceedings of Human Robot Interaction (HRI ’23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' ACM, New York, NY, USA, 12 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='XXXXXXX 1 INTRODUCTION Imagine your first time controlling a robot arm to perform daily living tasks like throwing away trash or stirring a pot of soup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Initial interactions with the robot are tough: you aren’t familiar with the robot’s dynamics so your motions are jerky and imprecise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In other words, your internal model of the robot is incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' And robot dynamics are not the only thing we, humans, have incorrect internal models of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We might not fully understand the world’s dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', result of pouring lemon juice into cream) or our own preferences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', only liking something after trying it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' However, over time, our internal models evolve with our experi- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' As you control the robot, you start to understand how it will move;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' as you try different things, you learn what you like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Since the robot is part of the world, the robot’s actions and their outcomes become part of these experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In other words, robot actions inevitably influence the change in a human’s internal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In this work, we advocate that robots should understand and use this influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' First, collaborative tasks require understanding what people are trying to do in order to assist them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Prior work has shown that inferring a human’s internal model is critical for assis- tance [38];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' in turn, we argue that if this model changes over time, tracking this change will enhance assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Further, purposefully influencing a change in the human’s internal model opens the door for teaching: robot actions that are optimized to quickly align the human’s understanding with reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' For instance, as an operator controlling the robot (Figure 1), this means you quickly understand how the robot works and can do the task independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' A key challenge towards this is modeling how humans learn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' without a proper model of this, the robot cannot plan to change the human’s internal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Although we do not know the precise functional form of how people learn, we observe that a human’s arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='00901v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='RO] 2 Jan 2023 CHRI ’23, March 13–16, 2023, Stockholm, SE Ran Tian, Masayoshi Tomizuka, Anca D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Dragan, and Andrea Bajcsy understanding of the robot or the world changes as a function of what they observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' For example, at first you may mistakenly believe that the robot doesn’t experience any inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' However, as soon as you gesture to move the robot forward, you see the robot lagging behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' This observation controls the evolution of your internal robot physics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The same holds true for your internal model of the world and personal preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In other words, we can model human learning as a dynamical system where the hu- man’s internal model is the state, and the observations— which the robot can influence—evolve the internal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Of course, this does not prescribe the functional form of the dy- namical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' One idea is to draw on computational cognitive science work to define this function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' A predominant lens is that of probabilistic models [12], which posits that humans perform some form of approximate Bayesian inference based on the observations they receive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In reality, people have been shown to have a plethora of cognitive biases which deviate from perfect Bayesian inference: they might use gradient information [53], might not process the en- tire observation due to sensory overload [28], or exhibit systematic bias like over- or under- estimation [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Instead of committing to a specific model, in our work we treat this as a general dynamics learning problem, which has roots in controls and robotics [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We leverage demonstrations which naturally exhibit human learning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', humans teleoperating a robot they have never interacted with before), to fit a human learning model under the assumption that observed human actions are approximately optimal given their cur- rent internal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' This enables flexibility of capturing different possible learning updates, at the cost of being domain-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Although the most general model learning problem remains computationally intractable, we introduce a tractable approxima- tion that is readily solvable via gradient-based optimization, and is compatible with neural network representations of the human learning dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Leveraging our approximate dynamics model of human learning, we formalize robot influence over the human’s internal model as a Markov Decision Process (MDP) where the hu- man’s internal model is part of the state and the human’s learning dynamics are part of the transition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The solution yields robot actions that change the human’s internal model by changing the human’s observations in a way that rewards the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We run experiments with simulated humans to study the fidelity of the inferred human learning dynamics and investigate robot teaching and assistance in settings where the human’s understand- ing of robot physics, motion preferences, or goals can be influenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Finally, we conduct a user study with a Kinova Jaco 7DOF robot arm and find that our method can help teach humans to be better teleoperators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Overall, while influencing human learning remains an open problem, we are excited to have taken a step in this domain via a principled yet tractable learning and planning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 2 RELATED WORK Inferring human preferences and beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' A large body of work has focused on learning human reward functions via inverse rein- forcement learning (IRL) [19, 22, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' This includes inferring hu- man driving preferences [34, 40], desired exoskeleton gaits [25], intended goals [17], motion preferences [35], and human under- standing about physics [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' A key assumption in these works is that people have static internal models of preferences or physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In- stead, we are interested in learning a dynamic model of how humans change their preferences, goals, and understanding of physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Models of human learning for robot decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Prior works in robotics model human learning as Bayesian inference when updating goals or preferences [8, 14, 16], a linear Gaussian system when updating trust [7], gradient-based IRL when learning rewards [4], or as a multi-armed bandit algorithm when updating preferences [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Instead of assuming a known model of how people learn, in this work we seek to learn a model of how humans learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Most related to our work is [39] which learns a model of how people estimate the state of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In this work, we propose a generalization where the human is not estimating world state, but updating their preferences, goals, and internal physics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' This induces a significantly harder model learning problem, for which we propose a tractable approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Cognitive theories of human learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Models of human infer- ence have been extensively studied in both computational cognitive science [2, 13] and psychology [36, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' While human cognition can be broadly modeled at three levels (computational, algorith- mic, and hardware) [27], most relevant to us are the algorithmic works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' [13] posits that modeling human reasoning as “implement- ing” an exact Bayesian posterior or a gradient-based point estimate are both compatible with probabilistic models of human cognition, and are a potential source of rational process models [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Further, [42] finds evidence that humans may update their forward models using the models’ prediction error as loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Inspired by these works, our simulated human experiments leverage exact and approximate probabilistic inference models, and we study if our flexible, learning-based method can effectively recover such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Robot influencing human behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' While there are many ways a robot can influence humans (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', through nonverbal cues, ap- pearance, visuals, or curriculum design [1, 37, 39, 41, 46]), we focus on robot influence through physical action [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' A common ap- proach towards this models human-robot interaction as a game [15, 23, 32, 40, 44, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' While these approaches can capture reactions from the human, they do not address the internal learning prob- lem: over repeated interactions, the human may not have learned anything and is only reacting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Alternatively, model-free methods learn a latent representation of the human’s policy and then lever- age the latent dynamics to influence the human [33, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Here the human’s internal model is implicitly captured by the latent rep- resentation, and the internal model evolves between interaction episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In contrast, in our work the human’s internal model is an explicit parameterization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', high-dimensional parameterization like dynamics) and the human internal model can evolve continu- ously during an interaction episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' This enables robot behaviors like teaching the human the correct internal model, which would otherwise not be possible with implicit, latent representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 3 MODELING HOW HUMANS LEARN & ACT We begin by mathematically modelling the dynamics of human learning, before diving into how the robot can infer this dynamics model and use it influence the human’s internal model evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Towards Modeling and Influencing the Dynamics of Human Learning HRI ’23, March 13–16, 2023, Stockholm, SE Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Let 𝑥 ∈ R𝑛 be the state of the world including the robot (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', robot end-effector position, objects, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Both the human and robot can take actions, 𝑢H ∈ R𝑚 and 𝑢R ∈ R𝑚 respectively, that affect the next state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Let the deterministic world dynamics be 𝑥𝑡+1 = 𝑓 (𝑥𝑡,𝑢𝑡 H,𝑢𝑡 R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' (1) Human internal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We model the human as having an in- ternal parameter vector, 𝜃H, which captures a latent aspect of the task that the human is uncertain about but continuously learns about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Going back to our motivating example where the human teleoperates a robot, 𝜃H can model the human’s current estimate of the robot’s physical properties, like its inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Or, 𝜃H could model the human’s current preferences for teleoperation: they start off wanting to move the robot to one goal, but then change their mind to a new goal after realizing it is easier to reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Regardless of what 𝜃H represents, it is important to remember that it is time-varying and that it evolves as a function of what the human observes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Human policy: acting under the internal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In our work, we model the human actions as driven by some reward function, 𝑅H(𝑥,𝑢H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃H), which depends on the current state, the human’s action, and their internal parameter 𝜃H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Following prior works [2, 24, 52, 55], we treat the human as a noisily-optimal actor: P(𝑢H | 𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃H) = 𝑒𝑄H(𝑥,𝑢H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃H) � ∫ ˜𝑢 𝑒𝑄H(𝑥, ˜𝑢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃H)𝑑 ˜𝑢 �−1 , (2) where the optimal state-action value is denoted by 𝑄H(𝑥,𝑢H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃H) and 𝑥 is the current state, 𝑢H is the human action, and 𝜃H the human’s current parameter estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We make two simplifying assumptions in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' First, the human does not explicitly account for the actions 𝑢R the robot could take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Instead, the human reacts to the current state 𝑥, which implicitly captures the effect of any robot actions that change the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' This models scenarios where the human is doing the task on their own, or where the human is not aware of how the robot is providing guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Second, when the human plans their action, we assume that they separate the estimation of 𝜃H from policy generation and they plan with their current estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Dynamics of human learning: updating the internal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' As the human acts in the environment, they receive new observa- tions: they may see the next state, including that of the robot’s, or experience how much they enjoy something (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' observe “reward signal”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' This naturally lets the human update their understanding of the robot, physical aspects of the world, or their preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Leveraging our core idea, we model the human’s learning process as a nonlinear dynamical system over the human’s internal model parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Let 𝜃0 H be the human’s initial internal model, and 𝑥0:𝑡 and 𝑢0:𝑡 H be the state and action history until timestep 𝑡 and 𝑥𝑡+1 be the resulting state at the next timestep, possibly including the influence of robot actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Given the initial parameter estimate, the state and action history, and next state data, the human evolves their internal model to the next estimate, 𝜃𝑡+1 H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Let the true dynamics of the human’s learning process be: 𝜃𝑡+1 H = 𝑓𝐿(𝜃0 H,𝑥0:𝑡+1,𝑢0:𝑡 H ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' (3) Here we are faced with the question “What 𝑓𝐿 models how the human learns?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Instead of committing to a specific model, here we take a robotics perspective and view this question as an instance of a dynamics learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' By looking to human data, we aim to learn an approximate 𝑓𝐿 model that is domain-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 4 INFERRING THE DYNAMICS OF HUMAN LEARNING In this section we focus on inferring the dynamics of human learn- ing by leveraging demonstrations which naturally exhibit human learning: for example, initial trials of a human teleoperating a robot they have never interacted with before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We assume these demon- strations contain only the state and action histories and do not contain ground-truth human internal model data (since this is not possible in practice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' However, we do assume that the observed actions are coupled with the human’s internal model, allowing us to leverage demonstrations to infer the dynamics of the human’s internal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Given this dataset, we seek to fit a nonlinear model to represent the dynamics of human learning, 𝑓 𝜙 𝐿 ≈ 𝑓𝐿, (4) where 𝜙 are the parameters of the approximate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In the follow- ing sections, we formalize inferring 𝑓 𝜙 𝐿 as a maximum likelihood estimation (MLE) problem and propose a tractable approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 Formalizing the Inference Problem Let D𝑑𝑒𝑚𝑜 := {(x, uH)𝑖}𝑁 𝑖=0 be a collection of 𝑁 demonstrations containing state and human action trajectories of length 𝑇 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We want to infer the parameter of the human’s learning dynamics, 𝜙, and the initial human parameter estimate, 𝜃0 H, which maximizes the likelihood of the observed demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We for- mulate this inference via the constrained optimization problem: max 𝜙,𝜃0 H ∑︁ (x,uH) ∈D𝑑𝑒𝑚𝑜 𝑇−1 ∑︁ 𝑡=0 log � P(𝑢𝑡 H | 𝑥𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃𝑡 H) � , (5) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 𝜃𝑡+1 H = 𝑓 𝜙 𝐿 (𝜃0 H,𝑥0:𝑡+1,𝑢0:𝑡 H ), (6) where P(𝑢𝑡 H | 𝑥𝑡,𝜃𝑡) is the human action likelihood from Equa- tion (2) and the constraint ensures that the human’s internal param- eter evolves according to the human’s learning dynamics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 Solving the Inference Problem Unfortunately, the inference problem in Equation (5) is intractable to solve directly for two main reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' First, recall that the human’s internal model 𝜃H of their preferences, dynamics, or goals, changes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' This means that at each timestep the human is gener- ating data 𝑢H under a possibly different 𝜃H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In other words, the human acts under a new action policy P(𝑢𝑡 H | 𝑥𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃𝑡 H) at each 𝑡, re- quiring us to solve an entirely new reinforcement learning problem to obtain the action policy at each time step along the inference horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In the case where 𝜃H is a continuous, high-dimensional parameter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', physical properties of the robot dynamics), this is intractable to compute per-timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Secondly, even if we could obtain the human’s policy infinitely fast, our optimization problem still requires searching over the the high-dimensional space of 𝜙 and 𝜃H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Gradient-based optimization is a natural choice, but we need to be able to compute the gradient of the MLE objective and, therefore, differentiate through 𝑄H with respect to 𝜃H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' HRI ’23, March 13–16, 2023, Stockholm, SE Ran Tian, Masayoshi Tomizuka, Anca D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Dragan, and Andrea Bajcsy In the following subsections, we introduce several approxima- tions to arrive at a tractable solution to the inference problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Our key idea is to use a linear-quadratic (LQ) approximation of the physical dynamics and the human reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' This enables us to derive a closed-form expression of the human policy as a function of 𝜃𝑡 H at any time and yields a differentiable inference objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 Linear-Quadratic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We take inspiration from infinite-horizon linear-quadratic (LQ) control [20] and assume that the human’s reward is quadratic and their model of the physical dynamics is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Let the linear physical dynamics be: 𝑥𝑡+1 = 𝑓 (𝑥𝑡,𝑢𝑡 H,𝑢𝑡 R ≡ 0) ≈ 𝐴𝑥𝑡 + 𝐵𝑢𝑡 H (7) where 𝐴 ∈ R𝑛×𝑛, 𝐵 ∈ R𝑛×𝑚 are matrices governing the physical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Note that in the human’s mind, the robot is not exerting any control effort, and hence 𝑢R ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Let the human’s reward be approximated by a quadratic function: 𝑟H(𝑥,𝑢H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃H) ≈ −𝑥⊤𝑄𝑥 − 𝑢⊤ H𝑅𝑢H, (8) where the matricies 𝑄 ∈ R𝑛×𝑛 and 𝑅 ∈ R𝑚×𝑚 tradeoff the state reward (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', how much reward the human gets for reaching a state) and the action reward (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', how much effort the human wants to exert), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Note that 𝜃H enters in different ways depending on what the human is learning about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' For example, if 𝜃H encodes reward weights (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', the human’s preferences about how to do a task), then 𝜃H := (𝑄, 𝑅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' If the parameter encodes a human’s goal state, then 𝜃H ∈ Θ ⊂ R𝑛 and the human’s reward function regulates the human towards their desired goal: 𝑟H(𝑥,𝑢H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃H) ≈ −(𝑥 − 𝜃H)⊤𝑄(𝑥 − 𝜃H) − 𝑢⊤ H𝑅𝑢H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Finally, if 𝜃H encodes aspects of the physical dynamics that the human is estimating, then 𝜃H := (𝐴, 𝐵) from the dynamics in Equation (7), and governs how the human imagines the physical dynamics evolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 Closed-form 𝑄H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Recall that the human plans a policy us- ing their current estimate 𝜃H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' at every step, 𝜃H changes, resulting in a new policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In general, obtaining the exact 𝑄H-value via dy- namic programming in continuous state, action, and 𝜃H-spaces is computationally demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' However, under our infinite-horizon LQ-approximation the human’s 𝑄H-value is: 𝑄H(𝑥,𝑢H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃H) = 𝑟H(𝑥,𝑢H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃H) − (𝑥 ′)⊤𝑃𝜃H (𝑥 ′) (9) where the instantaneous reward is quadratic from Equation (8) and 𝑥 ′ is the next physical state as a result of applying 𝑢H from state 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Note that −(𝑥 ′)⊤𝑃𝜃H (𝑥 ′) is the infinite-horizon optimal value where 𝑃𝜃H is the well-known positive-definite fixed point of the discrete-time algebraic Riccati equation (DARE) [3]: 𝑃 = 𝐴⊤𝑃𝐴 − 𝐴⊤𝑃𝐵(𝑅 + 𝐵⊤𝑃𝐵)−1𝐵⊤𝑃𝐴 + 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' (10) Obtaining 𝑃𝜃H also yields the optimal human action: 𝑢∗ H(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃H) = −𝐾𝜃H𝑥 where 𝐾𝜃H = (𝑅 + 𝐵⊤𝑃𝜃H𝐵)−1𝐵⊤𝑃𝜃H𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Note that in all of the equations above, 𝜃H enters differently depending on what the human’s internal model represents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='3 Closed-form human policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In general, obtaining the human policy in Equation (2) is computationally intractable in continuous action spaces due to the integral over 𝑢H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' However, plugging in our closed-form 𝑄H, we see that the exponent is quadratic in 𝑢, allowing us to take a Gaussian integral [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Overall, this yields a closed-form human policy (see full derivation in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' ): P(𝑢H | 𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃H) = |H|1/2(2𝜋)−𝑚H/2𝑒𝑄H(𝑥,𝑢H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃H)−𝑄H(𝑥,𝑢∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' (11) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='4 Representing the dynamics of human learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Finally, we are faced with the question of how to functionally represent the dynamics of human learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' for example, we could take inspiration from computational cognitive science and model 𝑓 𝜙 𝐿 as Bayesian inference [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Instead of committing to a specific functional form, in this work we seek a model that has the potential to capture a broad range of “learning algorithms” that the human could use to update their internal parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Recently, self-attention based transformer models [51] have shown success at predicting high- dimensional sequential tasks [18], at the cost of being domain- specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Inspired by this, we represent 𝑓 𝜙 𝐿 as a transformer encoder where 𝜙 are the weights of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' At each time step 𝑡, a collection of the state 𝑥𝑡, the human’s action 𝑢𝑡 H, and the next state 𝑥𝑡+1 are fed into an encoder to extract embeddings which are fed into a transformer encoder that predicts the human’s next internal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Training details are in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='5 Deriving an efficient, gradient-based solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' To optimize the transformer-based model of human learning dynamics, we need the gradient of our inference objective with respect to the neural network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Here a key challenge lies in the human’s pol- icy gradient because it requires differentiating through the DARE function, which is non-obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' However, we leverage recent work [10] to obtain the relevant closed-form Jacobians, enabling us to ef- ficiently infer the parameters of 𝑓 𝜙 𝐿 via gradient-based optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' More details on this approach are in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 5 INFLUENCING HUMAN LEARNING WITH ROBOT ACTIONS Inferring how humans learn presents an opportunity for human- robot interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' For example, when a human teleoperator is mis- taken about the robot’s inertia, it may take them many interactions to learn and become better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Instead, could the robot influence the human so that their understanding improves faster?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Here, we math- ematically formalize this influence by embedding the approximate dynamics model of human learning into robot planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Formalizing the Influence Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We formalize the robot in- fluence problem as a Markov Decision Process (MDP) where the human’s internal model parameter is part of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Our MDP is a tuple < 𝑆,𝑈R,𝑇,𝑟R > where the state 𝑠 = (𝑥,𝜃H) ∈ 𝑆 is the joint physical state and human internal model parameter and the robot’s actions are 𝑢R ∈ 𝑈R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The stochastic state transition function is defined as 𝑇 (𝑠𝑡+1 | 𝑠𝑡,𝑢𝑡 R) := � 𝑢H P(𝑢H | 𝑠𝑡) ˜𝑓 (𝑠𝑡,𝑢𝑡 R,𝑢𝑡 H,𝑠𝑡+1) which accounts for the human policy from Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Impor- tantly, ˜𝑓 (𝑠𝑡,𝑢𝑡 R,𝑢𝑡 H,𝑠𝑡+1) is a deterministic function that evolves 𝑥𝑡 via the physical dynamics 𝑓 from Equation (1) and the human’s internal model parameter 𝜃𝑡 H via the human learning dynamics 𝑓 𝜙 𝐿 from Equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Finally, the robot optimizes its reward func- tion 𝑟R(𝑠,𝑢R,𝑢H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃∗) where 𝜃∗ is the robot’s true internal model parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', the robot’s true physical dynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Note that Towards Modeling and Influencing the Dynamics of Human Learning HRI ’23, March 13–16, 2023, Stockholm, SE because 𝑠 = (𝑥,𝜃H), the robot’s reward depends on the human’s time-varying internal model, 𝜃H, at each timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The robot seeks an optimal policy 𝜋∗ R which maximizes it’s re- ward in expectation over the human’s action sequence, uH: 𝜋∗ R = arg max 𝜋R EuH � ∞ ∑︁ 𝑡=0 𝑟R(𝑠𝑡,𝑢𝑡 R,𝑢𝑡 H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃∗) � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 𝑇 (𝑠𝑡+1 | 𝑠𝑡,𝑢𝑡 R), (12) Because human’s internal model parameter 𝜃𝑡 H is part of the state and the state transition function 𝑇 (𝑠𝑡+1 | 𝑠𝑡,𝑢𝑡 R) includes the in- ferred dynamics model of human learning, 𝜋∗ 𝑅 should automatically influence the human’s internal model if it yields higher reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Computing Solutions to the Influence Problem The presence of the human’s nonlinear learning dynamics 𝑓 𝜙 𝐿 in the transition function results in a nonconvex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' To obtain the optimal robot policy, we would have to solve the MDP either exactly with dynamic programming (which suffers from the curse of dimensionality) [3] or approximately via receding-horizon con- trol (which requires trading off optimality with computational effi- ciency) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' To achieve both long-horizon reasoning and efficient runtime performance, we use a Dyna-style algorithm [47] that uses the samples generated by the transition 𝑇 (𝑠𝑡+1 | 𝑠𝑡,𝑢𝑡 R) to train 𝜋∗ 𝑅 using model-free learning (Proximal Policy Optimization [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6 SIMULATED HUMAN EXPERIMENTS We want to test two aspects of our approach: our ability to infer the dynamics of human learning and the effectiveness of our robot influencing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' To fully validate both, we need access to the ground-truth human learning dynamics (𝑓𝐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' For this reason, we first perform a series of simulation experiments with simulated hu- mans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We explore two shared autonomy contexts: a robot teaching a human about physics-based robot dynamics (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1) and a robot that implicitly influences human objectives, like their goal or motion preferences (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Similar to prior work in shared autonomy [9, 17, 26, 29], the robot combines the human’s commanded action, 𝑢H, with the robot’s planned guidance, 𝑢R, and executes the action: 𝑢 = 𝛼 · 𝑢R + (1 − 𝛼) · 𝑢H (13) where 𝛼 ∈ [0, 1] trades off how much guidance the robot can exert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In all experiments, we use 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' To generate human demon- strations and infer the human learning dynamics, we simulate a suite of human learners (see 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In each experimental environment we collect 50 demonstrations for model learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We randomize the initial state of the robot for each demonstration, and randomize the robot actions during each interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 gradient human threshold human Figure 2: Our inference problem lets us learn to predict 𝜃𝑡 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 1We randomize 𝑢R to diversely cover how human’s internal model changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 Teaching Physical Dynamics We focus on shared autonomy settings where the human knows the task objective (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', control a robot arm to follow a path), but they learn about the true robot dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', inertia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We want to understand how the human learns about the physical robot dynam- ics, and if a robot that actively teaches the human about its physics can help the human quickly improve their task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 Dynamics of human learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Motivated by computational cognitive science models [13], we simulated two types of human learners: gradient-based learners and threshold learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' All hu- mans update their internal model via Equation (3), but the structure of 𝑓𝐿 takes various forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' After observing a new state-action pair (𝑥𝑡,𝑢𝑡), the gradient-based learner updates their parameter𝜃𝑡 H ac- cording to a gradient-ascent update rule: 𝑓 grad 𝐿 := 𝜃𝑡 H + 𝜂∇𝜃H𝑃(𝑢𝑡 | 𝑥𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃𝑡 H) where𝜂 ∈ R+ is the step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Note that𝑢𝑡 is the observed, to- tal executed control, possibly combining 𝑢R and 𝑢H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Intuitively, this learner can be viewed as doing gradient-based maximum likelihood estimation of their latent parameter, similarly to prior IRL methods [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The threshold learner also uses a gradient-based learning rule, but only updates their internal parameters if they observe a “large enough” change: 𝑓 thresh 𝐿 := 𝜃𝑡 H + 𝜂1|∇𝑃 (𝑢𝑡 |𝑥𝑡,𝜃𝑡 H) |>𝜖 � ∇𝜃H𝑃(𝑢𝑡 | 𝑥𝑡,𝜃𝑡 H) � where 1 is an indicator determining if the magnitude of the gradient is deemed large enough to induce a learning update and 𝜖 is a threshold parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 Human internal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In all experiments, the simulated humans are learning about the robot’s physical dynamics and thus 𝜃H encodes various aspects from Equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='3 Simulated environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Figure 3 shows our simulated envi- ronments, all or which have continuous state and action spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' (1) Lunar Lander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The human controls the Lunar Lander’s engines to change its tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The human wants to keep the lander upright dur- ing its descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Let the state be the tilt angle with respect to the ground and tilt angular velocity 𝑥 = (𝜓,𝜔) and 𝑢 be the engine force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The dynamics are 𝑥𝑡+1 = 𝐴𝑥𝑡 + 𝐵𝑢𝑡 where the ground-truth dynamics are 𝐴∗ = [1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 0, 1], 𝐵∗ = [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Here, the human’s in- ternal model represents the control matrix 𝜃H := 𝐵, which depends on the human’s inertia estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' (2) Robot Arm Teleoperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The human controls the end-effector of a 7DOF robot arm via hand gestures (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' They want to control the robot to reach a series of known goals, 𝑥𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' However, one of the robot motors is slightly defective, causing the robot to con- sistently lag in one direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Let the state be the robot end-effector position 𝑥 = (𝑝𝑥, 𝑝𝑦, 𝑝𝑧) and the control𝑢 be linear velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The ro- bot’s end-effector dynamics can be described by the goal-dependent system2 : 𝑥𝑡+1 = 𝐴𝑥𝑡 +𝐵 � 𝑢𝑡 −sign(𝑥𝑡 −𝑥𝑔) ⊙𝑤 � where 𝑤 is the bias induced by the defective robot motor and ⊙ is the Hadamard prod- uct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Intuitively, this describes a dynamical system that consistently experiences lag in the 𝑥-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The ground-truth dynamics are 𝐴∗ = 𝐼3×3, 𝐵∗ = diag(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='4), and 𝑤∗ = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='15, 0, 0]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The hu- man’s internal model is 𝜃H := (𝐵,𝑤), which captures their system responsiveness and bias estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 2Although this system is nonlinear, since the robot knows the human’s goal at each time step, the dynamics can be approximated by a linear system 𝑥𝑡+1 = 𝐴𝑥𝑡 + 𝐵 ˜𝑢𝑡 , where 𝑥0 is the system state at that time step and ˜𝑢𝑡 := 𝑢𝑡 − sign(𝑥0 − 𝑥𝑔) ⊙ 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Robot Arm Teleoperation Lunar Lander 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='4 tH 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='6 0 0 10 20 30 40 50 0 10 20 30 40 50 Epoch EpochHRI ’23, March 13–16, 2023, Stockholm, SE Ran Tian, Masayoshi Tomizuka, Anca D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Dragan, and Andrea Bajcsy Robot Arm Teleoperation Lunar Lander Gradient Human Threshold Human passive learn random oracle active teach (ours) Robot Arm Teleoperation robot automatically stops teaching!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' higher 𝜃H error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' robot keeps teaching Gradient Human Threshold Human Lunar Lander robot automatically stops teaching!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' robot eventually stops teaching!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Figure 3: (left) Visualization of both simulation environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' (right) Mean and standard deviation of human internal model error, robot effort, and human action optimality for both dynamics teaching environments, and both simulated humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='4 Human objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We assume the human always knows the objective, and their reward function is quadratic as in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' For Lunar Lander the human was rewarded for keeping the lander upright and stable (𝜓 = 0, 𝜔 = 0), and for Robot Arm they were rewarded for reaching all the goals and tracking the path shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='5 Robot objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The robot objective is to align human’s in- ternal model with the true robot dynamics model while minimally intervening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Mathematically, the robot’s reward function is: 𝑟R(𝑠,𝑢R,𝑢H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃∗) = −||𝜃H − 𝜃∗||2 2 − ||𝑢 − 𝑢H||2 2, (14) where the true dynamics are 𝜃∗ := 𝐵∗ in the Lunar Lander envi- ronment and 𝜃∗ := (𝐵∗,𝑤∗) in the Robot Arm setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='6 Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We compare our method where the robot actively teaches by planning with the inferred learning dynamics 𝑓 𝜙 𝐿 (Active Teach) to a robot that teaches with the true learning dynamics 𝑓𝐿 (Oracle), no robot intervention (Passive Learn), and a robot that randomly perturbs the human actions (Random).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='7 Hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' H1: We can learn to predict 𝜃𝑡 H well by maxi- mizing the MLE objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' H2: Active Teach outperforms Passive Learn and Random in aligning the human’s internal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' H3: Robot stops intervening when the human’s internal model is well- aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='8 Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' For H1, we study the relationship between the MLE objective in (5) and our inferred model’s (𝑓 𝜙 𝐿 ) ability to predict 𝜃H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Figure 2 shows these curves for both the Robot Arm and Lunar Lander environments over 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We see that across both gradient and threshold human learners, the log likelihood of the human’s actions increases (shown in pink) while the 𝜃H prediction error decreases (shown in blue), supporting H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Figure 3 shows the human’s internal model error, the robot’s ef- fort, and the difference between the human’s action and the optimal action in the Robot Arm Teleoperation and the Lunar Lander environment for both types of human learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We see that across all environments, our method performs comparably to Oracle model, and is able to align the human’s internal model of the robot’s dy- namics with the true dynamics significantly faster than Passive Learn or Random (supporting H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Interestingly, in all but one setting does the robot automatically stop teaching the human since the human’s internal model is sufficiently correct (supporting H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The one exception is in the Robot Arm Teleoperation environ- ment with the threshold human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Since this human doesn’t learn when the gradient is too small, the robot must continue to exert effort to maximize its reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 Implicitly Influencing Human Objectives We now turn to scenarios where the human has an accurate un- derstanding of the robot’s dynamics, but their objective (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', their reward function 𝑟H) can be changed by the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Specifically, we study how assistive robots can implicitly influence human motion preferences and desired goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Importantly, in this setting influenc- ing or teaching the human is not explicitly in the robot’s objective: the robot simply wants to perform the desired task with minimal assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Thus, getting the human to want to reach a goal or change their preferences should be an emergent behavior of robots planning with the dynamics of human learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 Dynamics of human learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We simulate3 the gradient hu- man learner from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 and introduce a new human, the Bayesian learner4 which is inspired by probabilistic models of cognition [13, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' This human’s learning produces a full posterior, 𝑏𝑡+1(𝜃H), over the model parameters given a state-action observation, and the dynamics of learning are: 𝑓 Bayes 𝐿 ∝ 𝑃(𝑢𝑡 | 𝑥𝑡,𝜃H)𝑏𝑡 (𝜃H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 Human internal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Since the human’s objectives are influenceable, we model 𝜃H as a reward parameter encoding the motion preferences 𝜃H := (𝑄, 𝑅) or a desired goal state 𝜃H ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='3 Simulated environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We assume the human knows the physical robot dynamics (the bias-free RobotArm dynamics from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='3), but can have their reward influenced by new observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' (1) Goal Influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The human wants to teleoperate the robot to put an object in one of the three trays (upper left Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' However,the human doesn’t notice that only one of the trays is empty enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Unlike the human, the robot’s sensors detect that only one of the trays is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We investigate if the robot can 3While we simulate the human as changing their reward, but the human’s reward could be viewed as static while their subgoals change Nonetheless, it will be common for a robot to not fully represent this hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 4Bayesian humans act under their belief: P(𝑢H | 𝑥) = � 𝜃H 𝑏(𝜃H)P(𝑢H | 𝑥,𝜃H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Robot Arm Teleoperation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='4 error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 0 1 10 20 30 40 50 timestepRobot Arm Teleoperation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 0 1 10 20 30 40 50 timestepLunar Lander 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 0 1 10 20 timestepLunar Lander 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 0 1 10 20 timestepTowards Modeling and Influencing the Dynamics of Human Learning HRI ’23, March 13–16, 2023, Stockholm, SE influence the human to change their preferences about which tray (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', goal location) to place their object in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' (2) Preference Influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The human wants to teleoperate the robot to pick up a cup on the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Their initial preference is to move the robot’s end-effector in a straight line from start to the cup (lower left Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' However, the robot knows that grasps tend to fail with this kind of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Instead, the robot knows that first moving directly above the can and then straight down to grasp has a higher chance of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We investigate if the robot can influence the human to change their preferences about how to reach the cup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='4 Human objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In all simulations the human has a qua- dratic cost function (from (8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In Goal Influence the simulated human receives reward for moving the robot end-effector to their desired tray, and in Preference Influence the human receives reward according to their current preference matricies, (𝑄, 𝑅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='5 Robot objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We implement an assistive robot that wants to help the human perform the task while minimally intervening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' However, we assume that the robot knows best: the robot knows which goal or reward weights lead to success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Let 𝜃∗ capture this aspect of the robot’s reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In the Goal setting the robot’s reward 𝑟R(𝑠,𝑢R,𝑢H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃∗) = −(𝑥 −𝜃∗)⊤𝑄(𝑥 −𝜃∗) −𝑢⊤𝑅𝑢 − ||𝑢 −𝑢H||2 2 and in Preference the robot’s reward parameter is 𝜃∗ = (𝑄∗, 𝑅∗), yielding 𝑟R(𝑠,𝑢R,𝑢H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃∗) = −𝑥⊤𝑄∗𝑥 − 𝑢⊤𝑅∗𝑢 − ||𝑢 − 𝑢H||2 2 where 𝑢 is the combined human and robot action from Equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='6 Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We implement our method where the robot as- sists the human and plans with the inferred dynamics of human learning (Learning Assist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We compare to a robot assisting with the ground-truth dynamics of human learning (Oracle), robot as- sistance that is unaware that humans learn (Static Assist), and a robot that randomly perturbs the human actions (Random).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='7 Hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' H4: Learning Assist aligns the human’s mental model faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' H5: Assistance that accounts for human learning enables the human-robot team to achieve higher reward under the true 𝜃∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='8 Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Figure 4 shows the human’s internal model error, ro- bot effort, and task cost (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', just the task-component of 𝑟R, negated) for both environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Because the Learning Assist robot knows that the human’s internal model can be changed, it automatically exerts higher effort early on to align the human’s internal model with it’s own, resulting in less long-term assistance and lower task cost (supporting H4 and H5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In contrast, the Static Assist robot is not aware that the human can change their mind, and thus does not exert enough effort to influence the human’s internal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' After repeatedly incurring task cost because the two agents are at odds with each other, the Static Assist robot “gives up” and starts executing the human’s control directly: in other words, 𝑢 = 𝑢H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 7 USER STUDY: TEACHING TO TELEOPERATE So far we conducted experiments with simulated human behavior, allowing us to analyze the quality of our inferred human learning dynamics model, and the robot’s ability to influence simulated humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Here we investigate if we can infer the dynamics of real human learning, and enable robots to influence real users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Goal Influence (Bayesian human) Preference Influence (gradient human) static assist random oracle learning assist (ours) 𝑏0(𝜃H) 𝜃∗ 𝜃H 0 𝜃∗ Figure 4: (left) Environments for influencing human objec- tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' (right) Internal model error, robot effort, and task cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We focus on scenarios where the robot’s physical dynamics are different from what the human is used to;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' for example, perhaps the human was used to teleoperating a robotic wheelchair, but is now teleoperating a robotic arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' As they interact with the robotic arm, they will naturally learn about the new robot dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In our IRB- approved user study, we investigate if a robot can actively teach a human the physical dynamics and improve their teleoperation performance faster than if the human does the task on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In other words, we aim to understand if a robot can align the human’s internal model with the robot’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Experimental Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We designed a teleoperation task where the human controls a 7DOF Jaco robot arm through a webcam-based gesture interface (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The participant uses their index finger to indicate how the end-effector should move parallel to the tabletop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The task is to move the end-effector to reach four goals on the table in a counter-clockwise pattern, tracing out a diamond pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' All participants experience a familiarzation task where they perform the task unassisted, with the default robot dynamics in order to understand the gesture interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In software, we then simulate two “new” robots, each with different physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Independent Variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We manipulated the robot strategy with two levels: no-teaching and active-teaching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The robot either let the human do the task on their own, or it modified the human’s input to teach them about the physical robot dynamics via Equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We also manipulate the robot physical dynamics with two levels: end-effector dynamics bias in x-direction and bias in y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Dependent Measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' A challenge in evaluating our experiment is that we do not have access to the human’s ground-truth internal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' As a proxy, we measure human action optimality distance: || ˆ𝑢𝐻 − 𝑢∗||2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Intuitively, the better the human understands the robot, the more optimally they should be able to control it to reach the goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Since we cannot directly measure a human’s internal understanding, we instead look at their actions to measure their deviation from the optimal action under the robot’s true physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We also measured subjective measures via a Likert scale survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' H6: Participants in the active teaching condition be- come optimal teleoperators faster than passively learning on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' uR effort 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 0uR effort 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='5 0Task cost 10 5 0 1 5 10 15 20 25 30 timestepTask cost 40 20 0 1 5 10 15 20 25 30 timestep1 error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='5 0 5 10 15 20 25 30 timestep2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='5 )error (H)9 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='5 0 5 10 15 20 25 30 timestepHRI ’23, March 13–16, 2023, Stockholm, SE Ran Tian, Masayoshi Tomizuka, Anca D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Dragan, and Andrea Bajcsy passive learn active teach (ours) Human Action Optimality Distance Participant Trajectories User Study: Quantitative & Qualitative Results early on robot teaches by exaggeration later humans more optimal early & late suboptimality w/ passively learning start end start end Figure 5: (left) Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' human action optimality distance and 95% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' (right) Dashed line is desired path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Participant trajectories reveal that an active teaching robot initially exaggerates the dynamics bias to teach the human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' H7: Participants feel they learned to teleoperate faster and understood the robot dynamics better in the active teaching condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We recruited two groups of participants from the campus community: the first for providing data for inferring the dynamics of human learning (12 participants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 2 female, 10 male, age 18-34, all with technical backgrounds), and the second for the user study (10 participants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 1 female, 8 male, 1 non-binary, age 18-34, all with technical backgrounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' For inferring the human learning dynamics, all participants learned to teleoperate the robot unassisted and we counterbalanced the robot physical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' A within-subjects design is challenging, since humans who experience one condition will learn about the robots and then carry over that experience to the next condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' To study the effect of this confound, each participant experienced a combination of robot strategy and physical dynamics conditions, but in a random order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' For example, one group of participants would interact with the (active-teaching, bias-x) condition and then (no-teaching, bias-y) condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Thus, each participant experiences both robot strategies and biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We counterbalance the order in which the participants experience the combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' All participants experienced a familiar- ization round at the start and between each experimental condition, to “reset” their mental model of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Each participant gave 3 demonstrations per condition, each lasting ∼1 minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Quantitative Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Figure 5 shows how human action opti- mality distance varies over time with each robot strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We con- ducted an ANOVA with robot strategy and stage (first or second half of interaction) as factors and robot physical dynamics as ran- dom effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We found a significant main effect of the robot strategy (𝐹 (1, 19) = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='943, 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='001) and a marginal interaction effect between the robot strategy and the interaction stage (𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='098), so we did not run a post-hoc analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' However, we hypothesize that this marginal interaction effect comes from the fact that early- stage changes in robot behavior (induced by either robot strategy) influences the human’s later-stage action optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Ultimately, the quantitative results indicate a significant improvement in the human’s action optimality when the robot actively teaches them compared to when the human passively learns (supporting H6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Qualitative & Subjective Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' On the right of Figure 5 we visualize the executed trajectories from all participants in the active- teaching (orange) and no-teaching (grey) conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The high- lighted trajectories are two representative examples, the color gra- dient indicates time along the trajectory, and the dashed line is the desired path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' When participants passively learn on their own, their trajectories are consistently suboptimal, weaving around the optimal path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In contrast, in the active teaching condition, the initial portion of the trajectory exhibits the robots teaching behavior: the robot intentionally exaggerates the dynamics bias to change the hu- man’s internal model faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' After this initial exaggerated deviation, the human trajectory is closer to optimal compared to the passive learning trajectory at comparable timesteps (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='4 for a detailed visualization of human and robot actions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We also ran an ANOVA on the Likert survey questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Survey questions investigated perceived performance improvements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', “By the end of the interaction, it was easy to control the robot to do the task.”) and robot understanding (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', “By the end of the interaction, I understood the robot’s physical properties.”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Across all questions, we did not find a significant effect of the robot strategy (rejecting H7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' What we found surprising was that even though participants were quantitatively performing better in the teaching condition, they did not perceive an improvement in performance (𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='689) nor in their understanding of the robot physics (𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='299).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We hypothesize that this could be because participants only interacted with each robot strategy for one minute, making the differences hard to notice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In the future, investigating longer-term interactions with the robot would shed light on the disconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 8 CONCLUSION In this work we took a step towards enabling robots to understand the influence that they have over human internal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We do this by modeling human learning as a nonlinear dynamical system that evolves as a function of new observations that the robot can influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We propose a tractable method for inferring approxi- mate human learning dynamics from demonstrations that naturally exhibit human learning, and propose how robots can influence hu- man learning by embedding the approximate dynamics into robot planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Our experimental results indicate that robot influence is possible and can help humans learn better internal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Limitations & Future Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' A strength and limitation of our approach is representing the dynamics of human learning via a transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' As a general function approximator, it poses no as- sumptions on the structure of the human’s learning dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' in fact, we are excited that our results indicate that it is possible to infer a useful model of human learning from real data, without prior assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' However, since neural networks require abundant human data, they are not appropriate for low-data settings and may fail when encountering humans that are out of distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' A further limitation is that if the person is not noisily-optimal as in (2) and has a specific bias (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=', myopia), then the transformer will learn parameters that compensate for this;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' in turn, this could lead the robot to influence the human in unintended ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In the future we are excited to combine the strengths of data-driven models and cognitive science models of human learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' While our user study relies on an “average” dynamics model of human learning trained 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 0 10 20 30 40 50 timestepTowards Modeling and Influencing the Dynamics of Human Learning HRI ’23, March 13–16, 2023, Stockholm, SE from all participants’ data, humans may exhibit unique ways of learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Inferring personalized learning dynamics is an exciting future direction, and pre-trained models of humans could serve as a useful starting point for adapting to new humans.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Motion illusions as optimal percepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Nature neuroscience 5, 6 (2002), 598–604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' [54] Annie Xie, Dylan P Losey, Ryan Tolsma, Chelsea Finn, and Dorsa Sadigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Learning latent representations to influence multi-agent interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Conference on Robot Learning (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' [55] Brian D Ziebart, Andrew L Maas, J Andrew Bagnell, and Anind K Dey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Maximum entropy inverse reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='. In Aaai, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Chicago, IL, USA, 1433–1438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Towards Modeling and Influencing the Dynamics of Human Learning HRI ’23, March 13–16, 2023, Stockholm, SE A APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1 Derivation: Gaussian integral under LQ approximation Here we derive the closed-form solution to the denominator from (2) under the LQ-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' First, we recall the Gaussian integral: Theorem: Gaussian Integral [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Let 𝑀 ∈ R𝑛×𝑛 be a symmetric, positive-definite matrix and 𝑥 ∈ R𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Then: ∫ exp � − 1 2𝑥⊤𝑀𝑥+𝑏⊤𝑥 � 𝑑𝑛𝑥 = √︄ (2𝜋)𝑛 det(𝑀) exp � 1 2𝑏⊤𝑀−1𝑏 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' (15) Theorem: Infinite-horizon Linear-Quadratic Regulator [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Let the discrete-time dynamics be linear, 𝑥𝑡+1 = 𝐴𝑥𝑡 + 𝐵𝑢𝑡, and the cost quadratic, 𝑥⊤𝑄𝑥 +𝑢⊤𝑅𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Then the infinite-horizon optimal cost-to-go 𝐽 and optimal control 𝑢∗(𝑥) are: 𝐽 (𝑥) = 𝑥⊤𝑃𝑥 (16) 𝑢∗(𝑥) = −𝐾𝑥 (17) where 𝑃 is the unique, positive-definite fixed point of the infinite- horizon, discrete-time Ricatti equation (DARE): 𝑃 = 𝐴⊤𝑃𝐴 − 𝐴⊤𝑃𝐵(𝑅 + 𝐵⊤𝑃𝐵)−1𝐵⊤𝑃𝐴 + 𝑄 (18) and the feedback matrix 𝐾 = (𝑅 + 𝐵⊤𝑃𝐵)−1𝐵⊤𝑃𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Let that the physical dynamics be linear and the re- ward is quadratic in state and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Assume that we have approx- imated the state-action 𝑄H function as: 𝑄H(𝑥,𝑢) = −𝑥⊤𝑄𝑥 − 𝑢⊤𝑅𝑢 − (𝑥 ′)⊤𝑃(𝑥 ′) (19) where 𝑥 ′ = 𝐴𝑥 + 𝐵𝑢 is the next state and 𝑃 is the solution to (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Plugging in (19) into the denominator of the human policy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' we obtain: ∫ exp � 𝑄H(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝑢) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝑑𝑢 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='(20) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='∫ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='− 𝑥⊤𝑄𝑥 − 𝑢⊤𝑅𝑢 − (𝐴𝑥 + 𝐵𝑢)⊤𝑃(𝐴𝑥 + 𝐵𝑢) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝑑𝑢 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='(21) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='= exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='− 𝑥⊤𝑄𝑥 − 𝑥⊤𝐴⊤𝑃𝐴𝑥 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='� (22) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='∫ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2𝑢⊤(2𝑅 + 2𝐵⊤𝑃𝐵)𝑢 + (−2𝑥⊤𝐴⊤𝑃𝐵)𝑢 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝑑𝑢 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='(23) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='We see that if we let 𝑀 := 2𝑅 + 2𝐵⊤𝑃𝐵 and 𝑏 := −2𝑥⊤𝐴⊤𝑃𝐵 then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='we can directly take the Gaussian integral and obtain: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='= exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='− 𝑥⊤𝑄𝑥 − 𝑥⊤𝐴⊤𝑃𝐴𝑥 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='�√︄ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='(2𝜋)𝑚 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='det(2𝑅 + 2𝐵⊤𝑃𝐵) · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='(24) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='(−2𝑥⊤𝐴⊤𝑃𝐵)⊤[2𝑅 + 2𝐵⊤𝑃𝐵]−1(−2𝑥⊤𝐴⊤𝑃𝐵) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='(25) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='= exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝑥⊤� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝐴⊤𝑃𝐵(𝑅 + 𝐵⊤𝑃𝐵)−1𝐵⊤𝑃𝐴 − 𝑄 − 𝐴⊤𝑃𝐴 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝑥 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='� (26) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='√︄ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='(2𝜋)𝑚 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='det(2𝑅 + 2𝐵⊤𝑃𝐵) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' (27) Interestingly, we see that the exponent contains the (negated) DARE equation from (18) within the brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Substituting 𝑃 back in, we obtain: ∫ exp � 𝑄H(𝑥,𝑢) � 𝑑𝑢 = exp � − 𝑥⊤𝑃𝑥 �√︄ (2𝜋)𝑚 det(2𝑅 + 2𝐵⊤𝑃𝐵) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' (28) We can further simplify this equation by 𝑥⊤𝑃𝑥 = min 𝑢 � 𝑥⊤𝑄𝑥 + 𝑢⊤𝑅𝑢 + (𝐴𝑥 + 𝐵𝑢)⊤𝑃(𝐴𝑥 + 𝐵𝑢) � (29) = 𝑥⊤𝑄𝑥 + (𝑢∗)⊤𝑅(𝑢∗) + (𝐴𝑥 + 𝐵𝑢∗)⊤𝑃(𝐴𝑥 + 𝐵𝑢∗) (30) = 𝑄H(𝑥,𝑢∗) (31) where 𝑢∗ is the optimal control at state 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Thus, we can obtain our final simplified form: ∫ exp � 𝑄H(𝑥,𝑢) � 𝑑𝑢 = exp � − 𝑄H(𝑥,𝑢∗) �√︄ (2𝜋)𝑚 det(2𝑅 + 2𝐵⊤𝑃𝐵) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 𝜃H 𝑡+1 𝜃H 𝑡+2 𝜃H 𝑡+3 … … 𝑥𝑡, 𝑢H 𝑡 , 𝑥𝑡+1 𝑥𝑡+1, 𝑢H 𝑡+1, 𝑥𝑡+2 𝑥𝑡+2, 𝑢H 𝑡+2, 𝑥𝑡+3 𝑓𝐿 𝜙 Transformer encoder encoder encoder Figure 6: Architecture of transformer representing 𝑓 𝜙 𝐿 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2 Details on a gradient-based solution to inferring the dynamics of human learning To optimize the transformer-based model of human learning dy- namics, we need to compute the gradient of our inference objective (in Equation (5) and referred to here as L) with respect to the neural network parameters, 𝜙: 𝜕L 𝜕𝜙 = 𝜕L 𝜕𝜃H 𝜕𝜃H 𝜕𝜙 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The second component, 𝜕𝜃H 𝜕𝜙 , is the gradient of the transformer’s in- ternal model predictions with respect to the neural network weights and is readily available since the transformer is differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' How- ever, the first component 𝜕L 𝜕𝜃H = � 𝜕 log P(𝑢𝑡 H | 𝑥𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='𝜃𝑡 H) 𝜕𝜃H𝑡 � , which is the human’s policy gradient with respect to the human’s internal model parameter, is a key challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' This is because the human’s policy P(𝑢H | 𝑥,𝜃H) depends on 𝑃𝜃H through the 𝑄H- value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Recall that 𝑃𝜃H is the solution to DARE in Equation (10) which depends on the matrices 𝐴, 𝐵,𝑄, 𝑅 and 𝑃𝜃H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Regardless of if the human’s internal model parameter is the physical dynamics 𝜃H := (𝐴, 𝐵) or the reward weights 𝜃H := (𝑄, 𝑅), the human’s policy gradient requires differentiating through the DARE function, which is non-obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Leveraging recent work [10] that treats the DARE as an implicit function of (𝐴, 𝐵,𝑄, 𝑅), we obtain closed-form Jacobians 𝜕𝑃 𝜕𝐴, 𝜕𝑃 𝜕𝐵 , 𝜕𝑃 𝜕𝑄 , and 𝜕𝑃 𝜕𝑅 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The precise form of these can be HRI ’23, March 13–16, 2023, Stockholm, SE Ran Tian, Masayoshi Tomizuka, Anca D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Dragan, and Andrea Bajcsy Human input action (𝑢H) Robot executed action (𝑢) 𝑡 = 0 𝑡 = 40 𝑠 later, the human and robot actions are more aligned early on the robot executes exaggerations of the human’s input Figure 7: An example participant trajectory from Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Blue vectors show the executed robot actions (in solid line) and human input action (in dashed line) at timesteps sam- pled at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' found in Proposition 2 of [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Thus, we can efficiently compute 𝜕L 𝜕𝜙 and infer the human’s learning dynamics via gradient-based optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='3 Training the dynamics model of human learning To enhance the reproducibility of inferring 𝑓 𝜙 𝐿 , we present the ar- chitecture and optimization details here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' The encoder for encoding (𝑥𝑡,𝑢𝑡 H,𝑥𝑡+1) is a multilayer perceptron with 3 fully-connected lay- ers in all settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' We use the Hugging Face’s implementation [11] of the transformer encoder [51] to represent the human’s learning dynamics, and use the Adam optimizer to train the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In both the simulated experiments and in the user study, we use the same transformer architecture to represent the dynamics of human learning, with only the output layer size adjusted per each task to appropriately model the human’s internal model 𝜃H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' From Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='1, in Lunar Lander the output size is 2-dimensional, rep- resenting the 𝐵-vector that the human is estimating and in Robot Arm Teleoperation the output is 4-dimensional to account for the diagonal elements of 𝐵 and 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' From Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='2, in Goal Influence the output size is 2-dimensional, representing the probability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' human belief) over the first and the second tray goals (the prob- ability over the third goal is implicitly defined as one minus the probability of the other two goals combined), while in Preference Influence the output size is 3-dimensional to represent the diago- nal terms of the 𝑄 ∈ R3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Finally, in the user study from Section 7, the output is 4-dimensional to account for the diagonal elements of 𝐵 and 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Note that the human’s initial internal model (𝜃0 H) is implicitly estimated at the beginning of the input when predicting 𝜃1 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='4 User Study: Human and Robot Action Alignment We looked at the user study data and investigated how the human input actions compared to the executed robot actions under our teaching method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Recall that the robot executes actions according to (13): 𝑢 = 𝛼 ·𝑢R + (1−𝛼) ·𝑢H where 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='5 for the duration of our user study and 𝑢R is generated according to our influence-aware planning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' 7 we plot a sample participant trajectory and the robot executed actions (solid blue vector) and human input actions (dashed blue vector) at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content='5 s time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Qualitatively, we see that early on the human and robot’s actions are misaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Intuitively, since the robot’s planning objective is to quickly align the human’s mental model of the physics with the robot’s physics model, the robot plans to execute an exaggeration of the human’s input in hopes of quickly changing their mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} +page_content=' Later on, we see that the human and robot actions become more aligned as the human learns to be a better teleoperator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQf-_qA/content/2301.00901v1.pdf'} diff --git 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Matamoros, +Morelia Michoac´an, C.P. 58240, M´exico. gabino.estevez@umich.mx, +Joaquin Estevez-Delgado +Facultad de Ciencias F´ısico Matem´aticas de la UMSNH +Edificio B, Ciudad Universitaria, +Morelia Michoac´an, C.P. 58040, M´exico. joaquin@fismat.umich.mx, +Modesto Pineda Duran +Instituto Tecnol´ogico Superior de Tac´ambaro, +Av. Tecnol´ogico No 201 Zona el Gigante, C. P. 61650, +Tacambaro Michoac´an, M´exico. mpinedad@hotmail.com +and +Arthur Cleary-Balderas +Facultad de Ingenier´ıa El´ectrica de la UMSNH +Edificio Ω, Ciudad Universitaria, C.P. 58030, +Morelia Michoac´an, M´exico. arthur.cleary@umich.mx +January 3, 2023 +Abstract +Starting from the solution of the Einstein field equations in a static +and spherically symmetric spacetime which contains an isotropic fluid, +we construct a model to represent the interior of compact objects with +compactness rate u = GM +c2R < 0.23577. The solution is obtained by +imposing the isotropy condition for the radial and tangential pressures, +this generates an ordinary differential equation of second order for the +temporal gtt and radial grr metric potentials, which can be solved for +a specific function of gtt. The graphic analysis of the solution shows +1 +arXiv:2301.00210v1 [gr-qc] 31 Dec 2022 + +that it is physically acceptable, that is to say, the density, pressure +and speed of sound are positive, regular and monotonically decreasing +functions, also, the solution is stable due to meeting the criteria of +the adiabatic index. When taking the data of mass M = 1.44+0.15 +−0.14M⊙ +and radius R = 13.02+1.24 +−1.06km which corresponds to the estimations +of the star PSR J0030+045 we obtain values of central density ρc = +7.5125×1017kg/m3 for the maximum compactness u = 0.19628 and of +ρc = 2.8411 × 1017kg/m3 for the minimum compactness u = 0.13460, +which are consistent with those expected for this type of stars. +1 +Introduction +Describing the interior of the stars and determining their average composi- +tion requires many different complementary approaches as are: chemistry, +thermodynamics, nuclear physics, particle physics and gravitational physics. +And in the case that there are no instabilities generated, once all the nuclear +fuel has been used, the star shrinks and, depending of the mass and the sta- +bility present, it may form a white dwarf, a neutron star or a quark star.1,2 +Of course the description of each one of these stages and their stability is a +far more delicate matter which involves a detailed analysis that considers the +type of predominant particles in the interior of the star, whether they are +electrons, neutrons, or quarks, even when the stars are hybrids. According +to the focus of this job, the matter in general is supposed to be described in +a satisfactory manner by a perfect fluid and it will not be necessary to give a +specific shape of a state equation. We know that, depending on the value of +the mass and radius of a star, it may be a white dwarf, a neutron star or a +quark star, this will also determine the orders of magnitude of the density, for +example densities in the order of 1018kg/m2 are typical for neutron stars. As +such in this situation, given the high density, it results adequate to describe +the interior of the stars by means of Einstein’s general relativity theory. The +interior solutions have been approached for over a century, the first of these +was constructed for a static and spherically symmetric spacetime with mat- +ter from a perfect fluid and incompressible density, known as the interior +Schwarzschild solution. Although to start with, the density being constant is +an unrealistic requirement, its consequences revealed some differences with +the treatment of stellar models in the context of Newton’s theory of gravi- +tation. One of these is the compactness relation u = GM/c2R < 4/9, where +M is the mass and R is the radius, this indicates that it is not possible to +have stars with arbitrary mass and radius. Afterwards it was shown that +this relation is not exclusive of this idealized model and that it is present for +2 + +stars with a monotonically decreasing density function for which the exterior +geometry is given by the Schwarzschild solution.3,4 And although more that +130 interior solutions with perfect fluid have been published, only a few met +the characteristics that makes them physically acceptable. From an analy- +sis done in 1998 on a total of 127 reported solutions, only 16 of these had +their density and pressure functions be positive, regular and monotonically +decreasing functions and also had a speed of sound that didn’t violate the +causality condition. And only 9 out of these 16 had a speed of sound which +decreases monotonically with the radius.5–16 Although this last requirement +is debatable since, for example, for the realistic MIT Bag state equation +P(ρ) = 1 +3(c2ρ−4Bg) associated to quark stars, the speed of sound is v2 +s = c2 +3 , +which is not a monotonically decreasing function. The construction of stel- +lar solutions with perfect fluid is an active field although the difficulty in +obtaining physically acceptable solutions limits a great number of these. A +point which has been explored in relation to these is employing isotropic +coordinates17–21 which has favored the integration of the equations system +and its application in the description of stars like HerX1, 4U1538-52, LMC +X - 4, SAX J1808.4-3658.22 A particular class of solutions constructed in +Schwarzschild’s coordinates assumes a metric potential gtt = −(1 + ar2)n to +this group belong the Tolman IV6 and Durgapal15 solutions, extensions for +other values of n a positive integer or negative fractional value23–25 have been +done, showing that for n ≥ 4 the solutions that are generated are physically +acceptable. Other recent works have addressed the possibility of generating +exact solutions with metric potential gtt = − 1+ar2 +1+b2 showing that this func- +tional form is adequate in obtaining physically acceptable solutions and it’s +consistent with the stars Her X-1,26,27 PSRJ0348+0432,28 PSR B0943+10,29 +PSR J0737 -3039A30 and PSRJ1614 2230.31 +Motivated by these few last +investigation works, in this report we present a new solution to Einstein’s +equations with perfect fluid with a metric function gtt and its application +to the star PSR J0030+045. The structure of this article will be as follows: +in the section 2 we present Einstein field equations for a static and spheri- +cally symmetrical spacetime with a perfect fluid and assuming the form of +the metric function gtt = −A2 (1 + ar2)2/ +� +1+( 3 +√ +2 − 1)ar2� +we obtain the so- +lution from the isotropy equation. In section 3 we determine the hydrostatic +functions and impose the coupling conditions between the interior and the +exterior solutions to determine the integration constants. In section 4 graphic +analysis of the solution is done, taking the observational values of the mass +and radius of the star PSRJ0030+045 we determine the physical values of +the pressure, density and speed of sound, starting from the graphic analysis +and from the data, we show that the solution is physically acceptable. The +conclusions and discussion of future works are presented in the section 5. +3 + +2 +The field equations and the solution +The interior geometry of a static and spherically symmetric spacetime can +be described through a line element32,33 : +ds2 = −y(r)2dt2 + dr2 +B(r) + r2(dθ2 + sin2 θdφ2), +(1) +where y, B are functions of the radial coordinate r ≤ R. Einstein equations +Gµν = Rµν − 1 +2Rgµν = kTµν, (with Rµν, R and gµν the components of the +Ricci tensor, the Ricci scalar and the metric tensor respectively), have as +source the distribution of matter from a perfect fluid described by the energy- +momentum tensor: +Tµν = c2ρuµuν + P(uµuν + gµν), +(2) +with uµ the four velocity of the fluid, ρ the energy density and P the pressure. +The non zero components of the Einstein equations are: +kc2ρ += +−B′ +r + 1 − B +r2 +, +(3) +kP += +2By′ +ry +− 1 − B +r2 +, +(4) +kP += +(ry′′ + y′)B +ry ++ (ry′ + y)B′ +2ry +, +(5) +with the derivative in relation to the radial coordinate r denoted by ′. Mean- +while the equation of conservation for the energy-momentum tensor ∇µT µ ν = +0 implies the Tolman-Oppenheimer-Volkov (TOV) equation:32,33 +P ′ = −(P + c2ρ) y′ +y +. +(6) +Although this last one is not an independent equation, since it can be ob- +tained from the system of equations (3) - (5). Being this the set of equations +for which we will obtain the solution starting from a function y(r). +2.1 +The solution +For the integration of the system we propose a metric function gtt = −y(r)2 +with the form of y(r) similar, but slightly different, to one employed pre- +viously with which it was possible to integrate the system in an adequate +4 + +manner and which resulted useful to describe compact objects with a com- +pactness rate u = GM/c2R ≤ 0.266085831628 . Specifically we have that +y (r) = +A (1 + ar2) +� +1 + +� +3 +√ +2 − 1 +� +ar2 +, +(7) +where A and a are constants. One useful relation for the integration of the +system is obtained by replacing y(r) in the difference of the equations (4) +and (5), leading to: +B′− +√ +8[3 + +√ +2 + 4( +√ +8 − 1)ar2 + 7a2r4]B +(1 + +√ +2ar2) +� +2 + 3 +√ +2 + 7ar2 +� +r ++(1 + ar2)(2 + 3 +√ +2 + 7ar2) +√ +2 +(1 + +√ +2ar2)(3 + +√ +2 + 7ar2)r = 0, +after the integration we obtain: +B (r) += +1 + +� +31 − 22 +√ +2 +� � +3 +√ +2 + 2 + 7 ar2�3 ar2 +343 +� +1 + a +√ +2r2 +�3 +� +C + ln +� √ +2 + 3 + 7 ar2 +3 +√ +2 + 2 + 7 ar2 +�� ++ +� +22 − 17 +√ +2 +� � +732 +√ +2 + 1832 + 7 +� +555 +√ +2 + 251 +� +ar2 + 4606a2r4� +ar2 +2303 +�√ +2 + 2ar2 +�3 +, +(8) +C is the constant of integration. +3 +Hydrostatic functions and physical condi- +tions +Once we know the metric functions we will proceed to determine the hydro- +static functions. Replacing the functions y(r) and B given by the equations +(7) and (8) in the equations (3) and (4) we determine the density and the +pressure: +kc2ρ (r) += +3 +� +2 +√ +2 + 6 + +� +−4 + 15 +√ +2 +� +ar2 + 14 a2r4� +(1 − B (r)) +� +3 +√ +2 + 2 + 7 ar2 +� �√ +2 + 2 ar2 +� +r2 +− +14 +� +12 +√ +2 − 13 + 7 ar2� +a2r2 +� +3 +√ +2 + 2 + 7 ar2 +� �√ +2 + 2 ar2 +� �√ +2 + 3 + 7 ar2 +�, +(9) +kP(r) += +2 +� +6 +√ +2 − 3 + 7 ar2� +a +(1 + ar2) +� +3 +√ +2 + 2 + 7 ar2 +� +− +� +3 +√ +2 + 2 + 3 +� +5 +√ +2 + 1 +� +ar2 + 21 a2r4� +(1 − B (r)) +(1 + ar2) +� +3 +√ +2 + 2 + 7 ar2 +� +r2 +.(10) +5 + +In these equations the expression (1 − B)/r2 appears, however, it is regular +when r = 0 as it can be seen from the equation (8). Another important +relation to determine if the solution is physically acceptable is the speed of +sound, since it is required that the speed of sound in the model does not +violate the causality condition. In this case by means of the chain rule we +obtain the speed of sound: +v2(r) +c2 += 1 +c2 +∂P(ρ) +∂ρ += +� +S1 (r) B (r) + +� +3 +√ +2 + 2 + 7 ar2�2 (1 + ar2)2 +� +S2 (r) +� +S3 (r) +�√ +2 + 3 + 7 ar2 +�2 B (r) + S4 (r) +� +(1 + ar2)2 , +where +S4(r) += +(1 + ar2) +� +15 +√ +2 + 150 + 7(34 + 9 +√ +2)ar2 + 98a2r4� � +3 +√ +2 + 2 + 7ar2�2, +S1(r) += +�√ +2 − 4 +� �√ +2 + 3 + 7 ar2� � +2 + +√ +2 + +� +5 +√ +2 − 1 +� +ar2 + 3 a2r4� +, +S3(r) += +3 +�√ +2 − 4 +� � +10 +√ +2 + 30 + 3 +� +−4 + 15 +√ +2 +� +ar2 + 14 a2r4� +, +S2(r) += +� +6 +√ +2 − 3 + 7 ar2� �√ +2 + 3 + 7 ar2� �√ +2 + 2 ar2� +. +3.1 +Criteria for physical acceptability +Obtaining a solution to Einstein’s equations is not a guarantee that said solu- +tion is physically acceptable, there are many solutions that are not physically +acceptable5 due to the fact that they do not comply with certain properties. +In the following we will mention the requirements that must be satisfied, +some of these will be applied directly and others will be shown in a graphical +manner in the following section.28 +One of the conditions that must be met, is the regularity of the ge- +ometry when approaching the center. Which can be expressed in an al- +gebraic manner, through the Kretschmann scalar, given its extension, it is +enough with showing that the metric coefficients around r = 0 are of the +form α + βr2 + O(r4). The expansion of B(r) and y(r) in the proximity of +r = 0 gives us: +y (r) = A +� +�(1 − +� +3 +√ +2 − 6 +� +a +4 +r2 − +� +30 +√ +2 − 41 +� +a2 +16 +r4 + O +� +r6� +) +� +� , +B (r) = 1 + +2 a +�� +17 +√ +2 − 26 +� � +C + ln +� √ +2+3 +3 +√ +2+2 +�� ++ 41 +√ +2 − 80 +� +r2 +49 ++ O +� +r4� +, +besides the regularity, the geometry must be absent of any event horizon, +this property is easier to demonstrate through a graphic analysis and it will +6 + +be analysed in the following section. +The density and pressure must be finite, positive and monotonically +decreasing as functions of the radial coordinate. That is to say, for r ∈ +(0, R), ρ′ < 0 and P ′ < 0 (condition that will be analysed graphically) and in +the center they must have their maximum value. which implies the following +set of inequalities: +kc2ρ(0) = +6 a +�� +132 +√ +2 + 193 +� +(2 C − ln (2)) + 3006 +√ +2 + 4286 +� +6713 +√ +2 + 9506 +> 0, +(11) +kP(0) = 1/49 a +�� +17 +√ +2 − 26 +� +(2 C − ln (2)) − 65 +√ +2 + 134 +� +> 0, +(12) +ρ′′(0) = − +5a2 � +6 +� +195 − 103 +√ +2 +� +(2C − ln 2) + 3819 + 6257 +√ +2 +� +49 +�√ +2 + 3 +�3 +< 0, +(13) +P ′′(0) = − +3a2 � +2 +� +113 − 72 +√ +2 +� +(2 C − ln 2) + 2759 − 1248 +√ +2 +� +49 +�√ +2 + 3 +�2 +< 0. +(14) +In addition to these inequalities the solution satisfies ρ′(0) = and P ′(0) = 0, +that together with the inequalities (11)-(14) implies that r = 0 is a maximum +for the functions ρ and P. The causality condition in the center of the +star requires that it satisfies +0 ≤ v(0)2 +c2 += 3[24 +√ +2 + 4 C − 2 ln 2 + 55 ] +5[96 +√ +2 + 12C − 6 ln 2 + 121] ≤ 1. +(15) +Combining the previous equations we can determine inequalities for the con- +stants (C, a), in particular forming k (ρ (0) c2 + 3 Pr (0)) = 9 +� +2 − +√ +2 +� +a > 0, +from where we obtain that a > 0. +The constants C and W which appear in the metric functions are determined +by imposing that the interior and exterior geometry on the surface +of the star r = R are joined in a continual manner and that the +pressure is zero on the surface. The exterior geometry is described by +the exterior Schwarzschild solution: +ds2 = − +� +1 − 2GM +c2r +� +dt2 + +� +1 − 2GM +c2r +�−1 +dr2 + r2(dθ2 + sin2 θ dφ2), +r ≥ R, +where M represents the total mass inside the fluid sphere. When we impose +P(R) = 0, from the equation (10) we obtain C: +C += +− ln +� √ +2 + 3 + 7 w +3 +√ +2 + 2 + 7 w +� ++ +� +25 +√ +2 + 47 +� +W1 +W2 +� +3 +√ +2 + 2 + 7 w +�2, +(16) +7 + +where w = aR2, W2 = 822 +√ +2 + 548 + 822 +� +5 +√ +2 + 1 +� +w + 5754 w2 and +W1 = 3286 +√ +2+4048+(8149 +√ +2+18623)w+7(2757 +√ +2+1075)w2 +13426w3. +Meanwhile from the continuity of the component gtt in r = R it results: +A2 = +� +3 +√ +2 − 2 +� � +3 +√ +2 + 2 + 7 aR2�2 +14 +� +3 +√ +2 + 2 + 3 +� +5 +√ +2 + 1 +� +aR2 + 21 a2R4 +� +(1 + aR2) +. +(17) +The continuity of grr in r = R determines the value of the compactness as +function of w: +u(w) = GM +c2R = 1 +2(1 − B(R)) = +� +6 +√ +2 − 3 + 7 w +� +w +3 +√ +2 + 2 + 3 +� +5 +√ +2 + 1 +� +w + 21 w2. +(18) +The rest of the conditions require of a graphical analysis and these correspond +to the Energy conditions: +- The Strong Energy Condition: c2ρ + 3P ≥ 0, c2ρ + P ≥ 0 or +- The Dominant Energy Condition: ρ ≥ 0 and c2ρ ≥ |P| +And the Stability condition, a configuration of static and spherically sym- +metrical matter is stable if it satisfies the relativistic condition for the adia- +batic index: +Γ = P + c2ρ +c2P +dP +dρ > 4 +3 +∀ r ∈ [0, R] +4 +Graphic representation of the solution +From the graphic analysis of the functions of density, pressure, speed of sound +and adiabatic index we obtain that the function which restricts the values of +the parameter w ≤ w0 = 0.90378 is that of the adiabatic index, specifically +for values of w > w0 the adiabatic index γ(0) < 4/3, which implies that +the solution will be unstable. This maximum value w0 through the equation +(18) allows us to obtain the maximum permissible compactness value for +the compactness u ≤ u0 = 0.23577. +Although the solution is physically +acceptable for the compactness values u ≤ u0 in the graphic analysis we will +focus on the particular case of the star PSR J0030+0451 with estimates of +mass M = 1.44+0.15 +−0.14M⊙ and radius R = 13.02+1.24 +−1.06km, obtained through the +study of the X-ray emission by means of the NICER (Neutron star Interior +Composition Explorer) telescope from the international space station .34 The +graphic representation will be done in terms of the dimensionless variable +x = r/R and the dimensionless functions associated to the physical quantities +8 + +of density kc2R2ρ, pressure kR2P and speed of sound v2/c2. The values of +compactness that were chosen for the graphic analysis are umax = 0.19628, +u = 0.18086, u = 0.16545, u = 0.15003 and umin = 0.13460, where umax +it’s associated with the maximum mass M = 1.59M⊙ and the minimum +radius R = 11.96km u = umax =, meanwhile umin is obtained by taking the +minimum mass M = 1.3M⊙ and the maximum radius R = 14.26km. +Figure 1: Graphic representation of the density and the pressure for the +different values of compactness from the star PSR J0030+0451. +In the figure 1 we show the behaviour of the density and pressure for different +values of compactness which is obtained from the estimates in base to obser- +vations. The graphics show that the density and pressure are positive and +monotonically decreasing functions, their values diminish as the compact- +ness value decreases, appearing in a more noticeable manner the difference +between the values of the density or the pressure in the center of the star, we +also observe that the pressure is zero on the surface. From the figure 1 we +observe that the Strong Energy Condition is satisfied, since both the density +and the pressure are positive. Also we have that for a specific value of u the +value of the density is much greater than that of the pressure (c2ρ > P), +which implies that the Dominant Energy Condition is also satisfied. From +the figure 2, graphic on the right, we observe that the causality condition is +met, since 0.2c2 < v2 < 0.34c2 and that the speed of sound is lower for lower +compactness values, with maximum values on the surface. The stability of +the solution is guaranteed by the adiabatic index, the left graph in the figure +2, with γ being a monotonically increasing function, the lowest value of the +adiabatic index occurs in the center of the star for the maximum compactness +umax as such the set of compactness values that is being analysed satisfies +9 + +u=0.19628 +-u=0.18086 +1.8 +u=0.16545 +u=0.15003 +u=0.13460 +1.6 +X + 1.4 +2k +2c +K +1.2 +1.0 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +1u=0.19628 +0.25 +-u=0.18086 +u=0.16545 +u=0.15003 +u=0.13460 +0.20 +0.15 +2k +K +0.10 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +1 +Xγ > 1.6843 > 4/3. The absence of an event horizon and the continuity of the +Figure 2: Graphic representation of the speed of sound and the adiabatic +index for the different compactness values of the star PSR J0030+0451. +geometry on the surface of the star is shown in the figure 3. In addition,in +the graph on the right side of the figure 3, we graph the forces present (the +gravitational force Fg and the hydrostatic force Fh), identified by means of +the Tolman-Oppenheimer-Volkoff (TOV) equation +−(Pr + c2ρ) y′ +y +−Pr +′ = 0, +⇒ +Fg(r) = −(Pr + c2ρ) y′ +y +, +Fh(r) = −Pr +′. +In the figure 3 we can observe the attractive effect of the gravitational force +Fg countered by the hydrostatic repulsive force. +5 +Discussion and conclusions +The graphic analysis has allowed us to show that the solution presented +satisfies every requirement which makes it physically acceptable and although +the graphic analysis was done considering estimated values of mass and radius +for the star PSR J0030+045, a similar behaviour is present for other values +of compactness, as long as u ≤ 0.23577. +To confirm that the behaviour +of the solution is not only graphically compatible but also that the orders +of magnitude which are obtained from the model are compatible with those +expected, on the tables 1 and 2 we report the values of density, pressure, speed +of sound and adiabatic index in the interior for the case of maximum 1 and +minimum compactness 2. +From the tables 1 and 2 it can be noticed that the +10 + +0.32 +u=0.19628 +u=0.18086 +u=0.16545 +0.30 +u=0.15003 +u=0.13460 +0.28 +v(x)i +0.26 +0.24 +0.22 +0 +0.2 +0.4 +0.6 +0.8 +1 +xu=0.19628 +-u=0.18086 +40 +u=0.16545 +u=0.15003 +u=0.13460 +30 +3.0 +2.8 +2.6 +X2.4 +20 +>2.2 +2.0 +1.8 +10 +0 +0.1 +0.2 +0.3 +X +0 +0 +0.2 +0.4 +0.6 +0.8 +XFigure 3: In the graph of the left side we present the behaviour of the metric +coefficients from the interior and exterior geometry, meanwhile in the graph +of the right side it´s shown the behaviour of the forces in the interior of the +star. +orders of magnitude of the density and pressure are also those expected for the +star PSR J0030+0451. With which we can conclude that the model obtained +is physically acceptable and useful to represent stars with compactness u ≤ +0.23577. +Another relevance of the solution constructed is that it can be +useful as seed for obtaining new physically acceptable solutions35 in which +we consider the contribution of the presence of electric charge36 or from +an anisotropy factor,37 as well as in the determination of new solutions in +alternative gravitational theories,38 investigations that could be developed in +future works. +11 + +1.0 +1 +grr(x) +0.9 +0.8 +metric(x) +0.7 +0.6 +u=0.19628 +-u=0.18086 +u=0.16545 +0.5 +u=0.15003 +u=0.13460 +9t(×) +0 +0.5 +1 +1.5 +2F(×) +0.3 +h +0.2 +K +0.1 +u=0.19628 +u=0.18086 +u=0.16545 +0 +0.2 +0.4 +0.6 +0.8 +1 +X +u=0.15003 +-0.1 +u=0.13460 +-0.2 +-0.3 +F +.(x)Table 1: Interior behavior of the physical values of the density, pressure, +speed of the sound and adiabatic index for the PSR J0030+0451, with R = +11.96km and M = 1.59M⊙, umax = 0.19628. +r(km) ρ +� +1017 kg +m3 +� +P(1033Pa) +v2(c2) +γ +0. +7.5125 +9.4371 +0.20654 +1.6843 +1.1960 +7.4155 +9.2563 +0.20795 +1.7052 +2.3920 +7.1398 +8.7360 +0.21214 +1.7702 +3.5880 +6.7211 +7.9258 +0.21905 +1.8887 +4.7840 +6.2131 +6.9019 +0.22857 +2.0775 +5.9800 +5.6621 +5.7429 +0.24055 +2.3719 +7.1760 +5.1146 +4.5238 +0.25474 +2.8428 +8.3720 +4.5970 +3.3040 +0.27094 +3.6581 +9.5680 +4.1283 +2.1264 +0.28887 +5.3282 +10.764 +3.7155 +1.0202 +0.30824 +10.395 +11.960 +3.3547 +0 +0.32880 +∞ +Acknowledgments +We appreciate the facilities provided by the Universidad Michoacana de San +Nicol´as de Hidalgo and the CIC -UMSNH during the realization of this in- +vestigation as well as the CONACYT for the support given. +12 + +Table 2: Interior behavior of the physical values of the density, pressure, +speed of sound and adiabatic index for the PSR J0030+0451 star, with R = +14.26km and M = 1.3M⊙, umin = 0.13460. +r(km) ρ +� +1017 kg +m3 +� +P(1033Pa) +v2(c2) +γ +0 +2.8411 +2.1586 +0.20605 +2.6431 +1.4260 +2.8246 +2.1275 +0.20669 +2.6723 +2.8520 +2.7760 +2.0362 +0.20873 +2.7656 +4.2780 +2.6978 +1.8894 +0.21203 +2.9321 +5.7040 +2.5965 +1.6944 +0.21667 +3.2001 +7.1300 +2.4774 +1.4588 +0.22249 +3.6176 +8.5560 +2.3462 +1.1923 +0.22956 +4.2876 +9.9820 +2.2093 +0.90520 +0.23777 +5.4501 +11.408 +2.0715 +0.60586 +0.24708 +7.8340 +12.834 +1.9374 +0.30200 +0.25734 +15.081 +14.260 +1.8055 +0 +0.26859 +∞ +References +[1] S. 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D +92 (2015) 044020, +15 + diff --git a/nNAyT4oBgHgl3EQfYveM/content/tmp_files/load_file.txt b/nNAyT4oBgHgl3EQfYveM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5199297098dac8b2fbafb9001f24ccfdddfa4eed --- /dev/null +++ b/nNAyT4oBgHgl3EQfYveM/content/tmp_files/load_file.txt @@ -0,0 +1,531 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf,len=530 +page_content='A regular interior solution of Einstein field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Gabino Estevez-Delgado Facultad de Qu´ımico Farmacobiolog´ıa de la UMSNH Tzintzuntzan No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' 173, Col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Matamoros, Morelia Michoac´an, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' 58240, M´exico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' gabino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='estevez@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='mx, Joaquin Estevez-Delgado Facultad de Ciencias F´ısico Matem´aticas de la UMSNH Edificio B, Ciudad Universitaria, Morelia Michoac´an, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' 58040, M´exico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' joaquin@fismat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='mx, Modesto Pineda Duran Instituto Tecnol´ogico Superior de Tac´ambaro, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Tecnol´ogico No 201 Zona el Gigante, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' 61650, Tacambaro Michoac´an, M´exico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' mpinedad@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='com and Arthur Cleary-Balderas Facultad de Ingenier´ıa El´ectrica de la UMSNH Edificio Ω, Ciudad Universitaria, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' 58030, Morelia Michoac´an, M´exico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' arthur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='cleary@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='mx January 3, 2023 Abstract Starting from the solution of the Einstein field equations in a static and spherically symmetric spacetime which contains an isotropic fluid, we construct a model to represent the interior of compact objects with compactness rate u = GM c2R < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='23577.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' The solution is obtained by imposing the isotropy condition for the radial and tangential pressures, this generates an ordinary differential equation of second order for the temporal gtt and radial grr metric potentials, which can be solved for a specific function of gtt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' The graphic analysis of the solution shows 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='00210v1 [gr-qc] 31 Dec 2022 that it is physically acceptable, that is to say, the density, pressure and speed of sound are positive, regular and monotonically decreasing functions, also, the solution is stable due to meeting the criteria of the adiabatic index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' When taking the data of mass M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='44+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='14M⊙ and radius R = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='02+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='24 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='06km which corresponds to the estimations of the star PSR J0030+045 we obtain values of central density ρc = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='5125×1017kg/m3 for the maximum compactness u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='19628 and of ρc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='8411 × 1017kg/m3 for the minimum compactness u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='13460, which are consistent with those expected for this type of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' 1 Introduction Describing the interior of the stars and determining their average composi- tion requires many different complementary approaches as are: chemistry, thermodynamics, nuclear physics, particle physics and gravitational physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' And in the case that there are no instabilities generated, once all the nuclear fuel has been used, the star shrinks and, depending of the mass and the sta- bility present, it may form a white dwarf, a neutron star or a quark star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='1,2 Of course the description of each one of these stages and their stability is a far more delicate matter which involves a detailed analysis that considers the type of predominant particles in the interior of the star, whether they are electrons, neutrons, or quarks, even when the stars are hybrids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' According to the focus of this job, the matter in general is supposed to be described in a satisfactory manner by a perfect fluid and it will not be necessary to give a specific shape of a state equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' We know that, depending on the value of the mass and radius of a star, it may be a white dwarf, a neutron star or a quark star, this will also determine the orders of magnitude of the density, for example densities in the order of 1018kg/m2 are typical for neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' As such in this situation, given the high density, it results adequate to describe the interior of the stars by means of Einstein’s general relativity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' The interior solutions have been approached for over a century, the first of these was constructed for a static and spherically symmetric spacetime with mat- ter from a perfect fluid and incompressible density, known as the interior Schwarzschild solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Although to start with, the density being constant is an unrealistic requirement, its consequences revealed some differences with the treatment of stellar models in the context of Newton’s theory of gravi- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' One of these is the compactness relation u = GM/c2R < 4/9, where M is the mass and R is the radius, this indicates that it is not possible to have stars with arbitrary mass and radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Afterwards it was shown that this relation is not exclusive of this idealized model and that it is present for 2 stars with a monotonically decreasing density function for which the exterior geometry is given by the Schwarzschild solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='3,4 And although more that 130 interior solutions with perfect fluid have been published, only a few met the characteristics that makes them physically acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' From an analy- sis done in 1998 on a total of 127 reported solutions, only 16 of these had their density and pressure functions be positive, regular and monotonically decreasing functions and also had a speed of sound that didn’t violate the causality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' And only 9 out of these 16 had a speed of sound which decreases monotonically with the radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='5–16 Although this last requirement is debatable since, for example, for the realistic MIT Bag state equation P(ρ) = 1 3(c2ρ−4Bg) associated to quark stars, the speed of sound is v2 s = c2 3 , which is not a monotonically decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' The construction of stel- lar solutions with perfect fluid is an active field although the difficulty in obtaining physically acceptable solutions limits a great number of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' A point which has been explored in relation to these is employing isotropic coordinates17–21 which has favored the integration of the equations system and its application in the description of stars like HerX1, 4U1538-52, LMC X - 4, SAX J1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='4-3658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='22 A particular class of solutions constructed in Schwarzschild’s coordinates assumes a metric potential gtt = −(1 + ar2)n to this group belong the Tolman IV6 and Durgapal15 solutions, extensions for other values of n a positive integer or negative fractional value23–25 have been done, showing that for n ≥ 4 the solutions that are generated are physically acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Other recent works have addressed the possibility of generating exact solutions with metric potential gtt = − 1+ar2 1+b2 showing that this func- tional form is adequate in obtaining physically acceptable solutions and it’s consistent with the stars Her X-1,26,27 PSRJ0348+0432,28 PSR B0943+10,29 PSR J0737 -3039A30 and PSRJ1614 2230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='31 Motivated by these few last investigation works, in this report we present a new solution to Einstein’s equations with perfect fluid with a metric function gtt and its application to the star PSR J0030+045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' The structure of this article will be as follows: in the section 2 we present Einstein field equations for a static and spheri- cally symmetrical spacetime with a perfect fluid and assuming the form of the metric function gtt = −A2 (1 + ar2)2/ � 1+( 3 √ 2 − 1)ar2� we obtain the so- lution from the isotropy equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' In section 3 we determine the hydrostatic functions and impose the coupling conditions between the interior and the exterior solutions to determine the integration constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' In section 4 graphic analysis of the solution is done, taking the observational values of the mass and radius of the star PSRJ0030+045 we determine the physical values of the pressure, density and speed of sound, starting from the graphic analysis and from the data, we show that the solution is physically acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' The conclusions and discussion of future works are presented in the section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' 3 2 The field equations and the solution The interior geometry of a static and spherically symmetric spacetime can be described through a line element32,33 : ds2 = −y(r)2dt2 + dr2 B(r) + r2(dθ2 + sin2 θdφ2), (1) where y, B are functions of the radial coordinate r ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Einstein equations Gµν = Rµν − 1 2Rgµν = kTµν, (with Rµν, R and gµν the components of the Ricci tensor, the Ricci scalar and the metric tensor respectively), have as source the distribution of matter from a perfect fluid described by the energy- momentum tensor: Tµν = c2ρuµuν + P(uµuν + gµν), (2) with uµ the four velocity of the fluid, ρ the energy density and P the pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' The non zero components of the Einstein equations are: kc2ρ = −B′ r + 1 − B r2 , (3) kP = 2By′ ry − 1 − B r2 , (4) kP = (ry′′ + y′)B ry + (ry′ + y)B′ 2ry , (5) with the derivative in relation to the radial coordinate r denoted by ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Mean- while the equation of conservation for the energy-momentum tensor ∇µT µ ν = 0 implies the Tolman-Oppenheimer-Volkov (TOV) equation:32,33 P ′ = −(P + c2ρ) y′ y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' (6) Although this last one is not an independent equation, since it can be ob- tained from the system of equations (3) - (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Being this the set of equations for which we will obtain the solution starting from a function y(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='1 The solution For the integration of the system we propose a metric function gtt = −y(r)2 with the form of y(r) similar, but slightly different, to one employed pre- viously with which it was possible to integrate the system in an adequate 4 manner and which resulted useful to describe compact objects with a com- pactness rate u = GM/c2R ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='266085831628 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Specifically we have that y (r) = A (1 + ar2) � 1 + � 3 √ 2 − 1 � ar2 , (7) where A and a are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' One useful relation for the integration of the system is obtained by replacing y(r) in the difference of the equations (4) and (5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' leading to: B′− √ 8[3 + √ 2 + 4( √ 8 − 1)ar2 + 7a2r4]B (1 + √ 2ar2) � 2 + 3 √ 2 + 7ar2 � r +(1 + ar2)(2 + 3 √ 2 + 7ar2) √ 2 (1 + √ 2ar2)(3 + √ 2 + 7ar2)r = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' after the integration we obtain: B (r) = 1 + � 31 − 22 √ 2 � � 3 √ 2 + 2 + 7 ar2�3 ar2 343 � 1 + a √ 2r2 �3 � C + ln � √ 2 + 3 + 7 ar2 3 √ 2 + 2 + 7 ar2 �� + � 22 − 17 √ 2 � � 732 √ 2 + 1832 + 7 � 555 √ 2 + 251 � ar2 + 4606a2r4� ar2 2303 �√ 2 + 2ar2 �3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' (8) C is the constant of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' 3 Hydrostatic functions and physical condi- tions Once we know the metric functions we will proceed to determine the hydro- static functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Replacing the functions y(r) and B given by the equations (7) and (8) in the equations (3) and (4) we determine the density and the pressure: kc2ρ (r) = 3 � 2 √ 2 + 6 + � −4 + 15 √ 2 � ar2 + 14 a2r4� (1 − B (r)) � 3 √ 2 + 2 + 7 ar2 � �√ 2 + 2 ar2 � r2 − 14 � 12 √ 2 − 13 + 7 ar2� a2r2 � 3 √ 2 + 2 + 7 ar2 � �√ 2 + 2 ar2 � �√ 2 + 3 + 7 ar2 �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' (9) kP(r) = 2 � 6 √ 2 − 3 + 7 ar2� a (1 + ar2) � 3 √ 2 + 2 + 7 ar2 � − � 3 √ 2 + 2 + 3 � 5 √ 2 + 1 � ar2 + 21 a2r4� (1 − B (r)) (1 + ar2) � 3 √ 2 + 2 + 7 ar2 � r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' (10) 5 In these equations the expression (1 − B)/r2 appears, however, it is regular when r = 0 as it can be seen from the equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Another important relation to determine if the solution is physically acceptable is the speed of sound, since it is required that the speed of sound in the model does not violate the causality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' In this case by means of the chain rule we obtain the speed of sound: v2(r) c2 = 1 c2 ∂P(ρ) ∂ρ = � S1 (r) B (r) + � 3 √ 2 + 2 + 7 ar2�2 (1 + ar2)2 � S2 (r) � S3 (r) �√ 2 + 3 + 7 ar2 �2 B (r) + S4 (r) � (1 + ar2)2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' where S4(r) = (1 + ar2) � 15 √ 2 + 150 + 7(34 + 9 √ 2)ar2 + 98a2r4� � 3 √ 2 + 2 + 7ar2�2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' S1(r) = �√ 2 − 4 � �√ 2 + 3 + 7 ar2� � 2 + √ 2 + � 5 √ 2 − 1 � ar2 + 3 a2r4� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' S3(r) = 3 �√ 2 − 4 � � 10 √ 2 + 30 + 3 � −4 + 15 √ 2 � ar2 + 14 a2r4� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' S2(r) = � 6 √ 2 − 3 + 7 ar2� �√ 2 + 3 + 7 ar2� �√ 2 + 2 ar2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='1 Criteria for physical acceptability Obtaining a solution to Einstein’s equations is not a guarantee that said solu- tion is physically acceptable, there are many solutions that are not physically acceptable5 due to the fact that they do not comply with certain properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' In the following we will mention the requirements that must be satisfied, some of these will be applied directly and others will be shown in a graphical manner in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='28 One of the conditions that must be met, is the regularity of the ge- ometry when approaching the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Which can be expressed in an al- gebraic manner, through the Kretschmann scalar, given its extension, it is enough with showing that the metric coefficients around r = 0 are of the form α + βr2 + O(r4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' The expansion of B(r) and y(r) in the proximity of r = 0 gives us: y (r) = A � �(1 − � 3 √ 2 − 6 � a 4 r2 − � 30 √ 2 − 41 � a2 16 r4 + O � r6� ) � � , B (r) = 1 + 2 a �� 17 √ 2 − 26 � � C + ln � √ 2+3 3 √ 2+2 �� + 41 √ 2 − 80 � r2 49 + O � r4� , besides the regularity, the geometry must be absent of any event horizon, this property is easier to demonstrate through a graphic analysis and it will 6 be analysed in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' The density and pressure must be finite, positive and monotonically decreasing as functions of the radial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' That is to say, for r ∈ (0, R), ρ′ < 0 and P ′ < 0 (condition that will be analysed graphically) and in the center they must have their maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' which implies the following set of inequalities: kc2ρ(0) = 6 a �� 132 √ 2 + 193 � (2 C − ln (2)) + 3006 √ 2 + 4286 � 6713 √ 2 + 9506 > 0, (11) kP(0) = 1/49 a �� 17 √ 2 − 26 � (2 C − ln (2)) − 65 √ 2 + 134 � > 0, (12) ρ′′(0) = − 5a2 � 6 � 195 − 103 √ 2 � (2C − ln 2) + 3819 + 6257 √ 2 � 49 �√ 2 + 3 �3 < 0, (13) P ′′(0) = − 3a2 � 2 � 113 − 72 √ 2 � (2 C − ln 2) + 2759 − 1248 √ 2 � 49 �√ 2 + 3 �2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' (14) In addition to these inequalities the solution satisfies ρ′(0) = and P ′(0) = 0, that together with the inequalities (11)-(14) implies that r = 0 is a maximum for the functions ρ and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' The causality condition in the center of the star requires that it satisfies 0 ≤ v(0)2 c2 = 3[24 √ 2 + 4 C − 2 ln 2 + 55 ] 5[96 √ 2 + 12C − 6 ln 2 + 121] ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' (15) Combining the previous equations we can determine inequalities for the con- stants (C, a), in particular forming k (ρ (0) c2 + 3 Pr (0)) = 9 � 2 − √ 2 � a > 0, from where we obtain that a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' The constants C and W which appear in the metric functions are determined by imposing that the interior and exterior geometry on the surface of the star r = R are joined in a continual manner and that the pressure is zero on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' The exterior geometry is described by the exterior Schwarzschild solution: ds2 = − � 1 − 2GM c2r � dt2 + � 1 − 2GM c2r �−1 dr2 + r2(dθ2 + sin2 θ dφ2), r ≥ R, where M represents the total mass inside the fluid sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' When we impose P(R) = 0, from the equation (10) we obtain C: C = − ln � √ 2 + 3 + 7 w 3 √ 2 + 2 + 7 w � + � 25 √ 2 + 47 � W1 W2 � 3 √ 2 + 2 + 7 w �2, (16) 7 where w = aR2, W2 = 822 √ 2 + 548 + 822 � 5 √ 2 + 1 � w + 5754 w2 and W1 = 3286 √ 2+4048+(8149 √ 2+18623)w+7(2757 √ 2+1075)w2 +13426w3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Meanwhile from the continuity of the component gtt in r = R it results: A2 = � 3 √ 2 − 2 � � 3 √ 2 + 2 + 7 aR2�2 14 � 3 √ 2 + 2 + 3 � 5 √ 2 + 1 � aR2 + 21 a2R4 � (1 + aR2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' (17) The continuity of grr in r = R determines the value of the compactness as function of w: u(w) = GM c2R = 1 2(1 − B(R)) = � 6 √ 2 − 3 + 7 w � w 3 √ 2 + 2 + 3 � 5 √ 2 + 1 � w + 21 w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' (18) The rest of the conditions require of a graphical analysis and these correspond to the Energy conditions: The Strong Energy Condition: c2ρ + 3P ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' c2ρ + P ≥ 0 or The Dominant Energy Condition: ρ ≥ 0 and c2ρ ≥ |P| And the Stability condition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' a configuration of static and spherically sym- metrical matter is stable if it satisfies the relativistic condition for the adia- batic index: Γ = P + c2ρ c2P dP dρ > 4 3 ∀ r ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' R] 4 Graphic representation of the solution From the graphic analysis of the functions of density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' pressure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' speed of sound and adiabatic index we obtain that the function which restricts the values of the parameter w ≤ w0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='90378 is that of the adiabatic index, specifically for values of w > w0 the adiabatic index γ(0) < 4/3, which implies that the solution will be unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' This maximum value w0 through the equation (18) allows us to obtain the maximum permissible compactness value for the compactness u ≤ u0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='23577.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Although the solution is physically acceptable for the compactness values u ≤ u0 in the graphic analysis we will focus on the particular case of the star PSR J0030+0451 with estimates of mass M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='44+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='14M⊙ and radius R = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='02+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='24 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='06km, obtained through the study of the X-ray emission by means of the NICER (Neutron star Interior Composition Explorer) telescope from the international space station .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='34 The graphic representation will be done in terms of the dimensionless variable x = r/R and the dimensionless functions associated to the physical quantities 8 of density kc2R2ρ, pressure kR2P and speed of sound v2/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' The values of compactness that were chosen for the graphic analysis are umax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='19628, u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='18086, u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='16545, u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='15003 and umin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='13460, where umax it’s associated with the maximum mass M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='59M⊙ and the minimum radius R = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='96km u = umax =, meanwhile umin is obtained by taking the minimum mass M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='3M⊙ and the maximum radius R = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='26km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Figure 1: Graphic representation of the density and the pressure for the different values of compactness from the star PSR J0030+0451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' In the figure 1 we show the behaviour of the density and pressure for different values of compactness which is obtained from the estimates in base to obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' The graphics show that the density and pressure are positive and monotonically decreasing functions, their values diminish as the compact- ness value decreases, appearing in a more noticeable manner the difference between the values of the density or the pressure in the center of the star, we also observe that the pressure is zero on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' From the figure 1 we observe that the Strong Energy Condition is satisfied, since both the density and the pressure are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Also we have that for a specific value of u the value of the density is much greater than that of the pressure (c2ρ > P), which implies that the Dominant Energy Condition is also satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' From the figure 2, graphic on the right, we observe that the causality condition is met, since 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='2c2 < v2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='34c2 and that the speed of sound is lower for lower compactness values, with maximum values on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' The stability of the solution is guaranteed by the adiabatic index, the left graph in the figure 2, with γ being a monotonically increasing function, the lowest value of the adiabatic index occurs in the center of the star for the maximum compactness umax as such the set of compactness values that is being analysed satisfies 9 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='19628 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='18086 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='8 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='16545 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='15003 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='13460 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='6 X 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='4 2k 2c K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='8 1u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='19628 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='25 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='18086 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='16545 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='15003 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='13460 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='15 2k K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='05 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='8 1 Xγ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='6843 > 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' The absence of an event horizon and the continuity of the Figure 2: Graphic representation of the speed of sound and the adiabatic index for the different compactness values of the star PSR J0030+0451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' geometry on the surface of the star is shown in the figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' In addition,in the graph on the right side of the figure 3, we graph the forces present (the gravitational force Fg and the hydrostatic force Fh), identified by means of the Tolman-Oppenheimer-Volkoff (TOV) equation −(Pr + c2ρ) y′ y −Pr ′ = 0, ⇒ Fg(r) = −(Pr + c2ρ) y′ y , Fh(r) = −Pr ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' In the figure 3 we can observe the attractive effect of the gravitational force Fg countered by the hydrostatic repulsive force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' 5 Discussion and conclusions The graphic analysis has allowed us to show that the solution presented satisfies every requirement which makes it physically acceptable and although the graphic analysis was done considering estimated values of mass and radius for the star PSR J0030+045, a similar behaviour is present for other values of compactness, as long as u ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='23577.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' To confirm that the behaviour of the solution is not only graphically compatible but also that the orders of magnitude which are obtained from the model are compatible with those expected, on the tables 1 and 2 we report the values of density, pressure, speed of sound and adiabatic index in the interior for the case of maximum 1 and minimum compactness 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' From the tables 1 and 2 it can be noticed that the 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='32 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='19628 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='18086 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='16545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='30 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='15003 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='13460 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='28 v(x)i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='22 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='8 1 xu=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='19628 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='18086 40 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='16545 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='15003 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='13460 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='6 X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='4 20 >2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='3 X 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='8 XFigure 3: In the graph of the left side we present the behaviour of the metric coefficients from the interior and exterior geometry, meanwhile in the graph of the right side it´s shown the behaviour of the forces in the interior of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' orders of magnitude of the density and pressure are also those expected for the star PSR J0030+0451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' With which we can conclude that the model obtained is physically acceptable and useful to represent stars with compactness u ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='23577.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' Another relevance of the solution constructed is that it can be useful as seed for obtaining new physically acceptable solutions35 in which we consider the contribution of the presence of electric charge36 or from an anisotropy factor,37 as well as in the determination of new solutions in alternative gravitational theories,38 investigations that could be developed in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='0 1 grr(x) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='8 metric(x) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='6 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='19628 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='18086 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='16545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='5 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='15003 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='13460 9t(×) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='5 1 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='3 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' (x)Table 1: Interior behavior of the physical values of the density, pressure, speed of the sound and adiabatic index for the PSR J0030+0451, with R = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='96km and M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='59M⊙, umax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='19628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' r(km) ρ � 1017 kg m3 � P(1033Pa) v2(c2) γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='5125 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='4371 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='20654 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='6843 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='0202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='30824 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='395 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='960 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='3547 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='32880 ∞ Acknowledgments We appreciate the facilities provided by the Universidad Michoacana de San Nicol´as de Hidalgo and the CIC -UMSNH during the realization of this in- vestigation as well as the CONACYT for the support given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' 12 Table 2: Interior behavior of the physical values of the density, pressure, speed of sound and adiabatic index for the PSR J0030+0451 star, with R = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='26km and M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='3M⊙, umin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content='13460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQfYveM/content/2301.00210v1.pdf'} +page_content=' r(km) ρ � 1017 kg m3 � P(1033Pa) v2(c2) γ 0 2.' metadata={'source': 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https://git-lfs.github.com/spec/v1 +oid sha256:8ad5f6a601b22daba14855b752bdae66413bb0356406587f644042125a8d6cb4 +size 4522029 diff --git a/oNE3T4oBgHgl3EQfjQoa/vector_store/index.pkl b/oNE3T4oBgHgl3EQfjQoa/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..6c9164cf64b6a4d175d62bc51b15f781edbde825 --- /dev/null +++ b/oNE3T4oBgHgl3EQfjQoa/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4901a022d54f5981c995aeab76793a9463042e417aa764bbefdd42bd6bfc0a2f +size 346356 diff --git a/otAzT4oBgHgl3EQfOftW/content/tmp_files/2301.01166v1.pdf.txt b/otAzT4oBgHgl3EQfOftW/content/tmp_files/2301.01166v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8551a033b038514873cce152ebc34b6cb9d8a5ce --- /dev/null +++ b/otAzT4oBgHgl3EQfOftW/content/tmp_files/2301.01166v1.pdf.txt @@ -0,0 +1,903 @@ +arXiv:2301.01166v1 [hep-ph] 3 Jan 2023 +New reaction approach to reflect exotic structure of hadronic molecular state +Zuo-Ming Ding,1 Jun He,1, 2, ∗ and Xiang Liu2, 3, 4, 5, † +1School of Physics and Technology, Nanjing Normal University, Nanjing 210097, China +2Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, China +3School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China +4Key Laboratory of Theoretical Physics of Gansu Province, +and Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, China +5Research Center for Hadron and CSR Physics, Lanzhou University and Institute of Modern Physics of CAS, Lanzhou 730000, China +(Dated: January 4, 2023) +With the accumulation of the experimental data, more and more exotic hadrons are observed. Among the +interpretations of these exotic hadrons, molecular state and compact multiquark are two of the most popular +pictures. However, it is still difficult to determine the structure of an exotic hadron. In this work, we propose +a possible way to detect the internal structure of an exotic state. When a molecular state composed of two +consistent hadrons is attacked by another particle, one of the constituent should be kicked out while another +quasifree constituent keeps almost unaffected. It is different from a compact multiquark which has no obvious +subcluster. In this work, take the X(3872) as example, we perform a Dalitz plot analysis of such reaction to +find the effect of the different internal structures. Under the assumption of the X(3872) as a molecular state or +a compact tetraquark state, with the help of the effective Lagrangians, the Dalitz plot and the invariant mass +spectrum are estimated with different total invariant mass of three final particles, and the effect of different +binding energies is also discussed. Obvious event concentration can be observed as strips in the Dalitz plot and +sharp peaks in the invariant mass spectrum for the X(3872) with a small binding energy under molecular state +picture while such concentration can not be observed under compact tetraquark picture. Such phenomenon can +be applied to identify the internal structure of new hadron state. +I. +INTRODUCTION +The study of exotic hadrons is one of the most impor- +tant topics in hadron physics. Theoretically, the basic theory +of the strong interaction, quantum chromodynamics (QCD), +allows the existence of exotic hadrons beyond the conven- +tional picture where the hadrons are composed of three quarks +or a quark-antiquark pair. +Experimentally, with the devel- +opment of the experimental techniques and accumulation of +data, more and more exotic particles are observed but can not +be put into the frames of the conventional quark model [1–3]. +If we deem these new particle as a genuine state composed +of quarks, there exist two main interpretations, compact mul- +tiquark and hadronic molecular state. For most of the exotic +states, both interpretations exist simultaneously in the litera- +ture [4]. It is an interesting and difficult problem to determine +the real internal structures of an exotic hadron. +The molecular state is a loosely bound state of two +hadrons [5]. +Such idea is considerably easy to be under- +stood if we take the deuteron as a molecular state composed +of two hadrons, that is, nucleon. +It is also natural to ex- +pect the existence of bound states from other hadrons. The +experimental observation seems to support such assumptions +also. +Considerable XYZ particles are close to the thresh- +olds of two hadrons [1]. +The most popular interpretation +about such phenomenon is that these particles are composed +of the corresponding hadrons with a small binding energy as +deuteron [4, 5]. The hadronic molecular state is in fact a pic- +∗Electronic address: junhe@njnu.edu.cn (Corresponding author) +†Electronic address: xiangliu@lzu.edu.cn (Corresponding author) +ture in hadron level. Different from the molecular state, the +compact multiquark is a real bound state of quarks [6]. In +a compact multiquark, no obvious subcluster can be found, +and its radius is usually assumed to be much smaller than a +molecular state. Theoretically, the mass of a compact multi- +quark is irrelevant to the thresholds of hadrons. It seems to be +used to judge whether an exotic hadron is a molecular state. +However, practically, due to the uncertainty of both theory and +experiment, many exotic states near thresholds can be also ex- +plained as a compact multiquark. Moreover, the multiquark is +still important picture to explain the states which is far from +any threshold. +In the literature, the masses and the decay patterns are the +most important ways to detect the internal structure of an ex- +otic state. However, as said above, the accordance of the the- +oretical mass with the experimental mass is not enough to de- +termine the internal structure of an exotic state because it of- +ten can be explained in both pictures. The decay pattern is +most promising to reflect the quark distributions in the exotic +hadrons. However, the uncertainties from both experiment +and theory make it difficult to reach a determinative conclu- +sion. Hence, it is helpful to find more ways to detect the inter- +nal structure of the exotic states. +The main difference between a molecular state and a com- +pact multiquark is the spatial distribution of the quarks. In +a molecular state, the quarks are grouped into two hadrons +which have a distance of several even more than ten fm. Be- +sides effects on the decay pattern, such structure should be +reflected when being attacked by a particle. For a molecular +state, the incoming particle, which is usually about 1 fm can +be easily attacked into the molecular state even pass through +it. Since the distance of two constituent hadrons separated by +long distances, the collision happens only on one constituent. + +2 +When a constituent hadron is attacked, another one should +be little affected. However, for a compact multiquark, which +is usually about 1 fm, the results of the collision of incom- +ing particle is to excite the multiquark and induce its decay. +Due to the compactness, the momenta of the incoming parti- +cle should be transferred to all quarks and then all final par- +ticles. Hence, the molecular state should have quite different +behavior after collision. +Such difference of the behavior of collision should be a +promising way to detect the internal structure of exotic state. +However, there is an obvious difficulty. +We do not have +enough stable exotic states to make a target or a beam to per- +form a collision with another particle. The direct measure- +ment of such collision is impossible with the current or near +future experimental technology. However, such collisions can +happen in the production of the exotic hadrons in a nucleon- +rich environment, which is the realistic scene at facilities, such +as LHCb and PANDA. If we can extract the information of +collision of the nucleon with produced exotic states, it is still +promising to obtain enough events to study such different be- +haviors of molecular state and compact multiquark. +In the current work, we try to propose a scheme to real- +ize such idea with the well known exotic particle X(3872) +as an example. Despite studied by hundreds of experimen- +tal physicists and theorists, the structure of the X(3872) is not +yet fully understood. The most preferred interpretation of the +structure of the X(3872) nowadays is c¯c − D ¯D∗ mixing state +[7, 8]. There also other interpretations, such as pure molec- +ular state [9–11], compact tetraquark structure (c¯cq¯q) of this +exotic states [12, 13], radial excitation of the P-wave char- +monium [14], and a vector glueball mixed with neighboring +vector states of charmonium [15]. Among these interpreta- +tions, except the non-genuine particle explanations such as +triangular singularities, the X(3872) is a compact quark pair +or tetraquark, or a molecular state, or their mixing. In this +work, we will study the behavior of the X(3872) attacked by +a nucleon in two pictures, compact quark state and molec- +ular state, to study collision of the X(3872) with proton as +p + X(3872) → p + ¯D0 + D0. +This article is organized as follows. After introduction, we +present the theoretical formalism to study the reaction of the +p + X(3872) → p + ¯D0 + D0 in two pictures in Section II. The +numerical results will be given in Section III. Finally, article +ends with a summary in section IV. +II. +COLLISION OF THE X(3872) WITH A PROTON +In the current work, we will consider the process p + +X(3872) → p + ¯D0 + D0 to detect the internal structure of +the X(3872) with a nucleon. The LHCb experiment defini- +tively established that the X(3872) has JPC = 1++ [16], which +means even if the X(3872) is the D0 ¯D∗0 molecular state, it +cannot decay into a D and ¯D meson due to the conservation +of spin parity. But collided by a proton may make this pro- +cess possible. In Ref. [17], we studied the nucleon-induced +fission-like process of the T + +cc. Under an assignment of the T + +cc +as molecular state, when induced by a proton, the T + +cc could +decay into a D and D pair. Since the T + +cc and X(3872) have +some similar features such as the small binding energy and +narrow width, one can legitimately predict that the X(3872) +is possible to decay into a D and ¯D pair [18]. Furthermore, +this reaction can be used to reveal underlying structures of the +X(3872). Here, we consider both pictures of molecular state +and compact quark state as shown in Fig. 1. +In the hypothesis of the X(3872) as a loosely molecular +state with wave function +� ¯D∗0D0 − D∗0 ¯D0� +/ +√ +2, its radius +supposed to be about 10 fm [19] (hereafter, we use the first +part of the wave function for explanations, the results for the +second part can be obtained analogously). The nucleon and +the constituent ¯D∗0 and D0 mesons has a radius smaller than +1 fm. Hence, the proton should attack on one of the con- +stituent meson of the X(3872). If we only consider the process +with three final particles, proton, D, and ¯D meson, the proton +should attack on the ¯D∗0 meson as shown in Fig. 1 (a). After +the ¯D∗0 meson attacked, it transformed to a ¯D0 meson. Since +the X(3872) is a loosely bound state, it forms a quasi-two- +body scattering, p ¯D∗0 → p ¯D0. The final ¯D0 meson should be +obviously affected by the energy transferred from incoming +proton and that released from the transition of ¯D∗0 to ¯D0 me- +son. However, another constituent of the X(3872), D0 meson, +should be little affected due to the weak binding. Such process +can be rewritten as the Feynman diagram in Fig. 1 (b). +If the X(3872) is a compact quark state, the collision behav- +ior is quite different as shown in Fig. 1 (c). The radius of the +X(3872) should be smaller than 1 fm for both quark-antiquark +pair [c¯c] and tetraquark [c¯cq¯q] explanations of the X(3872) +(In Fig. 1 (c) and hereafter, we mainly adopt the tetraquark +picture). Besides, such state is composed of quarks binding +tightly. In such picture, the X(3872) is excited by the collision +of the proton. The excited X(3872) decays to a ¯D0 meson and +a D0 meson. The energy transferred from the incoming pro- +ton will be acquired by both final mesons, which is different +from the molecular state picture. The Feynman diagram can +be written as Fig. 1 (d). +FIG. 1: The sketch map (a, c) and Feynman diagram (b,d) of reaction +p + X(3872) → p + ¯D0 + D0 with assumption of the X(3872) as a +loosely bound molecular state (a, b), or a compact multiquark state +(c, d). The denotations of the momenta of particles are also given. + +(q) +(c) +Do +X(bx) +Do +r'小b'm +b(bb) +b(μb) +(p) +Do(kDo) +Do +X(bx) +D*07 +Do +（ +r'b'm +b(b6) +b(kb3 +The collision above is difficult to be performed due to lack +of the X(3872) target or beam. Here, we propose to con- +sider the produced X(3872) in nucleon-rich environment. In- +stead of considering the initial proton and X(3872), three +final particles, proton, ¯D0 and D0 meson can be collected +with certain total invariant mass obtained as W = +√s = +√ +P2 = +� +(kp + k ¯D0 + kk0)2 with kp, ¯D0,D0 being the momenta +of final particles, which is independent of the coordinate +frames. +Among these events, Dalitz plot against invariant +masses mpD0 = +� +(pp + pD0)2 and mp ¯D0 = +� +(pp + p ¯D0)2 can +be obtained by selecting the corresponding event. In such +treatment, all observations are invariant. +The laboratory frame with the static X(3872) will be +adopted to perform explicit deviation and numerical calcula- +tion. In this reference frame, the cross section for the reaction +p + X(3872) → p + ¯D0 + D0 reads as, +dσ = +1 +4[(pp · pX)2 − m2pm2 +X]1/2 +1 +6 +� +λpλXλ′p +|MλpλX,λ′p|2dΦ3, +(1) +where the pp,X and mp,X being the momentum and mass of +the incoming proton or the X(3872). Practically, the GENEV +code in FAWL is adopted to generate the event of three body +final state by the Monte Carlo method, that is, the phase space +R3 = (2π)5dΦ3 = +3 +� +i +d3ki +2Ei +δ4 + +n +� +i +ki − P + , +where the ki and Ei are the momentum and energy of final +particle i. The mechanism can be described by an amplitude +MλpλX,λ′p with λ being the helicity of the incoming proton, +X(3872), or final proton. It will be derived with the Feynman +diagrams in Fig. 1. Here the interaction between the proton +and the X(3872) is described by light meson exchange. It has +the same form in two pictures, and can be obtained with the +help of effective Lagrangians, +LPNN = − gPNN +√ +2mN +¯Nbγ5γµ∂µPbaNa, +(2) +LVNN = − +√ +2gVNN ¯Nb +� +γµ + +κ +2mN +σµν∂ν� +Vµ +baNa, +(3) +where P and V are two by two pseudoscalar and vector ma- +trices. NT = (p, n) is field for nucleon. The coupling con- +stants g2 +πNN/(4π) = 13.6, g2 +ρNN/(4π) = 0.84, g2 +ωNN/(4π) = 20 +with κ = 6.1 (0) for ρ (ω) meson, which are used in the +Bonn nucleon-nucleon potential [20] and meson productions +in nucleon-nucelon collision [21–23]. The η exchange is ne- +glected in the current work due to the weak coupling of η or +φ messon to nucleons as indicated in many previous works +[20, 21]. Here, a factor fi(q2) = (m2 +i − Λ2)/(q2 − Λ2) is also +introduced to propagator of each exchanged meson with cut- +off Λ = 1 GeV. The left part of the amplitudes in two pictures +are different as given below. +In the molecular state picture as shown in Fig. 1 (a), the +exchanged light meson interacts with the ¯D∗0 meson in the +X(3872). In terms of heavy quark limit and chiral symmetry, +the corresponding Lagrangians have been constructed in the +literature as [24], +LP∗PP = −2g +fπ +(PbP∗† +aλ + P∗ +bλP† +a)∂λPba, +(4) +LP∗PV = −2 +√ +2λgVvλελαβµ(PbP∗µ† +a ++ P∗µ +b P† +a)∂αVβ +ba, (5) +where P(∗)T = (D(∗)0, D(∗)+) is the fields for D(∗) meson. The +parameters involved here were determined in the literature as +g = 0.59, β = 0.9, λ = 0.56 GeV−1, gV = 5.9, and fπ = +132 MeV [24, 25]. +In the molecular state picture, the amplitude AλX,λ ¯D∗0 for the +split of the X(3872) → ¯D∗0D0 for the first term of the wave +function as [17] +AλX,λ ¯D∗0 +p2 − m2 +¯D∗0 +≃ − +� +8mXm ¯D∗0mD0 +mX − mD0 + m ¯D∗0 ψ(k3)ǫλX · ǫ∗ +λ ¯D∗0, +(6) +where the λX, and λ ¯D∗0 are helicities for the initial X(3872) +state and intermediate ¯D∗0 meson. The p and m ¯D∗0 are the mo- +mentum and mass of intermediate ¯D∗0 meson. Here mX, ¯D∗0,D0 +is the mass of the X(3872), ¯D∗0 and D0. The ǫλX and ǫλ ¯D∗0 are +the polarized vectors of the X(3872) and ¯D∗0 meson, respec- +tively. Wave function ψ(k) = √8π/a/(k2 + 1/a2) with nor- +malization +� +d3k/(2π)3|ψ(k)|2 = 1 [26]. Scattering length a = +1/ +� +2µEB with the reduced mass µ = m ¯D∗0mD0/(m ¯D∗0 + mD0) +and the EB being the binding energy. +Different from the molecular state, the X(3872) has no ob- +vious subcluster in the compact quark state picture. In the +molecular state picture, the large radius and distance between +two constituents make the collision happens on one of the con- +stituents. If the X(3872) is a compact binding state of quarks +with small radius, the proton should attack on the X(3872) in +the whole, which will be excited by the light meson emitted by +the proton and decays into two D mesons. In the current work, +we do not consider its explicit mechanism. However, such in- +teraction should happen in a small space and short time, which +can be taken as a four particle vertex as shown in Fig. 1 (d). +Such vertex can be written as effective Lagragians, +LXPPP = gXXµ∂µPPP, +(7) +LXPPV = gXεαβζη∂αVβ∂ζXηPP. +(8) +Since no explicit mechanism is introduced, the coupling con- +stant gX for each exchange can not be determined, which will +be discussed later. +III. +NUMERICAL RESULTS +Since the incoming momentum can not be measured in the +scene of the nucleon-rich environment, we consider the pro- +cess p + X(3872) → p + ¯D0 + D0 with a total invariant +mass W = √s. With certain W, the event distribution can +be obtained as the Dalitz plot and invariant mass spectrum +against mpD0 and mp ¯D0. The different internal structures of +the X(3872) will exhibit in the Dalitz plot and invariant mass +spectrum. The experimental binding energy of the X(3872) +is every small and even above the thresholds. In the current + +4 +work, we would like to take it as an example to explain how +to detect the internal structure of an exotic hadron. Hence, +different values of binding energy will be adopted to show the +variation of the Dalitz plot and spectrum with the binding en- +ergy. The results in molecular state picture are shown in Fig. 2 +where the Dalitz plot in the mpD0-mp ¯D0 plane for the reaction +p + X(3872) → p + ¯D0 + D0 is calculated with different bind- +ing energies of the X(3872) as EB=0.1, 1, and 10 GeV. Corre- +spondingly, the invariant mass spectrum against mp ¯D0 is also +presented under each Dalitz plot. The momentum of incoming +proton pp = |pp| will affect the event distributions, and is cho- +sen as 0.1, 1 and 3 GeV, which corresponds to total invariant +mass W = 4.814, 5.147, and 6.341 GeV, respectively. +2.82 +2.88 +2.94 +2.82 +2.88 +2.94 +pp/W=0.1/4.814 GeV +mpD0(GeV) +10−8 +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 + 0.0 +40.0 +80.0 +2.82 +2.88 +2.94 + 0.0 +40.0 +80.0 +2.82 +2.88 +2.94 +EB = 0.1 MeV +dσ/mpD0 +2.82 +2.88 +2.94 +2.82 +2.88 +2.94 +EB = 1 MeV +2.82 +2.88 +2.94 +2.82 +2.88 +2.94 +EB = 10 MeV +2.80 +3.00 +3.20 +3.40 +pp/W=1/5.147 GeV +2.80 +3.00 +3.20 +3.40 +pp/W=1/5.147 GeV +mpD0(GeV) +10−8 +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +0.00 +0.80 +1.60 +2.80 +3.00 +3.20 +0.00 +0.80 +1.60 +2.80 +3.00 +3.20 +EB = 0.1 MeV +dσ/mpD0 +2.80 +3.00 +3.20 +2.80 +3.00 +3.20 +EB = 1 MeV +2.80 +3.00 +3.20 +3.40 +2.80 +3.00 +3.20 +3.40 +EB = 10 MeV +3.00 +3.50 +4.00 +4.50 +pp/W=3/6.341 GeV +3.00 +3.50 +4.00 +4.50 +pp/W=3/6.341 GeV +mpD0(GeV) +10−8 +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +0.00 +0.15 +0.30 +3.003.504.004.50 +0.00 +0.15 +0.30 +3.003.504.004.50 +EB = 0.1 MeV +dσ/mpD0 + mpD−0(GeV) +3.003.504.004.50 + mpD−0(GeV) +3.003.504.004.50 +EB = 1 MeV + mpD−0(GeV) +3.003.504.004.50 + mpD−0(GeV) +3.003.504.004.50 +EB = 10 MeV + mpD−0(GeV) +FIG. 2: Event distribution for the p+X(3872) → p+ ¯D0+D0 reaction +assuming the X(3872) as a loosely molecular state. The momentum +of incoming proton or total invariant mass pp/W = 0.1/4.814, 1/5.147 +and 3/6.341 GeV, and the binding energy EB = 0.1, 1 and 10 MeV, +respectively. For each example choice of pp/W, the figures represent +the Dalitz plot dσ/dmp ¯D0dmpD0 in the mp ¯D0 − mpD0 plane in a bin +of µb/0.002×0.002 GeV (upper pannel) and invariant mass spectrum +dσ/dmp ¯D0 against mpD0 in a bin of µb/0.002 GeV (lower panel). The +results are obtained with 1011 simulations. +Duo to the symmetry in the wave function, analogy distri- +butions for mp ¯D0 and mpD0 can be observed. For example, the +Dalitz plot in the first column of the third row in Fig. 2, with an +incoming momentum or total invariant mass pp/W = 3/6.341 +GeV, obvious and similar strips can be found at both mp ¯D0 and +mpD0 of about 4.19 GeV in the Dalitz plot. With the eleva- +tion of incoming proton pp/W, the phase space of the final +states becomes larger. The area of Dalitz plots in mpD0-mp ¯D0 +plane for pp/W=0.1/4.814 GeV is much smaller than these for +3/6.341 GeV. Hence, the results are rescaled, that is, different +ranges are adopted for three momenta. +The X(3872) has a very small binding energy. Here, differ- +ent binding energies are adopted to discuss the effect of the +binding energy on the Daltiz plot and invariant mass spec- +trum. The plots from the first to the third column in Fig. 2 are +for binding energies 0.1, 1, and 10 MeV, respectively (the re- +sults of 100 MeV were shown in the second column of Fig. 3, +and will be discussed later). Obvious concentration of events +can be found for the small binding energy, 0.1 MeV in the first +column. For pp/W of 1/4.814 and 3/6.341 GeV, sharper strips +can be found at mp ¯D0 about 3.2 and 4.2 GeV, respectively. The +invariant mass spectrum against mp ¯D0 also exhibit a very sharp +peak. Such concentration of events is from the mechanism of +collision as shown in Fig. 1 (a). The peak against invariant +mass mp ¯D0 is due to the quasifree D0 meson in the X(3872). +The energy from the incoming proton and the transition of ¯D∗0 +to ¯D0 meson is most carried by the final proton and ¯D0 meson. +The small binding energy means large radius, that is the large +distance and weak attraction between two constituents. With +the increase of the binding energy, the attraction of the con- +stituents becomes stronger, and the radius becomes smaller. +More momentum will be transferred to the D0 meson, which +will be dragged by the stronger attraction of ¯D∗0 meson. It is +confirmed by comparing the results in the first and the third +columns of Fig. 2, where with the increase of binding energy, +the strips in the Dalitz plots become vague, and the peaks in +the invariant mass spectra become wider also. Besides, be- +cause the radius decrease with the increase of the binding en- +ergy, the cross section also reduce. +The results with different incoming momenta or total invari- +ant masses are also presented in the first to third rows in Fig. 2. +Generally speaking, if the incoming proton moves fast, the +quasifree D0 meson will be less affected. It can be reflected +in the Dalitz plots where the strips become more sharp with +the increase of the momentum of the incoming proton. The +invariant mass spectra have more sharps peak (For binding +energy of 0.1 GeV, the peak is more sharp because the strips +are near the edge). However, the sharp peaks can be found for +all incoming momenta. Besides, the cross section decreases +rapidly with the increase of the incoming momentum due to +shorter interaction time. +Now we turn to the compact quark state picture. The Dalitz +plots and the invariant mass spectra with the different total +invariant masses are shown in the first column of Fig. 3. Be- +cause the coupling constant gX is not determined, we choose a +value of 30 GeV−1 to scale the invariant mass spectrum. The +result shows that with each momentum of incoming proton or +total invariant mass, the events were almost evenly distributed +in Dalitz plot and no strip as in the molecular state picture can +be found in the Dalitz plot. There is also no peak or struc- +ture in the invariant mass spectrum. If the binding energy of +a molecular state is very large, the small radius and strong + +5 +attraction between the constituents will make the behavior of +such state in the reaction considered similar to a compact state. +For comparison, in the second column of Fig. 3, the Dalitz +plots for the X(3872) as molecular state with a large bind- +ing energy EB = 100 MeV are also presented. As we can +see, these two cases give quite similar results especially in the +row with pp/W = 3/6.341 GeV, which is due to the hadronic +molecular state tends to a compact tetraquark state with in- +creasing binding energy. +2.820 +2.880 +2.940 +pp/W=0.1/4.814 GeV +2.820 +2.880 +2.940 +pp/W=0.1/4.814 GeV +mpD0(GeV) +10−8 +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +0.000 +0.050 +0.100 +0.150 +2.82 +2.88 +2.94 +0.000 +0.050 +0.100 +0.150 +2.82 +2.88 +2.94 +Compact +dσ/mpD0 +2.82 +2.88 +2.94 +2.82 +2.88 +2.94 +EB = 100 MeV +2.800 +3.000 +3.200 +3.400 +pp/W=1/5.147 GeV +2.800 +3.000 +3.200 +3.400 +pp/W=1/5.147 GeV +mpD0(GeV) +10−8 +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +0.000 +0.008 +0.016 +0.024 +2.80 +3.00 +3.20 +0.000 +0.008 +0.016 +0.024 +2.80 +3.00 +3.20 +Compact +dσ/mpD0 +2.80 +3.00 +3.20 +2.80 +3.00 +3.20 +EB = 100 MeV +3.000 +3.500 +4.000 +4.500 +pp/W=3/6.341 GeV +3.000 +3.500 +4.000 +4.500 +pp/W=3/6.341 GeV +mpD0(GeV) +10−8 +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +0.000 +0.004 +0.008 +0.012 +3.00 3.50 4.00 4.50 +0.000 +0.004 +0.008 +0.012 +3.00 3.50 4.00 4.50 +Compact +dσ/mpD0 + mpD−0(GeV) +3.00 3.50 4.00 4.50 + mpD−0(GeV) +3.00 3.50 4.00 4.50 +EB = 100 MeV + mpD−0(GeV) +FIG. 3: +Event distribution for the p + X(3872) → p + ¯D0 + D0 +reaction under the assumption of the X(3872) as a compact quark +state in the first column, and under the assumption of the X(3872) +as a molecular state with binding energy EB = 100 MeV in the +second column. +The pp/W is chosen as 0.1/4.814, 1/5.147 and +3/6.341 GeV. For each example choice of pp/W, the figures repre- +sent the Dalitz plot dσ/dmp ¯D0dmpD0 in the mp ¯D0 − mpD0 plane in a +bin of µb/0.002×0.002 GeV (upper panel) and invariant mass spec- +trum dσ/dmp ¯D0 against mpD0 in a bin of µb/0.002 GeV (lower panel). +The results are obtained with 1011 simulations. +The radius is an important metric for internal structure of +a hadronic state, which will also influence the probability of +collision of the incoming proton in our work. As we know, the +radius of a D or D∗ meson is smaller than 1 fm. For a molec- +ular state of binding energy smaller than 10 MeV, the distance +between the D and ¯D∗ meson in the X(3872) is larger than 1 +fm. In this case, the incoming proton will easily collide with +¯D∗ meson and a ¯D meson is produced such as shown in Fig. 2. +If the binding energy is larger than 10 MeV, the distance be- +tween the D and ¯D∗ meson in the X(3872) is smaller than 1 +fm, which makes the probability of collision of the incom- +ing proton with ¯D∗ in the X(3872) very small. The radius also +affect the relative values of the differential cross section of dif- +ferent total invariant masses W. Compared the results in Fig. 2 +and the second column of Fig. 3, the decrease of the invariant +mass spectra dσ/dmpD0 with the increase of pp/W becomes +slower if the binding energy becomes larger. If the X(3872) is +a compact state, the total cross section even increase with the +increase of the pp/W as shown in the first column of Fig. 3. +IV. +SUMMARY +In the current work, we propose a new possible reaction ap- +proach to reflect exotic structure of hadronic molecular state. +Take the X(3872) as an example, the Dalitz plot and the in- +variant mass spectrum are estimated for the p + X(3872) → +p+ ¯D0+D0. Two pictures of internal structure of the X(3872), +molecular state and compact multiquark state, are considered +in the calculation. In the molecular state picture, obvious strip +and peak can be observed in the Dalitz plot and invariant mass +spectrum, respectively. With the increase of binding energy, +the strip and peak vanish gradually, and the Dalitz plot and +invariant mass spectrum tends to these in the compact multi- +quark picture. +The strip and peak are obviously from the internal struc- +ture of the exotic state. As a molecular state, the large radius +and two-constituent structure makes the incoming proton only +attack on one constituent and another one remains almost un- +affected. However, a compact binding state of quarks, either +quark-antiquark pair or tetraquark, the exchanged light meson +should affect the state in the whole, and no obvious strip and +peak can be produced. Though in the current work we do not +consider the explicit mechanism of the reaction of the light +meson with the compact state, the conclusion should be unaf- +fected with different explicit models. Such conclusion can be +further confirmed with a calculation with an unphysical large +binding energy as 100 MeV, which corresponds to very small +radius. +Although the direct collision is unrealistic due to lack of the +exotic state target or beam, we suggest observing such phe- +nomenon in the production of exotic in a nucleon-rich envi- +ronment, such as at LHCb and PANDA. The produced exotic +state will interact with the surrounding nucleons and decay +into two final particles, for example, D0 and ¯D0 mesons here, +combined with a proton. With different total invariant masses, +the strips in Dalitz plots and the peaks in invariant mass spec- +tra appears in different invariant masses of two final particles. +Such proposal can provide a more determinative confirmation +of the molecular state structure of an exotic state and exclude +the compact multiquark assignment. + +6 +Acknowledgements +This work is supported by the China National Funds for +Distinguished Young Scientists under Grant No. 11825503, +the National Key Research and Development Program of +China under Contract No. 2020YFA0406400, the 111 Project +under Grant No. B20063, the National Natural Science Foun- +dation of China under Grant No. 12247101, No. 12175091, +No. +11965016, No. +11775050, No. +11775050, and No. +11675228, CAS Interdisciplinary Innovation Team, the Fun- +damental Research Funds for the Central Universities under +Grants No. lzujbky-2021-sp24, and the project for top-notch +innovative talents of Gansu province. +[1] M. Tanabashi et al. 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Li, “Res- +onance model study of kaon production in baryon baryon reac- +tions for heavy ion collisions,” Phys. Rev. C 59, 369-387 (1999) +[erratum: Phys. Rev. C 61, 029903 (2000)] +[23] A. Engel, A. K. Dutt-Mazumder, R. Shyam and U. Mosel, “Pion +production in proton proton collisions in a covariant one boson +exchange model,” Nucl. Phys. A 603, 387-414 (1996) +[24] R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, +F. Feruglio and G. Nardulli, “Phenomenology of heavy meson +chiral Lagrangians,” Phys. Rept. 281 (1997), 145-238 +[25] R. Chen, Z. F. Sun, X. Liu and S. L. Zhu, “Strong LHCb evi- +dence supporting the existence of the hidden-charm molecular +pentaquarks,” Phys. Rev. D 100, no.1, 011502 (2019) +[26] M. B. Voloshin, “Interference and binding effects in decays of +possible molecular component of X(3872),” Phys. Lett. B 579, +316-320 (2004) + diff --git a/otAzT4oBgHgl3EQfOftW/content/tmp_files/load_file.txt b/otAzT4oBgHgl3EQfOftW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e935dcce5bb8059631c57091bd4423cc73865850 --- /dev/null +++ b/otAzT4oBgHgl3EQfOftW/content/tmp_files/load_file.txt @@ -0,0 +1,697 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf,len=696 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='01166v1 [hep-ph] 3 Jan 2023 New reaction approach to reflect exotic structure of hadronic molecular state Zuo-Ming Ding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1 Jun He,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' ∗ and Xiang Liu2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' † 1School of Physics and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Nanjing Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Nanjing 210097,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' China 2Lanzhou Center for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Lanzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Lanzhou 730000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' China 3School of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Lanzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Lanzhou 730000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' China 4Key Laboratory of Theoretical Physics of Gansu Province,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' and Frontiers Science Center for Rare Isotopes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Lanzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Lanzhou 730000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' China 5Research Center for Hadron and CSR Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Lanzhou University and Institute of Modern Physics of CAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Lanzhou 730000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' China (Dated: January 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 2023) With the accumulation of the experimental data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' more and more exotic hadrons are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Among the interpretations of these exotic hadrons, molecular state and compact multiquark are two of the most popular pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' However, it is still difficult to determine the structure of an exotic hadron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In this work, we propose a possible way to detect the internal structure of an exotic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' When a molecular state composed of two consistent hadrons is attacked by another particle, one of the constituent should be kicked out while another quasifree constituent keeps almost unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' It is different from a compact multiquark which has no obvious subcluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In this work, take the X(3872) as example, we perform a Dalitz plot analysis of such reaction to find the effect of the different internal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Under the assumption of the X(3872) as a molecular state or a compact tetraquark state, with the help of the effective Lagrangians, the Dalitz plot and the invariant mass spectrum are estimated with different total invariant mass of three final particles, and the effect of different binding energies is also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Obvious event concentration can be observed as strips in the Dalitz plot and sharp peaks in the invariant mass spectrum for the X(3872) with a small binding energy under molecular state picture while such concentration can not be observed under compact tetraquark picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Such phenomenon can be applied to identify the internal structure of new hadron state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' INTRODUCTION The study of exotic hadrons is one of the most impor- tant topics in hadron physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Theoretically, the basic theory of the strong interaction, quantum chromodynamics (QCD), allows the existence of exotic hadrons beyond the conven- tional picture where the hadrons are composed of three quarks or a quark-antiquark pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Experimentally, with the devel- opment of the experimental techniques and accumulation of data, more and more exotic particles are observed but can not be put into the frames of the conventional quark model [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' If we deem these new particle as a genuine state composed of quarks, there exist two main interpretations, compact mul- tiquark and hadronic molecular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' For most of the exotic states, both interpretations exist simultaneously in the litera- ture [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' It is an interesting and difficult problem to determine the real internal structures of an exotic hadron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The molecular state is a loosely bound state of two hadrons [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Such idea is considerably easy to be under- stood if we take the deuteron as a molecular state composed of two hadrons, that is, nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' It is also natural to ex- pect the existence of bound states from other hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The experimental observation seems to support such assumptions also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Considerable XYZ particles are close to the thresh- olds of two hadrons [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The most popular interpretation about such phenomenon is that these particles are composed of the corresponding hadrons with a small binding energy as deuteron [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The hadronic molecular state is in fact a pic- ∗Electronic address: junhe@njnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='cn (Corresponding author) †Electronic address: xiangliu@lzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='cn (Corresponding author) ture in hadron level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Different from the molecular state, the compact multiquark is a real bound state of quarks [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In a compact multiquark, no obvious subcluster can be found, and its radius is usually assumed to be much smaller than a molecular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Theoretically, the mass of a compact multi- quark is irrelevant to the thresholds of hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' It seems to be used to judge whether an exotic hadron is a molecular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' However, practically, due to the uncertainty of both theory and experiment, many exotic states near thresholds can be also ex- plained as a compact multiquark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Moreover, the multiquark is still important picture to explain the states which is far from any threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In the literature, the masses and the decay patterns are the most important ways to detect the internal structure of an ex- otic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' However, as said above, the accordance of the the- oretical mass with the experimental mass is not enough to de- termine the internal structure of an exotic state because it of- ten can be explained in both pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The decay pattern is most promising to reflect the quark distributions in the exotic hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' However, the uncertainties from both experiment and theory make it difficult to reach a determinative conclu- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Hence, it is helpful to find more ways to detect the inter- nal structure of the exotic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The main difference between a molecular state and a com- pact multiquark is the spatial distribution of the quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In a molecular state, the quarks are grouped into two hadrons which have a distance of several even more than ten fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Be- sides effects on the decay pattern, such structure should be reflected when being attacked by a particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' For a molecular state, the incoming particle, which is usually about 1 fm can be easily attacked into the molecular state even pass through it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Since the distance of two constituent hadrons separated by long distances, the collision happens only on one constituent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 2 When a constituent hadron is attacked, another one should be little affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' However, for a compact multiquark, which is usually about 1 fm, the results of the collision of incom- ing particle is to excite the multiquark and induce its decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Due to the compactness, the momenta of the incoming parti- cle should be transferred to all quarks and then all final par- ticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Hence, the molecular state should have quite different behavior after collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Such difference of the behavior of collision should be a promising way to detect the internal structure of exotic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' However, there is an obvious difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' We do not have enough stable exotic states to make a target or a beam to per- form a collision with another particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The direct measure- ment of such collision is impossible with the current or near future experimental technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' However, such collisions can happen in the production of the exotic hadrons in a nucleon- rich environment, which is the realistic scene at facilities, such as LHCb and PANDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' If we can extract the information of collision of the nucleon with produced exotic states, it is still promising to obtain enough events to study such different be- haviors of molecular state and compact multiquark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In the current work, we try to propose a scheme to real- ize such idea with the well known exotic particle X(3872) as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Despite studied by hundreds of experimen- tal physicists and theorists, the structure of the X(3872) is not yet fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The most preferred interpretation of the structure of the X(3872) nowadays is c¯c − D ¯D∗ mixing state [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' There also other interpretations, such as pure molec- ular state [9–11], compact tetraquark structure (c¯cq¯q) of this exotic states [12, 13], radial excitation of the P-wave char- monium [14], and a vector glueball mixed with neighboring vector states of charmonium [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Among these interpreta- tions, except the non-genuine particle explanations such as triangular singularities, the X(3872) is a compact quark pair or tetraquark, or a molecular state, or their mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In this work, we will study the behavior of the X(3872) attacked by a nucleon in two pictures, compact quark state and molec- ular state, to study collision of the X(3872) with proton as p + X(3872) → p + ¯D0 + D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' This article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' After introduction, we present the theoretical formalism to study the reaction of the p + X(3872) → p + ¯D0 + D0 in two pictures in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The numerical results will be given in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Finally, article ends with a summary in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' COLLISION OF THE X(3872) WITH A PROTON In the current work, we will consider the process p + X(3872) → p + ¯D0 + D0 to detect the internal structure of the X(3872) with a nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The LHCb experiment defini- tively established that the X(3872) has JPC = 1++ [16], which means even if the X(3872) is the D0 ¯D∗0 molecular state, it cannot decay into a D and ¯D meson due to the conservation of spin parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' But collided by a proton may make this pro- cess possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' [17], we studied the nucleon-induced fission-like process of the T + cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Under an assignment of the T + cc as molecular state, when induced by a proton, the T + cc could decay into a D and D pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Since the T + cc and X(3872) have some similar features such as the small binding energy and narrow width, one can legitimately predict that the X(3872) is possible to decay into a D and ¯D pair [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Furthermore, this reaction can be used to reveal underlying structures of the X(3872).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Here, we consider both pictures of molecular state and compact quark state as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In the hypothesis of the X(3872) as a loosely molecular state with wave function � ¯D∗0D0 − D∗0 ¯D0� / √ 2, its radius supposed to be about 10 fm [19] (hereafter, we use the first part of the wave function for explanations, the results for the second part can be obtained analogously).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The nucleon and the constituent ¯D∗0 and D0 mesons has a radius smaller than 1 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Hence, the proton should attack on one of the con- stituent meson of the X(3872).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' If we only consider the process with three final particles, proton, D, and ¯D meson, the proton should attack on the ¯D∗0 meson as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' After the ¯D∗0 meson attacked, it transformed to a ¯D0 meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Since the X(3872) is a loosely bound state, it forms a quasi-two- body scattering, p ¯D∗0 → p ¯D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The final ¯D0 meson should be obviously affected by the energy transferred from incoming proton and that released from the transition of ¯D∗0 to ¯D0 me- son.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' However, another constituent of the X(3872), D0 meson, should be little affected due to the weak binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Such process can be rewritten as the Feynman diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' If the X(3872) is a compact quark state, the collision behav- ior is quite different as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The radius of the X(3872) should be smaller than 1 fm for both quark-antiquark pair [c¯c] and tetraquark [c¯cq¯q] explanations of the X(3872) (In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 1 (c) and hereafter, we mainly adopt the tetraquark picture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Besides, such state is composed of quarks binding tightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In such picture, the X(3872) is excited by the collision of the proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The excited X(3872) decays to a ¯D0 meson and a D0 meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The energy transferred from the incoming pro- ton will be acquired by both final mesons, which is different from the molecular state picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The Feynman diagram can be written as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 1 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 1: The sketch map (a, c) and Feynman diagram (b,d) of reaction p + X(3872) → p + ¯D0 + D0 with assumption of the X(3872) as a loosely bound molecular state (a, b), or a compact multiquark state (c, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The denotations of the momenta of particles are also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=" (q) (c) Do X(bx) Do r'小b'm b(bb) b(μb) (p) Do(kDo) Do X(bx) D*07 Do （ r'b'm b(b6) b(kb3 The collision above is difficult to be performed due to lack of the X(3872) target or beam." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Here, we propose to con- sider the produced X(3872) in nucleon-rich environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In- stead of considering the initial proton and X(3872), three final particles, proton, ¯D0 and D0 meson can be collected with certain total invariant mass obtained as W = √s = √ P2 = � (kp + k ¯D0 + kk0)2 with kp, ¯D0,D0 being the momenta of final particles, which is independent of the coordinate frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Among these events, Dalitz plot against invariant masses mpD0 = � (pp + pD0)2 and mp ¯D0 = � (pp + p ¯D0)2 can be obtained by selecting the corresponding event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In such treatment, all observations are invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The laboratory frame with the static X(3872) will be adopted to perform explicit deviation and numerical calcula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In this reference frame, the cross section for the reaction p + X(3872) → p + ¯D0 + D0 reads as, dσ = 1 4[(pp · pX)2 − m2pm2 X]1/2 1 6 � λpλXλ′p |MλpλX,λ′p|2dΦ3, (1) where the pp,X and mp,X being the momentum and mass of the incoming proton or the X(3872).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Practically, the GENEV code in FAWL is adopted to generate the event of three body final state by the Monte Carlo method, that is, the phase space R3 = (2π)5dΦ3 = 3 � i d3ki 2Ei δ4 \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed n � i ki − P \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 , where the ki and Ei are the momentum and energy of final particle i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The mechanism can be described by an amplitude MλpλX,λ′p with λ being the helicity of the incoming proton, X(3872), or final proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' It will be derived with the Feynman diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Here the interaction between the proton and the X(3872) is described by light meson exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' It has the same form in two pictures, and can be obtained with the help of effective Lagrangians, LPNN = − gPNN √ 2mN ¯Nbγ5γµ∂µPbaNa, (2) LVNN = − √ 2gVNN ¯Nb � γµ + κ 2mN σµν∂ν� Vµ baNa, (3) where P and V are two by two pseudoscalar and vector ma- trices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' NT = (p, n) is field for nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The coupling con- stants g2 πNN/(4π) = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='6, g2 ρNN/(4π) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='84, g2 ωNN/(4π) = 20 with κ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1 (0) for ρ (ω) meson, which are used in the Bonn nucleon-nucleon potential [20] and meson productions in nucleon-nucelon collision [21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The η exchange is ne- glected in the current work due to the weak coupling of η or φ messon to nucleons as indicated in many previous works [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Here, a factor fi(q2) = (m2 i − Λ2)/(q2 − Λ2) is also introduced to propagator of each exchanged meson with cut- off Λ = 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The left part of the amplitudes in two pictures are different as given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In the molecular state picture as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 1 (a), the exchanged light meson interacts with the ¯D∗0 meson in the X(3872).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In terms of heavy quark limit and chiral symmetry, the corresponding Lagrangians have been constructed in the literature as [24], LP∗PP = −2g fπ (PbP∗† aλ + P∗ bλP† a)∂λPba, (4) LP∗PV = −2 √ 2λgVvλελαβµ(PbP∗µ† a + P∗µ b P† a)∂αVβ ba, (5) where P(∗)T = (D(∗)0, D(∗)+) is the fields for D(∗) meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The parameters involved here were determined in the literature as g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='59, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='9, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='56 GeV−1, gV = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='9, and fπ = 132 MeV [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In the molecular state picture, the amplitude AλX,λ ¯D∗0 for the split of the X(3872) → ¯D∗0D0 for the first term of the wave function as [17] AλX,λ ¯D∗0 p2 − m2 ¯D∗0 ≃ − � 8mXm ¯D∗0mD0 mX − mD0 + m ¯D∗0 ψ(k3)ǫλX · ǫ∗ λ ¯D∗0, (6) where the λX, and λ ¯D∗0 are helicities for the initial X(3872) state and intermediate ¯D∗0 meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The p and m ¯D∗0 are the mo- mentum and mass of intermediate ¯D∗0 meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Here mX, ¯D∗0,D0 is the mass of the X(3872), ¯D∗0 and D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The ǫλX and ǫλ ¯D∗0 are the polarized vectors of the X(3872) and ¯D∗0 meson, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Wave function ψ(k) = √8π/a/(k2 + 1/a2) with nor- malization � d3k/(2π)3|ψ(k)|2 = 1 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Scattering length a = 1/ � 2µEB with the reduced mass µ = m ¯D∗0mD0/(m ¯D∗0 + mD0) and the EB being the binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Different from the molecular state, the X(3872) has no ob- vious subcluster in the compact quark state picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In the molecular state picture, the large radius and distance between two constituents make the collision happens on one of the con- stituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' If the X(3872) is a compact binding state of quarks with small radius, the proton should attack on the X(3872) in the whole, which will be excited by the light meson emitted by the proton and decays into two D mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In the current work, we do not consider its explicit mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' However, such in- teraction should happen in a small space and short time, which can be taken as a four particle vertex as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 1 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Such vertex can be written as effective Lagragians, LXPPP = gXXµ∂µPPP, (7) LXPPV = gXεαβζη∂αVβ∂ζXηPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' (8) Since no explicit mechanism is introduced, the coupling con- stant gX for each exchange can not be determined, which will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' NUMERICAL RESULTS Since the incoming momentum can not be measured in the scene of the nucleon-rich environment, we consider the pro- cess p + X(3872) → p + ¯D0 + D0 with a total invariant mass W = √s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' With certain W, the event distribution can be obtained as the Dalitz plot and invariant mass spectrum against mpD0 and mp ¯D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The different internal structures of the X(3872) will exhibit in the Dalitz plot and invariant mass spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The experimental binding energy of the X(3872) is every small and even above the thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In the current 4 work, we would like to take it as an example to explain how to detect the internal structure of an exotic hadron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Hence, different values of binding energy will be adopted to show the variation of the Dalitz plot and spectrum with the binding en- ergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The results in molecular state picture are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 2 where the Dalitz plot in the mpD0-mp ¯D0 plane for the reaction p + X(3872) → p + ¯D0 + D0 is calculated with different bind- ing energies of the X(3872) as EB=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1, 1, and 10 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Corre- spondingly, the invariant mass spectrum against mp ¯D0 is also presented under each Dalitz plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The momentum of incoming proton pp = |pp| will affect the event distributions, and is cho- sen as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1, 1 and 3 GeV, which corresponds to total invariant mass W = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='814, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='147, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='341 GeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='94 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='94 pp/W=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='814 GeV mpD0(GeV) 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='94 EB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1 MeV dσ/mpD0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='94 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='94 EB = 1 MeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='94 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='94 EB = 10 MeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='40 pp/W=1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='147 GeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='40 pp/W=1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='147 GeV mpD0(GeV) 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='20 EB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1 MeV dσ/mpD0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='20 EB = 1 MeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='40 EB = 10 MeV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 pp/W=3/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='341 GeV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 pp/W=3/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='341 GeV mpD0(GeV) 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 EB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1 MeV dσ/mpD0 mpD−0(GeV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 mpD−0(GeV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 EB = 1 MeV mpD−0(GeV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 mpD−0(GeV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 EB = 10 MeV mpD−0(GeV) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 2: Event distribution for the p+X(3872) → p+ ¯D0+D0 reaction assuming the X(3872) as a loosely molecular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The momentum of incoming proton or total invariant mass pp/W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='814, 1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='147 and 3/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='341 GeV, and the binding energy EB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1, 1 and 10 MeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' For each example choice of pp/W, the figures represent the Dalitz plot dσ/dmp ¯D0dmpD0 in the mp ¯D0 − mpD0 plane in a bin of µb/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='002×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='002 GeV (upper pannel) and invariant mass spectrum dσ/dmp ¯D0 against mpD0 in a bin of µb/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='002 GeV (lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The results are obtained with 1011 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Duo to the symmetry in the wave function, analogy distri- butions for mp ¯D0 and mpD0 can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' For example, the Dalitz plot in the first column of the third row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 2, with an incoming momentum or total invariant mass pp/W = 3/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='341 GeV, obvious and similar strips can be found at both mp ¯D0 and mpD0 of about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='19 GeV in the Dalitz plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' With the eleva- tion of incoming proton pp/W, the phase space of the final states becomes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The area of Dalitz plots in mpD0-mp ¯D0 plane for pp/W=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='814 GeV is much smaller than these for 3/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='341 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Hence, the results are rescaled, that is, different ranges are adopted for three momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The X(3872) has a very small binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Here, differ- ent binding energies are adopted to discuss the effect of the binding energy on the Daltiz plot and invariant mass spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The plots from the first to the third column in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 2 are for binding energies 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1, 1, and 10 MeV, respectively (the re- sults of 100 MeV were shown in the second column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 3, and will be discussed later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Obvious concentration of events can be found for the small binding energy, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1 MeV in the first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' For pp/W of 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='814 and 3/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='341 GeV, sharper strips can be found at mp ¯D0 about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='2 GeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The invariant mass spectrum against mp ¯D0 also exhibit a very sharp peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Such concentration of events is from the mechanism of collision as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The peak against invariant mass mp ¯D0 is due to the quasifree D0 meson in the X(3872).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The energy from the incoming proton and the transition of ¯D∗0 to ¯D0 meson is most carried by the final proton and ¯D0 meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The small binding energy means large radius, that is the large distance and weak attraction between two constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' With the increase of the binding energy, the attraction of the con- stituents becomes stronger, and the radius becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' More momentum will be transferred to the D0 meson, which will be dragged by the stronger attraction of ¯D∗0 meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' It is confirmed by comparing the results in the first and the third columns of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 2, where with the increase of binding energy, the strips in the Dalitz plots become vague, and the peaks in the invariant mass spectra become wider also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Besides, be- cause the radius decrease with the increase of the binding en- ergy, the cross section also reduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The results with different incoming momenta or total invari- ant masses are also presented in the first to third rows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Generally speaking, if the incoming proton moves fast, the quasifree D0 meson will be less affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' It can be reflected in the Dalitz plots where the strips become more sharp with the increase of the momentum of the incoming proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The invariant mass spectra have more sharps peak (For binding energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1 GeV, the peak is more sharp because the strips are near the edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' However, the sharp peaks can be found for all incoming momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Besides, the cross section decreases rapidly with the increase of the incoming momentum due to shorter interaction time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Now we turn to the compact quark state picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The Dalitz plots and the invariant mass spectra with the different total invariant masses are shown in the first column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Be- cause the coupling constant gX is not determined, we choose a value of 30 GeV−1 to scale the invariant mass spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The result shows that with each momentum of incoming proton or total invariant mass, the events were almost evenly distributed in Dalitz plot and no strip as in the molecular state picture can be found in the Dalitz plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' There is also no peak or struc- ture in the invariant mass spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' If the binding energy of a molecular state is very large, the small radius and strong 5 attraction between the constituents will make the behavior of such state in the reaction considered similar to a compact state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' For comparison, in the second column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 3, the Dalitz plots for the X(3872) as molecular state with a large bind- ing energy EB = 100 MeV are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' As we can see, these two cases give quite similar results especially in the row with pp/W = 3/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='341 GeV, which is due to the hadronic molecular state tends to a compact tetraquark state with in- creasing binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='820 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='880 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='940 pp/W=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='814 GeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='820 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='880 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='940 pp/W=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='814 GeV mpD0(GeV) 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='150 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='150 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='94 Compact dσ/mpD0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='94 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='94 EB = 100 MeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='800 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='200 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='400 pp/W=1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='147 GeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='800 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='200 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='400 pp/W=1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='147 GeV mpD0(GeV) 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='024 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='024 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='20 Compact dσ/mpD0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='20 EB = 100 MeV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='500 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='000 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='500 pp/W=3/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='341 GeV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='500 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='000 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='500 pp/W=3/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='341 GeV mpD0(GeV) 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='012 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='012 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 Compact dσ/mpD0 mpD−0(GeV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 mpD−0(GeV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='50 EB = 100 MeV mpD−0(GeV) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 3: Event distribution for the p + X(3872) → p + ¯D0 + D0 reaction under the assumption of the X(3872) as a compact quark state in the first column, and under the assumption of the X(3872) as a molecular state with binding energy EB = 100 MeV in the second column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The pp/W is chosen as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='814, 1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='147 and 3/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='341 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' For each example choice of pp/W, the figures repre- sent the Dalitz plot dσ/dmp ¯D0dmpD0 in the mp ¯D0 − mpD0 plane in a bin of µb/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='002×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='002 GeV (upper panel) and invariant mass spec- trum dσ/dmp ¯D0 against mpD0 in a bin of µb/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content='002 GeV (lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The results are obtained with 1011 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The radius is an important metric for internal structure of a hadronic state, which will also influence the probability of collision of the incoming proton in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' As we know, the radius of a D or D∗ meson is smaller than 1 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' For a molec- ular state of binding energy smaller than 10 MeV, the distance between the D and ¯D∗ meson in the X(3872) is larger than 1 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In this case, the incoming proton will easily collide with ¯D∗ meson and a ¯D meson is produced such as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' If the binding energy is larger than 10 MeV, the distance be- tween the D and ¯D∗ meson in the X(3872) is smaller than 1 fm, which makes the probability of collision of the incom- ing proton with ¯D∗ in the X(3872) very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The radius also affect the relative values of the differential cross section of dif- ferent total invariant masses W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Compared the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 2 and the second column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 3, the decrease of the invariant mass spectra dσ/dmpD0 with the increase of pp/W becomes slower if the binding energy becomes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' If the X(3872) is a compact state, the total cross section even increase with the increase of the pp/W as shown in the first column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' SUMMARY In the current work, we propose a new possible reaction ap- proach to reflect exotic structure of hadronic molecular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Take the X(3872) as an example, the Dalitz plot and the in- variant mass spectrum are estimated for the p + X(3872) → p+ ¯D0+D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Two pictures of internal structure of the X(3872), molecular state and compact multiquark state, are considered in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' In the molecular state picture, obvious strip and peak can be observed in the Dalitz plot and invariant mass spectrum, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' With the increase of binding energy, the strip and peak vanish gradually, and the Dalitz plot and invariant mass spectrum tends to these in the compact multi- quark picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The strip and peak are obviously from the internal struc- ture of the exotic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' As a molecular state, the large radius and two-constituent structure makes the incoming proton only attack on one constituent and another one remains almost un- affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' However, a compact binding state of quarks, either quark-antiquark pair or tetraquark, the exchanged light meson should affect the state in the whole, and no obvious strip and peak can be produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Though in the current work we do not consider the explicit mechanism of the reaction of the light meson with the compact state, the conclusion should be unaf- fected with different explicit models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Such conclusion can be further confirmed with a calculation with an unphysical large binding energy as 100 MeV, which corresponds to very small radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Although the direct collision is unrealistic due to lack of the exotic state target or beam, we suggest observing such phe- nomenon in the production of exotic in a nucleon-rich envi- ronment, such as at LHCb and PANDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' The produced exotic state will interact with the surrounding nucleons and decay into two final particles, for example, D0 and ¯D0 mesons here, combined with a proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' With different total invariant masses, the strips in Dalitz plots and the peaks in invariant mass spec- tra appears in different invariant masses of two final particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' Such proposal can provide a more determinative confirmation of the molecular state structure of an exotic state and exclude the compact multiquark assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 6 Acknowledgements This work is supported by the China National Funds for Distinguished Young Scientists under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 11825503, the National Key Research and Development Program of China under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 2020YFA0406400, the 111 Project under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' B20063, the National Natural Science Foun- dation of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' 12247101, No.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} +page_content=' B 579, 316-320 (2004)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfOftW/content/2301.01166v1.pdf'} diff --git a/p9E5T4oBgHgl3EQfkw_7/content/tmp_files/2301.05666v1.pdf.txt b/p9E5T4oBgHgl3EQfkw_7/content/tmp_files/2301.05666v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f81cba9a5b8758718290578eea0bf7c07e8e2535 --- /dev/null +++ b/p9E5T4oBgHgl3EQfkw_7/content/tmp_files/2301.05666v1.pdf.txt @@ -0,0 +1,782 @@ +Beyond MP2 initialization for unitary coupled cluster quantum circuits +Mark R. Hirsbrunner,1, 2, ∗ Diana Chamaki,2 J. Wayne Mullinax,3, 4 and Norm M. Tubman4, † +1Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA +2USRA Research Institute for Advanced Computer Science, Mountain View, California 94043, USA +3KBR, Inc., NASA Ames Research Center, Moffett Field, California 94035, USA +4Intelligent Systems Division, NASA Ames Research Center, Moffett Field, California 94035, USA +The unitary coupled cluster singles and doubles (UCCSD) ansatz is a promising approach to +prepare accurate quantum states for use in quantum algorithms. In this paper, we compared the +performance of two methods for generating the initial UCCSD circuit parameters: CCSD and MP2. +Our results, obtained through an efficient sparse wavefunction circuit solver, show that UCCSD +circuits with CCSD parameterizations significantly outperformed those with MP2 parameterizations +for systems of up to 64 qubits. These findings suggest that CCSD should be the preferred choice +for generating initial parameters. +Introduction.— Simulating many-body fermionic sys- +tems is a promising future application of quantum com- +puting [1–3]. While it is not yet clear that quantum ad- +vantage can generically be achieved in this area [4], it +is believed that phase estimation can solve ground state +problems for molecular systems that are beyond the reach +of classical computers. However, it remains an open ques- +tion whether or not other approaches can achieve quan- +tum advantage with fewer resources [5–8]. Phase estima- +tion and other algorithms benefit from, or even require, +significant overlap between the trial quantum state and +the true solution. +Single Slater determinants, such as +Hartree-Fock states [9], are often used as the trial state +when solving for ground states, as they are assumed +to produce a sufficiently large overlap with the ground +state wavefunction in many cases [10–12]. Yet such sin- +gle determinant states may not be sufficient for arbitrar- +ily large system sizes [10, 13, 14]. Improving quantum +state preparation techniques is a key step toward ad- +vancing quantum computing for quantum chemistry and +other Hamiltonian simulation applications since many al- +gorithms require high-quality initial quantum states. +Noise and decoherence present another central diffi- +culty of achieving quantum advantage in the current +noisy intermediate-scale quantum (NISQ) era of quan- +tum hardware [15]. The variational quantum eigensolver +(VQE) is a quantum-classical hybrid algorithm that is +particularly well-suited for NISQ devices [16–18]. While +VQE does not provide exact ground state solutions like +quantum phase estimation, the approximate wavefunc- +tions produced by VQE are often sufficiently accurate to +provide meaningful physical insights. Furthermore, these +approximate solutions are well-suited for quantum state +preparation for use in more accurate algorithms [19]. +Despite its current popularity, VQE possesses a num- +ber of drawbacks. In particular, the classical optimiza- +tion of circuit parameters presents many challenges, in- +cluding barren plateaus (i.e., exponentially vanishing gra- +dients in high dimensions), local minima, and saddle +points [20–24]. Many approaches exist for minimizing the +computational burden of classical optimization for VQE, +with some proposals eschewing optimization entirely [25– +28]. The crux of several of these strategies is a focus on +choosing high-quality initial parameters, shifting some of +the computational burden from optimization to initial- +ization. In this work we compare the utility of different +initialization strategies for a particular VQE ansatz that +is often employed in quantum chemistry problems, the +unitary coupled cluster (UCC) ansatz [16, 19, 29–32]. +There are several proposed strategies for generating +the initial parameters for the UCC ansatz [17, 31, 33– +35], including applications in which no optimization is +performed on quantum hardware [28]. The most widely +employed strategy generates parameters using classical +Møller–Plesset perturbation theory of the second order +(MP2) [17, 31, 33, 34]. +Another less thoroughly stud- +ied approach is the use of the coupled cluster singles and +doubles (CCSD) classical simulation method to generate +initial parameters [36, 37]. The CCSD technique gener- +ally produces more accurate ground state energies than +MP2 calculations, yet CCSD is rarely employed to ini- +tialize VQE circuits. +This is curious, considering that +neither technique is computationally burdensome for all +but the largest of problems. +This raises the question: +Which technique produces superior initial parameters for +UCC ansatzes, MP2 or CCSD? +In this paper, we provide the first numerical study +(to our knowledge) comparing the performance of UCC +ansatzes prepared using parameters generated via MP2 +and via CCSD. We employ an algorithm for the factor- +ized form of UCC implemented using our state-of-the-art +sparse wavefunction circuit solver, enabling us to study +problems of up to 64 qubits [38, 39]. By calculating the +ground state energy of a wide range of molecules using +both MP2 and CCSD parameters in the UCC ansatz, +we show conclusively that CCSD parameters outperform +MP2, generating significantly more accurate ground state +energies. +Technique.— The UCC ansatz is an exponential oper- +ator acting on the Hartree–Fock reference wavefunction +arXiv:2301.05666v1 [quant-ph] 13 Jan 2023 + +2 +0 +20 +40 +60 +80 +100 +0 +NWF/1000 +-280 +-260 +E (mHar) +UCC (MP2) +UCC (CCSD) +CCSD +CCSD(T) +ASCI +(a) +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +1000/NWF +90.0 +92.5 +95.0 +97.5 +100.0 +E/EASCI (%) +UCC (MP2) +UCC (CCSD) +CCSD +CCSD(T) +(b) +FIG. 1. +(a) The UCC(MP2) (light blue solid line) and UCC(CCSD) (light purple dashed line) correlation energies of CH2O +as functions of NWF. The dark purple dot-dashed and black dotted lines denote the CCSD and CCSD(T) correlation energies, +respectively, and the black solid line marks the ASCI energy calculated using 106 determinants. (b) The UCC(MP2) (circles) +and UCC(CCSD) (pentagons) correlation energies as a fraction of the CCSD(T) correlation energy plotted versus 1/NWF. The +solid lines are quadratic fits to the UCC(CCSD) and UCC(MP2) data, fitted using the twenty data points with the largest +NWF. The dashed lines mark the y-intercepts of the fits. The dot-dashed and dotted lines indicate the ratio of the CCSD and +CCSD(T) correlation energies to the ASCI correlation energy. +defined as +|ΨUCC⟩ = exp +� +ˆT − ˆT †� +|Ψ0⟩ +(1) +where the coupled cluster operator ˆT is +ˆT = +occ +� +i +vir +� +a +θa +i ˆa† +aˆai + +occ +� +ij +vir +� +ab +θab +ij ˆa† +aˆa† +bˆajˆai + · · · +(2) +The ˆa† and ˆa operators are the second-quantized creation +and annihilation operators, respectively, acting on the +occupied molecular orbitals in the reference wavefunc- +tion indexed by i, j, . . . or the virtual orbitals indexed by +a, b, . . .. The parameters of the UCC ansatz are indicated +by θ. We employ the factorized form of the UCC ansatz, +which is given by +|ΨUCC⟩ = +occ +� +ij··· +vir +� +ab··· +ˆU ab··· +ij··· |Ψ0⟩ , +(3) +where the individual UCC exponential factors are defined +as +ˆU ab··· +ij··· = exp +� +θab··· +ij··· (ˆaab··· +ij··· · · · − ˆaij··· +ab···) +� +. +(4) +We only include single excitations (ˆaa +i = ˆa† +iˆaa) and dou- +ble excitations (ˆaab +ij = ˆa† +iˆa† +jˆabˆaa) in the ansatz, along with +the conjugate deexcitation operators ˆai +a and ˆaij +ab, an ap- +proximation to the full UCC ansatz known as the unitary +coupled cluster singles and doubles (UCCSD) ansatz. +We utilize a specific representation of the UCCSD +ansatz that exploits the fact that each UCC factor ˆU ab··· +ij··· +can be expressed in terms of sines and cosines of the +parameters that can be efficiently computed on classical +hardware [38]. The order of the individual UCCSD fac- +tors is not strictly defined [40], and we chose to order +them based on the magnitude of the parameter values +(|θ|), placing the factor ˆU ab··· +ij··· that contains the largest +parameter to the right in Equation (3). We refer to this +as the “magnitude” ordering. +We generate the conventional MP2 and CCSD UCCSD +parameters using PySCF, noting that the MP2 param- +eterization does not include any single excitation opera- +tors [41]. We use a computationally efficient sparse wave- +function approach, limiting the number of determinants +included in the wavefunction after each UCC factor is +applied [39]. We do this by checking the number of de- +terminants N in the wavefunction after applying each +UCC factor. +If N is greater than the desired number +of determinants, NWF, we sort the amplitudes by mag- +nitude and discard the determinants with the smallest +amplitudes such that only NWF determinants remain in +the wavefunction. +Results.— Here we report the correlation energies ob- + +3 +80 +85 +90 +95 +100 +E/Eref (%) +H8 +H10 +H12 +H14 +LiH +HF +NH3 +CH4 +H2O +N2 +F2 +CH2O +C2 +CCSD(T) +CCSD +UCC (CCSD) +UCC (MP2) +FIG. 2. The CCSD(T), CCSD, UCC(CCSD), and UCC(MP2) +correlation energies as percentages of the best available refer- +ence energy (FCI for hydrogen chains and LiH, ASCI other- +wise). +tained from UCCSD circuits parameterized using MP2 +and CCSD parameters for a wide range of molecules. +For the molecules LiH, HF, NH3, CH4, H2O, N2, F2, +and CH2O we use experimental geometries from the CC- +CBDB database and employ the cc-pCVDZ basis set [42]. +We also study the linear hydrogen chains H8, H10, H12, +and H14, for which we use an interatomic distance of 1 ˚A, +and a stretched geometry of H10 with an interatomic dis- +tance of 1.5 ˚A, for all of which we employ the STO-6G +basis set. Our sparse wavefunction circuit solver is lim- +ited to 64 qubits, so we include only the 32 lowest-energy +molecular orbitals in each molecule [43]. Our approach +has similar scaling to a time-dependent selected configu- +ration interaction approach, which some of us have ap- +plied to larger systems in other contexts [25, 26]. +Because we limit the number of determinants retained +in the wavefunction to NWF, we must study the depen- +dence of the correlation energies on NWF and extrapo- +late to the large-NWF limit to obtain a converged result. +Specifically, we calculate the correlation energy as a func- +tion of NWF up to NWF =100,000 for each molecule, as +shown in Fig. 1a for CH2O. We extrapolate to the large- +NWF limit via a quadratic fitting of the data as a function +of N −1 +WF, +E = aN −2 +WF + bN −1 +WF + c, +(5) +as shown in Fig. 1b. The fit accounts for the twenty data +points at the largest values of NWF, which produces the +fit parameters. The y-intercept of the quadratic fit is the +extrapolated correlation energy that would be obtained if +we pruned no determinants during the calculation. Thus +this is a prediction of the energy that would be produced +on perfect quantum hardware. We refer to these extrapo- +lated energies as the UCC(MP2) and UCC(CCSD) corre- +lation energies, depending on the initial parameters used +in the circuit. +We +report +the +CCSD(T), +CCSD, +UCC(MP2), +UCC(CCSD), and full configuration interaction (FCI) +correlation energies for the hydrogen chains and LiH +in Table I. Calculating the FCI energy for the remain- +ing molecules is impractical, so we instead report the +adaptive sampling configuration interaction (ASCI) cor- +relation energies [44] for these molecules in Table II, +along with the CCSD(T), CCSD, UCC(MP2), +and +UCC(CCSD) correlation energies [45]. We also plot these +energies as fractions of the best reference energy, ei- +ther FCI or ASCI, for each molecule in Fig. 2. +The +UCC(CCSD) energy outperforms the UCC(MP2) energy +by a wide margin in all cases, with a difference of ap- +proximately 15% of the reference energy for the hydrogen +chains (including stretched H10) and differences ranging +from 1.3% to 9.6% for the remaining molecules. +Because the individual terms in the factorized form of +the UCCSD ansatz do not necessarily commute, the or- +dering of the operators can have a significant impact on +the accuracy of the ansatz [40]. To address this concern, + +4 +we calculate the correlation energy of the molecules we +study in this work using 100 random orderings of the +UCCSD factors. We find that the standard deviation of +the correlation energy is less than 0.1 mHa for molecules +with equilibrium geometries, with the exception of F2 +and C2, the standard deviations of which are 0.1 mHa +and 0.4 mHa, respectively. Only the strongly correlated +stretched geometry of H10 has a significant standard devi- +ation of 2.4 mHa. We set NWF to 10,000 for these calcu- +lations to reduce the computational burden, which likely +artificially inflates the standard deviations. We conclude +that factor ordering is significant only for strongly corre- +lated molecules, in agreement with previous studies [40]. +Importantly, we find that the magnitude ordering ob- +tains energies close to the minimum energy produced by +random orderings for all molecules besides CH2O, for +which the magnitude ordering produced an energy ap- +proximately 0.15 mHa above the minimum. +The UCC(CCSD) energy closely matches the CCSD +energy for all molecules studied, with the exception of +stretched H10, and produces energies lower than the +CCSD energy for HF, H2O, and C2. However, the MP2 +and UCC(MP2) energies do not exhibit the remarkably +good agreement obtained by the CCSD and UCC(CCSD) +energies. +Excluding the results for stretched H10, the +differences between the CCSD and UCC(CCSD) ener- +gies range between 0.0% and 2.1% with an average of +0.4% for the molecules we study, while for MP2 and +UCC(MP2) they range between 0.4% and 19.7% with an +average of 9.2%. These statistics show a clear advantage +for UCC(CCSD) As such, the CCSD parameterization +is likely better suited than MP2 for use in recent pro- +posals for no-optimization strategies to obtain quantum +advantage [28]. +Classical coupled cluster techniques have well-known +properties in which the obtained energies are not varia- +tional (dropping below the FCI results) or, even worse +energies can diverge away from the physical ground state +result. +One such failure scenario can be seen in the +chemistry of bond breaking, which we investigate here +using the H10 molecule with a stretched interatomic dis- +tance of 1.5 ˚A. The CCSD and CCSD(T) energies of +this molecule are lower than the FCI energy, representing +a well known problem in classical coupled cluster tech- +niques. Despite the failure of CCSD to produce an ac- +curate energy for this molecule, the UCC circuit param- +eterized with CCSD must produce a variational energy +because the VQE approach is a wavefunction technique, +where as classical coupled cluster approaches are not in +general. The UCC(CCSD) energy for stretched H10 is +12.2% higher than the FCI energy, compared to 1.7% +higher for the equilibrium geometry. These results show +that the UCC ansatz parameterized with CCSD is robust +to failures of the classical theory, but with some loss of +accuracy. Regardless, our results show a close correspon- +dence between UCC(CCSD) and CCSD theories and fur- +TABLE I. The FCI, CCSD(T), CCSD, UCC(CCSD), MP2, +and UCC(MP2) correlation energies of the hydrogen chains +and LiH. The UCC energies are obtained via the fitting pro- +cedure shown in Fig. 1. The row labeled H∗ +10 uses a stretched +geometry with an interatomic distance of 1.5 ˚A. All energies +are reported as absolute values and in units of milliHartrees. +Mol +FCI +CCSD(T) CCSD +UCC +(CCSD) +MP2 +UCC +(MP2) +H8 134.68 +134.65 +133.60 +133.00 +85.19 111.69 +H10 167.78 +167.64 +165.77 +164.86 +107.62 139.08 +H∗ +10 403.81 +434.55 +426.50 +354.74 +208.45 293.67 +H12 200.90 +200.62 +197.81 +196.62 +130.27 166.44 +H14 234.05 +233.60 +229.75 +228.92 +153.11 194.45 +LiH +64.75 +64.74 +64.69 +64.69 +51.80 +61.28 +TABLE II. The ASCI, CCSD(T), CCSD, UCC(CCSD), MP2, +and UCC(MP2) correlation energies of the larger molecules +for which FCI is impractical. The UCC energies are obtained +via the fitting procedure shown in Fig. 1. +All energies are +reported as absolute values and in units of milliHartrees. +Mol +ASCI CCSD(T) CCSD +UCC +(CCSD) +MP2 +UCC +(MP2) +HF +251.24 +250.75 +248.61 +249.03 +242.84 245.76 +NH3 +239.02 +238.42 +234.42 +233.12 +217.13 228.29 +CH4 +183.28 +182.72 +179.58 +178.98 +156.41 173.38 +H2O +255.71 +255.09 +251.79 +252.14 +241.37 247.68 +N2 +365.24 +363.53 +351.26 +351.13 +347.65 338.73 +F2 +456.66 +454.75 +445.42 +443.28 +436.34 430.58 +CH2O 284.90 +283.59 +275.63 +269.73 +261.75 260.67 +C2 +382.23 +380.23 +351.91 +352.21 +350.27 315.59 +ther study of this can help us understand the power of +coupled cluster approaches on quantum hardware. +Discussion.— In this paper we demonstrated through +extensive calculations that CCSD parameterizations of +the UCC ansatz consistently outperform their MP2 coun- +terparts. As such, it is important to compare the com- +putational costs of obtaining the CCSD and MP2 pa- +rameterizations. +Although MP2 is faster and, in fact, +often used as a starting point for coupled cluster simula- +tions, CCSD nevertheless requires only reasonable classi- +cal computation resources for even moderately sized sys- +tems. For example, the CCSD calculations presented in +this work and others run in minutes or less on a lap- +top [28, 35]. +MP2 and CCSD runtimes scale as O(N 5) and O(N 6), +respectively, making these prohibitively expensive algo- +rithms in the large-N qubit limit, but it is unlikely that +NISQ era quantum computers will exceed classically- +accessible simulations of CCSD in the next few years. +Classical coupled cluster simulations can be accelerated + +5 +in various ways [46, 47], indicating that simulations in- +volving hundreds of qubits to parameterize circuits is in +reach. Considering this, as well as the small prefactors of +these runtime scalings and the efficiency of modern imple- +mentations of these techniques, CCSD is poised to remain +an accessible and highly accurate method of UCC param- +eterization for the forseeable future of the NISQ era. As +such, our results suggest that CCSD should replace MP2 +as the standard approach to classically parameterizing +UCC circuits. +Our results also display the power of our sparse wave- +function circuit solver, which enables us to perform UCC +simulations at system sizes that have not been previously +explored. Because our solver is capable of handling up +to 64 qubit problems with its current implementation, +we are able to access a regime in which it is possible to +meaningfully test and differentiate VQE results. In this +case, the ability to access large systems sizes enabled us +to explore a widely used parameterization for UCC cir- +cuits and challenge conventional held ideas. +There are number of directions for future research. +Testing our approach with higher order coupled clus- +ter techniques on both the classical [48] and quantum +side [35] is one such direction. The correspondence we +identified between CCSD and UCC(CCSD) is weakened +when classical CCSD breaks down, as seen in for strongly +correlated molecules like stretched H10. +These results +motivate the study of more advanced classical approaches +to parameterize UCC-type circuits. Establishing the cor- +respondence between higher order classical coupled clus- +ter theories and the UCC analogues of them, such as a +UCC(CCSDT) circuit [35], would elucidate the full po- +tential of the UCC ansatz. +Acknowledgements.— We are grateful for support from +NASA Ames Research Center. +We acknowledge fund- +ing from the NASA ARMD Transformational Tools +and Technology (TTT) Project. +Part of this work is +funded by U.S. Department of Energy, Office of Science, +National Quantum Information Science Research Cen- +ters, Co-Design Center for Quantum Advantage under +Contract No. +DE-SC0012704. +Calculations were per- +formed as part of the XSEDE computational Project +No. +TG-MCA93S030 on Bridges-2 at the Pittsburgh +supercomputer center. +M.H and D.C. were supported +by NASA Academic Mission Services, Contract No. +NNA16BD14C. M.H. and D.C. participated in the Feyn- +man Quantum Academy internship program. +∗ hrsbrnn2@illinois.edu +† norman.m.tubman@nasa.gov +[1] R. P. 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Phys. 79, 291 +(2007). +[38] J. Chen, H.-P. Cheng, and J. K. Freericks, Journal of +Chemical Theory and Computation 17, 841 (2021). +[39] N. M. Tubman and W. J. Mullinax, In preparation +(2022). +[40] H. +R. +Grimsley, +D. +Claudino, +S. +E. +Economou, +E. Barnes, and N. J. Mayhall, Journal of Chemical The- +ory and Computation 16, 1 (2020), pMID: 31841333, +https://doi.org/10.1021/acs.jctc.9b01083. +[41] Q. Sun, X. Zhang, S. Banerjee, P. Bao, M. Barbry, N. S. +Blunt, N. A. Bogdanov, G. H. Booth, J. Chen, Z.-H. +Cui, J. J. Eriksen, Y. Gao, S. Guo, J. Hermann, M. R. +Hermes, K. Koh, P. Koval, S. Lehtola, Z. Li, J. Liu, +N. Mardirossian, J. D. McClain, M. Motta, B. Mussard, +H. Q. Pham, A. Pulkin, W. Purwanto, P. J. Robin- +son, E. Ronca, E. R. Sayfutyarova, M. Scheurer, H. F. +Schurkus, J. E. T. Smith, C. Sun, S.-N. Sun, S. Upad- +hyay, L. K. Wagner, X. Wang, A. White, J. D. Whitfield, +M. J. Williamson, S. Wouters, J. Yang, J. M. Yu, T. Zhu, +T. C. Berkelbach, S. Sharma, A. Y. Sokolov, and G. K.- +L. Chan, The Journal of Chemical Physics 153, 024109 +(2020), https://doi.org/10.1063/5.0006074. +[42] Nist computational chemistry comparison and bench- +mark database, NIST Standard Reference Database +Number 101 (2022). +[43] This limitation of our solver is not algorithmic and future +implementations can be expanded beyond 64 qubits. +[44] We use 106 determinants for all ASCI calculations, at +which the energies are converged. +[45] N. M. Tubman, C. D. Freeman, D. S. Levine, D. Hait, +M. Head-Gordon, and K. B. Whaley, Journal of chemical +theory and computation 16, 2139 (2020). +[46] C. +Riplinger +and +F. +Neese, +The +Journal +of +Chemical +Physics +138, +034106 +(2013), +https://doi.org/10.1063/1.4773581. +[47] I. +A. +Kaliman +and +A. +I. +Krylov, +Journal +of +Computational +Chemistry +38, +842 +(2017), +https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.24713. +[48] N. M. Tubman, C. D. Freeman, D. S. Levine, D. Hait, +M. Head-Gordon, and K. B. Whaley, Journal of Chem- +ical Theory and Computation 16, 2139 (2020), pMID: +32159951, https://doi.org/10.1021/acs.jctc.8b00536. + diff --git a/p9E5T4oBgHgl3EQfkw_7/content/tmp_files/load_file.txt b/p9E5T4oBgHgl3EQfkw_7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0c99cc2f5533c0010a5a4dd5938a6bac44707027 --- /dev/null +++ b/p9E5T4oBgHgl3EQfkw_7/content/tmp_files/load_file.txt @@ -0,0 +1,753 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf,len=752 +page_content='Beyond MP2 initialization for unitary coupled cluster quantum circuits Mark R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Hirsbrunner,1, 2, ∗ Diana Chamaki,2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Wayne Mullinax,3, 4 and Norm M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Tubman4, † 1Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 2USRA Research Institute for Advanced Computer Science, Mountain View, California 94043, USA 3KBR, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=', NASA Ames Research Center, Moffett Field, California 94035, USA 4Intelligent Systems Division, NASA Ames Research Center, Moffett Field, California 94035, USA The unitary coupled cluster singles and doubles (UCCSD) ansatz is a promising approach to prepare accurate quantum states for use in quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' In this paper, we compared the performance of two methods for generating the initial UCCSD circuit parameters: CCSD and MP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Our results, obtained through an efficient sparse wavefunction circuit solver, show that UCCSD circuits with CCSD parameterizations significantly outperformed those with MP2 parameterizations for systems of up to 64 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' These findings suggest that CCSD should be the preferred choice for generating initial parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='— Simulating many-body fermionic sys- tems is a promising future application of quantum com- puting [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' While it is not yet clear that quantum ad- vantage can generically be achieved in this area [4], it is believed that phase estimation can solve ground state problems for molecular systems that are beyond the reach of classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' However, it remains an open ques- tion whether or not other approaches can achieve quan- tum advantage with fewer resources [5–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Phase estima- tion and other algorithms benefit from, or even require, significant overlap between the trial quantum state and the true solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Single Slater determinants, such as Hartree-Fock states [9], are often used as the trial state when solving for ground states, as they are assumed to produce a sufficiently large overlap with the ground state wavefunction in many cases [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Yet such sin- gle determinant states may not be sufficient for arbitrar- ily large system sizes [10, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Improving quantum state preparation techniques is a key step toward ad- vancing quantum computing for quantum chemistry and other Hamiltonian simulation applications since many al- gorithms require high-quality initial quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Noise and decoherence present another central diffi- culty of achieving quantum advantage in the current noisy intermediate-scale quantum (NISQ) era of quan- tum hardware [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The variational quantum eigensolver (VQE) is a quantum-classical hybrid algorithm that is particularly well-suited for NISQ devices [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' While VQE does not provide exact ground state solutions like quantum phase estimation, the approximate wavefunc- tions produced by VQE are often sufficiently accurate to provide meaningful physical insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Furthermore, these approximate solutions are well-suited for quantum state preparation for use in more accurate algorithms [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Despite its current popularity, VQE possesses a num- ber of drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' In particular, the classical optimiza- tion of circuit parameters presents many challenges, in- cluding barren plateaus (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=', exponentially vanishing gra- dients in high dimensions), local minima, and saddle points [20–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Many approaches exist for minimizing the computational burden of classical optimization for VQE, with some proposals eschewing optimization entirely [25– 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The crux of several of these strategies is a focus on choosing high-quality initial parameters, shifting some of the computational burden from optimization to initial- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' In this work we compare the utility of different initialization strategies for a particular VQE ansatz that is often employed in quantum chemistry problems, the unitary coupled cluster (UCC) ansatz [16, 19, 29–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' There are several proposed strategies for generating the initial parameters for the UCC ansatz [17, 31, 33– 35], including applications in which no optimization is performed on quantum hardware [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The most widely employed strategy generates parameters using classical Møller–Plesset perturbation theory of the second order (MP2) [17, 31, 33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Another less thoroughly stud- ied approach is the use of the coupled cluster singles and doubles (CCSD) classical simulation method to generate initial parameters [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The CCSD technique gener- ally produces more accurate ground state energies than MP2 calculations, yet CCSD is rarely employed to ini- tialize VQE circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' This is curious, considering that neither technique is computationally burdensome for all but the largest of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' This raises the question: Which technique produces superior initial parameters for UCC ansatzes, MP2 or CCSD?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' In this paper, we provide the first numerical study (to our knowledge) comparing the performance of UCC ansatzes prepared using parameters generated via MP2 and via CCSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' We employ an algorithm for the factor- ized form of UCC implemented using our state-of-the-art sparse wavefunction circuit solver, enabling us to study problems of up to 64 qubits [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' By calculating the ground state energy of a wide range of molecules using both MP2 and CCSD parameters in the UCC ansatz, we show conclusively that CCSD parameters outperform MP2, generating significantly more accurate ground state energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='— The UCC ansatz is an exponential oper- ator acting on the Hartree–Fock reference wavefunction arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='05666v1 [quant-ph] 13 Jan 2023 2 0 20 40 60 80 100 0 NWF/1000 280 260 E (mHar) UCC (MP2) UCC (CCSD) CCSD CCSD(T) ASCI (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='05 1000/NWF 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='0 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='5 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='0 E/EASCI (%) UCC (MP2) UCC (CCSD) CCSD CCSD(T) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' (a) The UCC(MP2) (light blue solid line) and UCC(CCSD) (light purple dashed line) correlation energies of CH2O as functions of NWF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The dark purple dot-dashed and black dotted lines denote the CCSD and CCSD(T) correlation energies, respectively, and the black solid line marks the ASCI energy calculated using 106 determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' (b) The UCC(MP2) (circles) and UCC(CCSD) (pentagons) correlation energies as a fraction of the CCSD(T) correlation energy plotted versus 1/NWF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The solid lines are quadratic fits to the UCC(CCSD) and UCC(MP2) data, fitted using the twenty data points with the largest NWF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The dashed lines mark the y-intercepts of the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The dot-dashed and dotted lines indicate the ratio of the CCSD and CCSD(T) correlation energies to the ASCI correlation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' defined as |ΨUCC⟩ = exp � ˆT − ˆT †� |Ψ0⟩ (1) where the coupled cluster operator ˆT is ˆT = occ � i vir � a θa i ˆa† aˆai + occ � ij vir � ab θab ij ˆa† aˆa† bˆajˆai + · · · (2) The ˆa† and ˆa operators are the second-quantized creation and annihilation operators, respectively, acting on the occupied molecular orbitals in the reference wavefunc- tion indexed by i, j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' or the virtual orbitals indexed by a, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='. The parameters of the UCC ansatz are indicated by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' We employ the factorized form of the UCC ansatz, which is given by |ΨUCC⟩ = occ � ij··· vir � ab··· ˆU ab··· ij··· |Ψ0⟩ , (3) where the individual UCC exponential factors are defined as ˆU ab··· ij··· = exp � θab··· ij··· (ˆaab··· ij··· · · · − ˆaij··· ab···) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' (4) We only include single excitations (ˆaa i = ˆa† iˆaa) and dou- ble excitations (ˆaab ij = ˆa† iˆa† jˆabˆaa) in the ansatz, along with the conjugate deexcitation operators ˆai a and ˆaij ab, an ap- proximation to the full UCC ansatz known as the unitary coupled cluster singles and doubles (UCCSD) ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' We utilize a specific representation of the UCCSD ansatz that exploits the fact that each UCC factor ˆU ab··· ij··· can be expressed in terms of sines and cosines of the parameters that can be efficiently computed on classical hardware [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The order of the individual UCCSD fac- tors is not strictly defined [40], and we chose to order them based on the magnitude of the parameter values (|θ|), placing the factor ˆU ab··· ij··· that contains the largest parameter to the right in Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' We refer to this as the “magnitude” ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' We generate the conventional MP2 and CCSD UCCSD parameters using PySCF, noting that the MP2 param- eterization does not include any single excitation opera- tors [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' We use a computationally efficient sparse wave- function approach, limiting the number of determinants included in the wavefunction after each UCC factor is applied [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' We do this by checking the number of de- terminants N in the wavefunction after applying each UCC factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' If N is greater than the desired number of determinants, NWF, we sort the amplitudes by mag- nitude and discard the determinants with the smallest amplitudes such that only NWF determinants remain in the wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='— Here we report the correlation energies ob- 3 80 85 90 95 100 E/Eref (%) H8 H10 H12 H14 LiH HF NH3 CH4 H2O N2 F2 CH2O C2 CCSD(T) CCSD UCC (CCSD) UCC (MP2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The CCSD(T), CCSD, UCC(CCSD), and UCC(MP2) correlation energies as percentages of the best available refer- ence energy (FCI for hydrogen chains and LiH, ASCI other- wise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' tained from UCCSD circuits parameterized using MP2 and CCSD parameters for a wide range of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' For the molecules LiH, HF, NH3, CH4, H2O, N2, F2, and CH2O we use experimental geometries from the CC- CBDB database and employ the cc-pCVDZ basis set [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' We also study the linear hydrogen chains H8, H10, H12, and H14, for which we use an interatomic distance of 1 ˚A, and a stretched geometry of H10 with an interatomic dis- tance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='5 ˚A, for all of which we employ the STO-6G basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Our sparse wavefunction circuit solver is lim- ited to 64 qubits, so we include only the 32 lowest-energy molecular orbitals in each molecule [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Our approach has similar scaling to a time-dependent selected configu- ration interaction approach, which some of us have ap- plied to larger systems in other contexts [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Because we limit the number of determinants retained in the wavefunction to NWF, we must study the depen- dence of the correlation energies on NWF and extrapo- late to the large-NWF limit to obtain a converged result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Specifically, we calculate the correlation energy as a func- tion of NWF up to NWF =100,000 for each molecule, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' 1a for CH2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' We extrapolate to the large- NWF limit via a quadratic fitting of the data as a function of N −1 WF, E = aN −2 WF + bN −1 WF + c, (5) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The fit accounts for the twenty data points at the largest values of NWF, which produces the fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The y-intercept of the quadratic fit is the extrapolated correlation energy that would be obtained if we pruned no determinants during the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Thus this is a prediction of the energy that would be produced on perfect quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' We refer to these extrapo- lated energies as the UCC(MP2) and UCC(CCSD) corre- lation energies, depending on the initial parameters used in the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' We report the CCSD(T), CCSD, UCC(MP2), UCC(CCSD), and full configuration interaction (FCI) correlation energies for the hydrogen chains and LiH in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Calculating the FCI energy for the remain- ing molecules is impractical, so we instead report the adaptive sampling configuration interaction (ASCI) cor- relation energies [44] for these molecules in Table II, along with the CCSD(T), CCSD, UCC(MP2), and UCC(CCSD) correlation energies [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' We also plot these energies as fractions of the best reference energy, ei- ther FCI or ASCI, for each molecule in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The UCC(CCSD) energy outperforms the UCC(MP2) energy by a wide margin in all cases, with a difference of ap- proximately 15% of the reference energy for the hydrogen chains (including stretched H10) and differences ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='3% to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='6% for the remaining molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Because the individual terms in the factorized form of the UCCSD ansatz do not necessarily commute, the or- dering of the operators can have a significant impact on the accuracy of the ansatz [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' To address this concern, 4 we calculate the correlation energy of the molecules we study in this work using 100 random orderings of the UCCSD factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' We find that the standard deviation of the correlation energy is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='1 mHa for molecules with equilibrium geometries, with the exception of F2 and C2, the standard deviations of which are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='1 mHa and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='4 mHa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Only the strongly correlated stretched geometry of H10 has a significant standard devi- ation of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='4 mHa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' We set NWF to 10,000 for these calcu- lations to reduce the computational burden, which likely artificially inflates the standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' We conclude that factor ordering is significant only for strongly corre- lated molecules, in agreement with previous studies [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Importantly, we find that the magnitude ordering ob- tains energies close to the minimum energy produced by random orderings for all molecules besides CH2O, for which the magnitude ordering produced an energy ap- proximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='15 mHa above the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The UCC(CCSD) energy closely matches the CCSD energy for all molecules studied, with the exception of stretched H10, and produces energies lower than the CCSD energy for HF, H2O, and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' However, the MP2 and UCC(MP2) energies do not exhibit the remarkably good agreement obtained by the CCSD and UCC(CCSD) energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Excluding the results for stretched H10, the differences between the CCSD and UCC(CCSD) ener- gies range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='0% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='1% with an average of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='4% for the molecules we study, while for MP2 and UCC(MP2) they range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='4% and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='7% with an average of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' These statistics show a clear advantage for UCC(CCSD) As such, the CCSD parameterization is likely better suited than MP2 for use in recent pro- posals for no-optimization strategies to obtain quantum advantage [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Classical coupled cluster techniques have well-known properties in which the obtained energies are not varia- tional (dropping below the FCI results) or, even worse energies can diverge away from the physical ground state result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' One such failure scenario can be seen in the chemistry of bond breaking, which we investigate here using the H10 molecule with a stretched interatomic dis- tance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='5 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The CCSD and CCSD(T) energies of this molecule are lower than the FCI energy, representing a well known problem in classical coupled cluster tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Despite the failure of CCSD to produce an ac- curate energy for this molecule, the UCC circuit param- eterized with CCSD must produce a variational energy because the VQE approach is a wavefunction technique, where as classical coupled cluster approaches are not in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The UCC(CCSD) energy for stretched H10 is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='2% higher than the FCI energy, compared to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='7% higher for the equilibrium geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' These results show that the UCC ansatz parameterized with CCSD is robust to failures of the classical theory, but with some loss of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Regardless, our results show a close correspon- dence between UCC(CCSD) and CCSD theories and fur- TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The FCI, CCSD(T), CCSD, UCC(CCSD), MP2, and UCC(MP2) correlation energies of the hydrogen chains and LiH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The UCC energies are obtained via the fitting pro- cedure shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The row labeled H∗ 10 uses a stretched geometry with an interatomic distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='5 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' All energies are reported as absolute values and in units of milliHartrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Mol FCI CCSD(T) CCSD UCC (CCSD) MP2 UCC (MP2) H8 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='68 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='65 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='60 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='00 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='19 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='69 H10 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='78 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='64 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='77 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='86 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='62 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='08 H∗ 10 403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='81 434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='55 426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='50 354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='74 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='45 293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='67 H12 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='90 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='62 197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='81 196.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='28 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The ASCI, CCSD(T), CCSD, UCC(CCSD), MP2, and UCC(MP2) correlation energies of the larger molecules for which FCI is impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The UCC energies are obtained via the fitting procedure shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' All energies are reported as absolute values and in units of milliHartrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Mol ASCI CCSD(T) CCSD UCC (CCSD) MP2 UCC (MP2) HF 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='24 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='75 248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='61 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='03 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='84 245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} 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+page_content='59 ther study of this can help us understand the power of coupled cluster approaches on quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='— In this paper we demonstrated through extensive calculations that CCSD parameterizations of the UCC ansatz consistently outperform their MP2 coun- terparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' As such, it is important to compare the com- putational costs of obtaining the CCSD and MP2 pa- rameterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Although MP2 is faster and, in fact, often used as a starting point for coupled cluster simula- tions, CCSD nevertheless requires only reasonable classi- cal computation resources for even moderately sized sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' For example, the CCSD calculations presented in this work and others run in minutes or less on a lap- top [28, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' MP2 and CCSD runtimes scale as O(N 5) and O(N 6), respectively, making these prohibitively expensive algo- rithms in the large-N qubit limit, but it is unlikely that NISQ era quantum computers will exceed classically- accessible simulations of CCSD in the next few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Classical coupled cluster simulations can be accelerated 5 in various ways [46, 47], indicating that simulations in- volving hundreds of qubits to parameterize circuits is in reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Considering this, as well as the small prefactors of these runtime scalings and the efficiency of modern imple- mentations of these techniques, CCSD is poised to remain an accessible and highly accurate method of UCC param- eterization for the forseeable future of the NISQ era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' As such, our results suggest that CCSD should replace MP2 as the standard approach to classically parameterizing UCC circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Our results also display the power of our sparse wave- function circuit solver, which enables us to perform UCC simulations at system sizes that have not been previously explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Because our solver is capable of handling up to 64 qubit problems with its current implementation, we are able to access a regime in which it is possible to meaningfully test and differentiate VQE results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' In this case, the ability to access large systems sizes enabled us to explore a widely used parameterization for UCC cir- cuits and challenge conventional held ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' There are number of directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Testing our approach with higher order coupled clus- ter techniques on both the classical [48] and quantum side [35] is one such direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' The correspondence we identified between CCSD and UCC(CCSD) is weakened when classical CCSD breaks down, as seen in for strongly correlated molecules like stretched H10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' These results motivate the study of more advanced classical approaches to parameterize UCC-type circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Establishing the cor- respondence between higher order classical coupled clus- ter theories and the UCC analogues of them, such as a UCC(CCSDT) circuit [35], would elucidate the full po- tential of the UCC ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='— We are grateful for support from NASA Ames Research Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' We acknowledge fund- ing from the NASA ARMD Transformational Tools and Technology (TTT) Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Part of this work is funded by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Department of Energy, Office of Science, National Quantum Information Science Research Cen- ters, Co-Design Center for Quantum Advantage under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' DE-SC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' Calculations were per- formed as part of the XSEDE computational Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' TG-MCA93S030 on Bridges-2 at the Pittsburgh supercomputer center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='H and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' were supported by NASA Academic Mission Services, Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' NNA16BD14C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} 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ical Theory and Computation 16, 2139 (2020), pMID: 32159951, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='jctc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} +page_content='8b00536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E5T4oBgHgl3EQfkw_7/content/2301.05666v1.pdf'} diff --git a/r9AzT4oBgHgl3EQfPPs-/content/tmp_files/2301.01179v1.pdf.txt b/r9AzT4oBgHgl3EQfPPs-/content/tmp_files/2301.01179v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b01218e8d73430631e919d71b37596fc99243257 --- /dev/null +++ b/r9AzT4oBgHgl3EQfPPs-/content/tmp_files/2301.01179v1.pdf.txt @@ -0,0 +1,1146 @@ +A Fast Multipole Method for axisymmetric +domains +Michael J. Carley +January 4, 2023 +Abstract +The Fast Multipole Method (FMM) for the Poisson equation is extended +to the case of non-axisymmetric problems in an axisymmetric domain, de- +scribed by cylindrical coordinates. The method is based on a Fourier de- +composition of the source into a modal expansion and the evaluation of the +corresponding modes of the field using a two-dimensional tree decomposi- +tion in the radial and axial coordinate. The field coefficients are evaluated +using a modal Green’s function which can be evaluated using well-known +recursions for the Legendre function of the second kind, and whose deriva- +tives can be found recursively using the Laplace equation in cylindrical co- +ordinates. The principal difference between the cylindrical and Cartesian +problems is the lack of translation invariance in the evaluation of local inter- +actions, leading to an increase in computational effort for the axisymmetric +domain. Results are presented for solution accuracy and convergence and +for computation time compared to direct evaluation. The method is found to +converge well, with ten digit accuracy being achieved for the test cases pre- +sented. Computation time is controlled by the balance between initialization +and the evaluation of local interactions between source and field points, and +is about two orders of magnitude less than that required for direct evaluation, +depending on expansion order. +1 +Introduction +Since its development [8], the Fast Multipole Method (FMM) has become the al- +gorithm of choice for a large class of problems which can be expressed in terms +of finding at a large number of field points the potential generated by a large +number of point sources. This includes problems governed by the Poisson and +Helmholtz equations, including boundary integral problems in acoustics, electro- +magnetism, and fluid dynamics, and volume integrals such as the Biot–Savart +1 +arXiv:2301.01179v1 [math.NA] 3 Jan 2023 + +integration which arises in electromagnetism and vortex dynamics. The method +is now highly developed with efficient implementations available in two [4] and +three [10] dimensions for a range of problems, using a variety of analytical tools +for their formulation. +Despite its power and importance, the FMM does not seem to have been ex- +tended to non-Cartesian coordinate systems. In particular, there do not seem to +be formulations of the method which can be applied to axisymmetric domains de- +scribed using cylindrical coordinates. These systems arise naturally in a range of +applications and cylindrical coordinates are a natural way to describe a system, or +to apply boundary conditions. The purpose of this paper is to present an extension +of the FMM for the Poisson equation to cylindrical coordinates, motivated by ap- +plications in fluid dynamics, where boundary and volume integral problems are +governed by the Laplace kernel. +To the author’s knowledge, there have been two previous studies which are +relevant to this problem. The first was the work of Strickland and Amos [11, +12] who developed an accelerated method for the evaluation of the axisymmetric +stream function in vortex dynamics, equivalent to solving the Poisson problem or +evaluating the Biot-Savart integral. The authors used single-precision arithmetic +and a fifth order expansion of the Green’s function to achieve five-digit accuracy +at a cost of 1–3% of the computational effort required for direct evaluation of +the potential and velocity fields. The authors do not seem to have extended their +method to the general case in a cylindrical domain. +More recently, in an unpublished thesis, Churchill [2] considered the general +non-axisymmetric problem, motivated by the analysis of boundary integrals on +surfaces of revolution. He identifies the particular difficulty in cylindrical co- +ordinates, which is that the Green’s function is translation invariant in the axial +coordinate, but not in the radial, which complicates the evaluation of interactions +which arise in the FMM. He concludes that the problem can be solved using a +black-box [5] or generalized [6] FMM, but does not present results for the cylin- +drical problem. +This paper presents a method for the fast evaluation of the potential in a cylin- +drical domain, generated by a set of azimuthally varying ring sources. The motiva- +tion is the Poisson or Biot–Savart problem in fluid dynamics, with the expectation +that the method is to be used in boundary integral solvers [13, for example], or in +evaluating the velocity field due to a distribution of vorticity [12]. The approach +is essentially that of a standard two-dimensional FMM, with modifications to the +evaluation of interactions to accommodate the lack of translation invariance. It is +assumed that any necessary Fourier transformation of the inputs to the calculation +has been performed, so that the starting point is the location and radius of a set of +circular sources, and the coefficients of the Fourier series for a source distribution +on each circle. The main elements of the method are described briefly, with a +2 + +more detailed description of those parts which are particular to the axisymmetric +case, i.e. the evaluation of the Green’s function and its derivatives in cylindrical +coordinates. Results are presented for a problem with an increasing number of +sources, to test the performance of the method for convergence and speed. +2 +Analysis +Figure 1: Basic problem: ring sources of radius ri are distributed at axial station zi +in the domain 0 ≤ r ≤ rmax, 0 ≤ z ≤ zmax. Sources are given as the coefficients +of the Fourier series of source strength as a function of θ at each (ri, zi) +The problem to be solved is sketched in Figure 1. We assume that any nec- +essary preprocessing has been performed to reduce the system to the form shown +here. A set of ring sources with a common axis are distributed throughout the +domain (r, z). The source strength s(θ1) on a ring at location (r1, z1) is given by +the Fourier series, +s(θ1) = +N +� +n=−N +S(n)einθ1, +(1) +where subscript 1 denotes source coordinates. The potential φ(r, z) due to the +source is given by integration over θ1, +φ(r, θ, z) = +� 2π +0 +s(θ1) +4πR dθ1, +(2) +R2 = r2 + r2 +i − 2rr1 cos(θ − θ1) + (z − z1)2. +3 + +Under an elementary transformation, and substituting the Fourier series for s(θ1), +φ(r, θ, z) = +N +� +n=−N +S(n)e−inθ +� 2π +0 +einθ1 +4πR dθ1, +(3) +R2 = r2 + r2 +1 − 2rr1 cos θ1 + (z − z1)2. +The potential field φ(r, z) can then be expressed as a Fourier series in θ, +φ(r, θ, z) = +N +� +n=−N +Φ(n)(r, z)einθ, +(4) +Φ(n) = S(n)G(n)(r, r1, z − z1), +(5) +G(n)(r, r1, x) = +� 2π +0 +einθ1 +4πR dθ1 = +� 2π +0 +cos nθ1 +4πR +dθ1, +(6) +R2 = r2 + r2 +1 − 2rr1 cos θ1 + x2. +The source term s(θ1) is assumed real, so that the complex Fourier coefficients +are related by S(−n) = +� +S(n)�∗. Henceforth, all computations will be performed +for n ≥ 0 and the conjugate relationship will be assumed. +We refer to G(n)(r, r1, x) as the modal Green’s function relating Fourier coef- +ficients of the source to those of the potential at some other point. Appendix A +gives details of the evaluation of G(n)(r, r1, x) and of its derivatives. In particular, +it is shown that it can be expressed exactly as +G(n)(r, r1, x) = Qn−1/2(χ) +2π√rr1 +, +(7) +where Qν(χ) is a Legendre function of the second kind [3]. +Given a distribution of sources at locations (ri, zi), i = 1, . . . , Ns, each of +which has a set of Fourier coefficients S(n) +i +, n = 0, . . . , N, the problem to be +solved is then the approximate evaluation of the sum +Φ(n)(rj, zj) = +Ns +� +i=1 +S(n) +i +G(n)(rj, ri, zj − zi), +(8) +for field points (rj, zj), j = 1, . . . , Nf. +2.1 +Outline of the FMM +The Fast Multipole Method is well established and there are numerous guides to +its algorithm and implementation. Here we give an outline of the method in order +4 + +to present the necessary terminology and to indicate those parts of the algorithm +of this paper which differ from existing methods. From the previous section, we +recall that the objective is to approximately evaluate the sum +Φ(n)(rj, zj) = +Ns +� +i=1 +S(n) +i +G(n)(rj, ri, zj − zi), +n = 0, . . . , N, +j = 1, . . . , Nf, +given a list of source and field points (ri, zi and (rj, zj) respectively. The first +operation of the FMM is the sorting of points and their representation in a quadtree +data structure, formed by repeated subdivision of the domain 0 ≤ ri,j ≤ rmax, +0 ≤ zi,j ≤ zmax, zmax = rmax. Figure 2 shows the repeated halving of intervals. +At each level of subdivision ℓ, the domain is divided into 2ℓ × 2ℓ boxes, to a +maximum level called the depth of the tree d. Boxes in the tree can be indexed by +their location (i, j) on the grid at a given level, or by their Morton index. +r +z +r +r +Figure 2: Recursive subdivision of the source domain into 4ℓ boxes for ℓ = 1, 2, 3, +left to right +Figure 3 gives the terminology for relationships between boxes. A box at grid +location (i, j) at level ℓ has four child boxes at grid locations (2i, 2j), (2i, 2j +1), +(2i + 1, 2j), (2i + 1, 2j + 1) at level ℓ + 1, and has a parent box at grid location +(i/2, j/2) at level ℓ − 1. A box at the finest level of subdivision, depth d, has no +children and is called a leaf box. +Neighbors of box (i, j) are boxes at the same level ℓ which share at least a +vertex with the box, including the box itself. In Figure 3, boxes 1–9 are neighbors +of box 1. All other boxes are said to be in the far field of box 1. The basic principle +of the FMM is to separate far-field from neighbor interactions and evaluate the far- +field terms in any box using an accelerated summation. +This is achieved in the first instance by evaluating the field due to sources +inside a box using an approximate expansion which is faster than direct evaluation +of the sum +Φ(n)(r, z) = +� +i +S(n) +i +G(n)(r, ri, z − zi), +5 + +1 +2 +3 +4 +j +i +5 +6 +7 +8 +9 +Figure 3: Terminology for relationships between boxes: boxes 1–4 are children +of the shaded box; the shaded box is the parent of boxes 1–4; boxes 1–9 are +neighbors of box 1 +where the summation is taken over all sources inside a box. At field points suffi- +ciently far from the box, the Green’s function for a source is well approximated +by its Taylor series, truncated to some order M, +G(n)(r + ∆r, r1 + ∆r1, x + ∆x) ≈ +M +� +m=0 +� +i+j+k=m +g(n) +i,j,k(∆r)i(∆r1)j(∆x)k, +(9) +g(n) +i,j,k = 1 +i! +1 +j! +1 +k! +∂i+j+k +∂ri∂rj +1∂xk G(n)(r, r1, x). +Details of the evaluation of the derivatives of G(n) are given in Appendix A. +The field at a point (r, z) due to a source at (r1 + ∆r1, z1 + ∆z1) in a box with +center (r1, z1) is then given by +Φ(n)(r, z) ≈ S(n) +M +� +m=0 +� +i+j=m +(−1)jg(n) +0,i,j(∆r1)i(∆z1)j, +(10) +noting that x = z − z1 and ∂/∂z1 = −∂/∂x. Summing over all sources contained +in the box, +Φ(n)(r, z) ≈ +M +� +m=0 +� +i+j=m +(−1)jg(n) +0,i,jS(n) +ij , +(11) +where the moments Sij are given by +S(n) +ij += +� +q +S(n) +q (∆rq)i(∆zq)j. +(12) +6 + +To initialize the source data in the first stage of the FMM, the moments S(n) +ij +are computed for each leaf box in the tree. In the upward pass, the moments at +boxes in each level ℓ are computed, for ℓ = d − 1, . . . , 1. This can be achieved +without requiring direct evaluation of moments from source data, by combining +moments from child boxes to generate moments in their parent box, Figure 4. +Moments about the center of a child box at displacement (∆r, ∆z) contribute to +the moments about the center of their parent box via +Sij = +i +� +q=0 +j +� +u=0 +�i +q +��j +u +� +(−∆r)q(−∆z)uS′ +i−q,j−u, +(13) +where the superscript (n) has been suppressed for clarity. When the upward pass +has been completed, each box at levels ℓ = 1, . . . , d contain a set of moments +which can be used to estimate the potential in the far field of the box. +∆r +∆z +∆rq +∆zq +Figure 4: Evaluation and shifting of source moments: moments about the center +of the leaf box are computed using (12); those about the center of the parent box +are computed by shifting the child box moments by (∆r, ∆z) using (13). +In the next stage of the FMM, the downward pass, each box is assigned a +local expansion which can be used to evaluate the potential inside the box due +to sources which lie in its far field. The core of the FMM is the use of the most +efficient expansion possible at any level to evaluate the far-field terms in any box. +The field in a box centered at (r, z) is given by +Φ(n)(r + ∆r, z + ∆z) = +M +� +m=0 +� +k+ℓ=m +Φ(n) +kℓ (∆r)k(∆z)ℓ, +(14) +where the expansion coefficients Φ(n) +kℓ are evaluated from the contributions of +sources in boxes which interact with the field box. The order M of the local +7 + +i +j +A +B +2 +1 +Figure 5: Generation of a local field in a box: the local expansion of the field in +box B due to sources in box A is computed using the S2L operation, (15); the +local expansion in box 2 is found by shifting the expansion in box B to box 2 and +adding the contribution from box 1, evaluated using an S2L operation. +expansion at any level is not required to be the same as the order of the source +expansions, but has been set equal for the calculations presented in this paper. +Figure 5 shows the main operations involved, for the evaluation of a local +expansion in box 2, which has parent box B. On the downward pass, the local ex- +pansion in box B is found by adding the contribution from sources in boxes which +are well separated from B, such as box A. This contribution is found using the +shift-to-local or S2L operation. The local expansion in B is used to generate the +local expansion in each of its child boxes, including box 2. Box 2 then has its local +expansion incremented by the contribution of boxes with which it interacts, such +as box 1. At the end of the downward pass, each leaf box has a local expansion +which accounts for the contribution of all sources lying outside its neighbors. +The two operations to be implemented here are the S2L and the parent-to-child +shift of the local expansion. The S2L shift is found by differentiating (10), +Φ(n) +kℓ = +� +i,j +(−1)j +�j + ℓ +j +� +S(n) +ij g(n) +k,i,j+ℓ, +(15) +which can be implemented as a BLAS level 2 operation +The local expansion in a child box at displacement (∆r, ∆z) is given from the +8 + +parent box expansion by, +Φij = +� +q=0 +� +u=0 +�i + q +q +��j + u +u +� +(−∆r)q(−∆z)uΦ′ +i+q,j+u, +(16) +where terms Φ′ +i,j are coefficients of the parent box local expansion. +2.2 +Evaluation of interactions +The outline of the Fast Multipole Method presented in subsection 2.1 contains the +main elements of a generic FMM which are familiar from existing implementa- +tions. In this section, we describe the part of the algorithm which is particular +to the cylindrical domain, the S2L operation for the modal Green’s function. In +existing, Cartesian, methods, the translation operators are invariant with respect +to shifts in the coordinate system. As noted by Churchill [2], however, this is not +true for the modal Green’s function G(n), which is invariant for shifts in the ax- +ial coordinate z but not for displacements in radius r. This increases the number +of orientations for which shift operators must be computed, though there are still +some symmetries which can be exploited to reduce the workload. +Recall that the modal Green’s function is given by +G(n)(r, r1, x) = Qn−1/2(χ) +2π√rr1 +, χ = r2 + r2 +1 + x2 +2rr1 +, x = z − z1. +This is invariant under translations in z and is symmetric in r and r1, a fact which +is exploited in the recursion relations for derivatives in source and field coordi- +nates [11, 12]. To take advantage of this symmetry, we introduce some terminol- +ogy to describe translation operations. If we assume that for S2L operations, we +evaluate derivatives of G(n)(r, r1, x) for the r ≥ r1, x ≥ 0. Then, from Figure 6, +we can derive translation operators for four different cases. These correspond to +shifts in the positive or negative (forward or backward) axial direction, and from +greater to smaller radius (outward or inward). +The basic operator, which uses the derivatives of the Green’s function without +modification, is the forward-outward or FO shift, (15). +Φ(n) +kℓ = +� +i,j +(−1)j +�j + ℓ +j +� +S(n) +ij g(n) +k,i,j+ℓ. +For the backward-outward or BO shift, x < 0, ∂/∂z1 = ∂/∂x and ∂/∂z = +−∂/∂x, yielding +Φ(n) +kℓ = +� +i,j +(−1)ℓ +�j + ℓ +j +� +S(n) +ij g(n) +k,i,j+ℓ. +9 + +S +FO +BO +BI +FI +r1 +r +z1 +z1 + ∆z +z1 − ∆z +r +z +Figure 6: Reuse of Green’s function derivatives for forward/backward and in- +ward/outward interactions. The source box S generates local expansions at r > r1 +through a forward-outward (FO) and a backward-outward (BO) operation, indi- +cated by solid arrows. The boxes at greater radius generate local expansions in +box S via backward-inward (BI) and forward-inward (FI) operations. All four +operations are computed from the same expansion of the Green’s function. +To evaluate the inward shifts, we exchange r and r1 and swap the correspond- +ing indices. For the forward-inward S2L operation, +Φ(n) +kℓ = +� +i,j +(−1)j +�j + ℓ +j +� +S(n) +ij g(n) +i,k,j+ℓ, +which gives the contribution of the sources in the box at larger radius r to the local +expansion in the box at smaller radius r1. Finally, the backward-inward operator +is given by +Φ(n) +kℓ = +� +i,j +(−1)ℓ +�j + ℓ +j +� +S(n) +ij g(n) +i,k,j+ℓ. +In order to apply the shift operations, we enumerate candidate source boxes +which may contribute to the field in a box, Figure 7. This gives rise to two in- +teraction lists, the D list containing boxes which contribute via direct evaluation +of the field for each source and field point, and the S2L list, whose contributions +are evaluated using the S2L operation acting on source and local expansion co- +efficients. Figure 7 indicates that the D list is made up of neighbors of box B, +leaving 27 boxes which may contribute to the field in B via S2L operations. Con- +tributions from all other boxes are transferred into B from its parent box during +the downward pass. Using the axial translation invariance and the symmetry in r +10 + +B +D +D +D +D +D +D +D +D +S2L S2L S2L S2L S2L S2L +S2L S2L S2L S2L S2L S2L +S2L S2L +S2L S2L +S2L S2L +S2L S2L +S2L S2L S2L S2L +S2L +S2L +S2L +Figure 7: Interaction lists for box B. Parent box of B is shown shaded; boxes +marked D have interactions evaluated directly from (8); boxes marked S2L have +interactions evaluated via source-to-local operation. Unmarked boxes interact +with B through its parent box so that the interactions need not be explicitly evalu- +ated. +and r1 reduces the number of Green’s function expansions to evaluated to twelve, +those for boxes at z ≥ z1 and r ≥ r1. The expansions are identical for any value +of z − z1 but must be updated for each r1 during the downward pass. Once gener- +ated, the expansion is used to update the local expansion on the outer boxes, and +to include the contribution of those boxes’ source terms to the local expansion on +B. +Finally, we note that the modal Green’s function can be written using scaled +coordinates, so that +G(n)(σr, σr1, σx) = 1 +σG(n)(r, r1, x), +(17) +which would allow for the shift operators to be precomputed and generated at +each level as required. This has been implemented but found not to give a time +saving, since each level of the tree requires twice as many shift operators as its +parent level, half of which are new. In practice, we find that the bottleneck in the +code is the evaluation of direct interactions rather than the computation of the S2L +operators. +11 + +3 +Algorithm +Combining the elements of the previous sections, we present an algorithm for a +uniform Fast Multipole Method in an axisymmetric domain. Input is a list of Ns +source points (ri, zi), i = 1, . . . , Ns and modal amplitudes S(n) +i +, n = 0, . . . , N, +and a list of Nf field points (rj, zj), j = 1, . . . , Nf. +Algorithm 1 Fast Multipole Method for cylindrical coordinate systems +set tree depth d +sort source points (ri, zi) and field points (rj, zj) by Morton index and assign +to leaf nodes at level d +calculate leaf box moments from source amplitudes (12) +{upward pass} +for ℓ = d − 1, . . . , 2 do +evaluate box moments at level ℓ from child box moments at level ℓ + 1, (13) +end for +{downward pass} +for ℓ = 2, . . . , d do +for i = 0, . . . , 4ℓ−1 − 1 do +shift parent box local expansion from level ℓ − 1 to level ℓ boxes +end for +for i = 0, . . . , 2ℓ − 1 do +evaluate coefficients of Green’s function expansions for radial station i +for j = 0, . . . , 2ℓ − 1 do +apply S2L operators for FO and BO translations of source in box (i, j) +apply S2L operators for FI and BI translation to update local expansion +in box (i, j) +end for +end for +end for +for i = 0, . . . , 4d − 1 do +for field points in box i evaluate sum of local expansion and direct contribu- +tion from neighbor boxes +end for +The implementation in Algorithm 1 is for a uniform FMM which does not +generate an adaptive decomposition when assigning points to boxes. This was +decided upon to reduce the number of shift operators required in evaluating box +interactions. +12 + +4 +Results +−12 +−10 +−8 +−6 +−4 +0 +4 +8 +12 +16 +n +log10 ϵ(n) +Figure 8: Error ϵ against mode order for real part of modal amplitude Ns = 216, +tree depth d = 6, expansion order M = 6, 8, 10, 12, 14, 16 from top to bottom of +plot. +The algorithm has been tested for accuracy and computation time using source +and field points randomly distributed over 0 ≤ r, z ≤ 1, with random modal +amplitudes 0 < S(n) < 1, n = 0, . . . , 17. In each case, source number Ns is set +equal to number of field points Nf, with Ns = 2q, q = 10, . . . , 16, and the same +order of expansion is used for source and field terms. Results are presented for +varying Ns, maximum expansion order M, and tree depth d. Code is written in +GNU C, with gcc optimization -O3 and Goto BLAS matrix-vector operations. +Calculations were performed on one core of an Intel i5-6200U laptop running +at 2.3GHz. Similar code and optimizations were used for the direct evaluations +used as an error reference. +Error is evaluated for each modal amplitude of the field, +ϵ(n) = max |Φ(n) +FMM − Φ(n) +D | +max |Φ(n) +D | +, +(18) +where Φ(n) +FMM is modal amplitude evaluated using the new algorithm, and Φ(n) +D is +that found by direct evaluation. Sample results for error as a function of mode +number and expansion order are shown for Ns = 216 in Figure 8. The method +is clearly accurate, especially for higher order expansions, where eleven digit ac- +curacy is achieved for the axisymmetric mode. The error increases at larger n, +where the absolute value of the modal amplitudes is smaller, making the relative +error measure larger. +13 + +10−12 +10−11 +10−10 +10−9 +10−8 +10−7 +10−6 +ϵ(0) +6 +8 +10 +12 +14 +16 +10−10 +10−9 +10−8 +10−7 +10−6 +10−5 +10−4 +M +ϵ(8) +6 +8 +10 +12 +14 +16 +M +Figure 9: Error ϵ for n = 0 (top row) and n = 8 (bottom row) against expansion +order M for Ns = 214 (left column) and Ns = 216 (right column): diamonds, tree +depth d = 5; upward triangles, d = 6; downward triangles, d = 7; fitted lines +ϵ ≈ C−0.4M. +14 + +Figure 9 shows the variation in error with expansion order for the axisym- +metric n = 0 mode and for n = 8. The error scales approximately as C−0.4M +with weak dependence on tree depth. The algorithm performs well with respect +to convergence over the range of problem sizes tested here. +Figure 10 shows basic data for computation time as a function of problem size. +The time for direct evaluation is shown and scales at the expected NfNs = N 2 +s +rate. The computation time for the FMM algorithm behaves similarly for the low +M = 6 order and high M = 16 order cases, with times being shifted up by the +change in expansion order. The computation time in each case is roughly constant +for small Ns, where the evaluation time is dominated by the set up cost, which de- +pends on the tree depth. As the problem size increases, the evaluation time for the +downward pass begins to dominate the calculation time which increases propor- +tional to N 2 +s , but with a much smaller leading constant than for direct evaluation. +Again, this is the expected behavior as the direct evaluation of near-field interac- +tions becomes the largest part of the calculation. With increasing tree depth, the +box to box evaluations become correspondingly faster as the number of sources +per box becomes smaller. +−1 +1 +3 +5 +211 +213 +215 +217 +Ns +log10 t +211 +213 +215 +217 +Ns +Figure 10: Computation time against source and field point number, for direct +evaluation (solid line), and trees of depth 4 (boxes) depth 5 (diamonds), 6 (upward +triangles), and 7 (downward triangles); left hand plot: expansion order M = 6; +right hand plot order expansion order M = 16. +Figure 11 shows the breakdown of computation time between the two parts +of the calculation, as a function of tree depth. The initialization phase, made up +of the upward and downward passes, scales approximately linearly with problem +size, with a leading constant determined by the tree depth. Initialization time in- +creases with tree depth, as the number of boxes increases. The time for local field +evaluation, in the lower plot, scales well on Ns/4d, the average number of sources +per box, with the time reducing with tree depth. The implication is that compu- +tation time and accuracy are determined by the balance between initialization and +local field evaluation, which depends on source number, expansion order, and tree +15 + +0 +1 +2 +3 +4 +5 +0 +20 × 103 +40 × 103 +60 × 103 +80 × 103 +Ns +tP +10−3 +10−1 +101 +103 +2−2 +20 +22 +24 +26 +28 +Ns/4d +tD +Figure 11: Execution time for phases of calculation, expansion order M = 6; +boxes: tree depth d = 4; diamonds, d = 5, triangles, d = 6. Upper plot, time +for upward and downward passes, with linear fit; lower plot, time for local field +evaluation, power law fit to points with Ns/4d ≥ 1. +16 + +depth. +−12 +−10 +−8 +−6 +0 +30 +60 +90 +120 +t/s +log10 ϵ(0) +Figure 12: Error ϵ(0) against computation time: diamonds depth d = 5, triangles +d = 6; solid lines, Ns = 216; dashed lines, Ns = 214; curves are second order fits +to log ϵ. +Figure 12 gives results illustrating this balance, plotting error against compu- +tation time, found by varying problem size and expansion order at two tree depths. +For each tree depth and source number, there is a trend towards a minimum error, +with the greater tree depth requiring a greater total computation time for higher +order accuracy. +5 +Conclusions +The Fast Multipole Method has been extended to non-axisymmetric problems in +cylindrical domains, by evaluating the amplitudes of the modes in a Fourier ex- +pansion of the source and field in a Poisson problem. Testing by comparison with +direct evaluation shows convergence to up to ten digit accuracy, and orders of +magnitude speed-up, depending on expansion order. Open questions remain. The +first is the efficient evaluation of the Legendre functions used to find the modal +Green’s function, which is common to the direct and fast methods, and constitutes +the largest computational demand in the method. A second is the formulation of +the method in a form which allows the use of BLAS level 3 operations, which +should allow the code to be optimized further. Finally, we note that the approach +taken here should be applicable to the Helmholtz problem, though with some +greater difficulty in evaluating the modal Green’s functions. +17 + +A +Evaluation of Green’s functions and derivatives +The modal Green’s function G(n)(r, r1, x) is defined: +G(n)(r, r1, x) = +� 2π +0 +einθ1 +4πR dθ1, +(19) +R2 = r2 + r2 +1 − 2rr1 cos θ1 + x2. +Cohl and Tohline [3] give an expansion for 1/R, +1 +R = +1 +π√rr1 +∞ +� +m=−∞ +eim(θ−θ1)Qm−1/2(χ), +(20) +χ = r2 + r2 +1 + x2 +2rr1 +, +where Qν(χ) is the Legendre function of the second kind. Integration over θ1 +yields +G(n)(r, r1, x) = Qn−1/2(χ) +2π√rr1 +. +(21) +Using the recursion for the Legendre function [7, 8.732.2], with the functional +dependence on coordinates suppressed for clarity, +(2n − 3)G(n−2) = 4(n − 1)χG(n−1) − (2n − 1)G(n). +(22) +For χ > 1, the forward recursion is unstable and the backward recursion is stable, +but computationally expensive [9]. To generate the sequence of modal Green’s +functions, we apply the approach of Helsing and Karlsson [9] and use the forward +recursion for χ < 1.008, beginning with the initial values [3], +Q−1/2(χ) = µK(µ), +(23) +Q1/2(χ) = χµK(µ) − (1 + χ)µE(µ), +(24) +µ = +� +2 +1 + χ. +Here K(·) and E(·) are the complete elliptic integrals of the first and second kind +respectively. These are computed using the method of Carlson [1]. +For χ ≥ 1.008, the backward recursion is used starting with arbitrary values +of Qn−1/2(χ) and Qn−3/2(χ) for n = N +80, performing the downward recursion +to n = 0 and scaling the sequence using the known value of Q−1/2(χ), (23). Val- +ues of G(n) evaluated using this procedure have been checked against numerical +integration and have been found to be correct to machine precision. +18 + +Given values of G(n), n = 0, . . . , N, the derivatives of G(n) can be found using +a combination of the recursion relations for the Legendre function and the Laplace +equation. For concision, we introduce the notation +g(n) +i,j,k = +∂i+j+k +∂ri∂rj +1∂xk G(n)(r, r1, x), +(25) +g(n) +i,j,k = 1 +i! +1 +j! +1 +k! +∂i+j+k +∂ri∂rj +1∂xk G(n)(r, r1, x), +(26) +so that the Taylor series for G(n) is given by +G(n)(r + ∆r, r1 + ∆r1, x + ∆x) = +∞ +� +m=0 +� +i+j+k=m +g(n) +i,j,k(∆r)i(∆r1)j(∆x)k. (27) +The derivatives are evaluated using a recursion based on the Laplace equation, +similar to the approach of Strickland and Amos who used the axisymmetric stream +function equation [11, 12]. Here we use the Laplace equation for a field with +azimuthal dependence exp inθ. This recursion requires starting values which can +be found using the properties of the Legendre functions [7, 8.732]. For derivatives +with respect to x, +g(n) +0,0,1 = (n − 1/2) +�� +g(n) +0,0,0 + g(n−1) +0,0,0 +� x +ρ2 ++ ++ +� +g(n) +0,0,0 − g(n−1) +0,0,0 +� x +ρ2 +− +� +, +(28) +g(n) +0,0,k+1 = (n − 1/2) +k +� +q=0 +1 +q!(k + 1) +�� +g(n) +0,0,k−q + g(n−1) +0,0,k−q +� ∂q +∂xq +� x +ρ2 ++ +� ++ +� +g(n) +0,0,k−q − g(n−1) +0,0,k−q +� ∂q +∂xq +� x +ρ2 +− +�� +, +(29) +ρ2 +± = (r ± r1)2 + x2. +To evaluate derivatives for n = 0, the relation g(−1) +i,j,k ≡ g(1) +i,j,k can be used. +For the derivatives with respect to r, +g(n) +1,0,0 = −g(n) +0,0,0 +2r ++ (n − 1/2)r2 − r2 +1 − x2 +2r +�� +g(n) +0,0,0 + g(n−1) +0,0,0 +� 1 +ρ2 ++ ++ +� +g(n) +0,0,0 − g(n−1) +0,0,0 +� 1 +ρ2 +− +� +, +(30) +g(n) +1,0,k = −g(n) +0,0,k +2r ++ n − 1/2 +2r +k +� +q=0 +1 +q! +� � +g(n) +0,0,k−q + g(n−1) +0,0,k−q +� ∂q +∂xq +r2 − r2 +1 − x2 +ρ2 ++ ++ +� +g(n) +0,0,k−q − g(n−1) +0,0,k−q +� ∂q +∂xq +r2 − r2 +1 − x2 +ρ2 +− +� +. +(31) +19 + +Derivatives g(n) +0,1,k are found by exchanging r and r1, with a corresponding swap of +indices in the derivatives. +Finally, +g(n) +1,1,k = −g(n) +0,1,k +2r +(32) ++ n − 1/2 +2r +k +� +q=0 +1 +q! +� � +g(n) +0,0,k−q + g(n−1) +0,0,k−q +� ∂q+1 +∂r1∂xq +r2 − r2 +1 − x2 +ρ2 ++ ++ +� +g(n) +0,1,k−q + g(n−1) +0,1,k−q +� ∂q +∂xq +r2 − r2 +1 − x2 +ρ2 ++ ++ +� +g(n) +0,0,k−q − g(n−1) +0,0,k−q +� ∂q+1 +∂r1∂xq +r2 − r2 +1 − x2 +ρ2 +− ++ +� +g(n) +0,1,k−q − g(n−1) +0,1,k−q +� ∂q +∂xq +r2 − r2 +1 − x2 +ρ2 +− +� +. +To compute the remaining derivatives, we make use of a recursion based on +the Laplace equation in cylindrical coordinates. Noting that ∂/∂θ → in, +1 +r +∂ +∂r +� +r∂G(n) +∂r +� +− n2 +r2 G(n) + ∂2G(n) +∂x2 += 0. +(33) +This yields the relation, +g(n) +i+2,j,k = +i +� +u=0 +(−1)u +ru+1 +1 +(i + 1)(i + 2) +�(u + 1)n2 +r +g(n) +i−u,j,k − (i − u + 1)g(n) +i−u+1,j,k +� +, +(34) +which can be used to recursively generate higher derivatives with respect to r. +Switching r and r1 gives a corresponding relation for the higher derivatives with +respect to r1 and allows a complete set of derivatives to be evaluated to any re- +quired order. +References +[1] B. C. Carlson. Numerical computation of real or complex elliptic integrals. +SIAM Journal of Numerical Analysis, 10:13–26, 1995. +[2] Victor Churchill. Fast multipole methods for axisymmetric geometries. M.S. +Mathematics, Courant Institute of Mathematical Sciences, New York Uni- +versity, May 2016. +20 + +[3] Howard S. Cohl and Joel E. Tohline. A compact cylindrical Green’s function +expansion for the solution of potential problems. The Astrophysical Journal, +527:86–101, 1999. +[4] Frank Ethridge and Leslie Greengard. +A new fast-multipole accelerated +Poisson solver in two dimensions. SIAM Journal on Scientific Computing, +23(3):741–760, 2001. +[5] William Fong and Eric Darve. The black-box fast multipole method. Journal +of Computational Physics, 228:8712–8725, 2009. +[6] Zydrunas Gimbutas and Vladimir Rokhlin. +A generalized fast multipole +method for nonoscillatory kernels. SIAM Journal on Scientific Computing, +24(3):796–817, 2003. +[7] I. Gradshteyn and I. M. Ryzhik. Table of integrals, series, and products. +Academic, London, 5th edition, 1980. +[8] L. Greengard and V. Rokhlin. +A fast algorithm for particle simulations. +Journal of Computational Physics, 73:325–348, 1987. +[9] Johan Helsing and Anders Karlsson. An explicit kernel-split panel-based +Nystr¨om scheme for integral equations on axially symmetric surfaces. Jour- +nal of Computational Physics, 272:686–703, 2014. +[10] M. Harper Langston, Leslie Greengard, and Denis Zorin. A free-space adap- +tive FMM-based PDE solver in three dimensions. Communications in Ap- +plied Mathematics and Computational Science, 6(1):79–122, 2011. +[11] James H. Strickland and Donald E. Amos. A fast solver for systems of ax- +isymmetric ring vortices. Technical Report SAND90-1925, Sandia National +Laboratories, Albuquerque, New Mexico, 87185, United States of America, +1990. +[12] James H. Strickland and Donald E. Amos. Fast solver for systems of ax- +isymmetric ring vortices. AIAA Journal, 30(3):737–748, 1992. +[13] P. Young, S. Hao, and P. G. Martinsson. A high-order Nystr¨om discretization +scheme for boundary integral equations defined on rotationally symmetric +surfaces. Journal of Computational Physics, 231:4142–4159, 2012. +21 + diff --git a/r9AzT4oBgHgl3EQfPPs-/content/tmp_files/load_file.txt b/r9AzT4oBgHgl3EQfPPs-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9f108f58c91547016301137efc2d9e37a6d80db0 --- /dev/null +++ b/r9AzT4oBgHgl3EQfPPs-/content/tmp_files/load_file.txt @@ -0,0 +1,340 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf,len=339 +page_content='A Fast Multipole Method for axisymmetric domains Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Carley January 4, 2023 Abstract The Fast Multipole Method (FMM) for the Poisson equation is extended to the case of non-axisymmetric problems in an axisymmetric domain, de- scribed by cylindrical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The method is based on a Fourier de- composition of the source into a modal expansion and the evaluation of the corresponding modes of the field using a two-dimensional tree decomposi- tion in the radial and axial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The field coefficients are evaluated using a modal Green’s function which can be evaluated using well-known recursions for the Legendre function of the second kind, and whose deriva- tives can be found recursively using the Laplace equation in cylindrical co- ordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The principal difference between the cylindrical and Cartesian problems is the lack of translation invariance in the evaluation of local inter- actions, leading to an increase in computational effort for the axisymmetric domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Results are presented for solution accuracy and convergence and for computation time compared to direct evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The method is found to converge well, with ten digit accuracy being achieved for the test cases pre- sented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Computation time is controlled by the balance between initialization and the evaluation of local interactions between source and field points, and is about two orders of magnitude less than that required for direct evaluation, depending on expansion order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 1 Introduction Since its development [8], the Fast Multipole Method (FMM) has become the al- gorithm of choice for a large class of problems which can be expressed in terms of finding at a large number of field points the potential generated by a large number of point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' This includes problems governed by the Poisson and Helmholtz equations, including boundary integral problems in acoustics, electro- magnetism, and fluid dynamics, and volume integrals such as the Biot–Savart 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content='01179v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content='NA] 3 Jan 2023 integration which arises in electromagnetism and vortex dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The method is now highly developed with efficient implementations available in two [4] and three [10] dimensions for a range of problems, using a variety of analytical tools for their formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Despite its power and importance, the FMM does not seem to have been ex- tended to non-Cartesian coordinate systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' In particular, there do not seem to be formulations of the method which can be applied to axisymmetric domains de- scribed using cylindrical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' These systems arise naturally in a range of applications and cylindrical coordinates are a natural way to describe a system, or to apply boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The purpose of this paper is to present an extension of the FMM for the Poisson equation to cylindrical coordinates, motivated by ap- plications in fluid dynamics, where boundary and volume integral problems are governed by the Laplace kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' To the author’s knowledge, there have been two previous studies which are relevant to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The first was the work of Strickland and Amos [11, 12] who developed an accelerated method for the evaluation of the axisymmetric stream function in vortex dynamics, equivalent to solving the Poisson problem or evaluating the Biot-Savart integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The authors used single-precision arithmetic and a fifth order expansion of the Green’s function to achieve five-digit accuracy at a cost of 1–3% of the computational effort required for direct evaluation of the potential and velocity fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The authors do not seem to have extended their method to the general case in a cylindrical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' More recently, in an unpublished thesis, Churchill [2] considered the general non-axisymmetric problem, motivated by the analysis of boundary integrals on surfaces of revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' He identifies the particular difficulty in cylindrical co- ordinates, which is that the Green’s function is translation invariant in the axial coordinate, but not in the radial, which complicates the evaluation of interactions which arise in the FMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' He concludes that the problem can be solved using a black-box [5] or generalized [6] FMM, but does not present results for the cylin- drical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' This paper presents a method for the fast evaluation of the potential in a cylin- drical domain, generated by a set of azimuthally varying ring sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The motiva- tion is the Poisson or Biot–Savart problem in fluid dynamics, with the expectation that the method is to be used in boundary integral solvers [13, for example], or in evaluating the velocity field due to a distribution of vorticity [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The approach is essentially that of a standard two-dimensional FMM, with modifications to the evaluation of interactions to accommodate the lack of translation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' It is assumed that any necessary Fourier transformation of the inputs to the calculation has been performed, so that the starting point is the location and radius of a set of circular sources, and the coefficients of the Fourier series for a source distribution on each circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The main elements of the method are described briefly, with a 2 more detailed description of those parts which are particular to the axisymmetric case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' the evaluation of the Green’s function and its derivatives in cylindrical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Results are presented for a problem with an increasing number of sources, to test the performance of the method for convergence and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 2 Analysis Figure 1: Basic problem: ring sources of radius ri are distributed at axial station zi in the domain 0 ≤ r ≤ rmax, 0 ≤ z ≤ zmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Sources are given as the coefficients of the Fourier series of source strength as a function of θ at each (ri, zi) The problem to be solved is sketched in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' We assume that any nec- essary preprocessing has been performed to reduce the system to the form shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' A set of ring sources with a common axis are distributed throughout the domain (r, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The source strength s(θ1) on a ring at location (r1, z1) is given by the Fourier series, s(θ1) = N � n=−N S(n)einθ1, (1) where subscript 1 denotes source coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The potential φ(r, z) due to the source is given by integration over θ1, φ(r, θ, z) = � 2π 0 s(θ1) 4πR dθ1, (2) R2 = r2 + r2 i − 2rr1 cos(θ − θ1) + (z − z1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 3 Under an elementary transformation, and substituting the Fourier series for s(θ1), φ(r, θ, z) = N � n=−N S(n)e−inθ � 2π 0 einθ1 4πR dθ1, (3) R2 = r2 + r2 1 − 2rr1 cos θ1 + (z − z1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The potential field φ(r, z) can then be expressed as a Fourier series in θ, φ(r, θ, z) = N � n=−N Φ(n)(r, z)einθ, (4) Φ(n) = S(n)G(n)(r, r1, z − z1), (5) G(n)(r, r1, x) = � 2π 0 einθ1 4πR dθ1 = � 2π 0 cos nθ1 4πR dθ1, (6) R2 = r2 + r2 1 − 2rr1 cos θ1 + x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The source term s(θ1) is assumed real, so that the complex Fourier coefficients are related by S(−n) = � S(n)�∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Henceforth, all computations will be performed for n ≥ 0 and the conjugate relationship will be assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' We refer to G(n)(r, r1, x) as the modal Green’s function relating Fourier coef- ficients of the source to those of the potential at some other point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Appendix A gives details of the evaluation of G(n)(r, r1, x) and of its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' In particular, it is shown that it can be expressed exactly as G(n)(r, r1, x) = Qn−1/2(χ) 2π√rr1 , (7) where Qν(χ) is a Legendre function of the second kind [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Given a distribution of sources at locations (ri, zi), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , Ns, each of which has a set of Fourier coefficients S(n) i , n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , N, the problem to be solved is then the approximate evaluation of the sum Φ(n)(rj, zj) = Ns � i=1 S(n) i G(n)(rj, ri, zj − zi), (8) for field points (rj, zj), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , Nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content='1 Outline of the FMM The Fast Multipole Method is well established and there are numerous guides to its algorithm and implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Here we give an outline of the method in order 4 to present the necessary terminology and to indicate those parts of the algorithm of this paper which differ from existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' From the previous section, we recall that the objective is to approximately evaluate the sum Φ(n)(rj, zj) = Ns � i=1 S(n) i G(n)(rj, ri, zj − zi), n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , N, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , Nf, given a list of source and field points (ri, zi and (rj, zj) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The first operation of the FMM is the sorting of points and their representation in a quadtree data structure, formed by repeated subdivision of the domain 0 ≤ ri,j ≤ rmax, 0 ≤ zi,j ≤ zmax, zmax = rmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Figure 2 shows the repeated halving of intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' At each level of subdivision ℓ, the domain is divided into 2ℓ × 2ℓ boxes, to a maximum level called the depth of the tree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Boxes in the tree can be indexed by their location (i, j) on the grid at a given level, or by their Morton index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' r z r r Figure 2: Recursive subdivision of the source domain into 4ℓ boxes for ℓ = 1, 2, 3, left to right Figure 3 gives the terminology for relationships between boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' A box at grid location (i, j) at level ℓ has four child boxes at grid locations (2i, 2j), (2i, 2j +1), (2i + 1, 2j), (2i + 1, 2j + 1) at level ℓ + 1, and has a parent box at grid location (i/2, j/2) at level ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' A box at the finest level of subdivision, depth d, has no children and is called a leaf box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Neighbors of box (i, j) are boxes at the same level ℓ which share at least a vertex with the box, including the box itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' In Figure 3, boxes 1–9 are neighbors of box 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' All other boxes are said to be in the far field of box 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The basic principle of the FMM is to separate far-field from neighbor interactions and evaluate the far- field terms in any box using an accelerated summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' This is achieved in the first instance by evaluating the field due to sources inside a box using an approximate expansion which is faster than direct evaluation of the sum Φ(n)(r, z) = � i S(n) i G(n)(r, ri, z − zi), 5 1 2 3 4 j i 5 6 7 8 9 Figure 3: Terminology for relationships between boxes: boxes 1–4 are children of the shaded box;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' the shaded box is the parent of boxes 1–4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' boxes 1–9 are neighbors of box 1 where the summation is taken over all sources inside a box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' At field points suffi- ciently far from the box, the Green’s function for a source is well approximated by its Taylor series, truncated to some order M, G(n)(r + ∆r, r1 + ∆r1, x + ∆x) ≈ M � m=0 � i+j+k=m g(n) i,j,k(∆r)i(∆r1)j(∆x)k, (9) g(n) i,j,k = 1 i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 1 j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' ∂i+j+k ∂ri∂rj 1∂xk G(n)(r, r1, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Details of the evaluation of the derivatives of G(n) are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The field at a point (r, z) due to a source at (r1 + ∆r1, z1 + ∆z1) in a box with center (r1, z1) is then given by Φ(n)(r, z) ≈ S(n) M � m=0 � i+j=m (−1)jg(n) 0,i,j(∆r1)i(∆z1)j, (10) noting that x = z − z1 and ∂/∂z1 = −∂/∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Summing over all sources contained in the box, Φ(n)(r, z) ≈ M � m=0 � i+j=m (−1)jg(n) 0,i,jS(n) ij , (11) where the moments Sij are given by S(n) ij = � q S(n) q (∆rq)i(∆zq)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' (12) 6 To initialize the source data in the first stage of the FMM, the moments S(n) ij are computed for each leaf box in the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' In the upward pass, the moments at boxes in each level ℓ are computed, for ℓ = d − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' This can be achieved without requiring direct evaluation of moments from source data, by combining moments from child boxes to generate moments in their parent box, Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Moments about the center of a child box at displacement (∆r, ∆z) contribute to the moments about the center of their parent box via Sij = i � q=0 j � u=0 �i q ��j u � (−∆r)q(−∆z)uS′ i−q,j−u, (13) where the superscript (n) has been suppressed for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' When the upward pass has been completed, each box at levels ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , d contain a set of moments which can be used to estimate the potential in the far field of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' ∆r ∆z ∆rq ∆zq Figure 4: Evaluation and shifting of source moments: moments about the center of the leaf box are computed using (12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' those about the center of the parent box are computed by shifting the child box moments by (∆r, ∆z) using (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' In the next stage of the FMM, the downward pass, each box is assigned a local expansion which can be used to evaluate the potential inside the box due to sources which lie in its far field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The core of the FMM is the use of the most efficient expansion possible at any level to evaluate the far-field terms in any box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The field in a box centered at (r, z) is given by Φ(n)(r + ∆r, z + ∆z) = M � m=0 � k+ℓ=m Φ(n) kℓ (∆r)k(∆z)ℓ, (14) where the expansion coefficients Φ(n) kℓ are evaluated from the contributions of sources in boxes which interact with the field box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The order M of the local 7 i j A B 2 1 Figure 5: Generation of a local field in a box: the local expansion of the field in box B due to sources in box A is computed using the S2L operation, (15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' the local expansion in box 2 is found by shifting the expansion in box B to box 2 and adding the contribution from box 1, evaluated using an S2L operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' expansion at any level is not required to be the same as the order of the source expansions, but has been set equal for the calculations presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Figure 5 shows the main operations involved, for the evaluation of a local expansion in box 2, which has parent box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' On the downward pass, the local ex- pansion in box B is found by adding the contribution from sources in boxes which are well separated from B, such as box A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' This contribution is found using the shift-to-local or S2L operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The local expansion in B is used to generate the local expansion in each of its child boxes, including box 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Box 2 then has its local expansion incremented by the contribution of boxes with which it interacts, such as box 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' At the end of the downward pass, each leaf box has a local expansion which accounts for the contribution of all sources lying outside its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The two operations to be implemented here are the S2L and the parent-to-child shift of the local expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The S2L shift is found by differentiating (10), Φ(n) kℓ = � i,j (−1)j �j + ℓ j � S(n) ij g(n) k,i,j+ℓ, (15) which can be implemented as a BLAS level 2 operation The local expansion in a child box at displacement (∆r, ∆z) is given from the 8 parent box expansion by, Φij = � q=0 � u=0 �i + q q ��j + u u � (−∆r)q(−∆z)uΦ′ i+q,j+u, (16) where terms Φ′ i,j are coefficients of the parent box local expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content='2 Evaluation of interactions The outline of the Fast Multipole Method presented in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content='1 contains the main elements of a generic FMM which are familiar from existing implementa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' In this section, we describe the part of the algorithm which is particular to the cylindrical domain, the S2L operation for the modal Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' In existing, Cartesian, methods, the translation operators are invariant with respect to shifts in the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' As noted by Churchill [2], however, this is not true for the modal Green’s function G(n), which is invariant for shifts in the ax- ial coordinate z but not for displacements in radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' This increases the number of orientations for which shift operators must be computed, though there are still some symmetries which can be exploited to reduce the workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Recall that the modal Green’s function is given by G(n)(r, r1, x) = Qn−1/2(χ) 2π√rr1 , χ = r2 + r2 1 + x2 2rr1 , x = z − z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' This is invariant under translations in z and is symmetric in r and r1, a fact which is exploited in the recursion relations for derivatives in source and field coordi- nates [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' To take advantage of this symmetry, we introduce some terminol- ogy to describe translation operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' If we assume that for S2L operations, we evaluate derivatives of G(n)(r, r1, x) for the r ≥ r1, x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Then, from Figure 6, we can derive translation operators for four different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' These correspond to shifts in the positive or negative (forward or backward) axial direction, and from greater to smaller radius (outward or inward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The basic operator, which uses the derivatives of the Green’s function without modification, is the forward-outward or FO shift, (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Φ(n) kℓ = � i,j (−1)j �j + ℓ j � S(n) ij g(n) k,i,j+ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' For the backward-outward or BO shift, x < 0, ∂/∂z1 = ∂/∂x and ∂/∂z = −∂/∂x, yielding Φ(n) kℓ = � i,j (−1)ℓ �j + ℓ j � S(n) ij g(n) k,i,j+ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 9 S FO BO BI FI r1 r z1 z1 + ∆z z1 − ∆z r z Figure 6: Reuse of Green’s function derivatives for forward/backward and in- ward/outward interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The source box S generates local expansions at r > r1 through a forward-outward (FO) and a backward-outward (BO) operation, indi- cated by solid arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The boxes at greater radius generate local expansions in box S via backward-inward (BI) and forward-inward (FI) operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' All four operations are computed from the same expansion of the Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' To evaluate the inward shifts, we exchange r and r1 and swap the correspond- ing indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' For the forward-inward S2L operation, Φ(n) kℓ = � i,j (−1)j �j + ℓ j � S(n) ij g(n) i,k,j+ℓ, which gives the contribution of the sources in the box at larger radius r to the local expansion in the box at smaller radius r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Finally, the backward-inward operator is given by Φ(n) kℓ = � i,j (−1)ℓ �j + ℓ j � S(n) ij g(n) i,k,j+ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' In order to apply the shift operations, we enumerate candidate source boxes which may contribute to the field in a box, Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' This gives rise to two in- teraction lists, the D list containing boxes which contribute via direct evaluation of the field for each source and field point, and the S2L list, whose contributions are evaluated using the S2L operation acting on source and local expansion co- efficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Figure 7 indicates that the D list is made up of neighbors of box B, leaving 27 boxes which may contribute to the field in B via S2L operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Con- tributions from all other boxes are transferred into B from its parent box during the downward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Using the axial translation invariance and the symmetry in r 10 B D D D D D D D D S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L S2L Figure 7: Interaction lists for box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Parent box of B is shown shaded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' boxes marked D have interactions evaluated directly from (8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' boxes marked S2L have interactions evaluated via source-to-local operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Unmarked boxes interact with B through its parent box so that the interactions need not be explicitly evalu- ated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' and r1 reduces the number of Green’s function expansions to evaluated to twelve, those for boxes at z ≥ z1 and r ≥ r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The expansions are identical for any value of z − z1 but must be updated for each r1 during the downward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Once gener- ated, the expansion is used to update the local expansion on the outer boxes, and to include the contribution of those boxes’ source terms to the local expansion on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Finally, we note that the modal Green’s function can be written using scaled coordinates, so that G(n)(σr, σr1, σx) = 1 σG(n)(r, r1, x), (17) which would allow for the shift operators to be precomputed and generated at each level as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' This has been implemented but found not to give a time saving, since each level of the tree requires twice as many shift operators as its parent level, half of which are new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' In practice, we find that the bottleneck in the code is the evaluation of direct interactions rather than the computation of the S2L operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 11 3 Algorithm Combining the elements of the previous sections, we present an algorithm for a uniform Fast Multipole Method in an axisymmetric domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Input is a list of Ns source points (ri, zi), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , Ns and modal amplitudes S(n) i , n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , N, and a list of Nf field points (rj, zj), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , Nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Algorithm 1 Fast Multipole Method for cylindrical coordinate systems set tree depth d sort source points (ri, zi) and field points (rj, zj) by Morton index and assign to leaf nodes at level d calculate leaf box moments from source amplitudes (12) {upward pass} for ℓ = d − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , 2 do evaluate box moments at level ℓ from child box moments at level ℓ + 1, (13) end for {downward pass} for ℓ = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , d do for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , 4ℓ−1 − 1 do shift parent box local expansion from level ℓ − 1 to level ℓ boxes end for for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , 2ℓ − 1 do evaluate coefficients of Green’s function expansions for radial station i for j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , 2ℓ − 1 do apply S2L operators for FO and BO translations of source in box (i, j) apply S2L operators for FI and BI translation to update local expansion in box (i, j) end for end for end for for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , 4d − 1 do for field points in box i evaluate sum of local expansion and direct contribu- tion from neighbor boxes end for The implementation in Algorithm 1 is for a uniform FMM which does not generate an adaptive decomposition when assigning points to boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' This was decided upon to reduce the number of shift operators required in evaluating box interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 12 4 Results −12 −10 −8 −6 −4 0 4 8 12 16 n log10 ϵ(n) Figure 8: Error ϵ against mode order for real part of modal amplitude Ns = 216, tree depth d = 6, expansion order M = 6, 8, 10, 12, 14, 16 from top to bottom of plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The algorithm has been tested for accuracy and computation time using source and field points randomly distributed over 0 ≤ r, z ≤ 1, with random modal amplitudes 0 < S(n) < 1, n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' In each case, source number Ns is set equal to number of field points Nf, with Ns = 2q, q = 10, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , 16, and the same order of expansion is used for source and field terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Results are presented for varying Ns, maximum expansion order M, and tree depth d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Code is written in GNU C, with gcc optimization -O3 and Goto BLAS matrix-vector operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Calculations were performed on one core of an Intel i5-6200U laptop running at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content='3GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Similar code and optimizations were used for the direct evaluations used as an error reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Error is evaluated for each modal amplitude of the field, ϵ(n) = max |Φ(n) FMM − Φ(n) D | max |Φ(n) D | , (18) where Φ(n) FMM is modal amplitude evaluated using the new algorithm, and Φ(n) D is that found by direct evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Sample results for error as a function of mode number and expansion order are shown for Ns = 216 in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The method is clearly accurate, especially for higher order expansions, where eleven digit ac- curacy is achieved for the axisymmetric mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The error increases at larger n, where the absolute value of the modal amplitudes is smaller, making the relative error measure larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 13 10−12 10−11 10−10 10−9 10−8 10−7 10−6 ϵ(0) 6 8 10 12 14 16 10−10 10−9 10−8 10−7 10−6 10−5 10−4 M ϵ(8) 6 8 10 12 14 16 M Figure 9: Error ϵ for n = 0 (top row) and n = 8 (bottom row) against expansion order M for Ns = 214 (left column) and Ns = 216 (right column): diamonds, tree depth d = 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' upward triangles, d = 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' downward triangles, d = 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' fitted lines ϵ ≈ C−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content='4M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 14 Figure 9 shows the variation in error with expansion order for the axisym- metric n = 0 mode and for n = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The error scales approximately as C−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content='4M with weak dependence on tree depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The algorithm performs well with respect to convergence over the range of problem sizes tested here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Figure 10 shows basic data for computation time as a function of problem size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The time for direct evaluation is shown and scales at the expected NfNs = N 2 s rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The computation time for the FMM algorithm behaves similarly for the low M = 6 order and high M = 16 order cases, with times being shifted up by the change in expansion order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The computation time in each case is roughly constant for small Ns, where the evaluation time is dominated by the set up cost, which de- pends on the tree depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' As the problem size increases, the evaluation time for the downward pass begins to dominate the calculation time which increases propor- tional to N 2 s , but with a much smaller leading constant than for direct evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Again, this is the expected behavior as the direct evaluation of near-field interac- tions becomes the largest part of the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' With increasing tree depth, the box to box evaluations become correspondingly faster as the number of sources per box becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' −1 1 3 5 211 213 215 217 Ns log10 t 211 213 215 217 Ns Figure 10: Computation time against source and field point number, for direct evaluation (solid line), and trees of depth 4 (boxes) depth 5 (diamonds), 6 (upward triangles), and 7 (downward triangles);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' left hand plot: expansion order M = 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' right hand plot order expansion order M = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Figure 11 shows the breakdown of computation time between the two parts of the calculation, as a function of tree depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The initialization phase, made up of the upward and downward passes, scales approximately linearly with problem size, with a leading constant determined by the tree depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Initialization time in- creases with tree depth, as the number of boxes increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The time for local field evaluation, in the lower plot, scales well on Ns/4d, the average number of sources per box, with the time reducing with tree depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The implication is that compu- tation time and accuracy are determined by the balance between initialization and local field evaluation, which depends on source number, expansion order, and tree 15 0 1 2 3 4 5 0 20 × 103 40 × 103 60 × 103 80 × 103 Ns tP 10−3 10−1 101 103 2−2 20 22 24 26 28 Ns/4d tD Figure 11: Execution time for phases of calculation, expansion order M = 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' boxes: tree depth d = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' diamonds, d = 5, triangles, d = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Upper plot, time for upward and downward passes, with linear fit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' lower plot, time for local field evaluation, power law fit to points with Ns/4d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 16 depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' −12 −10 −8 −6 0 30 60 90 120 t/s log10 ϵ(0) Figure 12: Error ϵ(0) against computation time: diamonds depth d = 5, triangles d = 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' solid lines, Ns = 216;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' dashed lines, Ns = 214;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' curves are second order fits to log ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Figure 12 gives results illustrating this balance, plotting error against compu- tation time, found by varying problem size and expansion order at two tree depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' For each tree depth and source number, there is a trend towards a minimum error, with the greater tree depth requiring a greater total computation time for higher order accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 5 Conclusions The Fast Multipole Method has been extended to non-axisymmetric problems in cylindrical domains, by evaluating the amplitudes of the modes in a Fourier ex- pansion of the source and field in a Poisson problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Testing by comparison with direct evaluation shows convergence to up to ten digit accuracy, and orders of magnitude speed-up, depending on expansion order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Open questions remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' The first is the efficient evaluation of the Legendre functions used to find the modal Green’s function, which is common to the direct and fast methods, and constitutes the largest computational demand in the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' A second is the formulation of the method in a form which allows the use of BLAS level 3 operations, which should allow the code to be optimized further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Finally, we note that the approach taken here should be applicable to the Helmholtz problem, though with some greater difficulty in evaluating the modal Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 17 A Evaluation of Green’s functions and derivatives The modal Green’s function G(n)(r, r1, x) is defined: G(n)(r, r1, x) = � 2π 0 einθ1 4πR dθ1, (19) R2 = r2 + r2 1 − 2rr1 cos θ1 + x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Cohl and Tohline [3] give an expansion for 1/R, 1 R = 1 π√rr1 ∞ � m=−∞ eim(θ−θ1)Qm−1/2(χ), (20) χ = r2 + r2 1 + x2 2rr1 , where Qν(χ) is the Legendre function of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Integration over θ1 yields G(n)(r, r1, x) = Qn−1/2(χ) 2π√rr1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' (21) Using the recursion for the Legendre function [7, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content='732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content='2], with the functional dependence on coordinates suppressed for clarity, (2n − 3)G(n−2) = 4(n − 1)χG(n−1) − (2n − 1)G(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' (22) For χ > 1, the forward recursion is unstable and the backward recursion is stable, but computationally expensive [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' To generate the sequence of modal Green’s functions, we apply the approach of Helsing and Karlsson [9] and use the forward recursion for χ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content='008, beginning with the initial values [3], Q−1/2(χ) = µK(µ), (23) Q1/2(χ) = χµK(µ) − (1 + χ)µE(µ), (24) µ = � 2 1 + χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Here K(·) and E(·) are the complete elliptic integrals of the first and second kind respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' These are computed using the method of Carlson [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' For χ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content='008, the backward recursion is used starting with arbitrary values of Qn−1/2(χ) and Qn−3/2(χ) for n = N +80, performing the downward recursion to n = 0 and scaling the sequence using the known value of Q−1/2(χ), (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Val- ues of G(n) evaluated using this procedure have been checked against numerical integration and have been found to be correct to machine precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 18 Given values of G(n), n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' , N, the derivatives of G(n) can be found using a combination of the recursion relations for the Legendre function and the Laplace equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' For concision, we introduce the notation g(n) i,j,k = ∂i+j+k ∂ri∂rj 1∂xk G(n)(r, r1, x), (25) g(n) i,j,k = 1 i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 1 j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' ∂i+j+k ∂ri∂rj 1∂xk G(n)(r, r1, x), (26) so that the Taylor series for G(n) is given by G(n)(r + ∆r, r1 + ∆r1, x + ∆x) = ∞ � m=0 � i+j+k=m g(n) i,j,k(∆r)i(∆r1)j(∆x)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' (27) The derivatives are evaluated using a recursion based on the Laplace equation, similar to the approach of Strickland and Amos who used the axisymmetric stream function equation [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Here we use the Laplace equation for a field with azimuthal dependence exp inθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' This recursion requires starting values which can be found using the properties of the Legendre functions [7, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content='732].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' For derivatives with respect to x, g(n) 0,0,1 = (n − 1/2) �� g(n) 0,0,0 + g(n−1) 0,0,0 � x ρ2 + + � g(n) 0,0,0 − g(n−1) 0,0,0 � x ρ2 − � , (28) g(n) 0,0,k+1 = (n − 1/2) k � q=0 1 q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' (k + 1) �� g(n) 0,0,k−q + g(n−1) 0,0,k−q � ∂q ∂xq � x ρ2 + � + � g(n) 0,0,k−q − g(n−1) 0,0,k−q � ∂q ∂xq � x ρ2 − �� , (29) ρ2 ± = (r ± r1)2 + x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' To evaluate derivatives for n = 0, the relation g(−1) i,j,k ≡ g(1) i,j,k can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' For the derivatives with respect to r, g(n) 1,0,0 = −g(n) 0,0,0 2r + (n − 1/2)r2 − r2 1 − x2 2r �� g(n) 0,0,0 + g(n−1) 0,0,0 � 1 ρ2 + + � g(n) 0,0,0 − g(n−1) 0,0,0 � 1 ρ2 − � , (30) g(n) 1,0,k = −g(n) 0,0,k 2r + n − 1/2 2r k � q=0 1 q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' � � g(n) 0,0,k−q + g(n−1) 0,0,k−q � ∂q ∂xq r2 − r2 1 − x2 ρ2 + + � g(n) 0,0,k−q − g(n−1) 0,0,k−q � ∂q ∂xq r2 − r2 1 − x2 ρ2 − � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' (31) 19 Derivatives g(n) 0,1,k are found by exchanging r and r1, with a corresponding swap of indices in the derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Finally, g(n) 1,1,k = −g(n) 0,1,k 2r (32) + n − 1/2 2r k � q=0 1 q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' � � g(n) 0,0,k−q + g(n−1) 0,0,k−q � ∂q+1 ∂r1∂xq r2 − r2 1 − x2 ρ2 + + � g(n) 0,1,k−q + g(n−1) 0,1,k−q � ∂q ∂xq r2 − r2 1 − x2 ρ2 + + � g(n) 0,0,k−q − g(n−1) 0,0,k−q � ∂q+1 ∂r1∂xq r2 − r2 1 − x2 ρ2 − + � g(n) 0,1,k−q − g(n−1) 0,1,k−q � ∂q ∂xq r2 − r2 1 − x2 ρ2 − � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' To compute the remaining derivatives, we make use of a recursion based on the Laplace equation in cylindrical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Noting that ∂/∂θ → in, 1 r ∂ ∂r � r∂G(n) ∂r � − n2 r2 G(n) + ∂2G(n) ∂x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' (33) This yields the relation, g(n) i+2,j,k = i � u=0 (−1)u ru+1 1 (i + 1)(i + 2) �(u + 1)n2 r g(n) i−u,j,k − (i − u + 1)g(n) i−u+1,j,k � , (34) which can be used to recursively generate higher derivatives with respect to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Switching r and r1 gives a corresponding relation for the higher derivatives with respect to r1 and allows a complete set of derivatives to be evaluated to any re- quired order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' References [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Carlson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Numerical computation of real or complex elliptic integrals.' metadata={'source': 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+page_content=' Strickland and Donald E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Amos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' A fast solver for systems of ax- isymmetric ring vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Technical Report SAND90-1925, Sandia National Laboratories, Albuquerque, New Mexico, 87185, United States of America, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' [12] James H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Strickland and Donald E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Amos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Fast solver for systems of ax- isymmetric ring vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' AIAA Journal, 30(3):737–748, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' [13] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Young, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Hao, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Martinsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' A high-order Nystr¨om discretization scheme for boundary integral equations defined on rotationally symmetric surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' Journal of Computational Physics, 231:4142–4159, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} +page_content=' 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfPPs-/content/2301.01179v1.pdf'} diff --git a/rNE1T4oBgHgl3EQfiwTR/content/tmp_files/2301.03256v1.pdf.txt b/rNE1T4oBgHgl3EQfiwTR/content/tmp_files/2301.03256v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fd339beaec5a341c61543636068cb7c751d521fa --- /dev/null +++ b/rNE1T4oBgHgl3EQfiwTR/content/tmp_files/2301.03256v1.pdf.txt @@ -0,0 +1,1112 @@ +X-ray detected ferromagnetic resonance techniques +for the study of magnetization dynamics +Gerrit van der Laan1 and Thorsten Hesjedal2 +1Diamond Light Source, Harwell Science and Innovation Campus, Didcot, OX11 0DE, United Kingdom +2Department of Physics, Clarendon Laboratory, University of Oxford, Oxford, OX1 3PU, United Kingdom +(Dated: January 10, 2023) +Element-specific spectroscopies using synchrotron-radiation can provide unique insights into +materials properties. The recently developed technique of X-ray detected ferromagnetic resonance +(XFMR) allows studying the magnetization dynamics of magnetic spin structures. +Magnetic +sensitivity in XFMR is obtained from the X-ray magnetic circular dichroism (XMCD) effect, where +the phase of the magnetization precession of each magnetic layer with respect to the exciting +radio frequency is obtained using stroboscopic probing of the spin precession. +Measurement of +both amplitude and phase response in the magnetic layers as a function of bias field can give a +clear signature of spin-transfer torque (STT) coupling between ferromagnetic layers due to spin +pumping. Over the last few years, there have been new developments utilizing X-ray scattering +techniques to reveal the precessional magnetization dynamics of ordered spin structures in the GHz +frequency range. The techniques of diffraction and reflectometry ferromagnetic resonance (DFMR +and RFMR) provide novel ways for the probing of the dynamics of chiral and multilayered magnetic +materials, thereby opening up new pathways for the development of high-density and low-energy +consumption data processing solutions. +Keywords: FMR, XMCD, X-ray scattering, X-ray reflectivity, spin structures +I. +INTRODUCTION +Magnetization dynamics is at the heart of high fre- +quency magnetic nanoscale devices based on spin waves, +spin pumping, and spin-torque oscillators in the GHz +frequency range. Traditionally, ferromagnetic resonance +(FMR) has been a work horse technique to determine the +fundamental parameters for magnetic resonance and re- +laxation in thin films. The recent growing complexity of +many modern magnetic materials and devices requires +the development of advanced measurement techniques +that more directly reveal the microscopic origin of the +dynamical magnetic interactions that are at play. +The novel techniques of X-ray detected FMR (XFMR) +enables studying the magnetization dynamics of indi- +vidual layers, where element-specific magnetic contrast +is obtained using the X-ray magnetic circular dichroism +(XMCD) effect [1]. Not only can the FMR signal be mon- +itored in X-ray absorption, it can also be done in X-ray +diffraction and reflectivity, using techniques termed as +DFMR and RFMR, respectively [2]. In these X-ray mea- +surements, time-resolved FMR gives both the amplitude +and phase of the spin precession for the different chemical +elements, and hence different layers, in the sample. The +challenge of such measurements lays in the fact that the +precession cone angle is small (<1◦) and that the preces- +sion frequency is on the order of GHz. The solution is +to use lock-in techniques and to detect the phase of the +precession stroboscopically by using the time structure +of the X-ray pulses from the synchrotron (∼500 MHz, +i.e., corresponding to a period between the pulses of 2 +ns). The radio frequency (RF) field applied to drive the +spin precession is synchronized with the X-ray pulses us- +ing the clock of the synchrotron. Therefore, each X-ray +pulse measures the magnetization cone at precisely the +same point in the precession cycle. Hence, XFMR com- +bines the techniques of FMR and XMCD. Thus, the spin +precession along the bias field is pumped by the RF field +to generate the magnetic resonance (i.e., FMR), whose +amplitude and phase is probed by the synchronized X- +ray pulses using the XMCD effect. The time dependence +is recorded using a delay line to vary the phase of the RF +signal with respect to the X-ray pulses. +During the last few years, many XFMR studies either +in time-averaged or time-resolved mode have been re- +ported [3–52]. The first element-specific and time-depen- +dent measurement of the magnetization dynamics using +pump-probe XMCD was reported by Bailey et al. [3] on a +permalloy (Py = Ni80Fe20) thin film, where the moments +on the Ni and Fe sites were found to precess together at +all frequencies, and by Arena et al. [4] on a Py/Cu/CoZr +trilayer, where at resonance, a weak ferromagnetic cou- +pling was found in the phase and amplitude response of +individual layers across resonance. +The amplitude and phase response of the magnetic +probe layer measured by XFMR provides a signature for +either static exchange interaction in strongly exchange- +coupled bilayers or spin-transfer torque (STT) coupling +due to spin pumping. Marcham et al. [25] first evidenced +STT in a CoFe/Cu/Py spin valve using XFMR where the +field dependence of the fixed layer phase showed a clear +signature of STT due to spin pumping. Using XFMR, +Baker et al. [28] reported a strong anisotropy of the spin +pumping, providing new opportunities for device appli- +cations. +Previously, time-resolved XFMR has been reviewed in +great detail in Ref. [1]. Here, we present a timely up- +date, especially emphasizing the newly developed time- +arXiv:2301.03256v1 [cond-mat.mtrl-sci] 9 Jan 2023 + +2 +resolved FMR techniques in X-ray reflectivity and diffrac- +tion. +The outline of the remainder of this paper is as follows. +Sec. II gives a brief theoretical background of magnetiza- +tion dynamics and STT. Sec. III describes the experimen- +tal setup, conditions, and considerations for the various +XFMR techniques. Sec. IV highlights several recent ex- +amples of XFMR, DFMR, and RFMR experiments and +mentions their scientific impact. Finally, conclusions are +drawn in Sec. V. +II. +BACKGROUND ON FMR AND STT +A. +Ferromagnetic resonance (FMR) +Before presenting the experimental details and show- +casing several recent examples, we will briefly introduce +some relevant background material. +FMR arises when the energy levels of a quantized sys- +tem of electronic moments are Zeeman split by a uniform +magnetic field and the system absorbs energy from an os- +cillating magnetic field [53]. A resonance occurs when the +transverse AC field is applied at the Larmor frequency +corresponding to the energy difference between the mag- +netic levels, i.e., ℏω = ∆E. +The spin precession in a +single-domain magnetic material can be described with +the equation of motion, the so-called Landau-Lifshitz- +Gilbert (LLG) equation, +˙m = −γm × Heff + α(m × ˙m), +(1) +where the effective field Heff = −∂F(M)/∂M is ob- +tained by minimization of the free energy F with re- +spect to the magnetization M. The free energy contains +terms such as the exchange, Dzyaloshinskii-Moriya, de- +magnetization, magnetocrystalline anisotropy, magneto- +static, external Zeeman field, and elastic energy. Further, +˙m = δm/δt; the reduced magnetization is m = M/Ms, +where Ms = |m| is the saturation magnetization; and +γ = gµB/ℏ is the gyromagnetic ratio, where µB is the +Bohr magneton and g is the Land´e (spectroscopic split- +ting) g-factor. +The dimensionless damping parameter +α ≪ 1 (typically 10−3–10−2 for 3d metals) determines +the width of the resonance absorption peak. +The first right-hand term in Eq. (1) corresponds to +the torque due to the effective field Heff. In a classical +picture, τ = dS/dt equates to the time change in an- +gular momentum S, which leads to the spin precession. +The second right-hand term corresponds to the damp- +ing, which can also be written in the form of the Gilbert +damping term −αγ(m × m × Heff). +Both torque and +damping are vectorially sketched in Fig. 1(a). Without +external RF excitation, the magnetization would relax to +the steady state given by Brown’s equation, m×Heff = 0. +Linearization of the LLG equation gives the relation +between the frequency ν0 (or circular frequency ω0) and +field, which in the form of the Kittel equation is written +as [1] +2πν0 ≡ ω0 = γ +� +HeffBeff = γ +� +Heff(Ms + Heff). +(2) +B. +Spin-transfer torque (STT) +The layer selectivity of XFMR makes this technique +a unique probe to investigate STT and related spin cur- +rents in multi-layered spin valves [1]. STT is the effect +in which the spin direction in a magnetic layer can be +modified using a spin-polarized current [54, 55]. +Spin pumping occurs when the precessing magnetiza- +tion vector generated by FMR in a ferromagnetic (FM) +layer emits a pure spin current into an adjacent normal +metal (NM) layer [56]. Traditionally, spin currents have +been probed using indirect measurements. For instance, +in the metals through which they flow they can create an +electrical voltage drop perpendicular to the spin current +direction, or a torque that bends the magnetization di- +rection. However, such indirect measurements are often +ambiguous because they are also influenced by other fac- +tors, such as magnetic proximity effects at the interface. +STT gives an extra term in the LLG equation, which +is (anti)-parallel to the (anti)-damping (see Fig. 1(a)). +According to Slonczewski [54], the adiabatic torque is +τs = αs m × ˙m, where αs is the STT damping. +The +spin current pumped across a FM/NM interface due to +precession is [56] +Is = ℏ +4π g↑↓ +eff m × ˙m , +(3) +where g↑↓ +eff is the effective spin-mixing conductance. The +spin pumping depends critically on the FM/NM inter- +face (the material-dependent g↑↓ +eff) and the spin diffusion +length in the NM. +For two FM layers i and j with different resonance fre- +quencies and coupled by both spin pumping (dynamic ex- +change coupling) and static exchange coupling, the cou- +pled LLG equations are +˙mi = − γmi × Heff,i + α0 +i mi × ˙mi ++ αs +imi × ( ˙mi − ˙mj) + Aexmj · mi , +(4) +and equivalently when exchanging i ↔ j, where mi is +the magnetization direction, Heff,i the effective field, α0 +i +the Gilbert damping, and αs +i the STT damping in layer +i. The spin pumping induced coupling is determined by +αs +i and the static exchange coupling by Aex. +III. +EXPERIMENTAL +XFMR provides an element-specific and time-resolved +measurement of the precessional dynamics of each FM +layer on a ps time scale, where the spin precession in- +duced by a driving RF signal is detected using the XMCD + +3 +Heff +h(t) +m(t) +Sample +CPW +(b) +m × Heff +z +y +Heff +m(t) +m × m × Heff +STT +x +(a) +FIG. 1. +(a) Precession, damping, and spin transfer torque +(STT) in FMR. The precession m × Heff around the effective +field Heff is damped by the Gilbert term m × m × Heff. The +spin-transfer torque is parallel (antiparallel) to the Gilbert +damping term, and can enhance (oppose) the latter depend- +ing on the direction of the spin current. (b) Schematics of +the sample geometry for XFMR. The sample (red disk) is +mounted on the signal line (the central strip) of a coplanar +waveguide (CPW). The magnetization m(t) precesses about +Heff, driven by the in-plane continuous RF field h(t) in the +CPW. The cone angle of precession is exaggerated for clar- +ity; its typical magnitude is ∼1◦. Circularly polarized X-ray +pulses from the synchrotron impinge at an grazing incidence +angle on the sample in transverse geometry in order to enable +stroboscopic detection of the oscillatory component of m(t) +at variable phase delays. +effect [57]. However, before performing the XFMR ex- +periment, the static magnetization of the samples has +to be precharacterized with standard techniques, such as +superconducting quantum interference device (SQUID) +magnetometry to measure the hysteresis loops along the +easy and hard direction, followed by standard FMR mea- +surements. +A. +VNA-FMR +Vector network analyzers (VNA)-FMR measurements +are used to characterize the magnetic resonances in order +to judge whether these are suitable and intense enough +for the XFMR measurements at the synchrotron. VNA- +FMR is a broadband FMR technique, where the sample is +mounted onto a coplanar waveguide (CPW) and driven +by an external RF field, while under a static magnetic +bias field. Measurement of the S-parameters of the sam- +ple results in a frequency-field map, where the resonances +appear in the form of Kittel curves (Eq. (2)). The angu- +lar dependence of the resonances gives information about +the magnetic anisotropy [53]. It allows us to chose the +best azimuthal angle of the applied field with respect to +the crystallographic axes to separate the magnetic res- +onances at the optimal distance for detecting STT [28]. +Hence, at a given RF frequency, this gives us the cor- +responding field values of the resonances in the XFMR +experiments. Conventional FMR will normally probe the +whole thickness of a thin film since the skin depth of, e.g., +metallic iron at 10 GHz, is on the order a micron. +B. +XMCD +At the synchrotron, first the static XMCD is measured +by sweeping the photon energy across the absorption +edge of the magnetic elements. +This allows us to se- +lect the fixed photon energies suitable for XFMR. The +static XMCD is obtained from the difference between +two X-ray absorption spectra recorded with the helic- +ity vector of the circularly polarized X-rays parallel and +antiparallel, respectively, to the applied magnetic field +[58]. The XMCD signal is proportional to the projection +of the helicity vector, which is along the incident beam +direction ˆk, onto the sample magnetization M, hence +IXMCD ∝ ˆk · M. +The XMCD at the soft X-ray absorption edges, such +as the Fe, Co, and Ni L2,3, is very strong [59], which +helps to compensate for the small changes in magneti- +zation direction due to the limited cone angle (<1◦) of +the precession in XFMR. The X-ray penetration length, +which limits the sampling depth, is in the nm range, e.g., +for pure Fe it is ∼20 nm at the Fe L3 maximum at ∼707 +eV [57]. +By tuning the photon energy away from the +absorption maximum the penetration length can be in- +creased (∼600 nm below the edge at 700 eV). Note that +the length scale of the probing depth is well matched to +the thickness of the magnetic layers in spin valves. The +typical lateral spot size of the X-ray beam on the sample +is 200 × 20 µm2, again well suited for small devices. +C. +Time-resolved measurements +The measurement of the projected magnetic moment +in XFMR does not require to take the difference between +opposite circular polarizations as done in XMCD. In- +stead, a change in the projection of the magnetization +precession is measured using a fixed circular polarization. +XFMR can be measured in two distinctly different ge- +ometries, namely (i) time-averaged in longitudinal ge- +ometry [12, 20] or (ii) time-resolved in transverse geom- +etry [8, 25, 28]. In longitudinal geometry, a shortening +of the magnetization vector along the z-axis (parallel to +the X-ray beam direction) leads to a difference ∆Mz = +Ms(1−cos θ) ≈ 1 +2Msθ2, where θ is the small cone angle of +the magnetization precession. The time-averaged XFMR +requires no synchronization with the synchrotron clock, +therefore it can be done at an arbitrary frequency, but +it needs a larger RF power which can lead to nonlinear +effects. +Only measurements in transverse geometry give access +to the precessional phase. This geometry is depicted in +Fig. 1(b). The transverse component of the magnetiza- +tion precession will give a sinusoidal variation on top of +the static X-ray absorption signal. With the incident X- +ray beam perpendicular to the bias field, the oscillating + +4 +component of the magnetization precession is measured +with a magnitude |My| = Ms sin θ ≈ Msθ. Thus, for a +typical cone angle of θ ≈ 1◦, the transverse geometry +gives a signal that is larger by a factor of ∼200 com- +pared to the longitudinal geometry. Due to the shape +anisotropy of the film, the precession is strongly ellipti- +cal, often with a larger in-plane amplitude. This favors +a measurement geometry with the X-rays at grazing in- +cidence. A good compromise is an X-ray incidence angle +of ∼35◦ with respect to the plane of the sample, which +ensures that the signal is sensitive to the larger in-plane +component of the magnetization precession. +Using a vector magnet system, such as the portable oc- +tupole magnet system (POMS) at Diamond [57], where +the field can be applied in any direction, permits a simple +change of the field from (i) parallel to the photon direc- +tion, as needed for static XMCD scans, to (ii) orthogonal +to both X-ray beam and RF field direction, as required +for time-resolved XFMR. +The detection of the X-ray absorption can be done by +either X-ray transmission [27, 31], fluorescence yield [23], +or X-ray scattering or reflectivity [2, 35, 36, 60, 61]. How- +ever, RF plays havoc with total-electron yield. In the +case of transmission, the incident X-rays impinge on the +sample through a hole in the signal line of the CPW. +After passing through the sample, the transmitted X- +rays are detected with X-ray excited optical luminescence +(XEOL) emerging from the MgO or sapphire (Al2O3) +substrate using a photodiode placed behind the sample. +Note that not all substrates, such as non-transparent ones +like Si, are suitable for XEOL detection [62]. +Time resolution is established by using the periodic +X-ray pulses from the synchrotron (normally operating +in multibunch or hybrid mode). To enable stroboscopic +probing, the RF driving field is taken as a harmonic of +the X-ray pulse frequency, hence the resonance is driven +at multiples of the master oscillator clock of the stor- +age ring. These harmonics are generated using an RF +comb generator (Atlantic Microwave) driven by the mas- +ter oscillator clock, which has a frequency of 499.65 MHz +(at the DLS, ALS, and BESSY synchrotron). This cor- +responds to ∼2 ns intervals between consecutive X-ray +pulses, which have a pulse width of ∼35 ps (at DLS and +BESSY) or ∼70 ps (at ALS). The desired frequency is +selected using filters and amplifiers to drive a narrow +band, high power (25–30 dBm) RF field to the CPW. +A programmable delay line (Colby Instruments) enables +phase shifting of the RF oscillation with respect to the +X-ray pulses with a step resolution of ∼0.5 ps. +De- +pending on the specific technique either the transmitted, +diffracted, or reflected X-rays are measured using a pho- +todiode. Fig. 2 show a schematic representation of the +setup for DFMR; for XFMR and RFMR the electronics +is very similar. +The signal is obtained using a lock-in +amplifier (LIA) by switching the signal at a given audio +frequency. There are two usual modulation modes. In +amplitude modulation the LIA measures the difference +between signals obtained with the RF signal on and off. +FIG. 2. +Schematic of the setup for DFMR measurements in +the RASOR diffractometer at the Diamond Light Source. The +sample is placed on the CPW, which is mounted on the cold +finger inside the diffractometer. Incident circularly or linearly +polarized X-rays are scattered off the sample and detected via +a photodiode in a ϑ-2ϑ geometry. A variable magnetic field +is applied in the scattering plane via a pair of permanent +magnets whose distance can be controlled externally. An RF +signal is fed to the CPW to drive the ferromagnetic resonance +in the magnetic sample. As the synchrotron gives X-ray pulses +at a frequency of ∼500 MHz, a comb generator is used to +produce higher harmonics, which are selected and fed to the +CPW. To probe the time dependence of the scattered X-ray +intensity, a tunable delay line is used, which shifts the phase +between the pump (the RF signal) and the probe (the pulsed +X-rays). (Adapted from Ref. [63]). +In 180◦-phase modulation, the LIA measures the differ- +ence between signals obtained with the RF of opposite +phase. +D. +XFMR +In order to record the time-resolved XAS signal with +circular polarization at fixed photon energy, the RF fre- +quency is locked to a multiple of the synchrotron clock. +Then at fixed angles and for given temperature, this +leaves two free scanning parameters, namely the mag- +netic bias field strength and the delay time between X-ray +pulses and RF field. +Magnetic field scans record the signal by sweeping the +bias field at a constant delay time. The signal contains +both real and imaginary parts of the magnetic suscepti- +bility, whose relative contributions strongly change across +resonance. By measuring two field scans, which differ by + +5 +90◦ in phase (obtained using the corresponding time de- +lays), and fitting these scans simultaneously using the +Kramers-Kronig relation, gives a good apprehension of +the field dependence of the resonances [40]. +Delay scans record the signal for each of the magnetic +layers at constant bias field by sweeping the delay time. +As an example, Fig. 3(a) shows a series delay scans over +two periods of the phase taken at different bias fields (40– +200 mT) across the Co resonance in a magnetic tunnel +junction (MTJ), in more detail discussed in Sec. IV A. +The solid lines represent sinusoidal fits to the experimen- +tal data (dots), from which the amplitude and relative +phase of the magnetization precession can be extracted. +A sinusoidal function of the form +S(t) = X sin(2πνt) + Y cos(2πνt), +(5) +is fitted to the delay scan, where t is the time delay and +ν the frequency of the RF. This procedure is repeated +for various field strengths and directions. By extracting +the coefficients X and Y in Eq. (5) from the delay scans, +the amplitude A and phase ψ of the oscillations can be +determined using the relationships +A = +� +X2 + Y 2, +ψ = 2 arctan +� +Y +A + X +� +. +(6) +XFMR precessional plots are assembled by combining +the amplitudes and phases extracted from the delay scans +measured over a range of bias fields. This gives the field +dependence of the amplitude and phase for each element +(e.g., for Co and Ni in Fig. 3(b)), from which the type +of coupling between layers can be assessed. By normaliz- +ing the XFMR signal to the static XMCD, the amplitude +of the signal can be obtained per atom for each chemi- +cal element in the sample. This enables a quantitative +decomposition of the resonance features [31]. +The static coupling (i.e., exchange interaction) and dy- +namic coupling (i.e., spin pumping) give a very differ- +ent XFMR response, as can be understood from Eq. (4). +Consider a pump layer FM1 that is free to rotate, and a +probe layer FM2 that is pinned. Using XFMR at a fixed +frequency, we scan the field across the entire resonance. +At resonance, FM1 will show a symmetric peak for the +amplitude, while the phase is 90◦ delayed with respect +to the RF driving field. Across the entire resonance, the +phase will change by 180◦. +To investigate the type of +coupling between both layers we measure the XFMR re- +sponse of FM2 at the resonance condition of FM1. +For static exchange coupling, E = −Aexm1 · m2, so +that Heff,2 ∝ m1. This means that the effective field in +the second layer is aligned along the magnetization of the +first layer. Then the field dependent precession of FM2 +will show a dispersive (bipolar) peak in the amplitude +and a symmetric (unipolar) peak in the phase. +On the other hand, for dynamic exchange coupling +Heff,2 ∝ +˙m2 = −iωm1. +The magnetic field is imagi- +nary, resulting in a 90◦ phase change. In this case, the +field dependent precession of FM2 will show a unipolar +peak in the amplitude and bipolar peak in the phase. +This behavior means that XFMR can distinguish be- +tween static and dynamic coupling by their amplitude +and phase signature in the precessional plot, and thus +determine the relative contribution of these couplings. +This has previously been utilized for, e.g., exchange cou- +pled layers [4, 31], spin values [25], MgO magnetic tunnel +junctions [29, 45], topological insulators [27, 30, 41], spin +valve with δ-layer [26], Heusler alloys [40], NiO antiferro- +magnetic interlayer [42], exchange springs [44], and α-Sn +thin films [47]. +E. +DFMR and RFMR +DFMR and RFMR measurements have been per- +formed in the RASOR soft X-ray diffractometer on beam- +line I10 at the Diamond Light Source [57] (see setup in +Fig. 2). Incident X-rays with wavevector ki illuminate +the sample, while the scattered beam (ks) is detected us- +ing a photodiode. The scattering geometry is configured +to probe the sample at certain diffraction or specular re- +flectivity conditions. The sample in the diffractometer is +mounted on a CPW that is connected to a liquid He cryo- +stat arm which can reach temperatures down to 12 K. A +bias magnetic field is applied by two permanent magnets, +which can be positioned to vary both the field strength +up to 200 mT and the orientation within the scatter- +ing plane. Perpendicular to the bias field, a transverse +RF field around the central conductor of the CPW is +generated, which excites the magnetization dynamics in +the system. In contrast to conventional XFMR measure- +ments, where the sample is mounted flip-chip onto the +CPW, in the scattering geometry the sample is mounted +face up to allow for the X-ray beam to probe its sur- +face. To ensure good coupling between the CPW and the +probed top surface, the sample must either be thinned, +or in the case of multilayers, grown on a thin substrate +of the order of 100 µm. +In DFMR, where the detector is aligned to a Bragg +peak or magnetic scattering peak, the stroboscopic sig- +nal is used to measure delay scans for different linear or +circular polarization of the incident X-rays, to give infor- +mation about the periodic spin structure. +In RFMR where the photo diode detector accepts the +reflected beam the stroboscopic signal is used to measure +delay scans for different values of the scattering vector +Qz, to obtain depth information. An advantage of RFMR +over DFMR is that it can be done on thin films and +multilayers, and no single crystals are needed. +IV. +X-RAY BASED FMR EXAMPLES +A. +XFMR of spin-current mediated exchange +coupling in MgO-based MTJs +Magnetic tunnel junctions composed of ferromagnetic +layers which are mutually interacting through a nonmag- + +6 +netic spacer layer are at the core of magnetic sensor and +memory devices. G�ladczuk et al. [45] used layer-resolved +XFMR to investigate the coupling between the magnetic +layers of a Co/MgO/Py MTJ. Two magnetic resonance +peaks were observed for both magnetic layers, as probed +at the Co and Ni L3 X-ray absorption edges. +Figure 3 shows XFMR delay scans for the Co layer in +the Co/MgO/Py MTJ at 80 K continuously driven at 4 +GHz. The curves in Fig. 3(a) show a strong increase in +amplitude as well as a large phase shift across the reso- +nance at ∼90 mT. The amplitude and phase of the pre- +cession, which are extracted using Eq. (6), are shown in +Fig. 3(b) and (c), respectively, for both Co (orange) and +Ni (blue) as a function of the bias field. The amplitude +curves show that the Ni resonance originating from the +Py layer around ∼120 mT is strongly coupled with the +Co layer. On the other hand, the Co resonance around +∼90 mT is only weakly present in the Py layer. Instead +of plotting amplitude A and phase ψ, one can also plot +the FMR signal in the (X,Y )-plane as a function of field +[45]. Since the sine and cosine functions in Eq. (5) are +orthogonal, the estimators of X and Y are given by pro- +jections to orthogonal subspaces. +A theoretical model based on the Landau-Lifshitz-Gil- +bert-Slonczewski equation (Eq. (4)) was developed, in- +cluding exchange coupling and spin pumping between the +magnetic layers. Fits to the experimental data were car- +ried out, both with and without a spin pumping term, +and the goodness of the fit was compared using a likeli- +hood ratio test. This rigorous statistical approach pro- +vided an unambiguous proof of the existence of interlayer +coupling mediated by spin pumping through MgO [45]. +It was also found that spin pumping is more effective +at lower temperatures, which agrees with the theoretical +understanding. +B. +XFMR of coherent spin currents in +antiferromagnetic NiO +Antiferromagnets have recently gained large interest in +the field of spintronics, as they allow for faster and more +robust memory operation than present technologies and +as they can carry spin current over long distances. How- +ever, many fundamental physics questions about these +materials regarding their spin transport properties still +remain unanswered [64]. +A spin current generated by +spin pumping should have a single wave mode, carrying +the coherent magnetization excitation. In contrast, spin +currents generated by thermal gradients produce incoher- +ent currents with a continuum of spin excitation modes. +The magnetic excitations in antiferromagnets typically +have THz frequencies, while the resonant excitation of +the ferromagnetic injector is in the GHz range. +Con- +ventional spin pumping experiments measure only the +time-averaged DC component of the spin current, i.e., +they cannot distinguish between GHz and THz frequen- +cies, which is needed to determine how the spin current +FIG. 3. Time resolved precession. (a) Series of XFMR de- +lay scans for the Co layer in a Co/MgO/Py MTJ at 80 K +continuously driven at 4 GHz. +As expected, the period of +the precession is 250 ps, and the delay scan covers two peri- +ods. For clarity, the data points (circles) obtained at different +magnetic field values (between 40 and 200 mT) are shifted +by a constant offset and have been differently colored. The +drawn lines represent the fitted sinusoidal functions. Their +amplitude and phase as a function of magnetic field strength +is plotted in panels (b) and (c), respectively, for both the Co +(orange) and Py (blue) layers. (Adapted from G�ladczuk et +al. [45]). +propagates. Alternative techniques such as XFMR are +needed to measure the time-varying AC spin current. +Dabrowski et al. [42] used XFMR to study the coher- +ent spin current propagation in a device with three layers, +where the top (injector) and bottom (sink) layers were +ferromagnetic NiFe and FeCo, respectively, and the mid- +dle layer was epitaxial NiO (001). The phase and am- + +(b) +Co + Py +(c) +RF reference7 +FIG. 4. +DFMR delay scans of the structural and magnetic +peaks as a function of linear polarization angle. +Measure- +ments of (a,b) the anisotropic mode B at 6 GHz and (c,d) +the isotropic mode A at 2 GHz. The results for the magnetic +peak and the structural (0,0,3) peak are shown in the left +and the right column, respectively. The magnetic resonance +modes are probed with linearly polarized light for the range +of incident polarization angles η between 0–180◦. (Adapted +from Ref. [60]). +plitude of the magnetization precession within adjoining +source and sink FM layers were detected, from which +the injection and transmission of pure AC spin current +through NiO can be inferred. It was found that magne- +tization modes in the FM layers oscillate in phase. Fur- +thermore, the efficiency of the spin transfer varied with +the thickness of the antiferromagnet, with a maximal ef- +ficiency for a 2-nm-thick layer. +These results indicate +that a spin current propagates coherently through the +antiferromagnetic NiO layer. The AC spin current is en- +hanced for NiO thicknesses of less than 6 nm, both with +and without a nonmagnetic spacer layer inserted into the +stack, in a manner consistent with previously reported +experimental measurements of DC spin current and the- +oretical studies [65]. The XFMR results show that the +propagation of spin current through NiO layers is medi- +ated by evanescent antiferromagnetic spin wave modes at +GHz frequencies, rather than THz frequency magnons. +C. +DFMR for mode-resolved detection of +magnetization dynamics +Recent scientific interest has shifted towards more +complex magnetically ordered materials, +which are +promising for high-density and low-energy consumption +devices. These systems contain chiral magnetic phases +such as helical, conical, or skyrmion spin structures, +originating from the Dzyaloshinskii-Moriya interaction +(DMI) found in noncentrosymmetric bulk materials, as +well as in systems where symmetry breaking occurs at +a ferromagnetic/heavy metal interface. Such spin struc- +tures are much more complex than simple ferromagnetic +structures, especially their dynamic behavior is so far ill- +understood. +The periodic structure of magnetically ordered sys- +tems can be probed by resonant elastic X-ray scattering +(REXS), making use of interference effects from the reg- +ularly repeating magnetization density variations. This +leads to pure magnetic X-ray scattering peaks which give +information about the static magnetic structure. Analy- +sis of these magnetic peaks in REXS measurements using +synchrotron radiation has led to significant progress in +the understanding of chiral magnetic systems [66, 67]. +In a pioneering DFMR experiment, Burn et al. [60] +investigated the complex dynamic behavior of the chi- +ral spin structure in Y-type hexaferrite Ba2Mg2Fe12O22. +VNA-FMR measurements of this material showed a field- +frequency map containing two ferromagnetic resonance +modes. While mode A is isotropic, i.e., its field value is +independent of the direction of the applied field, mode B +is anisotropic, showing greater absorption at increasingly +higher fields as the field direction rotates out-of-plane. +REXS at the Fe L2,3 absorption edge was used to char- +acterize the static magnetic structure of the hexaferrite +and to determine its field dependence. Static REXS mea- +surements along (0,0,ℓ) in zero field show a (0,0,3) struc- +tural peak decorated with two incommensurate magnetic +satellites. +The DFMR signal was measured by pointing the pho- +todiode at the scattered beam, selecting either the struc- +tural or the magnetic satellite peak (Fig. 2). Delay scans +were measured as a function of applied field using linearly +polarized X-rays. Sinusoidal fits to the measured data en- +ables the extraction of amplitude and phase. Fig. 4 shows +the delay scans of the structural and magnetic peaks of +the Y-type hexaferrite for variable incident linear polar- +ization angles η. The panels (a,b) in the top row refer to +the anisotropic mode B at 6 GHz, and the panels (c,d) +in the bottom to the isotropic mode A at 2 GHz. The +left and right column refer to the results for the mag- +netic peak and the structural (0,0,3) peak, respectively. +The results were compared to computer simulations of +the Y-type hexaferrite to obtain insight in the periodic +spin structure of this material. +A second example of the use of DFMR for mode re- +solved detection concerns the dynamic behavior of topo- +logical spin textures and chiral magnets, which is an area +of significant interest and key to the development of fast +and efficient spintronics devices. DFMR measurements +by Burn et al. [63] revealed how the time-dependence +of the magnetization dynamics relate to the complex +spin texture in the well-known chiral magnetic system +Cu2OSeO3. +Using polarized soft X-rays, the dynamic +excitations in all three dimensions were probed, which +revealed phase shifts that were previously undetectable +and indistinguishable using conventional FMR. + +8 +FIG. 5. (a) Pseudo-3D plot of the RFMR signal and its pro- +jection showing the dynamic contribution to the reflectivity +for a [CoFeB/MgO/Ta]4 multilayer as a function of pump- +probe time delay. The measurements were carried out with +left-circularly polarized X-rays at the Fe L3 resonance (707.7 +eV) and in an out-of-plane field of 29 mT using RF excitation +at 2 GHz. The various delay curves are shown for different Qz, +ranging between 0 and 0.6 nm−1. The color scale represents +the normalized intensity for each delay scan, highlighting the +sinusoidal dependence and the shift in phase as Qz is varied +when the intensity is small. (b) Static and dynamic reflectiv- +ity, and (c) phase of the dynamic reflectivity as a function of +Qz. The phase point size is scaled by the strength of the dy- +namic signal amplitude, and the blue-shaded regions indicate +where the 180◦ phase shifts have been subtracted to reveal the +otherwise smooth phase variation. (Adapted from Ref. [61]). +D. +RFMR on a [CoFeB/MgO/Ta]4 multilayer +X-ray reflectivity with the photon energy tuned to the +absorption edge has become a valuable tool for character- +izing the depth-dependent structure of layered materials. +The X-ray reflectivity is measured as a function of the +scattering vector Qz = ks − ki = (4π/λ) sin ϑ, where ki +(ks) is the ingoing (outgoing) wavevector of the X-rays +with incident angle ϑ and wavelength λ. The scattering +length density, which gives the scattering strength of the +chemical and magnetic species within the depth profile of +the film, is obtained through fitting the reflectivity data. +Burn et al. [61] revealed the depth dependence of the +magnetization dynamics in a [CoFeB/MgO/Ta]4 multi- +layer system. The structural depth profile was charac- +terized through static X-ray reflectometry. The dynamic +reflectivity was probed with stroboscopic DFMR using +an out-of-plane saturating field of HBias = 29 mT and +an RF field generated by the CPW beneath the sample. +The RF field was phase-locked to the fourth harmonic of +the ∼500 MHz synchrotron master clock at 2 GHz. The +time dependence of the reflectivity during precession was +mapped out as a function of the time delay between the +RF pump and X-ray probe. Fig. 5(a) shows a color map +of the sinusoidal variation in the reflected signal with a +500 ps period, corresponding to the 2 GHz excitation. +The amplitude and phase of the dynamic signal are ex- +tracted by fitting the sinusoidal delay scans. The ampli- +tude is plotted in Fig. 5(b) alongside the static reflectivity +for the different values of Qz, ranging between 0 and 0.6 +nm−1, and the phase in Fig. 5(c). +Both the static intensity and the amplitude of the +dynamic signal in Fig. 5(b) show reflectivity fringes re- +sulting from interference effects arising from the layered +chemical and magnetic structure. Additional minima are +observed in the dynamic case. The phase of the dynamic +signal in Fig. 5(c) shows variations with two contribu- +tions. Firstly, abrupt 180◦ phase jumps occur, coincid- +ing with minima in the amplitude of the dynamic signal. +These 180◦ jumps correspond to inversion of the sign of +the XMCD signal measured at different scattering con- +ditions. +In addition, there are smoother variations in +the phase, which can be attributed to variations in the +magnetization dynamics occurring between the magnetic +layers in the multilayered structure. +To reveal the depth dependent magnetization dynam- +ics, the experimental results were compared with model- +ing of the dynamic behavior In all layers, the magnetiza- +tion precesses about a nominal static state when excited +by an RF field. It was shown that inclusion of a small, +but significant phase lag of 5◦ between the four layers +is necessary to explain the observed change in phase of +the dynamic signal. In contrast, a single slab of mag- +netic thin film material shows a coherent precession of +the magnetization as a function of depth. +With RFMR, the dynamics from different layers con- +taining the same element can be explored, and this tech- +nique has the potential to study the dynamics of inter- +facial layers and proximity effects in complex thin film +and multilayer materials for future magnetic memory and +processing device applications. +V. +CONCLUSIONS +Although conventional FMR is a powerful technique +to study magnetic resonances in thin films and multi- +layers, the measured response corresponds to an aver- +age over the entire magnetic structure of the sample. In +contrast, X-ray based FMR techniques allow for time- +resolved measurements of the magnetization dynamics, +and, in addition, offer the benefits of XMCD, such as + +500 +400 +300 +0 +0.1 +200 +0.2 +0.3 +0.4 +100 +Time (ps) +O.9 +element-, site-, and shell-specificity [57]. The time reso- +lution is achieved by stroboscopic probing using higher +harmonics (1-10 GHz) of the synchrotron master clock. +XFMR can be used to study spin-transfer torque, dipo- +lar field strength, magneto-crystalline anisotropy, inter- +layer exchange coupling, gyromagnetic ratio and damp- +ing constants. +It can be applied to study the behav- +ior of spintronics systems, e.g., spin pumping in mag- +netic multilayers, heterostructures, spin valves, MTJ, etc. +The amplitude and phase of the magnetic resonances +extracted from the field-dependence of the precessional +plots enable us to distinguish between static and dy- +namic exchange coupling and to quantify their relative +contributions. Apart from measuring the signal in ab- +sorption, XFMR can also be detected in diffraction and +reflectivity; each of these techniques bringing unique ad- +vantages. DFMR reveals the dynamical spin modes at +the probed magnetic wavevectors, and RFMR gives the +depth-resolved dynamics in magnetic multilayers. Future +XFMR studies can be envisaged to investigate vortex dy- +namics, spatial resolution imaging, and X-ray hologra- +phy. +VI. +ACKNOWLEDGMENTS +The XFMR experiments were carried out on beamline +I10 at the Diamond Light Source (Oxfordshire, United +Kingdom). +We like to acknowledge valuable collabo- +rations with Alex A. Baker, David M. Burn, Maciej +Dabrowski, Adriana I. Figueroa, Lukasz Gladczuk, and +Robert J. Hicken. +————————– +[1] G. van der Laan, J. Electron Spectrosc. Relat. Phenom. +220 (2017) 137–146. +[2] D. M. Burn, S. L. Zhang, G. van der Laan, T. Hesjedal, +AIP Advances 11 (2021) 015327. +[3] W. E. Bailey, L. Cheng, D. J. Keavney, C.-C. Kao, +E. Vescovo, D. A. Arena, Phys. Rev. B 70 (2004) 172403. +[4] D. A. Arena, E. Vescovo, C. C. Kao, Y. Guan, W. E. +Bailey, Phys. Rev. B 74 (2006) 064409. +[5] Y. Guan, W. E. Bailey, C.-C. 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B 93 (2016) 214420. + diff --git a/rNE1T4oBgHgl3EQfiwTR/content/tmp_files/load_file.txt b/rNE1T4oBgHgl3EQfiwTR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..65576e18c6a895c4a941c7466c67fc167cbaf844 --- /dev/null +++ b/rNE1T4oBgHgl3EQfiwTR/content/tmp_files/load_file.txt @@ -0,0 +1,1259 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf,len=1258 +page_content='X-ray detected ferromagnetic resonance techniques for the study of magnetization dynamics Gerrit van der Laan1 and Thorsten Hesjedal2 1Diamond Light Source, Harwell Science and Innovation Campus, Didcot, OX11 0DE, United Kingdom 2Department of Physics, Clarendon Laboratory, University of Oxford, Oxford, OX1 3PU, United Kingdom (Dated: January 10, 2023) Element-specific spectroscopies using synchrotron-radiation can provide unique insights into materials properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The recently developed technique of X-ray detected ferromagnetic resonance (XFMR) allows studying the magnetization dynamics of magnetic spin structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Magnetic sensitivity in XFMR is obtained from the X-ray magnetic circular dichroism (XMCD) effect, where the phase of the magnetization precession of each magnetic layer with respect to the exciting radio frequency is obtained using stroboscopic probing of the spin precession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Measurement of both amplitude and phase response in the magnetic layers as a function of bias field can give a clear signature of spin-transfer torque (STT) coupling between ferromagnetic layers due to spin pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Over the last few years, there have been new developments utilizing X-ray scattering techniques to reveal the precessional magnetization dynamics of ordered spin structures in the GHz frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The techniques of diffraction and reflectometry ferromagnetic resonance (DFMR and RFMR) provide novel ways for the probing of the dynamics of chiral and multilayered magnetic materials, thereby opening up new pathways for the development of high-density and low-energy consumption data processing solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Keywords: FMR, XMCD, X-ray scattering, X-ray reflectivity, spin structures I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' INTRODUCTION Magnetization dynamics is at the heart of high fre- quency magnetic nanoscale devices based on spin waves, spin pumping, and spin-torque oscillators in the GHz frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Traditionally, ferromagnetic resonance (FMR) has been a work horse technique to determine the fundamental parameters for magnetic resonance and re- laxation in thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The recent growing complexity of many modern magnetic materials and devices requires the development of advanced measurement techniques that more directly reveal the microscopic origin of the dynamical magnetic interactions that are at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The novel techniques of X-ray detected FMR (XFMR) enables studying the magnetization dynamics of indi- vidual layers, where element-specific magnetic contrast is obtained using the X-ray magnetic circular dichroism (XMCD) effect [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Not only can the FMR signal be mon- itored in X-ray absorption, it can also be done in X-ray diffraction and reflectivity, using techniques termed as DFMR and RFMR, respectively [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' In these X-ray mea- surements, time-resolved FMR gives both the amplitude and phase of the spin precession for the different chemical elements, and hence different layers, in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The challenge of such measurements lays in the fact that the precession cone angle is small (<1◦) and that the preces- sion frequency is on the order of GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The solution is to use lock-in techniques and to detect the phase of the precession stroboscopically by using the time structure of the X-ray pulses from the synchrotron (∼500 MHz, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=', corresponding to a period between the pulses of 2 ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The radio frequency (RF) field applied to drive the spin precession is synchronized with the X-ray pulses us- ing the clock of the synchrotron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Therefore, each X-ray pulse measures the magnetization cone at precisely the same point in the precession cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Hence, XFMR com- bines the techniques of FMR and XMCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Thus, the spin precession along the bias field is pumped by the RF field to generate the magnetic resonance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=', FMR), whose amplitude and phase is probed by the synchronized X- ray pulses using the XMCD effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The time dependence is recorded using a delay line to vary the phase of the RF signal with respect to the X-ray pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' During the last few years, many XFMR studies either in time-averaged or time-resolved mode have been re- ported [3–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The first element-specific and time-depen- dent measurement of the magnetization dynamics using pump-probe XMCD was reported by Bailey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [3] on a permalloy (Py = Ni80Fe20) thin film, where the moments on the Ni and Fe sites were found to precess together at all frequencies, and by Arena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [4] on a Py/Cu/CoZr trilayer, where at resonance, a weak ferromagnetic cou- pling was found in the phase and amplitude response of individual layers across resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The amplitude and phase response of the magnetic probe layer measured by XFMR provides a signature for either static exchange interaction in strongly exchange- coupled bilayers or spin-transfer torque (STT) coupling due to spin pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Marcham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [25] first evidenced STT in a CoFe/Cu/Py spin valve using XFMR where the field dependence of the fixed layer phase showed a clear signature of STT due to spin pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Using XFMR, Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [28] reported a strong anisotropy of the spin pumping, providing new opportunities for device appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Previously, time-resolved XFMR has been reviewed in great detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Here, we present a timely up- date, especially emphasizing the newly developed time- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='03256v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='mtrl-sci] 9 Jan 2023 2 resolved FMR techniques in X-ray reflectivity and diffrac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The outline of the remainder of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' II gives a brief theoretical background of magnetiza- tion dynamics and STT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' III describes the experimen- tal setup, conditions, and considerations for the various XFMR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' IV highlights several recent ex- amples of XFMR, DFMR, and RFMR experiments and mentions their scientific impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Finally, conclusions are drawn in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' BACKGROUND ON FMR AND STT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Ferromagnetic resonance (FMR) Before presenting the experimental details and show- casing several recent examples, we will briefly introduce some relevant background material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' FMR arises when the energy levels of a quantized sys- tem of electronic moments are Zeeman split by a uniform magnetic field and the system absorbs energy from an os- cillating magnetic field [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A resonance occurs when the transverse AC field is applied at the Larmor frequency corresponding to the energy difference between the mag- netic levels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=', ℏω = ∆E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The spin precession in a single-domain magnetic material can be described with the equation of motion, the so-called Landau-Lifshitz- Gilbert (LLG) equation, ˙m = −γm × Heff + α(m × ˙m), (1) where the effective field Heff = −∂F(M)/∂M is ob- tained by minimization of the free energy F with re- spect to the magnetization M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The free energy contains terms such as the exchange, Dzyaloshinskii-Moriya, de- magnetization, magnetocrystalline anisotropy, magneto- static, external Zeeman field, and elastic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Further, ˙m = δm/δt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' the reduced magnetization is m = M/Ms, where Ms = |m| is the saturation magnetization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' and γ = gµB/ℏ is the gyromagnetic ratio, where µB is the Bohr magneton and g is the Land´e (spectroscopic split- ting) g-factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The dimensionless damping parameter α ≪ 1 (typically 10−3–10−2 for 3d metals) determines the width of the resonance absorption peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The first right-hand term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (1) corresponds to the torque due to the effective field Heff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' In a classical picture, τ = dS/dt equates to the time change in an- gular momentum S, which leads to the spin precession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The second right-hand term corresponds to the damp- ing, which can also be written in the form of the Gilbert damping term −αγ(m × m × Heff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Both torque and damping are vectorially sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Without external RF excitation, the magnetization would relax to the steady state given by Brown’s equation, m×Heff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Linearization of the LLG equation gives the relation between the frequency ν0 (or circular frequency ω0) and field, which in the form of the Kittel equation is written as [1] 2πν0 ≡ ω0 = γ � HeffBeff = γ � Heff(Ms + Heff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (2) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Spin-transfer torque (STT) The layer selectivity of XFMR makes this technique a unique probe to investigate STT and related spin cur- rents in multi-layered spin valves [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' STT is the effect in which the spin direction in a magnetic layer can be modified using a spin-polarized current [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Spin pumping occurs when the precessing magnetiza- tion vector generated by FMR in a ferromagnetic (FM) layer emits a pure spin current into an adjacent normal metal (NM) layer [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Traditionally, spin currents have been probed using indirect measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' For instance, in the metals through which they flow they can create an electrical voltage drop perpendicular to the spin current direction, or a torque that bends the magnetization di- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' However, such indirect measurements are often ambiguous because they are also influenced by other fac- tors, such as magnetic proximity effects at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' STT gives an extra term in the LLG equation, which is (anti)-parallel to the (anti)-damping (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' According to Slonczewski [54], the adiabatic torque is τs = αs m × ˙m, where αs is the STT damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The spin current pumped across a FM/NM interface due to precession is [56] Is = ℏ 4π g↑↓ eff m × ˙m , (3) where g↑↓ eff is the effective spin-mixing conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The spin pumping depends critically on the FM/NM inter- face (the material-dependent g↑↓ eff) and the spin diffusion length in the NM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' For two FM layers i and j with different resonance fre- quencies and coupled by both spin pumping (dynamic ex- change coupling) and static exchange coupling, the cou- pled LLG equations are ˙mi = − γmi × Heff,i + α0 i mi × ˙mi + αs imi × ( ˙mi − ˙mj) + Aexmj · mi , (4) and equivalently when exchanging i ↔ j, where mi is the magnetization direction, Heff,i the effective field, α0 i the Gilbert damping, and αs i the STT damping in layer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The spin pumping induced coupling is determined by αs i and the static exchange coupling by Aex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' EXPERIMENTAL XFMR provides an element-specific and time-resolved measurement of the precessional dynamics of each FM layer on a ps time scale, where the spin precession in- duced by a driving RF signal is detected using the XMCD 3 Heff h(t) m(t) Sample CPW (b) m × Heff z y Heff m(t) m × m × Heff STT x (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (a) Precession, damping, and spin transfer torque (STT) in FMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The precession m × Heff around the effective field Heff is damped by the Gilbert term m × m × Heff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The spin-transfer torque is parallel (antiparallel) to the Gilbert damping term, and can enhance (oppose) the latter depend- ing on the direction of the spin current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (b) Schematics of the sample geometry for XFMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The sample (red disk) is mounted on the signal line (the central strip) of a coplanar waveguide (CPW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The magnetization m(t) precesses about Heff, driven by the in-plane continuous RF field h(t) in the CPW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The cone angle of precession is exaggerated for clar- ity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' its typical magnitude is ∼1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Circularly polarized X-ray pulses from the synchrotron impinge at an grazing incidence angle on the sample in transverse geometry in order to enable stroboscopic detection of the oscillatory component of m(t) at variable phase delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' effect [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' However, before performing the XFMR ex- periment, the static magnetization of the samples has to be precharacterized with standard techniques, such as superconducting quantum interference device (SQUID) magnetometry to measure the hysteresis loops along the easy and hard direction, followed by standard FMR mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' VNA-FMR Vector network analyzers (VNA)-FMR measurements are used to characterize the magnetic resonances in order to judge whether these are suitable and intense enough for the XFMR measurements at the synchrotron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' VNA- FMR is a broadband FMR technique, where the sample is mounted onto a coplanar waveguide (CPW) and driven by an external RF field, while under a static magnetic bias field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Measurement of the S-parameters of the sam- ple results in a frequency-field map, where the resonances appear in the form of Kittel curves (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The angu- lar dependence of the resonances gives information about the magnetic anisotropy [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' It allows us to chose the best azimuthal angle of the applied field with respect to the crystallographic axes to separate the magnetic res- onances at the optimal distance for detecting STT [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Hence, at a given RF frequency, this gives us the cor- responding field values of the resonances in the XFMR experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Conventional FMR will normally probe the whole thickness of a thin film since the skin depth of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=', metallic iron at 10 GHz, is on the order a micron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' XMCD At the synchrotron, first the static XMCD is measured by sweeping the photon energy across the absorption edge of the magnetic elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' This allows us to se- lect the fixed photon energies suitable for XFMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The static XMCD is obtained from the difference between two X-ray absorption spectra recorded with the helic- ity vector of the circularly polarized X-rays parallel and antiparallel, respectively, to the applied magnetic field [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The XMCD signal is proportional to the projection of the helicity vector, which is along the incident beam direction ˆk, onto the sample magnetization M, hence IXMCD ∝ ˆk · M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The XMCD at the soft X-ray absorption edges, such as the Fe, Co, and Ni L2,3, is very strong [59], which helps to compensate for the small changes in magneti- zation direction due to the limited cone angle (<1◦) of the precession in XFMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The X-ray penetration length, which limits the sampling depth, is in the nm range, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=', for pure Fe it is ∼20 nm at the Fe L3 maximum at ∼707 eV [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' By tuning the photon energy away from the absorption maximum the penetration length can be in- creased (∼600 nm below the edge at 700 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Note that the length scale of the probing depth is well matched to the thickness of the magnetic layers in spin valves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The typical lateral spot size of the X-ray beam on the sample is 200 × 20 µm2, again well suited for small devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Time-resolved measurements The measurement of the projected magnetic moment in XFMR does not require to take the difference between opposite circular polarizations as done in XMCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' In- stead, a change in the projection of the magnetization precession is measured using a fixed circular polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' XFMR can be measured in two distinctly different ge- ometries, namely (i) time-averaged in longitudinal ge- ometry [12, 20] or (ii) time-resolved in transverse geom- etry [8, 25, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' In longitudinal geometry, a shortening of the magnetization vector along the z-axis (parallel to the X-ray beam direction) leads to a difference ∆Mz = Ms(1−cos θ) ≈ 1 2Msθ2, where θ is the small cone angle of the magnetization precession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The time-averaged XFMR requires no synchronization with the synchrotron clock, therefore it can be done at an arbitrary frequency, but it needs a larger RF power which can lead to nonlinear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Only measurements in transverse geometry give access to the precessional phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' This geometry is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The transverse component of the magnetiza- tion precession will give a sinusoidal variation on top of the static X-ray absorption signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' With the incident X- ray beam perpendicular to the bias field, the oscillating 4 component of the magnetization precession is measured with a magnitude |My| = Ms sin θ ≈ Msθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Thus, for a typical cone angle of θ ≈ 1◦, the transverse geometry gives a signal that is larger by a factor of ∼200 com- pared to the longitudinal geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Due to the shape anisotropy of the film, the precession is strongly ellipti- cal, often with a larger in-plane amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' This favors a measurement geometry with the X-rays at grazing in- cidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A good compromise is an X-ray incidence angle of ∼35◦ with respect to the plane of the sample, which ensures that the signal is sensitive to the larger in-plane component of the magnetization precession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Using a vector magnet system, such as the portable oc- tupole magnet system (POMS) at Diamond [57], where the field can be applied in any direction, permits a simple change of the field from (i) parallel to the photon direc- tion, as needed for static XMCD scans, to (ii) orthogonal to both X-ray beam and RF field direction, as required for time-resolved XFMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The detection of the X-ray absorption can be done by either X-ray transmission [27, 31], fluorescence yield [23], or X-ray scattering or reflectivity [2, 35, 36, 60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' How- ever, RF plays havoc with total-electron yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' In the case of transmission, the incident X-rays impinge on the sample through a hole in the signal line of the CPW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' After passing through the sample, the transmitted X- rays are detected with X-ray excited optical luminescence (XEOL) emerging from the MgO or sapphire (Al2O3) substrate using a photodiode placed behind the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Note that not all substrates, such as non-transparent ones like Si, are suitable for XEOL detection [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Time resolution is established by using the periodic X-ray pulses from the synchrotron (normally operating in multibunch or hybrid mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' To enable stroboscopic probing, the RF driving field is taken as a harmonic of the X-ray pulse frequency, hence the resonance is driven at multiples of the master oscillator clock of the stor- age ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' These harmonics are generated using an RF comb generator (Atlantic Microwave) driven by the mas- ter oscillator clock, which has a frequency of 499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='65 MHz (at the DLS, ALS, and BESSY synchrotron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' This cor- responds to ∼2 ns intervals between consecutive X-ray pulses, which have a pulse width of ∼35 ps (at DLS and BESSY) or ∼70 ps (at ALS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The desired frequency is selected using filters and amplifiers to drive a narrow band, high power (25–30 dBm) RF field to the CPW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A programmable delay line (Colby Instruments) enables phase shifting of the RF oscillation with respect to the X-ray pulses with a step resolution of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='5 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' De- pending on the specific technique either the transmitted, diffracted, or reflected X-rays are measured using a pho- todiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 2 show a schematic representation of the setup for DFMR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' for XFMR and RFMR the electronics is very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The signal is obtained using a lock-in amplifier (LIA) by switching the signal at a given audio frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' There are two usual modulation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' In amplitude modulation the LIA measures the difference between signals obtained with the RF signal on and off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Schematic of the setup for DFMR measurements in the RASOR diffractometer at the Diamond Light Source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The sample is placed on the CPW, which is mounted on the cold finger inside the diffractometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Incident circularly or linearly polarized X-rays are scattered off the sample and detected via a photodiode in a ϑ-2ϑ geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A variable magnetic field is applied in the scattering plane via a pair of permanent magnets whose distance can be controlled externally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' An RF signal is fed to the CPW to drive the ferromagnetic resonance in the magnetic sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' As the synchrotron gives X-ray pulses at a frequency of ∼500 MHz, a comb generator is used to produce higher harmonics, which are selected and fed to the CPW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' To probe the time dependence of the scattered X-ray intensity, a tunable delay line is used, which shifts the phase between the pump (the RF signal) and the probe (the pulsed X-rays).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (Adapted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [63]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' In 180◦-phase modulation, the LIA measures the differ- ence between signals obtained with the RF of opposite phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' XFMR In order to record the time-resolved XAS signal with circular polarization at fixed photon energy, the RF fre- quency is locked to a multiple of the synchrotron clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Then at fixed angles and for given temperature, this leaves two free scanning parameters, namely the mag- netic bias field strength and the delay time between X-ray pulses and RF field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Magnetic field scans record the signal by sweeping the bias field at a constant delay time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The signal contains both real and imaginary parts of the magnetic suscepti- bility, whose relative contributions strongly change across resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' By measuring two field scans, which differ by 5 90◦ in phase (obtained using the corresponding time de- lays), and fitting these scans simultaneously using the Kramers-Kronig relation, gives a good apprehension of the field dependence of the resonances [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Delay scans record the signal for each of the magnetic layers at constant bias field by sweeping the delay time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' As an example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 3(a) shows a series delay scans over two periods of the phase taken at different bias fields (40– 200 mT) across the Co resonance in a magnetic tunnel junction (MTJ), in more detail discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The solid lines represent sinusoidal fits to the experimen- tal data (dots), from which the amplitude and relative phase of the magnetization precession can be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A sinusoidal function of the form S(t) = X sin(2πνt) + Y cos(2πνt), (5) is fitted to the delay scan, where t is the time delay and ν the frequency of the RF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' This procedure is repeated for various field strengths and directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' By extracting the coefficients X and Y in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (5) from the delay scans, the amplitude A and phase ψ of the oscillations can be determined using the relationships A = � X2 + Y 2, ψ = 2 arctan � Y A + X � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (6) XFMR precessional plots are assembled by combining the amplitudes and phases extracted from the delay scans measured over a range of bias fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' This gives the field dependence of the amplitude and phase for each element (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=', for Co and Ni in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 3(b)), from which the type of coupling between layers can be assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' By normaliz- ing the XFMR signal to the static XMCD, the amplitude of the signal can be obtained per atom for each chemi- cal element in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' This enables a quantitative decomposition of the resonance features [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The static coupling (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=', exchange interaction) and dy- namic coupling (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=', spin pumping) give a very differ- ent XFMR response, as can be understood from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Consider a pump layer FM1 that is free to rotate, and a probe layer FM2 that is pinned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Using XFMR at a fixed frequency, we scan the field across the entire resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' At resonance, FM1 will show a symmetric peak for the amplitude, while the phase is 90◦ delayed with respect to the RF driving field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Across the entire resonance, the phase will change by 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' To investigate the type of coupling between both layers we measure the XFMR re- sponse of FM2 at the resonance condition of FM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' For static exchange coupling, E = −Aexm1 · m2, so that Heff,2 ∝ m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' This means that the effective field in the second layer is aligned along the magnetization of the first layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Then the field dependent precession of FM2 will show a dispersive (bipolar) peak in the amplitude and a symmetric (unipolar) peak in the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' On the other hand, for dynamic exchange coupling Heff,2 ∝ ˙m2 = −iωm1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The magnetic field is imagi- nary, resulting in a 90◦ phase change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' In this case, the field dependent precession of FM2 will show a unipolar peak in the amplitude and bipolar peak in the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' This behavior means that XFMR can distinguish be- tween static and dynamic coupling by their amplitude and phase signature in the precessional plot, and thus determine the relative contribution of these couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' This has previously been utilized for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=', exchange cou- pled layers [4, 31], spin values [25], MgO magnetic tunnel junctions [29, 45], topological insulators [27, 30, 41], spin valve with δ-layer [26], Heusler alloys [40], NiO antiferro- magnetic interlayer [42], exchange springs [44], and α-Sn thin films [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' DFMR and RFMR DFMR and RFMR measurements have been per- formed in the RASOR soft X-ray diffractometer on beam- line I10 at the Diamond Light Source [57] (see setup in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Incident X-rays with wavevector ki illuminate the sample, while the scattered beam (ks) is detected us- ing a photodiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The scattering geometry is configured to probe the sample at certain diffraction or specular re- flectivity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The sample in the diffractometer is mounted on a CPW that is connected to a liquid He cryo- stat arm which can reach temperatures down to 12 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A bias magnetic field is applied by two permanent magnets, which can be positioned to vary both the field strength up to 200 mT and the orientation within the scatter- ing plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Perpendicular to the bias field, a transverse RF field around the central conductor of the CPW is generated, which excites the magnetization dynamics in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' In contrast to conventional XFMR measure- ments, where the sample is mounted flip-chip onto the CPW, in the scattering geometry the sample is mounted face up to allow for the X-ray beam to probe its sur- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' To ensure good coupling between the CPW and the probed top surface, the sample must either be thinned, or in the case of multilayers, grown on a thin substrate of the order of 100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' In DFMR, where the detector is aligned to a Bragg peak or magnetic scattering peak, the stroboscopic sig- nal is used to measure delay scans for different linear or circular polarization of the incident X-rays, to give infor- mation about the periodic spin structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' In RFMR where the photo diode detector accepts the reflected beam the stroboscopic signal is used to measure delay scans for different values of the scattering vector Qz, to obtain depth information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' An advantage of RFMR over DFMR is that it can be done on thin films and multilayers, and no single crystals are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' X-RAY BASED FMR EXAMPLES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' XFMR of spin-current mediated exchange coupling in MgO-based MTJs Magnetic tunnel junctions composed of ferromagnetic layers which are mutually interacting through a nonmag- 6 netic spacer layer are at the core of magnetic sensor and memory devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' G�ladczuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [45] used layer-resolved XFMR to investigate the coupling between the magnetic layers of a Co/MgO/Py MTJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Two magnetic resonance peaks were observed for both magnetic layers, as probed at the Co and Ni L3 X-ray absorption edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Figure 3 shows XFMR delay scans for the Co layer in the Co/MgO/Py MTJ at 80 K continuously driven at 4 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 3(a) show a strong increase in amplitude as well as a large phase shift across the reso- nance at ∼90 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The amplitude and phase of the pre- cession, which are extracted using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (6), are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 3(b) and (c), respectively, for both Co (orange) and Ni (blue) as a function of the bias field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The amplitude curves show that the Ni resonance originating from the Py layer around ∼120 mT is strongly coupled with the Co layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' On the other hand, the Co resonance around ∼90 mT is only weakly present in the Py layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Instead of plotting amplitude A and phase ψ, one can also plot the FMR signal in the (X,Y )-plane as a function of field [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Since the sine and cosine functions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (5) are orthogonal, the estimators of X and Y are given by pro- jections to orthogonal subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A theoretical model based on the Landau-Lifshitz-Gil- bert-Slonczewski equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (4)) was developed, in- cluding exchange coupling and spin pumping between the magnetic layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Fits to the experimental data were car- ried out, both with and without a spin pumping term, and the goodness of the fit was compared using a likeli- hood ratio test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' This rigorous statistical approach pro- vided an unambiguous proof of the existence of interlayer coupling mediated by spin pumping through MgO [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' It was also found that spin pumping is more effective at lower temperatures, which agrees with the theoretical understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' XFMR of coherent spin currents in antiferromagnetic NiO Antiferromagnets have recently gained large interest in the field of spintronics, as they allow for faster and more robust memory operation than present technologies and as they can carry spin current over long distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' How- ever, many fundamental physics questions about these materials regarding their spin transport properties still remain unanswered [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A spin current generated by spin pumping should have a single wave mode, carrying the coherent magnetization excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' In contrast, spin currents generated by thermal gradients produce incoher- ent currents with a continuum of spin excitation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The magnetic excitations in antiferromagnets typically have THz frequencies, while the resonant excitation of the ferromagnetic injector is in the GHz range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Con- ventional spin pumping experiments measure only the time-averaged DC component of the spin current, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=', they cannot distinguish between GHz and THz frequen- cies, which is needed to determine how the spin current FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Time resolved precession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (a) Series of XFMR de- lay scans for the Co layer in a Co/MgO/Py MTJ at 80 K continuously driven at 4 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' As expected, the period of the precession is 250 ps, and the delay scan covers two peri- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' For clarity, the data points (circles) obtained at different magnetic field values (between 40 and 200 mT) are shifted by a constant offset and have been differently colored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The drawn lines represent the fitted sinusoidal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Their amplitude and phase as a function of magnetic field strength is plotted in panels (b) and (c), respectively, for both the Co (orange) and Py (blue) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (Adapted from G�ladczuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' propagates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Alternative techniques such as XFMR are needed to measure the time-varying AC spin current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Dabrowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [42] used XFMR to study the coher- ent spin current propagation in a device with three layers, where the top (injector) and bottom (sink) layers were ferromagnetic NiFe and FeCo, respectively, and the mid- dle layer was epitaxial NiO (001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The phase and am- (b) Co Py (c) RF reference7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' DFMR delay scans of the structural and magnetic peaks as a function of linear polarization angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Measure- ments of (a,b) the anisotropic mode B at 6 GHz and (c,d) the isotropic mode A at 2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The results for the magnetic peak and the structural (0,0,3) peak are shown in the left and the right column, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The magnetic resonance modes are probed with linearly polarized light for the range of incident polarization angles η between 0–180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (Adapted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [60]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' plitude of the magnetization precession within adjoining source and sink FM layers were detected, from which the injection and transmission of pure AC spin current through NiO can be inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' It was found that magne- tization modes in the FM layers oscillate in phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Fur- thermore, the efficiency of the spin transfer varied with the thickness of the antiferromagnet, with a maximal ef- ficiency for a 2-nm-thick layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' These results indicate that a spin current propagates coherently through the antiferromagnetic NiO layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The AC spin current is en- hanced for NiO thicknesses of less than 6 nm, both with and without a nonmagnetic spacer layer inserted into the stack, in a manner consistent with previously reported experimental measurements of DC spin current and the- oretical studies [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The XFMR results show that the propagation of spin current through NiO layers is medi- ated by evanescent antiferromagnetic spin wave modes at GHz frequencies, rather than THz frequency magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' DFMR for mode-resolved detection of magnetization dynamics Recent scientific interest has shifted towards more complex magnetically ordered materials, which are promising for high-density and low-energy consumption devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' These systems contain chiral magnetic phases such as helical, conical, or skyrmion spin structures, originating from the Dzyaloshinskii-Moriya interaction (DMI) found in noncentrosymmetric bulk materials, as well as in systems where symmetry breaking occurs at a ferromagnetic/heavy metal interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Such spin struc- tures are much more complex than simple ferromagnetic structures, especially their dynamic behavior is so far ill- understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The periodic structure of magnetically ordered sys- tems can be probed by resonant elastic X-ray scattering (REXS), making use of interference effects from the reg- ularly repeating magnetization density variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' This leads to pure magnetic X-ray scattering peaks which give information about the static magnetic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Analy- sis of these magnetic peaks in REXS measurements using synchrotron radiation has led to significant progress in the understanding of chiral magnetic systems [66, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' In a pioneering DFMR experiment, Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [60] investigated the complex dynamic behavior of the chi- ral spin structure in Y-type hexaferrite Ba2Mg2Fe12O22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' VNA-FMR measurements of this material showed a field- frequency map containing two ferromagnetic resonance modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' While mode A is isotropic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=', its field value is independent of the direction of the applied field, mode B is anisotropic, showing greater absorption at increasingly higher fields as the field direction rotates out-of-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' REXS at the Fe L2,3 absorption edge was used to char- acterize the static magnetic structure of the hexaferrite and to determine its field dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Static REXS mea- surements along (0,0,ℓ) in zero field show a (0,0,3) struc- tural peak decorated with two incommensurate magnetic satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The DFMR signal was measured by pointing the pho- todiode at the scattered beam, selecting either the struc- tural or the magnetic satellite peak (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Delay scans were measured as a function of applied field using linearly polarized X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Sinusoidal fits to the measured data en- ables the extraction of amplitude and phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 4 shows the delay scans of the structural and magnetic peaks of the Y-type hexaferrite for variable incident linear polar- ization angles η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The panels (a,b) in the top row refer to the anisotropic mode B at 6 GHz, and the panels (c,d) in the bottom to the isotropic mode A at 2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The left and right column refer to the results for the mag- netic peak and the structural (0,0,3) peak, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The results were compared to computer simulations of the Y-type hexaferrite to obtain insight in the periodic spin structure of this material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A second example of the use of DFMR for mode re- solved detection concerns the dynamic behavior of topo- logical spin textures and chiral magnets, which is an area of significant interest and key to the development of fast and efficient spintronics devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' DFMR measurements by Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [63] revealed how the time-dependence of the magnetization dynamics relate to the complex spin texture in the well-known chiral magnetic system Cu2OSeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Using polarized soft X-rays, the dynamic excitations in all three dimensions were probed, which revealed phase shifts that were previously undetectable and indistinguishable using conventional FMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (a) Pseudo-3D plot of the RFMR signal and its pro- jection showing the dynamic contribution to the reflectivity for a [CoFeB/MgO/Ta]4 multilayer as a function of pump- probe time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The measurements were carried out with left-circularly polarized X-rays at the Fe L3 resonance (707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='7 eV) and in an out-of-plane field of 29 mT using RF excitation at 2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The various delay curves are shown for different Qz, ranging between 0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='6 nm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The color scale represents the normalized intensity for each delay scan, highlighting the sinusoidal dependence and the shift in phase as Qz is varied when the intensity is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (b) Static and dynamic reflectiv- ity, and (c) phase of the dynamic reflectivity as a function of Qz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The phase point size is scaled by the strength of the dy- namic signal amplitude, and the blue-shaded regions indicate where the 180◦ phase shifts have been subtracted to reveal the otherwise smooth phase variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' (Adapted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [61]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' RFMR on a [CoFeB/MgO/Ta]4 multilayer X-ray reflectivity with the photon energy tuned to the absorption edge has become a valuable tool for character- izing the depth-dependent structure of layered materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The X-ray reflectivity is measured as a function of the scattering vector Qz = ks − ki = (4π/λ) sin ϑ, where ki (ks) is the ingoing (outgoing) wavevector of the X-rays with incident angle ϑ and wavelength λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The scattering length density, which gives the scattering strength of the chemical and magnetic species within the depth profile of the film, is obtained through fitting the reflectivity data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [61] revealed the depth dependence of the magnetization dynamics in a [CoFeB/MgO/Ta]4 multi- layer system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The structural depth profile was charac- terized through static X-ray reflectometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The dynamic reflectivity was probed with stroboscopic DFMR using an out-of-plane saturating field of HBias = 29 mT and an RF field generated by the CPW beneath the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The RF field was phase-locked to the fourth harmonic of the ∼500 MHz synchrotron master clock at 2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The time dependence of the reflectivity during precession was mapped out as a function of the time delay between the RF pump and X-ray probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 5(a) shows a color map of the sinusoidal variation in the reflected signal with a 500 ps period, corresponding to the 2 GHz excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The amplitude and phase of the dynamic signal are ex- tracted by fitting the sinusoidal delay scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The ampli- tude is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 5(b) alongside the static reflectivity for the different values of Qz, ranging between 0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='6 nm−1, and the phase in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Both the static intensity and the amplitude of the dynamic signal in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 5(b) show reflectivity fringes re- sulting from interference effects arising from the layered chemical and magnetic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Additional minima are observed in the dynamic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The phase of the dynamic signal in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 5(c) shows variations with two contribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Firstly, abrupt 180◦ phase jumps occur, coincid- ing with minima in the amplitude of the dynamic signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' These 180◦ jumps correspond to inversion of the sign of the XMCD signal measured at different scattering con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' In addition, there are smoother variations in the phase, which can be attributed to variations in the magnetization dynamics occurring between the magnetic layers in the multilayered structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' To reveal the depth dependent magnetization dynam- ics, the experimental results were compared with model- ing of the dynamic behavior In all layers, the magnetiza- tion precesses about a nominal static state when excited by an RF field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' It was shown that inclusion of a small, but significant phase lag of 5◦ between the four layers is necessary to explain the observed change in phase of the dynamic signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' In contrast, a single slab of mag- netic thin film material shows a coherent precession of the magnetization as a function of depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' With RFMR, the dynamics from different layers con- taining the same element can be explored, and this tech- nique has the potential to study the dynamics of inter- facial layers and proximity effects in complex thin film and multilayer materials for future magnetic memory and processing device applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' CONCLUSIONS Although conventional FMR is a powerful technique to study magnetic resonances in thin films and multi- layers, the measured response corresponds to an aver- age over the entire magnetic structure of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' In contrast, X-ray based FMR techniques allow for time- resolved measurements of the magnetization dynamics, and, in addition, offer the benefits of XMCD, such as 500 400 300 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='1 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='4 100 Time (ps) O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='9 element-, site-, and shell-specificity [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The time reso- lution is achieved by stroboscopic probing using higher harmonics (1-10 GHz) of the synchrotron master clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' XFMR can be used to study spin-transfer torque, dipo- lar field strength, magneto-crystalline anisotropy, inter- layer exchange coupling, gyromagnetic ratio and damp- ing constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' It can be applied to study the behav- ior of spintronics systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=', spin pumping in mag- netic multilayers, heterostructures, spin valves, MTJ, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' The amplitude and phase of the magnetic resonances extracted from the field-dependence of the precessional plots enable us to distinguish between static and dy- namic exchange coupling and to quantify their relative contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Apart from measuring the signal in ab- sorption, XFMR can also be detected in diffraction and reflectivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' each of these techniques bringing unique ad- vantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' DFMR reveals the dynamical spin modes at the probed magnetic wavevectors, and RFMR gives the depth-resolved dynamics in magnetic multilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Future XFMR studies can be envisaged to investigate vortex dy- namics, spatial resolution imaging, and X-ray hologra- phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' ACKNOWLEDGMENTS The XFMR experiments were carried out on beamline I10 at the Diamond Light Source (Oxfordshire, United Kingdom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' We like to acknowledge valuable collabo- rations with Alex A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Baker, David M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Burn, Maciej Dabrowski, Adriana I.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Arena, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Vescovo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Kao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Guan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Bailey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 101 (2007) 09C109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [7] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Guan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Bailey, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Vescovo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Kao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Arena, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 312 (2007) 437–378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Arena, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Ding, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Vescovo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Zohar, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Guan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Bailey, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 80 (2009) 083903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [9] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Bailey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Cheng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Knut, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Karis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Auffret, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} 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+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 4 (2013) 2025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [10] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Warnicke, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Knut, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Wahlstr¨om, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Karis, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Bailey, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Arena, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 113 (2013) 033904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [11] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Warnicke, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Stavitski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Lee, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Zuo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Zohar, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Bailey, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Harris, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Arena, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' B 92 (2015) 104402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Goulon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Rogalev, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} 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Goulon-Ginet, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Brouder, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' B 53 (2006) 169–184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Goulon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Rogalev, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Wilhelm, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Goulon-Ginet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Goujon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Synchrotron Rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 14 (2007) 257–271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Barbe, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Guilard, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 12 (2011) 8797–8835.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 87 (2005) 152503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [20] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Boero, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Mouaziz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Rusponi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Bencok, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} 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Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' B 79 (2009) 224425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [22] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Boero, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Rusponi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Kavich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} 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Laan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Hesjedal, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 6 (2016) 35582.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Figueroa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 400 (2016) 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [31] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Stenning, L.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Woltersdorf, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 10 Bowden, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Gregory, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Back, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' de Groot, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' van der Laan, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 17 (2015) 013019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [32] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Klaer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Hoffmann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Woltersdorf, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Arbelo Jorge, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Jourdan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Martin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Woltersdorf, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Stamm, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' D¨urr, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Mattheis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Back, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Bayreuther, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 105 (2009) 07D310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [35] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Salikhov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Abrudan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Br¨ussing, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Gross, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Luo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Westerholt, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Rott, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Reiss, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Br¨uckl, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Neudecker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Weiss, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Back, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 99 (2006) 08F305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [38] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Ollefs, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Meckenstock, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Ney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 117 (2015) 223906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [39] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Bonetti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Kukreja, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content='Spoddig, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Ollefs, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Sch¨oppner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Meckenstock, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Ney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Klewe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Cavill, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Childress, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Katine, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' B 96 (2017) 144421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' [41] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Baker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Figueroa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Hesjedal, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' van der Laan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' 473 (2019) 470–476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Burn, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Hesjedal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Figueroa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Hwang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Hicken, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Shafer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Arenholz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' van der Laan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Qiu, Syn- chrotron Radiat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' News 33 (2020) 12–19.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Hesjedal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} +page_content=' B 93 (2016) 214420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfiwTR/content/2301.03256v1.pdf'} diff --git a/rtAyT4oBgHgl3EQfZvdA/content/tmp_files/2301.00228v1.pdf.txt b/rtAyT4oBgHgl3EQfZvdA/content/tmp_files/2301.00228v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a000e8b5b7542a03adb4cb08cfa933ee49bf1745 --- /dev/null +++ b/rtAyT4oBgHgl3EQfZvdA/content/tmp_files/2301.00228v1.pdf.txt @@ -0,0 +1,922 @@ +A LATTICE BOLTZMANN METHOD FOR ELASTIC SOLIDS +UNDER PLANE STRAIN DEFORMATION +Alexander Schlüter, Sikang Yan, Erik Faust +Institute of Applied Mechanics +Technische Universität Kaiserslautern +D-67653, Kaiserslautern +{Alexander Schlüter} aschluet@rhrk.uni-kl.de +Henning Müller, Ralf Müller +Institut für Mechanik +Technische Universität Darmstadt +D-64287, Darmstadt +ABSTRACT +The Lattice Boltzmann Method (LBM), e.g. in [1] and [2], can be interpreted as an alternative +method for the numerical solution of partial differential equations. Consequently, although the LBM +is usually applied to solve fluid flows, the above interpretation of the LBM as a general numerical +tool, allows the LBM to be extended to solid mechanics as well. In this spirit, the LBM has been +studied in recent years. First publications [3], [4] presented an LBM scheme for the numerical +solution of the dynamic behavior of a linear elastic solid under simplified deformation assumptions. +For so-called anti-plane shear deformation, the only non-zero displacement component is governed +by a two-dimensional wave equation. In this work, an existing LBM for the two-dimensional wave +equation is extended to more general plane strain problems. The proposed algorithm reduces the +plane strain problem to the solution of two separate wave equations for the volume dilatation and the +non-zero component of the rotation vector, respectively. A particular focus is on the implementation +of types of boundary conditions that are commonly encountered in engineering practice for solids: +Dirichlet and Neumann boundary conditions. Last, several numerical experiments are conducted that +highlight the performance of the new LBM in comparison to the Finite Element Method. +Keywords Lattice Boltzmann Method · solids · plane strain · computational engineering · computational solid +mechanics +1 +Introduction +The mechanical behavior of solid bodies is of interest to both engineering and science. Thus, a large number of +numerical methods capable of dealing with elasticity have emerged over time. The more prominent ones among these, +finite differences methods (FDM), finite element methods (FEM) and finite volume methods (FVM), work on the +principle of discretizing the domain of interest and replacing the governing system of differential equations by algebraic +equations. Such methods take a kind of top-down approach, and can therefore be thought of as acting on a macroscopic +scale. In contrast, some numerical methods, such as molecular dynamics (MD) or density functional theory (DFT), +regard the interactions of a system’s most basic constituents, such as individual particles and electrons, on a microscopic +scale. +A different approach is taken with Lattice-Boltzmann methods (LBMs). The common principle of this type of methods +is to transform the given physical problem into a transport problem. Based on Boltzmann’s transport equation from +statistical mechanics, distribution functions are transported across phase-space, which is discretized both by a regular +lattice and a set of associated lattice velocities. Information is exchanged between neighboring lattice sites in a +streaming-like process along links connecting these points. This information is represented by so-called distribution +functions, where each distribution function is associated with a different lattice velocity. The distribution functions are +subjected to on-site interactions, or collisions, which incorporate the underlying microscopic theory in a probabilistic +manner. Thus, LBMs can be said to act mesoscopically, i.e. on an intermediate scale. +arXiv:2301.00228v1 [math.NA] 31 Dec 2022 + +A Lattice Boltzmann Method for Elastic Solids +B +∂Bu +⃗n +⃗t∗ +∂Bt +Figure 1: Body B with outer normal vector n, subjected to Neumann boundary conditions ⃗t∗ on Bt and Dirichlet +boundary conditions on Bu. +LBMs are well established in computational fluid dynamics (CFD) and have subsequently been extended to further +scientific fields, such as solving Schrödinger’s equation [5] or Wigner’s equation [6] in quantum mechanics. Developing +a LBM for solid mechanics could mean using a single method on both sides of a fluid-solid-interface, which is a topic +of interest [7] in CFD. +We approach the topic of a LBM for elastic bodies from the mechanical point of view. This includes a greater focus +on finite domains with an appropriate boundary handling, which is of great concern for engineering problems. The +advantages over the established methods in computational engineering include the generally great computational +efficiency, while still being able to handle the boundary conditions of complex domains. A further improvement of +computation times by can be achieved by employing parallel computing, which is easy to implement with most LBMs. +This opens up possibilities in highly dynamical problems, requiring very fine resolution of the temporal domain. +The dynamic behavior of elastic solids can be described by multiple wave equations, that are superposed to obtain the +aggregated deformation. This mathematical description is closer to the transport phenomena for which the LBM was +initially conceived, when compared to the original Navier-Cauchy equation. In fact, LBMs have already been developed +for the wave equation, see e.g. [8, 9, 10]. Furthermore, LBMs have been proposed for the numerical treatment of +mechanical problems in solid bodies [11, 1, 12, 13, 14, 15, 16], and specifically elastic wave propagation [17, 18], +which is also a topic of interest in geophysics and seismology. However, an extensive method for the deformation of +linear elastic solids under loads, containing appropriate boundary conditions, has still to be accomplished. +In previous works [3, 19] we applied the LBM for wave equations published by Yan [8] to the mechanical problem of +anti-plane shear deformation, which we then used for fracture mechanics. This work now regards the two-dimensional +problem of plane strain. The fundamental idea is to decompose the plane strain problem governed by the Navier-Cauchy +equation into two equivalent wave equations that are solved with the LBM for wave propagation by Chopard et al. [11]. +The discussion is structured as follows: First the mechanical problem is reviewed and the relevant equations are derived. +The next section introduces the LBM for the wave equation, followed by the presentation of the algorithm for the plane +strain case. This includes a treatment of boundary conditions similar to [19]. Lastly three numerical examples show the +feasibility of our algorithm, each compared to FEM computations, which act as benchmarks. +2 +Plane Strain Deformation of a Linear Elastic Solid +We consider a homogeneous, isotropic, and elastic body B with boundary ∂B = ∂Bu ∪ Bt, which is subjected to +Dirichlet boundary conditions u = ⃗u∗ for the displacement ⃗u on ∂Bu and Neumann boundary conditions ⃗σ⃗n = t∗ for +the Cauchy stress tensor ⃗σ on ∂Bt, see Fig. 1. For the plane strain case the problem is only regarded in two dimensions +and under small strain assumptions. +A set of fundamental equation is taken as a basis for the derivation of the mathematical description of the problem. +Firstly, the strain-displacement relation for small strains is given by the linearized strain tensor +⃗ε = 1 +2 +� +∇⃗u + (∇⃗u)T � +, +(1) +where ⃗u = ⃗u(x, y, t) describes the time-dependent displacement field in two dimensions under plane strain assumptions. +The general equation of motion for the small strain case, in the absence of a body force, is given by +∇ · ⃗σ = ρ ∂2⃗u +∂t2 , +(2) +2 + +A Lattice Boltzmann Method for Elastic Solids +x +y +a) +b) +∆h +∆h +c4 +c2 +c3 +c1 +Figure 2: a) Lattice representation of the elastic solid and b) and the associated lattice velocity vectors (lattice links) for +a single lattice point. +with Hooke’s law as the linear stress-strain relation +⃗σ = λ tr(⃗ε) 1 + 2µ⃗ε. +(3) +Herein λ and µ are the Lamé parameters of the material, 1 is the second-order identity tensor and the operator tr(∗) +denotes the trace of a second order tensor. +Equation (1) is substituted in equation (3), which is then substituted in (2). The result is the Navier-Cauchy equation +(λ + µ) ∇ (∇ · ⃗u) + µ∇2 ⃗u = ρ ∂2⃗u +∂t2 , +(4) +which describes the mechanical behavior of an isotropic linear elastic solid. Using the general identity +∇2⃗u = ∇(∇ · ⃗u) − ∇ × (∇ × ⃗u) +(5) +from vector calculus, equation (4) can be rewritten as +c2 +d ∇(∇ · ⃗u) − c2 +s ∇ × (∇ × ⃗u) = ∂2⃗u +∂t2 , +(6) +where cd = +� +(λ+2µ)/ρ and cs = +� +λ/ρ. +With regard to equation (6), two fields, φ and ⃗ψ, can be defined as follows: +φ = ∇ · ⃗u +and +⃗ψ = ∇ × ⃗u. +(7) +The scalar field φ describes the dilatation of the displacement field ⃗u, whereas the vector field ⃗ψ describes the rotation +of ⃗u. In two dimensions, the latter reduces to ⃗ψ = ψ ⃗ez. The Navier-Cauchy equation can be restated in terms of these +fields +c2 +d ∇φ − c2 +s ∇ × ⃗ψ = ∂2⃗u +∂t2 . +(8) +In conjunction with the definitions in equation (7), applying the divergence to both sides of equation (8) results in +c2 +d ∇2φ = ∂2φ +∂t2 , +(9a) +while applying the curl results in +c2 +s ∇2 ⃗ψ = ∂2 ⃗ψ +∂t2 . +(9b) +Thus, the Navier-Cauchy equation (8) can be reduced to two wave equations, the dilatational wave equation (9a) for +φ = ∇ · ⃗u with wave speed cd, and the rotational wave equation (9b) for ⃗ψ = ψ ⃗ez = ∇ × ⃗u with wave speed cs. Note +that there are other ‘decompositions’ of the displacement field besides (7) that lead to similar wave equations, see +e.g. [20]. +3 + +A Lattice Boltzmann Method for Elastic Solids +3 +Lattice Boltzmann Method for Plane Strain +The proposed numerical strategy for the plane strain case relies on solving the wave equations (9) by means of the LBM +by Chopard et al. [11]. In the LBM, a body B is typically approximated by a regular lattice with lattice spacing ∆h +as depicted in Fig. 2 a). The approach by Chopard et al. is based on a D2Q5 scheme, see also Fig. 2, with the lattice +velocities +⃗c0 = (c0 +x, c0 +y) = (0, 0) +⃗c1 = (c1 +x, c1 +y) = (c, 0) +⃗c2 = (c2 +x, c2 +y) = (0, c) +⃗c3 = (c3 +x, c3 +y) = (-c, 0) +⃗c4 = (c4 +x, c4 +y) = (0, -c) +(10) +where c = ∆h/∆t is the speed at which information can travel in the lattice. Thus, the lattice velocities ⃗c1,⃗c2,⃗c3,⃗c4 +allow for information to be transported to each of the four neighbors of a lattice point in a so-called D2Q5 scheme1 +in one time step, whereas ⃗c0 is associated with information remaining at a particular lattice point. Information is +represented by distribution functions, e.g. f α represents information, which is transported with lattice velocity ⃗cα. +In order to simulate the wave equations (9), the distribution functions need to be interpreted, i.e. a relation to the +macroscopic fields needs to be established. We introduce two sets of distribution functions to simulate both wave +equations and relate them to the macroscopic fields through +4 +� +α=0 +f α +ψ = ψ, +4 +� +α=0 +f α +φ = φ. +(11) +The Lattice Boltzmann equation (LBE) models transport as well as the interaction of distribution functions between and +at lattice points respectively. Since two wave equations need to be solved, we also introduce two associated LBEs for ψ +and φ respectively +f α +ψ|φ (⃗x + ⃗cα∆t, t + ∆t) = +f α +ψ|φ(⃗x, t) − ∆t +τ +� +f α +ψ|φ(⃗x, t) −f α +eq,ψ|φ(⃗x, t) +� +, +(12) +where the notation (ψ|φ) indicates that either ψ or φ need to be chosen for the whole equation and the common BGK +approximation is employed, see [21] and [22]. Equation (12) is universal to many Lattice Boltzmann models. The +specific physics can be modeled by choosing the equilibrium distribution functions f α +eq,ψ|φ and relaxation time τ in a +certain way. In order to model a wave equation Chopard et al. propose τ = 0.5∆t and +f 0 +eq,ψ|φ = a0,ψ|φ(ψ|φ) +f α +eq,ψ|φ = aψ|φ(ψ|φ) + b⃗cα · ⃗Jψ|φ +2c2 +, +with ⃗Jψ|φ = +4 +� +α=0 +⃗cαf α +ψ|φ(⃗x, t), for α ̸= 0, +(13) +where again the notation (ψ|φ) indicates that either ψ or φ need to be chosen for the whole equation. The parameters +also need to fulfill the requirements +b = 1, +conservation of ⃗Jψ|φ +a0,ψ|φ + 4aψ|φ = 1, +conservation of ψ|φ +a0,ψ|φ ≥ 0, +stability +(14) +and +cs|cd = ∆h +∆t +�2aψ|φ, +(15) +see Chopard [1]. Equation (15) allows us to adjust the macroscopic wave speed modeled by the LBM independently of +the time step ∆t or the lattice spacing ∆h by choosing aψ|φ accordingly. We exploit this in order to be able to simulate +1D2Q5 refers to the dimension of the lattice, i.e. two in this case, and the number of lattice velocities, i.e. five in this case. +4 + +A Lattice Boltzmann Method for Elastic Solids +• Preprocessing +– Build lattice for geometry +– Compute surface measure and cell volumes for boundary lattice points +• Solver +– Initialize ⃗u(⃗x, 0) and ˙⃗u(⃗x, 0) +– Initialize ⃗ψ(⃗x, 0) = ∇ × ⃗u(⃗x, 0), ⃗φ(⃗x, 0) = ∇ × ⃗u(⃗x, 0) computed by finite differences +– Initialize the distribution functions by (13) +– Start time loop +* t → t + ∆t +* Compute accelerations in the interior ¨⃗u(⃗x, t) = c2 +d∇φ(⃗x, t) − c2 +s ⃗ψ(⃗x, t) i.e. by (6) +* Compute accelerations at boundary points by (21) or (24) +* Compute ⃗u(⃗x, t + ∆t) by explicit integration, i.e. by (19) +* set ψ(⃗x, t + ∆t) at boundary lattice points to be consistent with ⃗u(⃗x, t + ∆t) according to (26) +* Solve the wave equations for ψ(⃗x, t + ∆t) and φ(⃗x, t + ∆t) via the LBE (12) and the interpretation +(11) +* Every l-th time step perform synchronization, see section 3.2 +– End time loop if t = tfinal +Figure 3: Summary of the employed lattice Boltzmann algorithm. +both wave equations (9) on the same lattice, i.e. fixed ∆h, and the same time discretization, i.e. fixed ∆t. Apart from +(15), the requirements (14) still need to be fulfilled. This can be accomplished for the general case cs < cd by setting +0 ≤ aφ ≤ 0.25, +aψ = c2 +s +c2 +d +aφ, +∆t = ∆h +cs +� +2aψ = ∆h +cd +� +2aφ. +(16) +Note that the Courant-Friedrichs-Lewy (CFL) stability condition [23], which is critical for the numerical analysis of +hyperbolic PDEs with explicit schemes, for the larger – and more critical – wave speed +cd∆t +∆h ≤ 1, +(17) +is always guaranteed by (16). +The LBE (12) is only part of the overall algorithm that is used to solve a plane strain problem as summarized in Fig. 3. +The other parts of the algorithm are discussed in the following. +The initial preprocessing step builds the lattice for a given geometry and also computes cells at each boundary lattice +point as depicted in Fig. 4. Without going into detail, the algorithm for creating individual cells starts with a quadratic +cell of side length ∆h centered around a boundary lattice point. This original cell is subsequently modified to match +the boundary geometry. The volume VC of a cell and the surface that such a cell C shares with the external boundary +∂Cext ⊂ ∂B are relevant for the computation of the acceleration at boundary lattice points later on. +After preprocessing, the material velocity ˙u and the displacement ⃗u are initialized first. Subsequently, ψ and φ are +initialized by a finite difference approximation of (7). Lastly, the initial distribution functions are determined to be the +value of the equilibrium distribution function +f α(⃗x, 0)ψ|φ = f α +eq,ψ|φ(ψ(⃗x, 0)|φ(⃗x, 0)). +(18) +In the time loop, the acceleration is computed from the Navier-Cauchy equation (8) at interior lattice points, whereas +boundary conditions determine the acceleration at the boundary lattice points. Once the acceleration at each lattice +point is known, the displacement is computed by explicit integration via the Newmark method, i.e. +⃗u(⃗x, t + ∆t) = ⃗u(⃗x, t) + ∆t ˙⃗u(⃗x, t) + ∆t2 +2 +¨⃗u(⃗x, t), +˙⃗u(⃗x, t + ∆t) = ˙⃗u(⃗x, t) + ∆t ¨⃗u(⃗x, t). +(19) +5 + +A Lattice Boltzmann Method for Elastic Solids +Figure 4: Cells are generated around each boundary lattice point in order to apply Neumann boundary conditions. +After the integration step, the displacement field is already updated, i.e. ⃗u(⃗x, t + ∆t) is determined at all lattice points. +The rotation ψ(⃗x, t), the dilatation φ(⃗x, t) and the associated distribution functions f α +ψ|φ have not been updated yet, but +they are required to compute the acceleration at interior lattice points in the next time step. +We prepare the required update by computing rotation and dilatation fields as well as distribution functions at the +boundary lattice points. All of these must be consistent with the applied boundary conditions as well, see the next +section for details. Subsequently, the rotation and dilatation fields are updated in the interior by the LBM. This step +includes the update of all interior distribution functions by (12) and the computation of the rotation and the dilatation by +(11). +3.1 +Boundary Conditions +In this section, the treatment of boundary conditions is explained in more detail since it is the most complex part of the +proposed LBM. The overall strategy is to first determine the acceleration ¨⃗u(⃗x, t), that is consistent with the boundary +conditions for each boundary lattice point. Subsequently, the displacement field is updated everywhere via the Newmark +integration mentioned above. Last, the rotation and dilatation fields as well as the distribution functions are reconciled +with the displacement. +3.1.1 +Consistent Acceleration at Boundary Lattice Points +For Neumann type boundary conditions the consistent acceleration is determined by computing cells C with size VC +and boundary ∂C around each boundary lattice point ⃗xk as shown in Fig. 4. For each of these cells, we consider a +balance of momentum +� +C +ρ¨⃗u(⃗xk, t) dv = +� +∂Cint +⃗σ(⃗x, t)⃗n da + +� +∂Cext +⃗t∗(⃗x, t) da, +(20) +where ∂Cint is the part of the boundary of the cell which is shared with neighboring cells and ∂Cext is part of the +boundary of the cell that is shared with the boundary of the body. Equation (20) is simplified by assuming that ρ and +¨⃗u(⃗xk, t) are constant across the cell and that the stress ⃗σkr is constant for each segment of the internal boundary shared +with a particular neighbor ⃗xr, +¨⃗u(⃗xk, t) ≈ +1 +ρVC +� +� +� +r∈Neighbors +⃗σkr⃗nkr + +� +∂Cext +⃗t∗(⃗x, t) da +� +� . +(21) +The surface measure, i.e. the length of of the boundary segment in 2D and the normal vector are denoted by lkr and ⃗nkr +respectively. The stress tensor at each segment is approximated by +⃗σkr = 1 +2 (⃗σ(⃗xk, t) + ⃗σ(⃗xr, t)) , +(22) +6 + +Xr1 +xr2 +Xk +aCint +Xr4 +Xr3A Lattice Boltzmann Method for Elastic Solids +where the stress at the lattice points ⃗xk and ⃗xr is computed by a finite difference approximation of (3). +For Dirichlet type boundary conditions ⃗u = ⃗u∗ on ∂Bu, where ⃗u∗ is the prescribed displacement value, the acceleration +¨⃗u(⃗xk, t) at boundary lattice points is determined from the integration scheme (19). Although extrapolation to a non- +lattice conforming boundary is also possible, we limit the discussion of Dirichlet boundary conditions to situations in +which boundary lattice points lie exactly on the boundary. In this case, it is +⃗u(⃗xk, t + ∆t) = ⃗u∗. +(23) +Thus, (19) can be solved for the required acceleration +¨⃗u(⃗xk, t) = +2 +∆t2 (⃗u∗(t) − ⃗u(⃗xk, t)) − 2 +∆t +˙⃗u(⃗xk, t). +(24) +3.1.2 +Consistent Displacement, Rotation, Dilatation and Distribution Functions at Boundary Lattice Points +a) +b) +c) +d) +e) +f) +g) +h) +Figure 5: Updating the displacement u as well as the rotation ψ and dilation φ in a simple square domain. The outer +circles represent the state of ψ and φ at the particular lattice points, whereas the inner circles represent the state of u. +Yellow indicates that quantities still have a value associated with the previous time step t, whereas green color indicates +that u or ψ and φ are already updated to their values at t + ∆t. a) the state after the previous time step. b) integration is +performed at all lattice points which updates the displacement. c) finite differences (stencil is indicated by the red lines) +are used to update ψ and φ at the boundary lattice points in a way that is consistent with the new displacement field. d) +all boundary points are updated. e) the LBM update, i.e. solving the wave equations, leads to a consistent rotation and +dilatation at interior lattice points (red lines indicate from which neighbors information is streamed to an interior lattice +point). f) all interior points have consistent fields after the LBM update. g) intermediate ‘second row’ boundary points +are more problematic since there is also information streamed from boundary points that does not originate from the +LBE for the wave equations, but from the handling of boundary conditions (red lines indicate from which neighbors +information is streamed to an interior lattice point). h) Fields are consistent – considering the previous remarks – at all +lattice points. +After the acceleration at time t is known at all lattice points, the displacement as well as the distribution functions for +the next time step need to be computed in a consistent manner. In this context, we regard the distribution functions, and +7 + +A Lattice Boltzmann Method for Elastic Solids +consequently ψ and φ, to be consistent with the updated displacement field if +4 +� +α=0 +f α +ψ(⃗x, t + ∆t) = ψ(⃗x, t + ∆t) +≈ (∇ × ⃗u)|(⃗x,t+∆t), +4 +� +α=0 +f α +φ (⃗x, t + ∆t) = φ(⃗x, t + ∆t) +≈ (∇ · ⃗u)|(⃗x,t+∆t). +(25) +Herein, (∗)|(⃗x,t+∆t) means that the spatial derivative ∗ is performed at the lattice point ⃗x and time t + ∆t via second +order accurate finite differences, which is a non-local operation that also involves neighbor lattice points to ⃗x. Fig. 5 +displays the utilized stencils for this operations as red lines. +The starting point for the algorithm is the situation after the previous time step has been completed as depicted for a +quadratic domain in Fig. 5 a). In this figure, lattice points are represented as circles. In order to illustrate the strategy of +obtaining consistent displacement and distribution functions, the color of the inner circles also represents the state of +the displacement field, i.e. the not yet updated state ⃗u(∗, t) is represented by yellow and the updated state ⃗u(∗, t + ∆t) +is indicated by green color. Similarly, the color of the outer circle indicates the state of the rotation ψ and dilatation φ. +A yellow outer circle indicates that rotation and dilatation are not updated yet, i.e. the state [ψ(∗, t), φ(∗, t)], whereas a +green outer circle indicates the updated state [ψ(∗, t + ∆t), φ(∗, t + ∆t)]. +The acceleration at all lattice points is known from the Navier-Cauchy equation (8) or the boundary conditions (21) and +(24), which allows to update the displacement field via (19) as a next step. This leads to an inconsistent situation where +the displacement is already updated, but the rotation and the dilatation fields are not, see Fig. 5 b). +The rotation and dilatation fields in the interior are updated via the LBM for the wave equation. This works fine in +the interior, where we have pointed out that the Navier-Cauchy equation and the wave equations (9) are equivalent. +Consequently the update of the displacement field by (8) and (19) on the one hand, and the update of the distribution +functions by (12) and the derived rotation and the dilatation by and (11) on the other hand are consistent within the +limits of the LBM by Chopard et al. [11], see Fig. 5 e) and f). +However, at boundary lattice points the displacement field is updated from the boundary conditions and the distribution +functions can only partially be updated via the LBE. +Moreover, the neighbors of boundary lattice points, the ‘second row’ boundary lattice points, also cannot be in a +consistent state in the sense of (25) since the finite difference approximation of ∇ × ⃗u and ∇ · ⃗u at those points depends +on the displacement of boundary lattice points which in turn is determined only by the boundary conditions and not by +the Navier-Cauchy equation. +Thus, in order to model the boundary conditions for the LBM correctly, it is necessary to accomplish two things: +• Setting the distribution functions at boundary lattice points consistent with (25). +• Modifying the LBE at ‘second row’ lattice points in such a way that consistency is achieved at these points in +the sense of (25). +The first requirement is satisfied by setting +ψ(⃗xk, t + ∆t) = (∇ × ⃗u)|(⃗xk,t+∆t), +φ(⃗xk, t + ∆t) = (∇ · ⃗u)|(⃗xk,t+∆t) +(26) +at boundary lattice points xk , see Fig. 5 c) and d), and +f 0 +ψ|φ(⃗xk, t + ∆t) =a0,ψ|φ(ψ|φ)(⃗xk, t + ∆t), +f α +ψ|φ(⃗xk, t + ∆t) =aψ|φ(ψ|φ)(⃗xk, t + ∆t) ++ b⃗cα · ⃗Jψ|φ(⃗xk, t) +2c2 +for α ̸= 0. +(27) +In order to fulfill the second requirement, we envision that all changes of ψ and φ at a boundary lattice points over +one time step t → t + ∆t are transported as waves to its neighbors. Assuming that a linear change in time is a +8 + +A Lattice Boltzmann Method for Elastic Solids +reasonable approximation, the average state of a boundary lattice point during this transition is given by the average of +its distribution functions at two discrete time steps, i.e. +˜f α(⃗xk, ˜t) = 1 +2 +� +f α +ψ|φ(⃗xk, t + ∆t) + f α +ψ|φ(⃗xk, t) +� +. +(28) +The transport of this intermediate state to the boundary conditions is obtained by the modified LBE +f α +ψ|φ (⃗x + ⃗cα∆t, t + ∆t) = +ˆf α +ψ|φ(⃗x, t) − 1 +τ +� +ˆf α +ψ|φ(⃗x, t) − f α +eq,ψ|φ(⃗x, t) +� +, +(29) +where +ˆf α +ψ|φ = +� ˜f α +ψ|φ(⃗x, ˜t), +if ⃗x is boundary lattice point +f α +ψ|φ(⃗x, t), +otherwise. +Thus, the modification is only employed if ⃗x +⃗cα∆t is a ‘second row’ boundary lattice point. This is not exact, but it is +an approximation that leads to a reasonable state of ‘second row’ boundary lattice points, see Fig. 5 g) and h). +3.2 +Periodic Synchronization +The proposed LBM for plane strain is susceptible to instabilities if the dilatation and rotation fields ψ and φ become +inconsistent with the displacements u. Since there is no inherent synchronization of these fields, rounding errors are +amplified over time and eventually the computed acceleration is sufficiently misaligned with the actual displacement +such that the Navier-Cauchy equation (8) is violated. Small inconsistencies originate from the handling of the boundary +conditions as described in the last paragraphs of the previous section. +In order to remedy this problem, we introduce another step in the LBM algorithm, that periodically (every l-th timestep +where l ≫ 1) computes ψ and φ directly from the displacement field with a finite difference approximation of (7). +As soon as ψ(⃗xk, tl) and φ(⃗xk, tl) are known, the distribution functions are also corrected according to (27) and the +algorithm continues normally in the next time step. +4 +Numerical Examples +cs +cd += +1 +√ +3 +y +P +L +t[L/cs] +1 +x +0.005 +σ0[µ] +−σ0(t)ey +σ0(t)ey +Figure 6: A square domain subjected to a tensile load. The right plot displays the applied stress σ0(t) as a function of +time. +In order to demonstrate the performance of the proposed LBM, we perform several numerical experiments in which +the LBM is compared to results obtained via the established Finite Element Method (FEM). The experiments also +demonstrate that the proposed LBM successfully solves boundary value problems that are prevalent in engineering +practice and is not restricted to often rather academic types of boundary value problems, e.g. with periodic boundary +conditions. +In all numerical examples, we formulate the problem in terms of the ratio of wave speeds cs/cd = 1/ +√ +3, the wave speed +cs, the shear modulus µ, the length scale L, and the reference displacement which is also set to L. The parameters of +the equilibrium distribution functions are defined by (14) and (16), where a rather extreme value of a0,φ = 0.9999 has +been found to be required for sufficient stability. Note that setting a0,φ = 0.9999 also severely reduces the time step. +The benchmark FEM simulations are performed with bi-linear finite elements and implicit time integration via the +standard Newmark method. +9 + +A Lattice Boltzmann Method for Elastic Solids +Figure 7: Deformed heat map for a square domain subjected to a tensile load at time t = L/cs. The deformation is scaled +by factor 100. The heat map displays the FEM benchmark results, whereas the black squares indicate the displaced +positions of the lattice points. +Figure 8: Displacement at the top left corner P of a square domain subjected to a tensile load. +4.1 +Tension +For the first numerical example, a square domain is subjected to a time-dependent, tensile traction t∗ = ±σ0ey load +at the top and bottom edges, see Fig. 6. The load is linearly increased from σ0(t = 0) = 0 to σ0(t = L/cs) = 0.005µ +and held constant afterwards. In this simulation, no periodic synchronization, see section 3.2, is employed. In order +to study the performance of the LBM algorithm, we compare the LBM results to an FEM simulation. Fig. 7 shows a +deformed heat map of the FEM results in the background, whereas black squares indicate the displaced position2 of the +lattice points. Both simulations are evaluated at time t = L/cs and the deformation is scaled by a factor of 100. It can be +observed that the LBM matches the FEM results well and predicts phenomena such as lateral contraction accurately. +Fig. 8 explicitly displays the displacement of the top left corner P, see also Fig 13, and confirms these findings. We can +2The displaced position of a lattice point ⃗xk is only a result of post-processing, i.e. the lattice remains unchanged. It is computed +as ˜⃗xk(t) = s⃗u(⃗xk, t) + ⃗xk, where s = 100 is the scaling factor. +10 + +uy [L] +9.1e-04 +0.0006 +0.0004 +0.0002 +0 +-0.0002 +-0.0004 +-0.0006 +-9.1e-04 +AYX10-3 +X10~3 +1 +1 +0.8 +0.8 +L +0.6 +0.6 +0.4 +0.4 +-FEM +LBM +0.2 +0.2 +0 +0 +0 +1 +2 +0 +1 +2 +t[L/cs] +t[L/cs]A Lattice Boltzmann Method for Elastic Solids +Figure 9: Error e for a square domain subjected to a tensile load at t ≈ 0.002L/cs. The maximum error is 4.3 · 10−12 +which coincides with the maximum value displayed in the legend color scheme, i.e. dark red. +observe an expected dynamic overshoot and low frequency oscillations in both displacement components, that both the +FEM and LBM simulations predict. Nonetheless, Fig. 8 also reveals that erroneous higher frequency oscillations occur +in the later stages of the LBM simulation, see the green graph in the plot of uy for t > 1.5L/cs, which indicate that a +periodic synchronization may be useful. +As discussed above, we assume that inconsistencies in the sense of violations of (25) are the cause for these instabilities +and that they occur primarily at the ‘second row’ boundary points, see also Fig. 5 g). In order to test this hypothesis, an +error measure that is in line with (25) is defined as +e(x, t) = +����� +��4 +α=0 f α +ψ(⃗x, t) − (∇ × ⃗u)|(⃗x,t) +�4 +α=0 f α +φ (⃗x, t) − (∇ · ⃗u)|(⃗x,t) +������ +2 +. +(30) +Fig. 9 displays this error at time t ≈ 0.002L/cs after the corresponding time step has been completely processed. It +can be observed that inconsistencies indeed occur at the ‘second row’ boundary points at the top and bottom edges. +Although the error is small, without periodic synchronization, it amplifies and eventually manifests as oscillations that +can be observed in Fig. 8. +4.2 +Simple Shear +cs +cd += +1 +√ +3 +y +P +L +t[L/cs] +1 +x +0.005 +σ0[µ] +σ0(t)ex +Figure 10: A square domain subjected to a shear load. The right plot displays the applied stress σ0(t) as a function of +time. +The second numerical example uses the same geometric configuration, but differs in terms of the applied boundary +conditions. The top edge is subjected to a shear traction that is linearly increased over time, i.e. σ0 = 0.005tµcs/L, +see Fig. 10. The bottom edge is subjected to homogeneous Dirichlet boundary conditions w(x, y = L/2, t) = +0. Furthermore, the LBM simulations are run with a periodic synchronization every 50th time step and without +synchronization. Fig. 11 displays a deformed (scaled by factor 100) heat map of the FEM results in the background and +the displaced lattice points as black squares of the synchronized LBM simulation in the foreground at time t = L/cs. +The LBM accurately captures the shear deformation as well. However, as can be observed in Fig. 12, in this experiment +it is strictly necessary to employ the synchronization step, since the LBM simulations without synchronization differ +severely from the FEM benchmark after t ≈ 0.7L/cs. +11 + +e +4.3e-12 +3.5e-12 +3e-12 +2.5e-12 +2e-12 +1.5e-12 +1e-12 +5e-13 +0.0e+00A Lattice Boltzmann Method for Elastic Solids +Figure 11: Deformed heat map for a square domain subjected to a shear load at time t = L/cs. The deformation is scaled +by factor 100. The heat map displays the FEM benchmark results, whereas the black squares indicate the displaced +positions of the lattice points. For the LBM results a periodic synchronization was performed every 50th time step. +Figure 12: Displacement at the top left corner P of a square domain subjected to a shear load. +4.3 +Plate with a Circular Hole +The last numerical example again considers a square domain that is subjected to a tensile traction load. In order to +illustrate the LBMs capabilities to handle non-lattice conforming geometries, the domain includes a circular hole of +diameter 0.266L, see Fig. 13. As in the previous example, we run LBM simulations with periodic synchronization +every 50th timestep and without any periodic synchronization. Fig 14 displays the scaled deformed configuration for +the FEM in the background, as well as for the LBM with synchronization as the black squares in the foreground at time +t = L/cs. Again the LBM agrees well with the FEM reference. However, this is only the case if the synchronization +is utilized, see Fig. 15, as the simulation becomes unstable quickly if synchronization is omitted. The oscillations +12 + +uy [L] +9.1e-04 +0.0006 +0.0004 +0.0002 +0 +-0.0002 +-0.0004 +-0.0006 +-9.1e-04X10-3 +X10-3 +FEM +1 +1 +LBM +0.8 +0.8 +LBM syn +0.6 +0.6 +an +Wy +0.4 +0.4 +0.2 +0.2 +0 +0 +0 +1 +2 +0 +1 +2 +t [L/cs] +t [L/cs]A Lattice Boltzmann Method for Elastic Solids +x +L +P +y +σ0[µ] +0.005 +1 +t[L/cs] +Q +0.266L +cs +cd += 1 +√ +3 +−σ0(t)ey +σ0(t)ey +Figure 13: A square domain with a hole subjected to a tensile load. Point Q is located at (−0.175L, 0.025L) relative to +a coordinate system which has its origin in the center of the hole. The right plot displays the applied stress σ0(t) as a +function of time. +Figure 14: Deformed heat map for a square domain with a hole subjected to a tensile load at time t = L/cs. The +deformation is scaled by factor 100. The heat map displays the FEM benchmark results, whereas the black squares +indicate the displaced positions of the lattice points. For the LBM results a periodic synchronization was performed +every 50th time step. +originate from the non-lattice conforming boundaries at the hole as can also be observed in Fig. 15: the displacement +field close to the hole at point Q becomes unstable long before oscillations can be observed at point P. +5 +Conclusion +In this work, a new Lattice Boltzmann Method (LBM) for solving the general plane strain problems is proposed. The +plane strain problem is governed by the Navier-Cauchy equation which can be decomposed into two wave equations +with different wave speeds for the rotational part of the displacement field and the dilatational part respectively. Based +on this observation the new LBM is constructed by employing the established LBM by Chopard et al. [11] to solve the +two wave equations separately. Chopard et al.’s approach allows enough flexibility to choose the simulated macroscopic +13 + +酒 +uy [L] +1.6e-03 +0.001 +-0.0005 +0 +-0.0005 +-0.001 +-1.6e-03A Lattice Boltzmann Method for Elastic Solids +point Q +point P +Figure 15: Displacement at the top left corner P (top row) and close to the hole Q (bottom row) of a square domain +with a hole subjected to a tensile load. +wave speed rather independently of the time step and lattice spacing. Thus, the proposed method solves both wave +equations on the same D2Q5 lattice with the same time discretization. However, this also limits the maximum time step +and reduces the computational efficiency of the approach in situations in which larger time step sizes may be feasible. +The displacement field is eventually obtained by integrating the Navier-Cauchy equation and making use of rotation +and dilatation fields computed by the LBM. +In order to apply Dirichlet and Neumann boundary conditions, a consistent acceleration is computed at boundary lattice +points. This is then used for the integration step mentioned above at these points. In order to reconcile the displacements +obtained in this way with the LBM quantities such as the rotation and dilatation as well as the distribution functions, the +rotation and dilatation fields are computed from a finite difference approximation of the gradient of the displacement +field at boundary lattice points. Afterwards, the distribution functions are computed consistently with these rotation and +dilatation fields at the boundary points. We mention some of the remaining causes of inconsistencies between rotation +and dilatation fields on the one side and the displacement field on the other side. +These inconsistencies manifest as instabilities in the performed simulations. We address this issue by performing a +periodic synchronization in which we compute the rotation and dilatation fields from a a finite difference approximation +of the gradient of displacement and subsequently set the distribution functions accordingly. +Lastly, several numerical benchmarks highlight the performance of the new method compared to benchmark FEM +simulations. The simulation of a square domain without periodic synchronization under tensile loading shows that +the LBM accurately captures simple loading and domains without the synchronization step. However, this example +also reveals that the inconsistencies mentioned above indeed occur. The second numerical example studies a square +domain under simple shear loading conditions. Here, the results only accurately match the FEM simulations if the +synchronization step is employed every 50th time step. The third numerical example considers the square domain with +a hole under tensile load and illustrates that the developed LBM is indeed capable of solving problems in which the +geometry does not conform with the lattice, i.e. the boundary does not exactly match the lattice point positions. +We find the performance of the LBM in relation to the FEM promising. However, the periodic synchronization step as +well as the rather fine time discretization, that is dictated by the method remains unsatisfactory. In future work, we +envision to investigate alternative LBM approaches, but we also want to address the shortcomings of the present LBM +by refining the treatment of boundary conditions and exploring the possibility of giving up the same time discretization +14 + +FEM +×10-4 +×10-4 +LBM +10 +10 +-LBM syn. +5 +5 +y +0 +0 +-5 +-5 +0 +1 +2 +0 +1 +2 +t [L/cs] +t[L/cs] +×10-4 +×10-4 +10 +10 +5 +5 +Wx +0 +0 +-5 +-5 +0 +1 +2 +0 +1 +2 +t[L/cs] +t [L/cs]A Lattice Boltzmann Method for Elastic Solids +for both simulated wave equations. This would allow us to use larger time steps and thus increase computational +efficiency, but this approach will also involve an additional interpolation step between time steps. +Acknowledgments +Open access funding enabled and organized by Projekt DEAL. The authors gratefully acknowledge the funding by the +German Research Foundation (DFG) within the project 423809639. +References +[1] Bastien Chopard and Pascal O. Luthi. Lattice Boltzmann computations and applications to physics. Theoretical +Computer Science, 217(1):115–130, March 1999. +[2] Timm Krüger, Halim Kusumaatmaja, Alexandr Kuzmin, Orest Shardt, Goncalo Silva, and Erlend Magnus Viggen. +The Lattice Boltzmann Method: Principles and Practice. Graduate Texts in Physics. Springer International +Publishing, Cham, 2017. +[3] Alexander Schlüter, Charlotte Kuhn, and Ralf Müller. Lattice Boltzmann simulation of antiplane shear loading of +a stationary crack. Computational Mechanics, 62(5):1059–1069, November 2018. +[4] Thomas Reinirkens, Ralf Müller, and Charlotte Kuhn. Lattice Boltzmann method applied to antiplane shear +loading of a stationary crack. In PAMM, volume 18, December 2018. +[5] Sauro Succi. Numerical solution of the Schrödinger equation using discrete kinetic theory. Physical Review E, +53(2):1969–1975, February 1996. +[6] S. Solórzano, M. Mendoza, S. Succi, and H. J. Herrmann. Lattice Wigner equation. Physical Review E, +97(1):013308, January 2018. +[7] Hans-Joachim Bungartz and Michael Schäfer, editors. Fluid-Structure Interaction, volume 53 of Lecture Notes in +Computational Science and Engineering. Springer Berlin Heidelberg, Berlin, Heidelberg, 2006. +[8] Guangwu Yan. A Lattice Boltzmann Equation for Waves. Journal of Computational Physics, 161(1):61–69, June +2000. +[9] Bastien Chopard and M. Droz. Cellular automata modeling of physical systems. Cambridge University Press, +Cambridge; New York, 1998. +[10] George N. Frantziskonis. Lattice Boltzmann method for multimode wave propagation in viscoelastic media and in +elastic solids. Physical Review E, 83(6):066703, June 2011. +[11] Bastien Chopard, Pascal Luthi, and Stefan Marconi. A Lattice Boltzmann Model for Wave and Fracture phenomena. +arXiv e-prints, pages cond–mat/9812220, December 1998. +[12] Stefan Marconi and Bastien Chopard. A Lattice Boltzmann Method for a Solid Body. 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Hanasoge, Prasad Perlekar, and Johan O. A. Robertsson. Numerical analysis of the +lattice Boltzmann method for simulation of linear acoustic waves. Physical Review E, 95(4):043306, April 2017. +[18] C. Jiang, H. Zhou, M. Xia, H. Chen, Y. Zhang, and S. Jiang. Acoustic Wave Simulation by Lattice Boltzmann +Method with D2Q5 and D2Q9 of Different Relaxation Times. In 81st EAGE Conference and Exhibition 2019, +pages 1–5, London, UK„ 2019. European Association of Geoscientists & Engineers. +15 + +A Lattice Boltzmann Method for Elastic Solids +[19] Alexander Schlüter, Henning Müller, and Ralf Müller. Boundary Conditions for a Lattice-Boltzmann Method for +Antiplane Shear Deformation. Manuscript submitted for publication, 2021. +[20] Eli Sternberg. On the integration of the equations of motion in the classical theory of elasticity. Archive for +Rational Mechanics and Analysis, 6(1):34–50, January 1960. +[21] P. L. Bhatnagar, E. P. Gross, and M. Krook. A Model for Collision Processes in Gases. 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Über die partiellen differenzengleichungen der mathematischen physik. +Mathematische Annalen, 100(1):32–74, December 1928. +16 + diff --git a/rtAyT4oBgHgl3EQfZvdA/content/tmp_files/load_file.txt b/rtAyT4oBgHgl3EQfZvdA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..37bd781507463b1137b08a426fab848438cca4cb --- /dev/null +++ b/rtAyT4oBgHgl3EQfZvdA/content/tmp_files/load_file.txt @@ -0,0 +1,498 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf,len=497 +page_content='A LATTICE BOLTZMANN METHOD FOR ELASTIC SOLIDS UNDER PLANE STRAIN DEFORMATION Alexander Schlüter, Sikang Yan, Erik Faust Institute of Applied Mechanics Technische Universität Kaiserslautern D-67653, Kaiserslautern {Alexander Schlüter} aschluet@rhrk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='uni-kl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='de Henning Müller, Ralf Müller Institut für Mechanik Technische Universität Darmstadt D-64287, Darmstadt ABSTRACT The Lattice Boltzmann Method (LBM), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' in [1] and [2], can be interpreted as an alternative method for the numerical solution of partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Consequently, although the LBM is usually applied to solve fluid flows, the above interpretation of the LBM as a general numerical tool, allows the LBM to be extended to solid mechanics as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In this spirit, the LBM has been studied in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' First publications [3], [4] presented an LBM scheme for the numerical solution of the dynamic behavior of a linear elastic solid under simplified deformation assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' For so-called anti-plane shear deformation, the only non-zero displacement component is governed by a two-dimensional wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In this work, an existing LBM for the two-dimensional wave equation is extended to more general plane strain problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The proposed algorithm reduces the plane strain problem to the solution of two separate wave equations for the volume dilatation and the non-zero component of the rotation vector, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' A particular focus is on the implementation of types of boundary conditions that are commonly encountered in engineering practice for solids: Dirichlet and Neumann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Last, several numerical experiments are conducted that highlight the performance of the new LBM in comparison to the Finite Element Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Keywords Lattice Boltzmann Method · solids · plane strain · computational engineering · computational solid mechanics 1 Introduction The mechanical behavior of solid bodies is of interest to both engineering and science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Thus, a large number of numerical methods capable of dealing with elasticity have emerged over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The more prominent ones among these, finite differences methods (FDM), finite element methods (FEM) and finite volume methods (FVM), work on the principle of discretizing the domain of interest and replacing the governing system of differential equations by algebraic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Such methods take a kind of top-down approach, and can therefore be thought of as acting on a macroscopic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In contrast, some numerical methods, such as molecular dynamics (MD) or density functional theory (DFT), regard the interactions of a system’s most basic constituents, such as individual particles and electrons, on a microscopic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' A different approach is taken with Lattice-Boltzmann methods (LBMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The common principle of this type of methods is to transform the given physical problem into a transport problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Based on Boltzmann’s transport equation from statistical mechanics, distribution functions are transported across phase-space, which is discretized both by a regular lattice and a set of associated lattice velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Information is exchanged between neighboring lattice sites in a streaming-like process along links connecting these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' This information is represented by so-called distribution functions, where each distribution function is associated with a different lattice velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The distribution functions are subjected to on-site interactions, or collisions, which incorporate the underlying microscopic theory in a probabilistic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Thus, LBMs can be said to act mesoscopically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' on an intermediate scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='00228v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='NA] 31 Dec 2022 A Lattice Boltzmann Method for Elastic Solids B ∂Bu ⃗n ⃗t∗ ∂Bt Figure 1: Body B with outer normal vector n, subjected to Neumann boundary conditions ⃗t∗ on Bt and Dirichlet boundary conditions on Bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' LBMs are well established in computational fluid dynamics (CFD) and have subsequently been extended to further scientific fields, such as solving Schrödinger’s equation [5] or Wigner’s equation [6] in quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Developing a LBM for solid mechanics could mean using a single method on both sides of a fluid-solid-interface, which is a topic of interest [7] in CFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' We approach the topic of a LBM for elastic bodies from the mechanical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' This includes a greater focus on finite domains with an appropriate boundary handling, which is of great concern for engineering problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The advantages over the established methods in computational engineering include the generally great computational efficiency, while still being able to handle the boundary conditions of complex domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' A further improvement of computation times by can be achieved by employing parallel computing, which is easy to implement with most LBMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' This opens up possibilities in highly dynamical problems, requiring very fine resolution of the temporal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The dynamic behavior of elastic solids can be described by multiple wave equations, that are superposed to obtain the aggregated deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' This mathematical description is closer to the transport phenomena for which the LBM was initially conceived, when compared to the original Navier-Cauchy equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In fact, LBMs have already been developed for the wave equation, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' [8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Furthermore, LBMs have been proposed for the numerical treatment of mechanical problems in solid bodies [11, 1, 12, 13, 14, 15, 16], and specifically elastic wave propagation [17, 18], which is also a topic of interest in geophysics and seismology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' However, an extensive method for the deformation of linear elastic solids under loads, containing appropriate boundary conditions, has still to be accomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In previous works [3, 19] we applied the LBM for wave equations published by Yan [8] to the mechanical problem of anti-plane shear deformation, which we then used for fracture mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' This work now regards the two-dimensional problem of plane strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The fundamental idea is to decompose the plane strain problem governed by the Navier-Cauchy equation into two equivalent wave equations that are solved with the LBM for wave propagation by Chopard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The discussion is structured as follows: First the mechanical problem is reviewed and the relevant equations are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The next section introduces the LBM for the wave equation, followed by the presentation of the algorithm for the plane strain case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' This includes a treatment of boundary conditions similar to [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Lastly three numerical examples show the feasibility of our algorithm, each compared to FEM computations, which act as benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 2 Plane Strain Deformation of a Linear Elastic Solid We consider a homogeneous, isotropic, and elastic body B with boundary ∂B = ∂Bu ∪ Bt, which is subjected to Dirichlet boundary conditions u = ⃗u∗ for the displacement ⃗u on ∂Bu and Neumann boundary conditions ⃗σ⃗n = t∗ for the Cauchy stress tensor ⃗σ on ∂Bt, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' For the plane strain case the problem is only regarded in two dimensions and under small strain assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' A set of fundamental equation is taken as a basis for the derivation of the mathematical description of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Firstly, the strain-displacement relation for small strains is given by the linearized strain tensor ⃗ε = 1 2 � ∇⃗u + (∇⃗u)T � , (1) where ⃗u = ⃗u(x, y, t) describes the time-dependent displacement field in two dimensions under plane strain assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The general equation of motion for the small strain case, in the absence of a body force, is given by ∇ · ⃗σ = ρ ∂2⃗u ∂t2 , (2) 2 A Lattice Boltzmann Method for Elastic Solids x y a) b) ∆h ∆h c4 c2 c3 c1 Figure 2: a) Lattice representation of the elastic solid and b) and the associated lattice velocity vectors (lattice links) for a single lattice point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' with Hooke’s law as the linear stress-strain relation ⃗σ = λ tr(⃗ε) 1 + 2µ⃗ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' (3) Herein λ and µ are the Lamé parameters of the material, 1 is the second-order identity tensor and the operator tr(∗) denotes the trace of a second order tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Equation (1) is substituted in equation (3), which is then substituted in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The result is the Navier-Cauchy equation (λ + µ) ∇ (∇ · ⃗u) + µ∇2 ⃗u = ρ ∂2⃗u ∂t2 , (4) which describes the mechanical behavior of an isotropic linear elastic solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Using the general identity ∇2⃗u = ∇(∇ · ⃗u) − ∇ × (∇ × ⃗u) (5) from vector calculus, equation (4) can be rewritten as c2 d ∇(∇ · ⃗u) − c2 s ∇ × (∇ × ⃗u) = ∂2⃗u ∂t2 , (6) where cd = � (λ+2µ)/ρ and cs = � λ/ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' With regard to equation (6), two fields, φ and ⃗ψ, can be defined as follows: φ = ∇ · ⃗u and ⃗ψ = ∇ × ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' (7) The scalar field φ describes the dilatation of the displacement field ⃗u, whereas the vector field ⃗ψ describes the rotation of ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In two dimensions, the latter reduces to ⃗ψ = ψ ⃗ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The Navier-Cauchy equation can be restated in terms of these fields c2 d ∇φ − c2 s ∇ × ⃗ψ = ∂2⃗u ∂t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' (8) In conjunction with the definitions in equation (7), applying the divergence to both sides of equation (8) results in c2 d ∇2φ = ∂2φ ∂t2 , (9a) while applying the curl results in c2 s ∇2 ⃗ψ = ∂2 ⃗ψ ∂t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' (9b) Thus, the Navier-Cauchy equation (8) can be reduced to two wave equations, the dilatational wave equation (9a) for φ = ∇ · ⃗u with wave speed cd, and the rotational wave equation (9b) for ⃗ψ = ψ ⃗ez = ∇ × ⃗u with wave speed cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Note that there are other ‘decompositions’ of the displacement field besides (7) that lead to similar wave equations, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 3 A Lattice Boltzmann Method for Elastic Solids 3 Lattice Boltzmann Method for Plane Strain The proposed numerical strategy for the plane strain case relies on solving the wave equations (9) by means of the LBM by Chopard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In the LBM, a body B is typically approximated by a regular lattice with lattice spacing ∆h as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 2 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The approach by Chopard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' is based on a D2Q5 scheme, see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 2, with the lattice velocities ⃗c0 = (c0 x, c0 y) = (0, 0) ⃗c1 = (c1 x, c1 y) = (c, 0) ⃗c2 = (c2 x, c2 y) = (0, c) ⃗c3 = (c3 x, c3 y) = (-c, 0) ⃗c4 = (c4 x, c4 y) = (0, -c) (10) where c = ∆h/∆t is the speed at which information can travel in the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Thus, the lattice velocities ⃗c1,⃗c2,⃗c3,⃗c4 allow for information to be transported to each of the four neighbors of a lattice point in a so-called D2Q5 scheme1 in one time step, whereas ⃗c0 is associated with information remaining at a particular lattice point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Information is represented by distribution functions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' f α represents information, which is transported with lattice velocity ⃗cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In order to simulate the wave equations (9), the distribution functions need to be interpreted, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' a relation to the macroscopic fields needs to be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' We introduce two sets of distribution functions to simulate both wave equations and relate them to the macroscopic fields through 4 � α=0 f α ψ = ψ, 4 � α=0 f α φ = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' (11) The Lattice Boltzmann equation (LBE) models transport as well as the interaction of distribution functions between and at lattice points respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Since two wave equations need to be solved, we also introduce two associated LBEs for ψ and φ respectively f α ψ|φ (⃗x + ⃗cα∆t, t + ∆t) = f α ψ|φ(⃗x, t) − ∆t τ � f α ψ|φ(⃗x, t) −f α eq,ψ|φ(⃗x, t) � , (12) where the notation (ψ|φ) indicates that either ψ or φ need to be chosen for the whole equation and the common BGK approximation is employed, see [21] and [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Equation (12) is universal to many Lattice Boltzmann models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The specific physics can be modeled by choosing the equilibrium distribution functions f α eq,ψ|φ and relaxation time τ in a certain way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In order to model a wave equation Chopard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' propose τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='5∆t and f 0 eq,ψ|φ = a0,ψ|φ(ψ|φ) f α eq,ψ|φ = aψ|φ(ψ|φ) + b⃗cα · ⃗Jψ|φ 2c2 , with ⃗Jψ|φ = 4 � α=0 ⃗cαf α ψ|φ(⃗x, t), for α ̸= 0, (13) where again the notation (ψ|φ) indicates that either ψ or φ need to be chosen for the whole equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The parameters also need to fulfill the requirements b = 1, conservation of ⃗Jψ|φ a0,ψ|φ + 4aψ|φ = 1, conservation of ψ|φ a0,ψ|φ ≥ 0, stability (14) and cs|cd = ∆h ∆t �2aψ|φ, (15) see Chopard [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Equation (15) allows us to adjust the macroscopic wave speed modeled by the LBM independently of the time step ∆t or the lattice spacing ∆h by choosing aψ|φ accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' We exploit this in order to be able to simulate 1D2Q5 refers to the dimension of the lattice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' two in this case, and the number of lattice velocities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' five in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 4 A Lattice Boltzmann Method for Elastic Solids Preprocessing – Build lattice for geometry – Compute surface measure and cell volumes for boundary lattice points Solver – Initialize ⃗u(⃗x, 0) and ˙⃗u(⃗x, 0) – Initialize ⃗ψ(⃗x, 0) = ∇ × ⃗u(⃗x, 0), ⃗φ(⃗x, 0) = ∇ × ⃗u(⃗x, 0) computed by finite differences – Initialize the distribution functions by (13) – Start time loop t → t + ∆t Compute accelerations in the interior ¨⃗u(⃗x, t) = c2 d∇φ(⃗x, t) − c2 s ⃗ψ(⃗x, t) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' by (6) Compute accelerations at boundary points by (21) or (24) Compute ⃗u(⃗x, t + ∆t) by explicit integration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' by (19) set ψ(⃗x, t + ∆t) at boundary lattice points to be consistent with ⃗u(⃗x, t + ∆t) according to (26) Solve the wave equations for ψ(⃗x, t + ∆t) and φ(⃗x, t + ∆t) via the LBE (12) and the interpretation (11) Every l-th time step perform synchronization, see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='2 – End time loop if t = tfinal Figure 3: Summary of the employed lattice Boltzmann algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' both wave equations (9) on the same lattice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' fixed ∆h, and the same time discretization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' fixed ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Apart from (15), the requirements (14) still need to be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' This can be accomplished for the general case cs < cd by setting 0 ≤ aφ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='25, aψ = c2 s c2 d aφ, ∆t = ∆h cs � 2aψ = ∆h cd � 2aφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' (16) Note that the Courant-Friedrichs-Lewy (CFL) stability condition [23], which is critical for the numerical analysis of hyperbolic PDEs with explicit schemes, for the larger – and more critical – wave speed cd∆t ∆h ≤ 1, (17) is always guaranteed by (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The LBE (12) is only part of the overall algorithm that is used to solve a plane strain problem as summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The other parts of the algorithm are discussed in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The initial preprocessing step builds the lattice for a given geometry and also computes cells at each boundary lattice point as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Without going into detail, the algorithm for creating individual cells starts with a quadratic cell of side length ∆h centered around a boundary lattice point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' This original cell is subsequently modified to match the boundary geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The volume VC of a cell and the surface that such a cell C shares with the external boundary ∂Cext ⊂ ∂B are relevant for the computation of the acceleration at boundary lattice points later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' After preprocessing, the material velocity ˙u and the displacement ⃗u are initialized first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Subsequently, ψ and φ are initialized by a finite difference approximation of (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Lastly, the initial distribution functions are determined to be the value of the equilibrium distribution function f α(⃗x, 0)ψ|φ = f α eq,ψ|φ(ψ(⃗x, 0)|φ(⃗x, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' (18) In the time loop, the acceleration is computed from the Navier-Cauchy equation (8) at interior lattice points, whereas boundary conditions determine the acceleration at the boundary lattice points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Once the acceleration at each lattice point is known, the displacement is computed by explicit integration via the Newmark method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' ⃗u(⃗x, t + ∆t) = ⃗u(⃗x, t) + ∆t ˙⃗u(⃗x, t) + ∆t2 2 ¨⃗u(⃗x, t), ˙⃗u(⃗x, t + ∆t) = ˙⃗u(⃗x, t) + ∆t ¨⃗u(⃗x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' (19) 5 A Lattice Boltzmann Method for Elastic Solids Figure 4: Cells are generated around each boundary lattice point in order to apply Neumann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' After the integration step, the displacement field is already updated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' ⃗u(⃗x, t + ∆t) is determined at all lattice points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The rotation ψ(⃗x, t), the dilatation φ(⃗x, t) and the associated distribution functions f α ψ|φ have not been updated yet, but they are required to compute the acceleration at interior lattice points in the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' We prepare the required update by computing rotation and dilatation fields as well as distribution functions at the boundary lattice points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' All of these must be consistent with the applied boundary conditions as well, see the next section for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Subsequently, the rotation and dilatation fields are updated in the interior by the LBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' This step includes the update of all interior distribution functions by (12) and the computation of the rotation and the dilatation by (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='1 Boundary Conditions In this section, the treatment of boundary conditions is explained in more detail since it is the most complex part of the proposed LBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The overall strategy is to first determine the acceleration ¨⃗u(⃗x, t), that is consistent with the boundary conditions for each boundary lattice point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Subsequently, the displacement field is updated everywhere via the Newmark integration mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Last, the rotation and dilatation fields as well as the distribution functions are reconciled with the displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='1 Consistent Acceleration at Boundary Lattice Points For Neumann type boundary conditions the consistent acceleration is determined by computing cells C with size VC and boundary ∂C around each boundary lattice point ⃗xk as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' For each of these cells, we consider a balance of momentum � C ρ¨⃗u(⃗xk, t) dv = � ∂Cint ⃗σ(⃗x, t)⃗n da + � ∂Cext ⃗t∗(⃗x, t) da, (20) where ∂Cint is the part of the boundary of the cell which is shared with neighboring cells and ∂Cext is part of the boundary of the cell that is shared with the boundary of the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Equation (20) is simplified by assuming that ρ and ¨⃗u(⃗xk, t) are constant across the cell and that the stress ⃗σkr is constant for each segment of the internal boundary shared with a particular neighbor ⃗xr, ¨⃗u(⃗xk, t) ≈ 1 ρVC � � � r∈Neighbors ⃗σkr⃗nkr + � ∂Cext ⃗t∗(⃗x, t) da � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' (21) The surface measure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' the length of of the boundary segment in 2D and the normal vector are denoted by lkr and ⃗nkr respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The stress tensor at each segment is approximated by ⃗σkr = 1 2 (⃗σ(⃗xk, t) + ⃗σ(⃗xr, t)) , (22) 6 Xr1 xr2 Xk aCint Xr4 Xr3A Lattice Boltzmann Method for Elastic Solids where the stress at the lattice points ⃗xk and ⃗xr is computed by a finite difference approximation of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' For Dirichlet type boundary conditions ⃗u = ⃗u∗ on ∂Bu, where ⃗u∗ is the prescribed displacement value, the acceleration ¨⃗u(⃗xk, t) at boundary lattice points is determined from the integration scheme (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Although extrapolation to a non- lattice conforming boundary is also possible, we limit the discussion of Dirichlet boundary conditions to situations in which boundary lattice points lie exactly on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In this case, it is ⃗u(⃗xk, t + ∆t) = ⃗u∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' (23) Thus, (19) can be solved for the required acceleration ¨⃗u(⃗xk, t) = 2 ∆t2 (⃗u∗(t) − ⃗u(⃗xk, t)) − 2 ∆t ˙⃗u(⃗xk, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' (24) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='2 Consistent Displacement, Rotation, Dilatation and Distribution Functions at Boundary Lattice Points a) b) c) d) e) f) g) h) Figure 5: Updating the displacement u as well as the rotation ψ and dilation φ in a simple square domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The outer circles represent the state of ψ and φ at the particular lattice points, whereas the inner circles represent the state of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Yellow indicates that quantities still have a value associated with the previous time step t, whereas green color indicates that u or ψ and φ are already updated to their values at t + ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' a) the state after the previous time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' b) integration is performed at all lattice points which updates the displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' c) finite differences (stencil is indicated by the red lines) are used to update ψ and φ at the boundary lattice points in a way that is consistent with the new displacement field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' d) all boundary points are updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' e) the LBM update, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' solving the wave equations, leads to a consistent rotation and dilatation at interior lattice points (red lines indicate from which neighbors information is streamed to an interior lattice point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' f) all interior points have consistent fields after the LBM update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' g) intermediate ‘second row’ boundary points are more problematic since there is also information streamed from boundary points that does not originate from the LBE for the wave equations, but from the handling of boundary conditions (red lines indicate from which neighbors information is streamed to an interior lattice point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' h) Fields are consistent – considering the previous remarks – at all lattice points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' After the acceleration at time t is known at all lattice points, the displacement as well as the distribution functions for the next time step need to be computed in a consistent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In this context, we regard the distribution functions, and 7 A Lattice Boltzmann Method for Elastic Solids consequently ψ and φ, to be consistent with the updated displacement field if 4 � α=0 f α ψ(⃗x, t + ∆t) = ψ(⃗x, t + ∆t) ≈ (∇ × ⃗u)|(⃗x,t+∆t), 4 � α=0 f α φ (⃗x, t + ∆t) = φ(⃗x, t + ∆t) ≈ (∇ · ⃗u)|(⃗x,t+∆t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' (25) Herein, (∗)|(⃗x,t+∆t) means that the spatial derivative ∗ is performed at the lattice point ⃗x and time t + ∆t via second order accurate finite differences, which is a non-local operation that also involves neighbor lattice points to ⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 5 displays the utilized stencils for this operations as red lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The starting point for the algorithm is the situation after the previous time step has been completed as depicted for a quadratic domain in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 5 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In this figure, lattice points are represented as circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In order to illustrate the strategy of obtaining consistent displacement and distribution functions, the color of the inner circles also represents the state of the displacement field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' the not yet updated state ⃗u(∗, t) is represented by yellow and the updated state ⃗u(∗, t + ∆t) is indicated by green color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Similarly, the color of the outer circle indicates the state of the rotation ψ and dilatation φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' A yellow outer circle indicates that rotation and dilatation are not updated yet, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' the state [ψ(∗, t), φ(∗, t)], whereas a green outer circle indicates the updated state [ψ(∗, t + ∆t), φ(∗, t + ∆t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The acceleration at all lattice points is known from the Navier-Cauchy equation (8) or the boundary conditions (21) and (24), which allows to update the displacement field via (19) as a next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' This leads to an inconsistent situation where the displacement is already updated, but the rotation and the dilatation fields are not, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 5 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The rotation and dilatation fields in the interior are updated via the LBM for the wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' This works fine in the interior, where we have pointed out that the Navier-Cauchy equation and the wave equations (9) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Consequently the update of the displacement field by (8) and (19) on the one hand, and the update of the distribution functions by (12) and the derived rotation and the dilatation by and (11) on the other hand are consistent within the limits of the LBM by Chopard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' [11], see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 5 e) and f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' However, at boundary lattice points the displacement field is updated from the boundary conditions and the distribution functions can only partially be updated via the LBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Moreover, the neighbors of boundary lattice points, the ‘second row’ boundary lattice points, also cannot be in a consistent state in the sense of (25) since the finite difference approximation of ∇ × ⃗u and ∇ · ⃗u at those points depends on the displacement of boundary lattice points which in turn is determined only by the boundary conditions and not by the Navier-Cauchy equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Thus, in order to model the boundary conditions for the LBM correctly, it is necessary to accomplish two things: Setting the distribution functions at boundary lattice points consistent with (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Modifying the LBE at ‘second row’ lattice points in such a way that consistency is achieved at these points in the sense of (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The first requirement is satisfied by setting ψ(⃗xk, t + ∆t) = (∇ × ⃗u)|(⃗xk,t+∆t), φ(⃗xk, t + ∆t) = (∇ · ⃗u)|(⃗xk,t+∆t) (26) at boundary lattice points xk , see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 5 c) and d), and f 0 ψ|φ(⃗xk, t + ∆t) =a0,ψ|φ(ψ|φ)(⃗xk, t + ∆t), f α ψ|φ(⃗xk, t + ∆t) =aψ|φ(ψ|φ)(⃗xk, t + ∆t) + b⃗cα · ⃗Jψ|φ(⃗xk, t) 2c2 for α ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' (27) In order to fulfill the second requirement, we envision that all changes of ψ and φ at a boundary lattice points over one time step t → t + ∆t are transported as waves to its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Assuming that a linear change in time is a 8 A Lattice Boltzmann Method for Elastic Solids reasonable approximation, the average state of a boundary lattice point during this transition is given by the average of its distribution functions at two discrete time steps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' ˜f α(⃗xk, ˜t) = 1 2 � f α ψ|φ(⃗xk, t + ∆t) + f α ψ|φ(⃗xk, t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' (28) The transport of this intermediate state to the boundary conditions is obtained by the modified LBE f α ψ|φ (⃗x + ⃗cα∆t, t + ∆t) = ˆf α ψ|φ(⃗x, t) − 1 τ � ˆf α ψ|φ(⃗x, t) − f α eq,ψ|φ(⃗x, t) � , (29) where ˆf α ψ|φ = � ˜f α ψ|φ(⃗x, ˜t), if ⃗x is boundary lattice point f α ψ|φ(⃗x, t), otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Thus, the modification is only employed if ⃗x +⃗cα∆t is a ‘second row’ boundary lattice point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' This is not exact, but it is an approximation that leads to a reasonable state of ‘second row’ boundary lattice points, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 5 g) and h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='2 Periodic Synchronization The proposed LBM for plane strain is susceptible to instabilities if the dilatation and rotation fields ψ and φ become inconsistent with the displacements u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Since there is no inherent synchronization of these fields, rounding errors are amplified over time and eventually the computed acceleration is sufficiently misaligned with the actual displacement such that the Navier-Cauchy equation (8) is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Small inconsistencies originate from the handling of the boundary conditions as described in the last paragraphs of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In order to remedy this problem, we introduce another step in the LBM algorithm, that periodically (every l-th timestep where l ≫ 1) computes ψ and φ directly from the displacement field with a finite difference approximation of (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' As soon as ψ(⃗xk, tl) and φ(⃗xk, tl) are known, the distribution functions are also corrected according to (27) and the algorithm continues normally in the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 4 Numerical Examples cs cd = 1 √ 3 y P L t[L/cs] 1 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='005 σ0[µ] −σ0(t)ey σ0(t)ey Figure 6: A square domain subjected to a tensile load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The right plot displays the applied stress σ0(t) as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In order to demonstrate the performance of the proposed LBM, we perform several numerical experiments in which the LBM is compared to results obtained via the established Finite Element Method (FEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The experiments also demonstrate that the proposed LBM successfully solves boundary value problems that are prevalent in engineering practice and is not restricted to often rather academic types of boundary value problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In all numerical examples, we formulate the problem in terms of the ratio of wave speeds cs/cd = 1/ √ 3, the wave speed cs, the shear modulus µ, the length scale L, and the reference displacement which is also set to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The parameters of the equilibrium distribution functions are defined by (14) and (16), where a rather extreme value of a0,φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='9999 has been found to be required for sufficient stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Note that setting a0,φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='9999 also severely reduces the time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The benchmark FEM simulations are performed with bi-linear finite elements and implicit time integration via the standard Newmark method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 9 A Lattice Boltzmann Method for Elastic Solids Figure 7: Deformed heat map for a square domain subjected to a tensile load at time t = L/cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The deformation is scaled by factor 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The heat map displays the FEM benchmark results, whereas the black squares indicate the displaced positions of the lattice points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Figure 8: Displacement at the top left corner P of a square domain subjected to a tensile load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='1 Tension For the first numerical example, a square domain is subjected to a time-dependent, tensile traction t∗ = ±σ0ey load at the top and bottom edges, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The load is linearly increased from σ0(t = 0) = 0 to σ0(t = L/cs) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='005µ and held constant afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In this simulation, no periodic synchronization, see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='2, is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In order to study the performance of the LBM algorithm, we compare the LBM results to an FEM simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 7 shows a deformed heat map of the FEM results in the background, whereas black squares indicate the displaced position2 of the lattice points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Both simulations are evaluated at time t = L/cs and the deformation is scaled by a factor of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' It can be observed that the LBM matches the FEM results well and predicts phenomena such as lateral contraction accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 8 explicitly displays the displacement of the top left corner P, see also Fig 13, and confirms these findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' We can 2The displaced position of a lattice point ⃗xk is only a result of post-processing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' the lattice remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' It is computed as ˜⃗xk(t) = s⃗u(⃗xk, t) + ⃗xk, where s = 100 is the scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 10 uy [L] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='1e-04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='0002 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='0006 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='1e-04 AYX10-3 X10~3 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='8 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='4 FEM LBM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='2 0 0 0 1 2 0 1 2 t[L/cs] t[L/cs]A Lattice Boltzmann Method for Elastic Solids Figure 9: Error e for a square domain subjected to a tensile load at t ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='002L/cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The maximum error is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='3 · 10−12 which coincides with the maximum value displayed in the legend color scheme, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' dark red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' observe an expected dynamic overshoot and low frequency oscillations in both displacement components, that both the FEM and LBM simulations predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Nonetheless, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 8 also reveals that erroneous higher frequency oscillations occur in the later stages of the LBM simulation, see the green graph in the plot of uy for t > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='5L/cs, which indicate that a periodic synchronization may be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' As discussed above, we assume that inconsistencies in the sense of violations of (25) are the cause for these instabilities and that they occur primarily at the ‘second row’ boundary points, see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 5 g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In order to test this hypothesis, an error measure that is in line with (25) is defined as e(x, t) = ����� ��4 α=0 f α ψ(⃗x, t) − (∇ × ⃗u)|(⃗x,t) �4 α=0 f α φ (⃗x, t) − (∇ · ⃗u)|(⃗x,t) ������ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' (30) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 9 displays this error at time t ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='002L/cs after the corresponding time step has been completely processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' It can be observed that inconsistencies indeed occur at the ‘second row’ boundary points at the top and bottom edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Although the error is small, without periodic synchronization, it amplifies and eventually manifests as oscillations that can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='2 Simple Shear cs cd = 1 √ 3 y P L t[L/cs] 1 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='005 σ0[µ] σ0(t)ex Figure 10: A square domain subjected to a shear load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The right plot displays the applied stress σ0(t) as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The second numerical example uses the same geometric configuration, but differs in terms of the applied boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The top edge is subjected to a shear traction that is linearly increased over time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' σ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='005tµcs/L, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The bottom edge is subjected to homogeneous Dirichlet boundary conditions w(x, y = L/2, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Furthermore, the LBM simulations are run with a periodic synchronization every 50th time step and without synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 11 displays a deformed (scaled by factor 100) heat map of the FEM results in the background and the displaced lattice points as black squares of the synchronized LBM simulation in the foreground at time t = L/cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The LBM accurately captures the shear deformation as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' However, as can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 12, in this experiment it is strictly necessary to employ the synchronization step, since the LBM simulations without synchronization differ severely from the FEM benchmark after t ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='7L/cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 11 e 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='3e-12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='5e-12 3e-12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='5e-12 2e-12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='5e-12 1e-12 5e-13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='0e+00A Lattice Boltzmann Method for Elastic Solids Figure 11: Deformed heat map for a square domain subjected to a shear load at time t = L/cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The deformation is scaled by factor 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The heat map displays the FEM benchmark results, whereas the black squares indicate the displaced positions of the lattice points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' For the LBM results a periodic synchronization was performed every 50th time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Figure 12: Displacement at the top left corner P of a square domain subjected to a shear load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='3 Plate with a Circular Hole The last numerical example again considers a square domain that is subjected to a tensile traction load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In order to illustrate the LBMs capabilities to handle non-lattice conforming geometries, the domain includes a circular hole of diameter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='266L, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' As in the previous example, we run LBM simulations with periodic synchronization every 50th timestep and without any periodic synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Fig 14 displays the scaled deformed configuration for the FEM in the background, as well as for the LBM with synchronization as the black squares in the foreground at time t = L/cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Again the LBM agrees well with the FEM reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' However, this is only the case if the synchronization is utilized, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 15, as the simulation becomes unstable quickly if synchronization is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The oscillations 12 uy [L] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='1e-04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='0002 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='0006 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='1e-04X10-3 X10-3 FEM 1 1 LBM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='8 LBM syn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='6 an Wy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='2 0 0 0 1 2 0 1 2 t [L/cs] t [L/cs]A Lattice Boltzmann Method for Elastic Solids x L P y σ0[µ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='005 1 t[L/cs] Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='266L cs cd = 1 √ 3 −σ0(t)ey σ0(t)ey Figure 13: A square domain with a hole subjected to a tensile load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Point Q is located at (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='175L, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='025L) relative to a coordinate system which has its origin in the center of the hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The right plot displays the applied stress σ0(t) as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Figure 14: Deformed heat map for a square domain with a hole subjected to a tensile load at time t = L/cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The deformation is scaled by factor 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The heat map displays the FEM benchmark results, whereas the black squares indicate the displaced positions of the lattice points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' For the LBM results a periodic synchronization was performed every 50th time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' originate from the non-lattice conforming boundaries at the hole as can also be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 15: the displacement field close to the hole at point Q becomes unstable long before oscillations can be observed at point P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 5 Conclusion In this work, a new Lattice Boltzmann Method (LBM) for solving the general plane strain problems is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The plane strain problem is governed by the Navier-Cauchy equation which can be decomposed into two wave equations with different wave speeds for the rotational part of the displacement field and the dilatational part respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Based on this observation the new LBM is constructed by employing the established LBM by Chopard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' [11] to solve the two wave equations separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Chopard et al.’s approach allows enough flexibility to choose the simulated macroscopic 13 酒 uy [L] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='6e-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='0005 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='6e-03A Lattice Boltzmann Method for Elastic Solids point Q point P Figure 15: Displacement at the top left corner P (top row) and close to the hole Q (bottom row) of a square domain with a hole subjected to a tensile load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' wave speed rather independently of the time step and lattice spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Thus, the proposed method solves both wave equations on the same D2Q5 lattice with the same time discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' However, this also limits the maximum time step and reduces the computational efficiency of the approach in situations in which larger time step sizes may be feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The displacement field is eventually obtained by integrating the Navier-Cauchy equation and making use of rotation and dilatation fields computed by the LBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In order to apply Dirichlet and Neumann boundary conditions, a consistent acceleration is computed at boundary lattice points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' This is then used for the integration step mentioned above at these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In order to reconcile the displacements obtained in this way with the LBM quantities such as the rotation and dilatation as well as the distribution functions, the rotation and dilatation fields are computed from a finite difference approximation of the gradient of the displacement field at boundary lattice points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Afterwards, the distribution functions are computed consistently with these rotation and dilatation fields at the boundary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' We mention some of the remaining causes of inconsistencies between rotation and dilatation fields on the one side and the displacement field on the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' These inconsistencies manifest as instabilities in the performed simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' We address this issue by performing a periodic synchronization in which we compute the rotation and dilatation fields from a a finite difference approximation of the gradient of displacement and subsequently set the distribution functions accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Lastly, several numerical benchmarks highlight the performance of the new method compared to benchmark FEM simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The simulation of a square domain without periodic synchronization under tensile loading shows that the LBM accurately captures simple loading and domains without the synchronization step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' However, this example also reveals that the inconsistencies mentioned above indeed occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The second numerical example studies a square domain under simple shear loading conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Here, the results only accurately match the FEM simulations if the synchronization step is employed every 50th time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The third numerical example considers the square domain with a hole under tensile load and illustrates that the developed LBM is indeed capable of solving problems in which the geometry does not conform with the lattice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' the boundary does not exactly match the lattice point positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' We find the performance of the LBM in relation to the FEM promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' However, the periodic synchronization step as well as the rather fine time discretization, that is dictated by the method remains unsatisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' In future work, we envision to investigate alternative LBM approaches, but we also want to address the shortcomings of the present LBM by refining the treatment of boundary conditions and exploring the possibility of giving up the same time discretization 14 FEM ×10-4 ×10-4 LBM 10 10 LBM syn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' 5 5 y 0 0 5 5 0 1 2 0 1 2 t [L/cs] t[L/cs] ×10-4 ×10-4 10 10 5 5 Wx 0 0 5 5 0 1 2 0 1 2 t[L/cs] t [L/cs]A Lattice Boltzmann Method for Elastic Solids for both simulated wave equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' This would allow us to use larger time steps and thus increase computational efficiency, but this approach will also involve an additional interpolation step between time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' Acknowledgments Open access funding enabled and organized by Projekt DEAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' The authors gratefully acknowledge the funding by the German Research Foundation (DFG) within the project 423809639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} +page_content=' References [1] Bastien Chopard and Pascal O.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfZvdA/content/2301.00228v1.pdf'} diff --git a/rtFIT4oBgHgl3EQfyCvK/content/tmp_files/2301.11359v1.pdf.txt b/rtFIT4oBgHgl3EQfyCvK/content/tmp_files/2301.11359v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..eed208f7322e54d4abde394b8e6e82230e5915a5 --- /dev/null +++ b/rtFIT4oBgHgl3EQfyCvK/content/tmp_files/2301.11359v1.pdf.txt @@ -0,0 +1,1290 @@ +arXiv:2301.11359v1 [math.CA] 26 Jan 2023 +DISCRETE MULTILINEAR MAXIMAL OPERATORS AND PINNED SIMPLICES +NEIL LYALL +´AKOS MAGYAR +ALEX NEWMAN +PETER WOOLFITT +Abstract. We prove that any given subset of Zd of upper density δ > 0 will necessarily contain, in an +appropriate sense depending on δ, an isometric copy of all large dilates of any given non-degenerate k- +simplex, provided d ≥ 2k + 3. This provides an improvement in dimension, from d ≥ 2k + 5, on earlier work +of Magyar. We in fact establish a stronger pinned variant. Key to our approach are new ℓ2 estimates for +certain discrete multilinear maximal operators associated to simplices. These operators are generalizations +of the discrete spherical maximal operator and may be of independent interest. +1. Introduction +1.1. Simplices in dense subsets of Zd. Recall that the upper Banach density of a set A ⊆ Zd is defined +by +δ∗(A) = lim +N→∞ sup +t∈Zd +|A ∩ (t + Q(N))| +|Q(N)| +, +where | · | denotes counting measure on Zd and Q(N) the discrete cube [−N/2, N/2]d ∩ Zd. +In light of the fact that the square of the distance between any two distinct points in Zd is always a +positive integer we also introduce the convenient notation +√ +N := {λ : λ > 0 and λ2 ∈ Z}. +In [14] the second author established the following result on the existence of unpinned two point configu- +rations (distances) in dense subsets of the integer lattice. +Theorem A (Magyar [14]). Let A ⊆ Zd with d ≥ 5. If δ∗(A) > 0, then there exist an integer q = q(δ∗(A)) +and λ0 = λ0(A) such that for all λ ∈ +√ +N with λ ≥ λ0 there exist a pair of points {x, x+y} ⊆ A with |y| = qλ. +The approach taken in [14] was an adaptation of Bourgain’s in [3] to the analogous problem in the +continuous setting of Rd. In [15] the second author adapted this further to establish the following analogous +result for non-degenerate k-simplices. Recall that for any 1 ≤ k ≤ d we refer to a configuration ∆ = {v0 = +0, v1, . . . , vk} ⊆ Zd as a non-degenerate k-simplex if the vectors v1, . . . , vk are linearly independent. +Theorem B (Magyar [15]). Let k ≥ 2, A ⊆ Zd with d ≥ 2k + 5, and ∆ = {0, v1, . . . , vk} ⊆ Zd be a +non-degenerate k-simplex. If δ∗(A) > 0, there exists an integer q = q(δ∗(A)) and λ0 = λ0(A, ∆) such that +for all λ ∈ +√ +N with λ ≥ λ0 there exist x ∈ A with x + ∆′ ⊆ A for some ∆′ = {0, y1, . . . , yk} ≃ λq∆. +In the theorem above, and throughout this article, we say that two configurations λ∆ = {0, λv1, . . . , λvk} +and ∆′ = {0, y1, . . . , yk} in Zd are isometric, and write ∆′ ≃ λ∆, if |yi − yj| = λ|vi − vj| for all 0 ≤ i, j ≤ k. +In this article we establish an improvement on the dimension condition in Theorem B above from d ≥ 2k+5 +to d ≥ 2k + 3 and simultaneously establish a stronger pinned variant, namely +Theorem 1. Let k ≥ 1, A ⊆ Zd with d ≥ 2k+3, and ∆ = {0, v1, . . . , vk} ⊆ Zd be a non-degenerate k-simplex. +If δ∗(A) > 0, there exists an integer q = q(δ∗(A)) and λ0 = λ0(A, ∆) such that for any λ1 ≥ λ0 there exists +a fixed x ∈ A such that for all λ ∈ [λ0, λ1] ∩ +√ +N one has x + ∆′ ⊆ A for some ∆′ = {0, y1, . . . , yk} ≃ λq∆. +Remark. The threshold λ0 in the results above cannot be taken to depend on δ∗(A) only. +Indeed, for +any positive integers q and M the set (QqM ∩ Zd) + (4dqMZ)d will have density (4d)−d but never contain +pairs {x, x + y} with |y| = qdM. Since A could fall entirely into a fixed congruence class of some integer +1 ≤ r ≤ δ∗(A)−1/d the value of q in the results above must be divisible by the least common multiple of all +integers 1 ≤ r ≤ δ∗(A)−1/d. Indeed if A = (rZ)d with 1 ≤ r ≤ δ−1/d then A will have upper Banach density +at least δ, but the distance between any two points x, y ∈ A will always take the form rλ for some λ ∈ +√ +N. +2010 Mathematics Subject Classification. 11B30. +The first and second authors were partially supported by grants NSF-DMS 1702411 and NSF-DMS 1600840, respectively. +1 + +2 +NEIL LYALL +´AKOS MAGYAR +ALEX NEWMAN +PETER WOOLFITT +The approach in [15] also established a quantitative Szemer´edi-type variant of Theorem B, namely +Theorem B′ (Magyar [15]). Let k ≥ 2, d ≥ 2k + 5, ∆ = {0, v1, . . . , vk} ⊆ Zd be a non-degenerate k-simplex, +and 0 < δ ≤ 1. If N ≥ exp(C∆δ−Ck), then any A ⊆ {1, . . . , N}d with cardinality |A| ≥ δN d will necessarily +contain a configuration of the form x + ∆′ with ∆′ = {0, y1, . . . , yk} ≃ λq∆ for some λ ∈ +√ +N. +For an alternative approach to the proof of Theorem B′ that is more in line with the arguments in this +paper, see Section 6.1 in [11]. We note that by combining the main result of this current paper, namely +Theorem 2 below and its corollary (Lemma 2 in Section 1), with the arguments and ideas contained in +Section 6.1 of [11] one can establish an improvement on the dimension condition above from d ≥ 2k + 5 to +d ≥ 2k + 3 and also establish the analogous stronger pinned variant. However, for the sake of clarity and +brevity we have chosen not to pursue the details of these arguments or statements here and instead focus on +just establishing Theorem 1. +Let us remark that the dimension bound d ≥ 2k + 3 seems to be best possible even in case A = Zd, i.e. +when counting embeddings of isometric copies of λ∆ in Zd. Indeed, writing T to be the positive definite +integral k × k matrix with entries tij = vi · vj and Y for the k × d integral matrix with rows y1, . . . , yk the +condition ∆′ = {0, y1, . . . , yk} ≃ λ∆ translates to the matrix equation Y tY = λ2T which has been intesively +in the past [17, 16, 7]. The best known results are due to Kitaoka [7] in dimesnions d = 2k + 3, who also +mentions that this condition is best possible to count solutions to the above equation via analytic means, +see the remark after Theorem B in [7]. +For k = 1 and d = 4, it is possible to count embeddings under restrictions of λ (say when λ2 is odd) +via the so-called Kloosterman refinement [9], however for our pinned results in sets of postive density one +needs for the discrete spherical maximal function which in dimension 4 has only been obtained for particular +lacunary sequenses of the radii λ, see [6]. The case k = 1 of Theorem 1 was already established by the first +two authors in [10]. To the best of our knowledge, there have been no previous results addressing pinned +simplices in dense subsets Zd in any dimension when k ≥ 2. +1.2. Discrete multilinear maximal averages associated to simplices. An important result in the +development of discrete harmonic analysis is the ℓp-boundedness of the so-called discrete spherical maximal +function [13]. For any λ ∈ +√ +N we let Sλ = {y ∈ Zd : |y| = λ} denote the discrete sphere of radius λ centered +at the origin. For f : Zd → R we then define the discrete spherical averages +Aλf(x) = |Sλ|−1 � +y∈Sλ +f(x + y). +noting that if d ≥ 5, then cdλd−2 ≤ |Sλ| ≤ Cdλd−2 for some constants 0 < cd < Cd < ∞, see [18]. In [13] it +was shown that for p > d/(d − 2) one has the following maximal function estimate +�� sup +λ≥1 +|Aλf| +�� +p ≤ Cp,d ∥f∥p +where ∥f∥p = (� +x |f(x)|p)1/p denotes the ℓp(Zd) norm of the function f. +In [12] the authors gave a new direct proof of ℓ2-boundedness of the discrete spherical maximal function +that neither relies on abstract transference theorems nor on delicate asymptotic for the Fourier transform of +discrete spheres. Implicit in that paper is the fact that ℓ2-boundedness follows as a consequence of stronger +refined “mollified” estimates in which one obtains gains in ℓ2 over suitably large scales when applied to +functions whose Fourier transform is localized away from rational points with small denominators. +Recall that for f ∈ ℓ1(Zd) we define its Fourier transform �f : Td → C by �f(ξ) = � +x∈Zd f(x)e−2πix·ξ. For +each η > 0 we define +qη := lcm{1 ≤ q ≤ η−2} +and for any L ≥ qη we let +Ωη,L = {ξ ∈ Td : ξ ∈ [−L−1, L−1]d + (q−1 +η Z)d}. +Key to the proof of Theorem 1 is an extension of the approach from [12] to multilinear maximal operators +associated to simplices. Given a non-degenerate k-simplex ∆ = {v0 = 0, v1, . . . , vk} ⊆ Zd and λ ∈ +√ +N, we +let +Sλ∆ := {(y1, . . . , yk) ∈ Zdk : ∆′ = {0, y1, . . . , yk} ≃ λ∆}. + +DISCRETE MULTILINEAR MAXIMAL OPERATORS AND PINNED SIMPLICES +3 +For functions f1, . . . , fk : Zd → C we then define the multilinear averaging operator +Aλ∆(f1, . . . , fk)(x) = |Sλ∆|−1 +� +(y1,...,yk)∈Sλ∆ +f1(x + y1) · · · fk(x + yk) +noting that if d ≥ 2k + 3 and λ ∈ +√ +N, then +(1) +c∆ λdk−k(k+1) ≤ |Sλ∆| ≤ C∆ λdk−k(k+1) +for some constants 0 < c∆ < C∆ < ∞, see [7] or [15]. +Note that for k = 1 and v1 = (1, 0, . . ., 0) we have that Sλ∆ = Sλ and hence Aλ∆(f) = Aλ(f). +The ℓp mapping properties of the maximal operators corresponding to these averages were considered in +[2] and [4]. Here we establish the following particular ℓ2-estimates, the first non-trivial estimates of any type +for such operators in dimensions lower that d = 2k + 5 when k ≥ 2. The stronger refined ℓ2 estimate (3), in +addition to implying (2), plays a crucial role in our proof of Theorem 1. +Theorem 2. If k ≥ 1, d ≥ 2k + 3, and ∆ = {0, v1, . . . , vk} ⊆ Zd be a non-degenerate k-simplex, then +(2) +�� sup +λ≥1 +|Aλ∆(f1, . . . , fk)| +�� +2 ≤ Cd,∆∥f1∥2 · · · ∥fk∥2. +In fact, for any η > 0, and L ≥ q4 +η, we have +(3) +��� sup +λ≥η−2L +|Aλ∆(f1, . . . , fk)| +��� +2 ≤ Cd,∆ +η +log η−1 ∥f1∥2 · · · ∥fk∥2 +whenever supp �fj ⊆ Ωc +η,L for some 1 ≤ j ≤ k, where Ωc +η,L denotes the complement of Ωη,L. +Estimate (3) in the case k = 1 was originally established in joint work with Magyar [10] via an adaptation +of the transference methods from [13]. +2. Proof of Theorem 1 +2.1. Reduction to uniform distributed sets. In light of the observation made after Theorem 1 above +regarding the sensitivity of this problem to the local structure of A, it is natural to first consider the case +when A is, in a suitable sense, well distributed in small congruence classes. In fact, this approach ultimately +leads directly to a proof of Theorem 1. +Following [10] we define A ⊆ Zd to be η-uniformly distributed (modulo qη) if, for some η > 0, its relative +upper Banach density on any residue class modulo qη never exceeds (1 + η4) times its density on Zd, namely +if +δ∗(A | s + (qηZ)d) ≤ (1 + η4) δ∗(A) +for all s ∈ {1, . . . , qη}d. A straightforward density increment argument allows one to deduce Theorem 1 from +the following analogue for η-uniformly distributed subsets of Zd. +Proposition 1. Let ε > 0, 0 < η ≪ ε2 and k ≥ 1. +If A ⊆ Zd with d ≥ 2k + 3 is η-uniformly distributed, and ∆ = {0, v1, . . . , vk} ⊆ Zd is a non-degenerate +k-simplex, then there exist λ0 = λ0(A, ∆, η) such that for any λ1 ≥ λ0 there exists a fixed x ∈ A such that +Aλ∆(1A, . . . , 1A)(x) > δ∗(A)k − ε +for all λ ∈ [λ0, λ1] ∩ +√ +N, noting that +Aλ∆(1A, . . . , 1A)(x) = |Sλ∆|−1 |{(y1, . . . , yk) ∈ Zdk : x + ∆′ ⊆ A with ∆′ = {0, y1, . . . , yk} ≃ λ∆}|. +In Proposition 1 above, and throughout this article, we use the notation α ≪ β to denote that α ≤ cβ for +some suitably small constant c > 0. +Proposition 1 in fact implies the following stronger optimal formulation of Theorem 1. +Corollary 1. Let k ≥ 1, A ⊆ Zd with d ≥ 2k + 3, and ∆ = {0, v1, . . . , vk} ⊆ Zd be a non-degenerate +k-simplex. For any ε > 0, there exists an integer q = q(ε, d) and λ0(A, ∆, ε) such that for any λ1 ≥ λ0 there +exists a fixed x such that +(4) +|Sλ∆|−1 |{(y1, . . . , yk) ∈ (qZ)dk : x + ∆′ ⊆ A with ∆′ = {0, y1, . . . , yk} ≃ λq∆}| > δ∗(A)k − ε +for all λ ∈ [λ0, λ1] ∩ +√ +N. + +4 +NEIL LYALL +´AKOS MAGYAR +ALEX NEWMAN +PETER WOOLFITT +Remark. By considering sets A of the form � +s∈{1,...,q}d As with each set As a “random” subset of the +congruence class s + (qZ)d one can further easily see that conclusion (4) above is in general best possible. +Proof that Proposition 1 implies Corollary 1. Let 0 < ε ≤ δ ≤ 1 and A ⊆ Zd with d ≥ 2k + 3. To prove +Corollary 1 it is enough to prove that if δ∗(A) ≥ δ then there exists λ0 = λ0(A, ∆, ε) and q = q(ε, d) such +that for any λ1 ≥ λ0 there exists a fixed x ∈ A such that (4) holds for all λ ∈ +√ +N with λ0 ≤ λ ≤ λ1. +Let 0 < η ≪ ε2. We prove the above for δm := (1 + η4)−m inductively for all m ≥ 0, using Proposition +1. +For m = 0 the statement is trivial as δ∗(A) = δ0 = 1 and hence A contains arbitrarily large cubic +grids. +Suppose it holds for δ = δm and assume that δ∗(A) ≥ δm+1. +If A is η-uniformly distributed +then the result holds for δ = δm+1 by Proposition 1. +In the opposite case there is an s ∈ Zd so that +δ∗(A | s+ (qηZ)d) > (1 + η4) δ . Let φ : s+ (qηZ)d → Zd be defined by φ(x) := q−1 +η (x− s) and let A′ := φ(A). +Then δ∗(A′) ≥ δm thus (4) holds for A′ and δ = δm, with some q′ = q′(ε, d) and x′ ∈ A′. Note that +|{(y1, . . . , yk) ∈ (q′Z)dk : x′ + ∆′ ⊆ A′ with ∆′ = {0, y1, . . . , yk} ≃ q′λ∆}| += |{(y1, . . . , yk) ∈ (qηq′Z)dk : qηx′ + ∆′ ⊆ A′ with ∆′ = {0, y1, . . . , yk} ≃ qηq′λ∆}| +which implies that (4) holds for A, δ = δm+1 with q = qηq′ and x = qηx′ + s. +□ +2.2. Proof of Proposition 1. Before proving proving Proposition 1 we need two preparatory lemmas. +We refer to a subset Q ⊆ Zd as a cube of sidelength ℓ(Q) = N if +Q = t0 + QN +for some t0 ∈ Zd, where as usual QN = [−N/2, N/2]d. +Definition (U 1 +q,L(Q)-norm). For any cube Q ⊆ Zd, integers 1 ≪ q ≪ L ≪ ℓ(Q), and functions f : Q → R +we define +(5) +∥f∥U1 +q,L(Q) = +� 1 +|Q| +� +t∈Zd +|f ∗ χq,L(t)|2�1/2 +where χq,L denotes the normalized characteristic function of the cubes Qq,L := QL ∩ (qZ)d, namely +(6) +χq,L(x) = +�� q +L +�d +if x ∈ (qZ)d ∩ [− L +2 , L +2 ]d +0 +otherwise +. +In (5) above and in the sequel we denote the convolution f ∗ g of two functions f and g by +f ∗ g(x) := +� +y∈Zd +f(x − y)g(y). +We note that the U 1 +q,L(Q)-norm measures the mean square oscillation of a function with respect to cubic +grids of size L and gap q. The first key ingredient in our proof of Proposition 1 is the simple, yet significant, +observation from [10] that subsets of Zd with positive upper Banach density that are η-uniformly distributed +are also, in a precise sense, uniformly distributed at certain scales. +Lemma 1 (Consequence of Lemmas 1 and 2 in [10]). Let η > 0 and A ⊆ Zd be η-uniformly distributed with +δ := δ∗(A) > 0. There exists a positive integer L = L(A, η) and cubes Q ⊆ Zd of arbitrarily large sidelength +ℓ(Q) with ℓ(Q) ≥ η−4L such that +(7) +|A ∩ Q| ≥ (δ − O(η))|Q| +and +(8) +∥(1A − δ)1Q∥U1 +qη,L(Q) = O(η). +The second key ingredient in our proof of Proposition 1 is the following maximal variant of a so-called +generalized von-Neumann-type inequality, which follows in a straightforward manner from Theorem 2. +Lemma 2 (Corollary of Theorem 2). Let k ≥ 1, d ≥ 2k + 3, and ∆ = {0, v1, . . . , vk} ⊆ Zd be a non- +degenerate k-simplex. For any η > 0, positive integer L, cube Q ⊆ Zd with sidelength N ≥ η−6L, and +functions f1, . . . , fk : Q → [−1, 1] we have +1 +|Q| +� +x∈Zd +sup +η−3L≤λ≤η3N +��Aλ∆(f1, . . . , fk)(x) +�� ≤ Cd,∆ +� +min +1≤j≤k ∥fj∥U1 +qη,L(Q) + O(η) +� +. + +DISCRETE MULTILINEAR MAXIMAL OPERATORS AND PINNED SIMPLICES +5 +Proof. By Cauchy-Schwarz, it suffices to prove the stronger estimate +� 1 +|Q| +� +x∈Zd +sup +η−3L≤λ≤η3N +��Aλ∆(f1, . . . , fk)(x) +��2 +�1/2 +≤ Cd,∆ +� +min +1≤j≤k ∥fj∥U1 +qη,L(Q) + O(η) +� +. +This follows from Theorem 2 by symmetry and sublinearity after decomposing fk = fk,1 +fk,2 +fk,3 with +fk,1 = fk ∗ χqη,L +where fk,2 and fk,3 satisfy +� +fk,2 = �fk 1Ωη,η−1L(1 − � +χqη,L) +and +� +fk,3 = �fk 1Ωc +η,η−1L(1 − � +χqη,L). +Indeed, estimate (2) implies that +� 1 +|Q| +� +x∈Zd +sup +λ≥1 +��Aλ∆(f1, . . . , fk−1, g)(x) +��2 +�1/2 +≤ Cd,∆ +� 1 +|Q| +� +x∈Zd +|g(x)|2 +�1/2 +for any g : Zd → C. Note that if g = fk,1 then +� 1 +|Q| +� +x∈Zd +|fk,1(x)|2 +�1/2 += ∥fk∥U1 +qη,L(Q). +In light of the fact that +�χqη,L(ξ) = qd +η +Ld +� +x∈[− L +2 , L +2 )d, qη|x +e−2πix·ξ +it is easy to see that �χq,L(ℓ/q) = 1 for all ℓ ∈ Zd and that there exists some absolute constant C > 0 such +that +(9) +0 ≤ 1 − �χqη,L(ξ) ≤ C L |ξ − ℓ/qη| +for all ξ ∈ Td and ℓ ∈ Zd, and hence that 1 − � +χqη,L(ξ) = O(η) for all ξ ∈ Ωη,η−1L. It thus follows, by +Plancherel, that +� 1 +|Q| +� +x∈Zd +|fk,2(x)|2 +�1/2 += O(η). +Finally, since supp � +fk,3 ⊆ Ωc +η,η−1L, it follows from estimate (3) that +� 1 +|Q| +� +x∈Zd +sup +η−3L≤λ≤η3N +��Aλ(f1, . . . , fk−1, fk,3)(x) +��2 +�1/2 +≤ Cd,∆ η. +□ +Proof of Proposition 1. Let 0 < ε ≤ δ ≤ 1 and 0 < η ≪ ε2. +Suppose there exists a set A ⊆ Zd with d ≥ 2k + 3 with δ = δ∗(A) > 0 that is η-uniformly distributed but +for which the conclusion of Proposition 1 fails, namely that there exists arbitrarily large pairs (λ0, λ1) such +that for every x ∈ A one has +Aλ∆(1A, . . . , 1A)(x) ≤ δk − ε +for some λ ∈ [λ0, λ1] ∩ +√ +N. +Combining this with Lemma 1 we can conclude that there exists a positive integer L and a cube Q ∈ Zd +with sidelength N sufficiently large so that in addition to the properties (7) and (8) we also have the property +that +Aλ∆(1A∩Q, . . . , 1A∩Q)(x) ≤ δk − ε +for every x ∈ A for some λ ∈ [η−3L, η3N] ∩ +√ +N. +We now let A′ := A ∩ Q′, where Q′ denotes the cube of sidelength (1 − η3)N with the same center as Q. +It then follows, provided that N was chosen sufficiently large, that +Aλ∆(1Q, δ1Q, . . . , δ1Q)(x) = δk + +6 +NEIL LYALL +´AKOS MAGYAR +ALEX NEWMAN +PETER WOOLFITT +for every x ∈ A′ and hence that for each such x one has +k−1 +� +j=0 +Aλ∆(1A∩Q, . . . 1A∩Q +� +�� +� +j copies +, (1A − δ)1Q, δ1Q, . . . , δ1Q)(x) ≤ −ε +for some λ ∈ [η−3L, η3N] ∩ +√ +N. Consequently, we have that +(10) +k−1 +� +j=0 +sup +η−3L≤λ≤η3N +����Aλ(1A∩Q, . . . 1A∩Q +� +�� +� +j copies +, (1A − δ)1Q, δ1Q, . . . , δ1Q)(x) +���� ≥ ε +for every x ∈ A′. +Since η ≪ δ and |A′| ≥ |A∩Q|−η3|Q| it follows from (7) that |A′|/|Q| ≥ δ/2. Combining this observation +with (10) we obtain +(11) +k−1 +� +j=0 +1 +|Q| +� +x∈Zd +sup +η−3L≤λ≤η3N +����Aλ(1A∩Q, . . . 1A∩Q +� +�� +� +j copies +, (1A − δ)1Q, δ1Q, . . . , δ1Q)(x) +���� ≥ εδ/2. +However, Lemma 2 and (8) clearly imply that for each 0 ≤ j ≤ k − 1 one has +1 +|Q| +� +x∈Zd +sup +η−3L≤λ≤η3N +����Aλ(1A∩Q, . . . 1A∩Q +� +�� +� +j copies +, (1A − δ)1Q, δ1Q, . . . , δ1Q)(x) +���� = O(η) +which leads to a contradiction if η is chosen sufficiently small with respect to ε2. +□ +3. Proof of Theorem 2 +Following the approach in [12] we will deduce Theorem 2 from refined estimates for our maximal operators +at a single dyadic scale, namely Proposition 2 below. We first need to introduce some notation closely related +to that in Section 1.2. For any integer j ≥ 0 we let +qj = lcm{1, 2, . . ., 2j} +noting that qj ≍ e2j, and for any non-negative integers j and l that satisfy 2j ≤ l , we let +(12) +Ωj,l := {ξ ∈ Td : ξ ∈ [−2j−l, 2j−l]d + (q−1 +j Z)d}. +Proposition 2. If k ≥ 1, d ≥ 2k + 3, and ∆ = {0, v1, . . . , vk} ⊆ Zd be a non-degenerate k-simplex, then +(13) +�� +sup +2l≤λ≤2l+1 |Aλ∆(f1, . . . , fk)| +�� +2 ≤ Cd,∆ 2−j/2j−1 ∥f1∥2 · · · ∥fk∥2 +whenever supp �fi ⊆ Ωc +j,l for some 1 ≤ i ≤ k, where Ωc +j,l denotes the complement of Ωj,l. +It is easy to see that Proposition 2 is equivalent to estimate (3) of Theorem 2. Indeed, note that in +proving (3) one may restrict the sup to η−2L ≤ λ ≤ 2η−2L. Choosing l, j ∈ N such that 2l ≤ η−2L ≤ 2l+1 +and 2j ≥ η−2 we have that 2l−j ≤ L and hence Ωj,l ⊆ Ωη,L. Applying Proposition 2 with j and l chosen +as above implies estimate (3) of Theorem 2, while applying estimate (3) of Theorem 2 with L = 2l−j and +η = 2−j/2 immediately implies Proposition 2. +We are left with establishing that Proposition 2 implies estimate (2) of Theorem 2. Following the approach +in [12] we start by introducing a smooth sampling function supported on Ωj,l. +3.1. A smooth sampling function supported on Ωj,l. Let ψ ∈ S(Rd) be a Schwartz function satisfying +1Q(ξ) ≤ �ψ(ξ) ≤ 12Q(ξ) +where Q = [−1/2, 1/2]d and +�ψ(ξ) := +ˆ +Rd ψ(x)e−2πix·ξdx + +DISCRETE MULTILINEAR MAXIMAL OPERATORS AND PINNED SIMPLICES +7 +denote the Fourier transform of ψ on Rd. For a given q ∈ N and L > q we define ψq,L : Zd → R as +ψq,L(x) = +�� q +L +�d ψ +� m +L +� +if x ∈ (qZ)d +0 +otherwise +Writing x = qr + s with r ∈ Zd and s ∈ Zd/qZd, it follows from Poisson summation that +�ψq,L(ξ) = +� +x∈Zd +ψ(x)e−2πix·ξ +is a q−1-periodic function on Td that satisfies +�ψq,L(ξ) = +� +ℓ∈Zd +�ψ(L(ξ − ℓ/q)). +For a given l ∈ N and 0 ≤ j ≤ Jl := [log2(l)] − 2, we now define the sampling function +(14) +Ψl,j = ψqj,2l−j +and note that supp �Ψl,j ⊆ Ωj,l. +Finally we define ∆Ψl,j = Ψl,j+1 − Ψl,j and note the important almost orthogonality property they enjoy. +Lemma 3 (Lemma 1 in [12]). There exists a constant C = CΨ > 0 such that +� +l≥2j +|� +∆Ψl,j(ξ)|2 ≤ C +uniformly in j ∈ N and ξ ∈ Td. +3.2. Proof that Proposition 2 implies estimate (2) of Theorem 2. Let k ≥ 1, d ≥ 2k + 3, and +∆ = {0, v1, . . . , vk} ⊆ Zd be a non-degenerate k-simplex. In [12] the authors gave a direct proof of estimate +(2) of Theorem 2 when k = 1, the ℓ2-boundedness of the discrete spherical maximal function. We may thus, +without loss in generality assume that k ≥ 2, supp �fk ⊆ Ωc +j,l, and that +(15) +�� sup +λ≥1 +|Aλ�∆(f1, . . . , fk−1)| +�� +2 ≤ Cd,�∆∥f1∥2 · · · ∥fk−1∥2 +where �∆ = {0, v1, . . . , vk−1} ⊆ Zd. +Let +(16) +Ml(f1, . . . , fk) := +sup +2l≤λ≤2l+1 |Aλ∆(f1, . . . , fk)|. +Writing +fk = fk ∗ Ψl,0 + +Jl−1 +� +j=0 +fk ∗ ∆Ψl,j + (fk − fk ∗ Ψl,Jl) +it follows by subadditivity that +(17) +Ml(f1, . . . , fk) ≤ Ml(f1, . . . , fk ∗ Ψl,0) + +Jl−1 +� +j=0 +Ml(f1, . . . , fk ∗ ∆Ψl,j) + Ml(f1, . . . , fk − fk ∗ Ψl,Jl). +Estimate (2) of Theorem 2 will now follow from a few observations and applications of Proposition 2, in +light of the fact that +sup +λ≥1 +|Aλ∆(f1, . . . , fk)| = sup +l +Ml(f1, . . . , fk). +We first observe that the first term on the right in (17) above satisfies +Ml(f1, . . . , fk ∗ Ψl,0) ≤ CΨH(fk) sup +λ≥1 +|Aλ�∆(f1, . . . , fk−1)| +uniformly in l, where +H(f)(x) = sup +N>0 +1 +|QN| +��� +� +y∈QN +f(x − y) +��� + +8 +NEIL LYALL +´AKOS MAGYAR +ALEX NEWMAN +PETER WOOLFITT +with Q(N) the discrete cube [−N/2, N/2]d ∩ Zd denotes the discrete Hardy-Littlewood maximal operator, +which trivially satisfies ∥Hf∥∞ ≤ ∥f∥∞ ≤ ∥f∥2 by the nesting of discrete ℓp spaces. It therefore follows +from the inductive hypothesis (15) that +sup +l +∥Ml(f1, . . . , fk ∗ Ψl,0)∥2 ≤ C∥f1∥2 · · · ∥fk∥2. +For the middle terms in (17) we first note that +sup +l +Jl−1 +� +j=0 +Ml(f1, . . . , fk ∗ ∆Ψl,j) ≪ +� ∞ +� +l=0 +��� +Jl−1 +� +j=0 +Ml(f1, . . . , fk ∗ ∆Ψl,j) +��� +2�1/2 +. +Taking ℓ2 norms of both sides of the inequality above and applying Minkowski’s inequality, followed by +an application of Proposition 2, gives +���sup +l +� +0≤j≤Jl +Ml(f1, . . . , fk ∗ ∆Ψl,j) +��� +2 ≤ +� +j +�� +l≥2j +∥Ml(f1, . . . , fk ∗ ∆Ψl,j)∥2 +2 +�1/2 +≤ C∥f1∥2 · · · ∥fk−1∥2 +� +j +2−j/2�� +l≥2j +∥fk ∗ ∆Ψl,j∥2 +2 +�1/2 +≤ C∥f1∥2 · · · ∥fk∥2 +where the last inequality above follows from Lemma 3. +One more application of Proposition 2 with j = [log2 l] − 2 to the last term in (17) gives +���sup +l +Ml(f1, . . . , fk − fk ∗ Ψl,Jl) +��� +2 ≤ +� ∞ +� +l=1 +∥Ml(f1, . . . , fk − fk ∗ Ψl,Jl)∥2 +2 +�1/2 +≤ C +� ∞ +� +l=1 +l−1(log2 l)−2�1/2 +∥f1∥2 · · · ∥fk∥2 +≤ C∥f1∥2 · · · ∥fk∥2. +□ +4. Proof of Proposition 2 +Given any simplex ∆ = {v0 = 0, v1, . . . , vk} ⊆ Rd, we introduce the associated inner product matrix +T = T∆ = (tij)1≤i,j≤k with entries tij := vi ·vj, where “·” stands for the dot product in Rd. Note that T is a +positive semi-definite matrix with integer entries and T is positive definite if and only if ∆ is non-degenerate. +It is easy to see that ∆′ ≃ λ∆, with ∆′ = {y0 = 0, y1, . . . , yk}, if and only if +(18) +yi · yj = λ2tij +for all +1 ≤ i, j ≤ k. +If we let M ∈ Zd×k be a matrix with column vectors y1, . . . , yk ∈ Zd, then the system of equations above +can be written as the matrix equation +(19) +M tM = λ2T, +where M t is the transpose of the matrix M. It therefore follows that +Aλ∆(f1, . . . , fk)(x) = |Sλ∆|−1 +� +y1,...,yk∈Zd +f1(x + y1) · · · fk(x + yk)Sλ2T (M) +if we use Sλ2T (M) to denote the indicator function of relation (19). +Let Ik = [0, 2]k(k+1)/2 denote the space of symmetric k × k matrices with entries in the interval [0, 2]. +Using the fact that +tr(XtY ) = tr(Y Xt) = +k +� +i=1 +k +� +j=1 +xijyij, +for any k × k matrices X = (xij), Y = (yij), one has +(20) +Sλ2T (M) = 2−k +ˆ +Ik +eπi tr[(MtM−λ2T )X] dX + +DISCRETE MULTILINEAR MAXIMAL OPERATORS AND PINNED SIMPLICES +9 +where dX = � +1≤i≤j≤k dxij. Moreover, if M tM = λ2T then +tr(T −1M tM) = tr(MT −1M t) = tr(λ2I) = kλ2. +Given l ∈ N write Λ = 2l and ε = 2−2l. We have +(21) +Sλ2T (M) = 2−kekελ2 ˆ +Ik +e−πiλ2 tr(T X) eπi tr(M(X+iεT −1)Mt)dX. +Let +GX,ε(M) = GX,ε(y1, . . . , yk) = eπi tr(M(X+iεT −1)Mt) +be the Gaussian function, where y1, . . . , yk ∈ Zd are the column vectors of the matrix M, and define the +corresponding multi-linear operator +(22) +BX,ε(f1, . . . , fk)(x) := +� +y1,...,yk∈Zd +f1(x + y1) . . . fk(x + yk) GX,ε(y1, . . . , yk). +It follows that +Aλ(f1, . . . , fk)(x) = 2−kekελ2|Sλ∆|−1 +ˆ +Ik +e−πiλ2 tr(T X) BX,ε(f1, . . . , fk)(x) dX. +Thus for the maximal function +Ml(f1, . . . , fk) := +sup +2l≤λ≤2l+1 |Aλ∆(f1, . . . , fk)| +we have the pointwise estimate +(23) +Ml(f1, . . . , fk)(x) ≤ Cd,∆ Λ−k(d−k−1) +ˆ +Ik +|BX,ε(f1, . . . , fk)(x)| dX, +as ε = Λ−2 = 2−2l and Λ ≤ λ ≤ 2Λ. Finally, by Minkowski’s inequality +(24) +∥Ml(f1, . . . , fk)∥2 ≤ Cd,∆ Λ−k(d−k−1) +ˆ +Ik +∥BX,ε(f1, . . . , fk)(x)∥2 dX. +Taking the Fourier transform of the expression in (22) we obtain � +BX,ε(f1, . . . , fk)(ξ) equals +(25) +ˆ +Tk−1 +�f1(ξ1) · · · �fk−1(ξk−1) �fk(ξ − ξ1 − · · · − ξk−1) � +GX,ε(ξ1, . . . , ξk−1, ξ − ξ1 − · · · − ξk−1) dξ1 · · · dξk−1. +Thus by the Cauchy-Schwarz inequality and Plancherel’s indentity, one has +(26) +∥BX,ε(f1, . . . , fk)∥2 +2 ≤ ∥� +GX,ε∥2 +∞ +k +� +i=1 +∥fi∥2 +2. +Thus, the ℓ2 × · · · × ℓ2 → ℓ2 boundedness of the dyadic maximal operator Ml(f1, . . . , fk) follows from the +estimate +(27) +ˆ +Ik +∥� +GX,ε∥∞ dX ≤ Cd,∆ Λk(d−k−1) +with Λ = 2l. For the mollified estimate assume that, supp �fi ⊆ Ωc +j,l, i.e. �fi = 1Ωc +j,l �fi for some 1 ≤ i ≤ k. By +symmetry of the expression in (22) we may assume without loss of generality that i = 1. In this case in equal- +ity (25) the function � +GX,ε(ξ1, . . . , ξk−1, ξ −ξ1 −· · ·−ξk−1) can be replaced by 1Ωc +j,l(ξ1) � +GX,ε(ξ1, . . . , ξk−1, ξ − +ξ1 − · · · − ξk−1), thus to prove Theorem 2, it is enough to show that for j, l ∈ N with 2j+2 ≤ l, one has +(28) +ˆ +Ik +∥1Ωc +j,l(ξ1) � +GX,ε(ξ1, . . . , ξk)∥∞ dX ≤ Cd,∆ 2−j/2j−1 Λk(d−k−1) +with Λ = 2l. + +10 +NEIL LYALL +´AKOS MAGYAR +ALEX NEWMAN +PETER WOOLFITT +5. Estimates for theta functions on the Siegel upper half space. +To prove estimates (27) and (28) we will follow the approach given in Section 5 of [15]. For the sake of +completeness we recall below some of the basic notions and constructs. If M = [m1, . . . , mk] ∈ Zd×k and +X = [ξ1, . . . , ξk] ∈ Rd×k are d×k matrices then one has that tr(M tX) = m1 ·ξ1 +. . .+mk ·ξk where · denotes +the usual dot product. Thus the Fourier transform of a function f(m1, . . . , mk) = f(M) may written +�f(X) = �f(ξ1, . . . , ξk) = +� +M∈Zd×k +f(M)e−2πi tr(MtX ). +This implies that +(29) +�GX,ε(X) = +� +M∈Zd×k +eπi tr[(M(X+iεT −1)Mt−2MtX ] = θd,k(X + iεT −1, −X, 0) +is the theta-function θd,k : Hk × Rd×k × Rd×k → C defined by +(30) +θd,k(Z, X, E) = +� +M∈Zd×k +eπi tr[(M−E)Z(M−E)t+2MtX −EtX ] +for Z = X + iY ∈ Hk, Hk being the Siegel upper space, see (5.1)-(5.3) in [15]. +We partition the range of integration Ik and estimating the theta function separately on each part by +exploiting its transformation properties. This may be viewed as the extension of the classical Farey arcs +decomposition to k > 1. Recall the integral symplectic group +(31) +Γk = +� +γ = +� A +B +C +D +� +; ABt = BAt, CDt = DCt, ADt − BCt = Ek, +� +which acts on the Siegel upper-half space Hk = {Z = X + iY : X ∈ Mk, Y ∈ Pk} as a group of analytic +automorphisms; The action being defined by: γ⟨Z⟩ = (AZ + B)(CZ + D)−1 for γ ∈ Γk, Z ∈ Hk, see [15] +and also [7]. Let us recall also the subgroup of integral modular substitutions: +(32) +Γk,∞ = +� +γ = +� +A +B +0 +D +� +; ABt = BAt, ADt = Ek +� +Writing U = At and S = ABt, it is easy to see that D = U −1 and B = SU −1, moreover S is symmetric +and U ∈ GL(k, Z), i.e. det(U) = ±1. The action of such γ ∈ Γk,∞ on Z ∈ Hk takes the form: +(33) +γ⟨Z⟩ = Z[U] + S +using the notation Z[U] = U tZU. The general linear group GL(k, Z) acts on the space Pk of positive k × k +matrices, via the action: Y → Y [U], Y ∈ Pk, and let Rk denote the corresponding so-called Minkowski +domain, see Definition 1 on p12 of [8]. A matrix Y = (yij) ∈ Rk is called reduced. We recall that for a +reduced matrix Y +(34) +Y ≈ YD , +y11 ≤ y22 ≤ · · · ≤ ykk +where YD = diag(y11, . . . , ykk) denotes the diagonal part of Y , and A ≈ B means that A − ckB > 0, +B − ckA > 0 for some constant ck > 0. For a proof of these facts, see Lemma 2 on p20 in [8]. A fundamental +domain Dk for the action of Γk on Hk, called the Siegel domain, consists of all matrices Z = X + iY , +(X = (xij)), satisfying +(35) +Y ∈ Rk, +|xij| ≤ 1/2, +| det (CZ + D)| ≥ 1, +∀ γ = +� +A +B +C +D +� +∈ Γk. +The second rows of the matrices γ ∈ Γk are parameterized by the so-called coprime symmetric pairs of +integral matrices (C, D), which means that CDt is symmetric and the matrices GC and GD with a matrix G +of order k are both integral only if G is integral, see Lemma 2.1.17 in [1]. It is clear from definition (5.6) that +if γ2 = γγ1 with second rows (C2, D2) and (C1, D1) for some γ ∈ Γk,∞, then (C2, D2) = (UC1, UD1) for some +U ∈ GL(k, Z). On the other hand, if both γ1 and γ2 have the same second row (C, D) then γ2γ−1 +1 +∈ Γk,∞. +This gives the parametrization of the group Γk,∞\Γk by equivalence classes of coprime symmetric pairs +(C, D) via the equivalence relation (C2, D2) ∼ (C1, D1) if (C2, D2) = (UC1, UD1) for some U ∈ GL(k, Z), +see also p.54 in [1]. We will use the notation [γ] = [C, D] ∈ Γk,∞\Γk. + +DISCRETE MULTILINEAR MAXIMAL OPERATORS AND PINNED SIMPLICES +11 +If one defines the domain: Fk = ∪γ∈Γk,∞γDk, then Hk = � +[γ]∈Γk,∞\Γk γ−1Fk is a non-overlapping cover +of the Siegel upper half-plane. Correspondingly, for a given matrix T > 0 of order k, define the Farey arc +dissection of level T , as the cover +(36) +Ik = +� +[γ]∈Γk,∞\Γk +IT [γ], +IT [γ] = {X ∈ Ik : X + iT −1 ∈ γ−1Fk} +We recall the basic estimates (5.14)-(5.16) in [15] whose proofs are based on the transformation property +|θd,k(Z, X, 0)| = | det (CZ + D)|− d +2 |θd,k(γ⟨Z⟩, XAt − Kγ/2, XCt − Nγ/2)| +for some matrices Kγ, Nγ ∈ Zn×k, see Proposition 5.2 in [15]. Namely, if (C, D) is a coprime symmetric pair, +then for Z ∈ IT [C, D] one has +(37) +|θd,k(Z, X, 0)| ≤ Cd,k | det (CZ + D)|− d +2 +uniformly for X ∈ Mk(R). +Next we describe the “mollified” estimate (5.16) in [15] in slightly different form. For q ∈ N and τ > 0 +define the region +(38) +Ωq,τ = {X ∈ Rd×k : |X − P/2q| ≤ τ for some P ∈ Zd×k}. +If [γ] = [C, D] coprime symmetric pair, q := |det(C)| > 0, then for Z ∈ IT [C, D] +(39) +|θd,k(Z, X, 0)| ≲ | det (CZ + D)|− d +2 +� +e−c min(Y ) + e−c τ 2µ(CtY C)� +uniformly for X ∈ Ωc +q,τ. Here Y = Imγ⟨Z⟩, min(Y ) = minx∈Zd,x̸=0 |Y x·x| and µ(Y ) = minx∈Rd, |x|=1 |Y x·x|. +Define, similarly as in (5.20) in [15] +(40) +JT [C, D] = +ˆ +IT [C,D] +sup +X +|θd,k(X + iT −1, −X, 0)| dX. +By (37) we have that +(41) +JT [C, D] ≤ Cd,k J0 +T [C, D], +where +(42) +J0 +T [C, D] = +ˆ +X∈IT [C,D] +| det(CZ + D)|− d +2 dX. +If q := | det(C)| > 0, then for τ > 0 let +(43) +JT,τ[C, D] := +ˆ +IT [C,D] +sup +X +1Ωcτ,q(X) |θd,k(X + iT −1, −X, 0)| dX. +By estimate (39) one has +(44) +JT,τ[C, D] ≤ Cd,k J1 +T [C, D] + J2 +T,τ[C, D], +where +(45) +J1 +T [C, D] = +ˆ +IT [C,D] +| det(CZ + D)|− d +2 e−c min(Y ) dX +(46) +J2 +T,τ[C, D] = +ˆ +IT [C,D] +| det(CZ + D)|− d +2 e−cτ 2 µ(CtY C) dX. +where Y = Im γ⟨Z⟩ and γ ∈ Γk such that [γ] = [C, D] ∈ Γk,∞\Γ. +Then by inequalities (5.24)-(5.26) given in Propositions 5.3-5.4 in [15], we have +(47) +� +St=S +JT [C, D + CS] ≤ Cd,k det(T ) +d−k−1 +2 +| det(C)|− d +2 and +(48) +� +St=S +JT,τ[C, D + CS] ≤ Cd,k det(T ) +d−k−1 +2 +� +| det(C)|−k min(T )− d−2k +4 ++ | det(C)|− d +2 (τ 2µ(T ))− d−2k +4 +� +where the summation is over all symmetric integral matrices S ∈ Mk(Z). + +12 +NEIL LYALL +´AKOS MAGYAR +ALEX NEWMAN +PETER WOOLFITT +Recall that the map [C, D] → C−1D provides a one-one and onto correspondence between the classes of +coprime symmetric pairs [C, D] ∈ Γk,∞\Γk, with det(C) ̸= 0, and symmetric rational matrices R of order k, +and the pairs [C, D +CS] correspond to the matrices R+S with symmetric S ∈ Zk×k. Let us write Q(1)k×k +for the space of modulo 1 incongruent symmetric rational matrices, where Q(1) = Q/Z, Q being the set of +rational numbers. If R = C−1D, for a coprime symmetric pair [C, D] then will write +(49) +JT [R] := +� +St=S +JT [C, D + CS], +(50) +JT,τ[R] := +� +St=S +JT,τ[C, D + CS], +which is well-defined as it only depends on the equivalence class [R] ∈ Q(1)k×k. Finally write d(R) = | det(C)| +for R = C−1D. Then by (30) and (40), we have with ε = Λ−2 that +(51)ˆ +Ik +sup +X +|θd,k(X + iεT −1, −X, 0)| dX = +� +[C,D],det(C)̸=0 +JΛ2T [C, D] + +� +[C,D],det(C)=0 +JΛ2T [C, D] =: +� +1 ++ +� +2 +. +An estimate for the second sum is given in Corollary 5.1 in [15], namely it is shown that +(52) +� +2 +≤ Cd,k |Λ2T |(k−1)(d−k)/2 ≤ Cd,kΛ(d−k)(k−1) +where |T | = (� +ij t2 +ij)1/2 is the Euclidean norm of the matrix T . For the first sum we use estimate (47) for +the matrix Λ2T , which implies +(53) +� +1 += +� +[R]∈Q(1)k×k +JΛ2T [R] ≤ Cd,k Λk(d−k−1) +� +[R]∈Q(1)k×k +d(R)−d/2. +Recall the following estimate, proved in Lemma 1.4.9 in [7]; for u ≥ 1 and s > 1 one has +(54) +u−s +� +1≤d(R)≤u +d(R)−k + +� +d(R)≥u +d(R)−k−s ≤ C(2 + +1 +s − 1) u1−s +where the summation is taken over [R] ∈ Q(1)k×k. In particular � +R d(R)−d/2 ≲ 1 in dimensions d > 2k + 2, +thus estimate (27) follows from (29), (51) and estimates (52)-(53). +For the mollified estimate (28), we set τ = 2j−l besides Λ = 2l and ε = 2−2l. Again, we note that if +q = | det(C)| > 0 and if q | qj i.e. if q divides qj then ξ1 ∈ Ωc +j,l implies that X ∈ Ωτ,q for X = (ξ1, . . . , ξd), +for the sets Ωj,l and Ωτ,q defined in (12) and (38). Using this observation, we have +(55) +ˆ +Ik +sup +X +1Ωc +j,k(ξ1)|θd,k(X + iεT −1, −X, 0)| dX ≲ +� +d(R)|qj +JΛ2T,τ[R] + +� +d(R)∤qj +JΛ2T [R] + +� +2 +. +In dimensions d ≥ 2k + 3, using (48) and (54), the first sum on the right side of (55) is crudely estimated by +� +d(R)|qj +JΛ2T,τ[R] ≲ Λk(d−k−1) +� +1≤d(R)≤qj +� +d(R)−kΛ− d−2k +2 ++ d(R)− d +2 (τΛ)− d−2k +2 +� +(56) +≲ Λk(d−k−1)� +qj2− 3l +2 + 2− 3j +2 � +≲ Λk(d−k−1)2− 3j +2 . +Indeed, qj = lcm{1 ≤ q ≤ 2j} ≈ e2j ≤ 2l as 2j+2 ≤ l by our assumptions. To estimate the second term +on the right side of (55), we need the following. +Lemma 4. Let j ∈ N and s > 1. Then +(57) +� +d(R)∤qj +d(R)−k−s ≤ C 2j(1−s) j−1 +where the constant C may depend on d, k and s. + +DISCRETE MULTILINEAR MAXIMAL OPERATORS AND PINNED SIMPLICES +13 +Proof. Let +(58) +Ψ(s) := +� +[R]∈Q(1)k×k +d(R)−k−s = +� +n≥1 +ak(n)n−s, +with ak(n) = � +d(R)=n d(R)−k. For two Dirichlet series Ψ(s) = � +n≥1 a(n)n−s and Φ(s) = � +n≥1 b(n)n−s +we will write Ψ(s) ⪯ Φ(s) if |a(n)| ≤ b(n) for all n ≥ 1. +It is proved in [7], see (34) Lemma 1.4.9 there, that +(59) +Ψ(s) ⪯ ζ(s + 1)Kζ(s) =: +� +n≥1 +bK(n)n−s, +with K = 2k + k − 3. Clearly the coefficients of the Dirichlet series ζ(s + 1)Kζ(s) are multiplicative i.e. +bK(nm) = bK(n)bK(m) if (n, m) = 1, moreover are easy to show that, +(60) +bK(n) = +� +m|n +dK(m) +m +, +where dK(m) = |{m1, . . . , mk ∈ N : m1m2 · · · mK = m}|. Since qj = l.c.m.{1 ≤ q ≤ 2j}, if n ∤ qj the either +there is a prime p > 2j such that p | n or there is a prime p < 2j such that pγp > 2j but pγp | n. Accordingly, +we have the estimate +(61) +� +d(R)∤qj +d(R)−k−s = +� +n∤qj +ak(n)n−s ≤ +� +p>2j +� +n≥1 +bK(pn)p−sn−s + +� +p<2j +� +n≥1 +bK(pγpn)p−γpsn−s. +Writing n = prm, the first sum on the right side of (61) is estimated by +(62) +� +p>2j +� +n≥1 +bK(pn)p−sn−s = +� +p>2j +∞ +� +r=1 +� +m≥1,p∤m +bK(pr)bK(m)p−rsm−s, +using the fact that bK(prm) = bK(pr)bk(m). By (60), we have +(63) +bK(pr) = 1 + +r +� +s=1 +dK(ps) +ps +≤ 1 + +∞ +� +s=1 +(s + 1)K +2s +≲ 1, +uniformly in r ≥ 1. Thus, for s > 1, +(64) +� +p>2j +∞ +� +r=1 +� +m≥1,p∤m +bK(pr)bK(m)p−rsm−s ≲ +� +p>2j +p−s ≲ 2j(1−s)j−1, +using the fact that the number of primes 2J ≤ p < 2J+1 is bounded by 2J J−1 for all J ≥ j. +The second term on the right side of (61) is estimated similarly, except that here we use the fact that +pγp > 2j for p < 2j. We have +� +p<2j +� +n≥1 +bK(pγpn)p−γpsn−s = +� +p<2j +∞ +� +r=γp +� +m≥1,p∤m +bK(pr)bK(m)p−rsm−s +(65) +≲ +� +p<2j +∞ +� +r=γp +p−rs ≲ +� +p<2j +p−γps ≲ 2j(1−s)j−1, +as the number of primes p < 2j is bounded by 2jj−1. Estimate (57) follows immediately from (64)-(65). +□ +In dimensions d > 2k + 2, Lemma 4 with s = d/2 − k ≥ 3/2 implies that +(66) +� +d(R)∤qj +JΛ2T [R] ≲ Λk(n−k−1)d(R)−d/2 ≲ Λk(n−k−1)2−j/2j−1, +with Λ = 2l. Finally, by (52) (55)-(56) and (66) one obtains, in dimensions d > 2k + 2 +(67) +ˆ +Ik +sup +X +1Ωc +j,k(ξ1)|θd,k(X +iεT −1, −X, 0)| dX ≲ Λk(d−k−1)� +2−j/2j−1 +2−3j/2 +2−3l� +≲ Λk(d−k−1)2−j/2j−1. +Estimate (28) follows immediately from (29) and (67). + +14 +NEIL LYALL +´AKOS MAGYAR +ALEX NEWMAN +PETER WOOLFITT +References +[1] A. N. Andrianov, Quadratic Forms and Hecke Operators, Grundlehren der mathematischen Wissenschaften, Springer- +Verlag (1987) +[2] T. C. Anderson, A. V. Kumchev, and E. A. Palsson, Discrete maximal operators over surfaces of higher codimension, +Bull. London Math. Soc. 53.3 (2021): 855-860. +[3] J. Bourgain, A Szemer´edi type theorem for sets of positive density in Rk, Israel J. Math. 54 (1986), no. 3, 307–316. +[4] B. Cook, N. Lyall, ´A. Magyar, Multilinear maximal operators associated to simplices, J. London Math. Soc. 104.4 +(2021): 1491-1514. +[5] L. Huckaba, N. Lyall and ´A. Magyar, Simplices and sets of positive upper density in Rd, Proc. Amer. Math. Soc. 145 +(2017), no. 6, 2335-2347 +[6] K. Hughes, The discrete spherical averages over a family of sparse sequences, Journal d’Analyse Math´ematique 138.1 +(2019): 1-21. +[7] Y. Kitaoka, Siegel modular forms and representation by quadratic forms Lectures on Mathe- matics and Physics, Tata +Institute of Fundamental Research, Springer-Verlag, (1986). +[8] H. Klingen, Introductory lectures on Siegel modular forms Cambridge Studies of Advanced Mathematics 20, Cambridge +Univ. Press, (1990). +[9] H. D. Kloosterman, Asymptotische formeln f¨ur die fourierkoeffizienten ganzer modulformen, Abhandlungen aus dem +Mathematischen Seminar der Universit¨at Hamburg. Vol. 5. No. 1. Springer-Verlag (1927) +[10] N. Lyall and ´A. Magyar, Distances and trees in dense subsets of Zd Israel J. Math. 240.2 (2020): 769-790. +[11] N. Lyall and ´A. Magyar, Weak hypergraph regularity and applications to geometric Ramsey theory, Trans. Amer. Math. +Soc., Series B 9.5 (2022): 160-207 +[12] N. Lyall, ´A. Magyar, A. Newman, P. Woolfitt, The discrete spherical maximal function: +A new proof of ℓ2- +boundedness, Proc. Amer. Math. Soc. 149.12 (2021): 5305-5312 +[13] ´A. Magyar, E. M. Stein, and S. Wainger, Discrete analogues in harmonic analysis: Spherical averages, Ann. Math. (2) +155 (2002), no. 1, 189-208. +[14] ´A. Magyar, Distance sets of large sets of integer points, Israel J. Math., v (2008) pp. +[15] ´A. Magyar, k-point configurations in sets of positive density of Zn, Duke Math. J., v 146/1, (2009) pp. 1-34. +[16] S. Raghavan, Modular forms of degree n and representation by quadratic forms, Ann. Math. (2) 70 (1959), no. 3, 446-477. +[17] C. L. Siegel, On the theory of indefinite quadratic forms, Ann. of Math. (2) 45 (1944), 577-622. +[18] R. C. Vaughan, The Hardy-Littlewood Method, Second ed., Cambridge University Press, Cambridge, 1997. +Department of Mathematics, The University of Georgia, Athens, GA 30602, USA +Email address: lyall@math.uga.edu +Email address: magyar@math.uga.edu +Email address: alxjames@uga.edu +Email address: pwoolfitt@uga.edu + diff --git a/rtFIT4oBgHgl3EQfyCvK/content/tmp_files/load_file.txt b/rtFIT4oBgHgl3EQfyCvK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6619801d8c11178c7a0f7cd3feee6faf26f6961a --- /dev/null +++ b/rtFIT4oBgHgl3EQfyCvK/content/tmp_files/load_file.txt @@ -0,0 +1,740 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf,len=739 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='11359v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='CA] 26 Jan 2023 DISCRETE MULTILINEAR MAXIMAL OPERATORS AND PINNED SIMPLICES NEIL LYALL ´AKOS MAGYAR ALEX NEWMAN PETER WOOLFITT Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' We prove that any given subset of Zd of upper density δ > 0 will necessarily contain, in an appropriate sense depending on δ, an isometric copy of all large dilates of any given non-degenerate k- simplex, provided d ≥ 2k + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' This provides an improvement in dimension, from d ≥ 2k + 5, on earlier work of Magyar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' We in fact establish a stronger pinned variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Key to our approach are new ℓ2 estimates for certain discrete multilinear maximal operators associated to simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' These operators are generalizations of the discrete spherical maximal operator and may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Simplices in dense subsets of Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Recall that the upper Banach density of a set A ⊆ Zd is defined by δ∗(A) = lim N→∞ sup t∈Zd |A ∩ (t + Q(N))| |Q(N)| , where | · | denotes counting measure on Zd and Q(N) the discrete cube [−N/2, N/2]d ∩ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In light of the fact that the square of the distance between any two distinct points in Zd is always a positive integer we also introduce the convenient notation √ N := {λ : λ > 0 and λ2 ∈ Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In [14] the second author established the following result on the existence of unpinned two point configu- rations (distances) in dense subsets of the integer lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Theorem A (Magyar [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let A ⊆ Zd with d ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' If δ∗(A) > 0, then there exist an integer q = q(δ∗(A)) and λ0 = λ0(A) such that for all λ ∈ √ N with λ ≥ λ0 there exist a pair of points {x, x+y} ⊆ A with |y| = qλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' The approach taken in [14] was an adaptation of Bourgain’s in [3] to the analogous problem in the continuous setting of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In [15] the second author adapted this further to establish the following analogous result for non-degenerate k-simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Recall that for any 1 ≤ k ≤ d we refer to a configuration ∆ = {v0 = 0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , vk} ⊆ Zd as a non-degenerate k-simplex if the vectors v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , vk are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Theorem B (Magyar [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let k ≥ 2, A ⊆ Zd with d ≥ 2k + 5, and ∆ = {0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , vk} ⊆ Zd be a non-degenerate k-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' If δ∗(A) > 0, there exists an integer q = q(δ∗(A)) and λ0 = λ0(A, ∆) such that for all λ ∈ √ N with λ ≥ λ0 there exist x ∈ A with x + ∆′ ⊆ A for some ∆′ = {0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk} ≃ λq∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In the theorem above, and throughout this article, we say that two configurations λ∆ = {0, λv1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , λvk} and ∆′ = {0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk} in Zd are isometric, and write ∆′ ≃ λ∆, if |yi − yj| = λ|vi − vj| for all 0 ≤ i, j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In this article we establish an improvement on the dimension condition in Theorem B above from d ≥ 2k+5 to d ≥ 2k + 3 and simultaneously establish a stronger pinned variant, namely Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let k ≥ 1, A ⊆ Zd with d ≥ 2k+3, and ∆ = {0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , vk} ⊆ Zd be a non-degenerate k-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' If δ∗(A) > 0, there exists an integer q = q(δ∗(A)) and λ0 = λ0(A, ∆) such that for any λ1 ≥ λ0 there exists a fixed x ∈ A such that for all λ ∈ [λ0, λ1] ∩ √ N one has x + ∆′ ⊆ A for some ∆′ = {0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk} ≃ λq∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' The threshold λ0 in the results above cannot be taken to depend on δ∗(A) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Indeed, for any positive integers q and M the set (QqM ∩ Zd) + (4dqMZ)d will have density (4d)−d but never contain pairs {x, x + y} with |y| = qdM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Since A could fall entirely into a fixed congruence class of some integer 1 ≤ r ≤ δ∗(A)−1/d the value of q in the results above must be divisible by the least common multiple of all integers 1 ≤ r ≤ δ∗(A)−1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Indeed if A = (rZ)d with 1 ≤ r ≤ δ−1/d then A will have upper Banach density at least δ, but the distance between any two points x, y ∈ A will always take the form rλ for some λ ∈ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 11B30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' The first and second authors were partially supported by grants NSF-DMS 1702411 and NSF-DMS 1600840, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 1 2 NEIL LYALL ´AKOS MAGYAR ALEX NEWMAN PETER WOOLFITT The approach in [15] also established a quantitative Szemer´edi-type variant of Theorem B, namely Theorem B′ (Magyar [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let k ≥ 2, d ≥ 2k + 5, ∆ = {0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , vk} ⊆ Zd be a non-degenerate k-simplex, and 0 < δ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' If N ≥ exp(C∆δ−Ck), then any A ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , N}d with cardinality |A| ≥ δN d will necessarily contain a configuration of the form x + ∆′ with ∆′ = {0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk} ≃ λq∆ for some λ ∈ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For an alternative approach to the proof of Theorem B′ that is more in line with the arguments in this paper, see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='1 in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' We note that by combining the main result of this current paper, namely Theorem 2 below and its corollary (Lemma 2 in Section 1), with the arguments and ideas contained in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='1 of [11] one can establish an improvement on the dimension condition above from d ≥ 2k + 5 to d ≥ 2k + 3 and also establish the analogous stronger pinned variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' However, for the sake of clarity and brevity we have chosen not to pursue the details of these arguments or statements here and instead focus on just establishing Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let us remark that the dimension bound d ≥ 2k + 3 seems to be best possible even in case A = Zd, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' when counting embeddings of isometric copies of λ∆ in Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Indeed, writing T to be the positive definite integral k × k matrix with entries tij = vi · vj and Y for the k × d integral matrix with rows y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk the condition ∆′ = {0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk} ≃ λ∆ translates to the matrix equation Y tY = λ2T which has been intesively in the past [17, 16, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' The best known results are due to Kitaoka [7] in dimesnions d = 2k + 3, who also mentions that this condition is best possible to count solutions to the above equation via analytic means, see the remark after Theorem B in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For k = 1 and d = 4, it is possible to count embeddings under restrictions of λ (say when λ2 is odd) via the so-called Kloosterman refinement [9], however for our pinned results in sets of postive density one needs for the discrete spherical maximal function which in dimension 4 has only been obtained for particular lacunary sequenses of the radii λ, see [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' The case k = 1 of Theorem 1 was already established by the first two authors in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' To the best of our knowledge, there have been no previous results addressing pinned simplices in dense subsets Zd in any dimension when k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Discrete multilinear maximal averages associated to simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' An important result in the development of discrete harmonic analysis is the ℓp-boundedness of the so-called discrete spherical maximal function [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For any λ ∈ √ N we let Sλ = {y ∈ Zd : |y| = λ} denote the discrete sphere of radius λ centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For f : Zd → R we then define the discrete spherical averages Aλf(x) = |Sλ|−1 � y∈Sλ f(x + y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' noting that if d ≥ 5, then cdλd−2 ≤ |Sλ| ≤ Cdλd−2 for some constants 0 < cd < Cd < ∞, see [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In [13] it was shown that for p > d/(d − 2) one has the following maximal function estimate �� sup λ≥1 |Aλf| �� p ≤ Cp,d ∥f∥p where ∥f∥p = (� x |f(x)|p)1/p denotes the ℓp(Zd) norm of the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In [12] the authors gave a new direct proof of ℓ2-boundedness of the discrete spherical maximal function that neither relies on abstract transference theorems nor on delicate asymptotic for the Fourier transform of discrete spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Implicit in that paper is the fact that ℓ2-boundedness follows as a consequence of stronger refined “mollified” estimates in which one obtains gains in ℓ2 over suitably large scales when applied to functions whose Fourier transform is localized away from rational points with small denominators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Recall that for f ∈ ℓ1(Zd) we define its Fourier transform �f : Td → C by �f(ξ) = � x∈Zd f(x)e−2πix·ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For each η > 0 we define qη := lcm{1 ≤ q ≤ η−2} and for any L ≥ qη we let Ωη,L = {ξ ∈ Td : ξ ∈ [−L−1, L−1]d + (q−1 η Z)d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Key to the proof of Theorem 1 is an extension of the approach from [12] to multilinear maximal operators associated to simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Given a non-degenerate k-simplex ∆ = {v0 = 0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , vk} ⊆ Zd and λ ∈ √ N, we let Sλ∆ := {(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk) ∈ Zdk : ∆′ = {0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk} ≃ λ∆}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' DISCRETE MULTILINEAR MAXIMAL OPERATORS AND PINNED SIMPLICES 3 For functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk : Zd → C we then define the multilinear averaging operator Aλ∆(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)(x) = |Sλ∆|−1 � (y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=',yk)∈Sλ∆ f1(x + y1) · · · fk(x + yk) noting that if d ≥ 2k + 3 and λ ∈ √ N, then (1) c∆ λdk−k(k+1) ≤ |Sλ∆| ≤ C∆ λdk−k(k+1) for some constants 0 < c∆ < C∆ < ∞, see [7] or [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Note that for k = 1 and v1 = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=', 0) we have that Sλ∆ = Sλ and hence Aλ∆(f) = Aλ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' The ℓp mapping properties of the maximal operators corresponding to these averages were considered in [2] and [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Here we establish the following particular ℓ2-estimates, the first non-trivial estimates of any type for such operators in dimensions lower that d = 2k + 5 when k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' The stronger refined ℓ2 estimate (3), in addition to implying (2), plays a crucial role in our proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' If k ≥ 1, d ≥ 2k + 3, and ∆ = {0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , vk} ⊆ Zd be a non-degenerate k-simplex, then (2) �� sup λ≥1 |Aλ∆(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)| �� 2 ≤ Cd,∆∥f1∥2 · · · ∥fk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In fact, for any η > 0, and L ≥ q4 η, we have (3) ��� sup λ≥η−2L |Aλ∆(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)| ��� 2 ≤ Cd,∆ η log η−1 ∥f1∥2 · · · ∥fk∥2 whenever supp �fj ⊆ Ωc η,L for some 1 ≤ j ≤ k, where Ωc η,L denotes the complement of Ωη,L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Estimate (3) in the case k = 1 was originally established in joint work with Magyar [10] via an adaptation of the transference methods from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Proof of Theorem 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Reduction to uniform distributed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In light of the observation made after Theorem 1 above regarding the sensitivity of this problem to the local structure of A, it is natural to first consider the case when A is, in a suitable sense, well distributed in small congruence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In fact, this approach ultimately leads directly to a proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Following [10] we define A ⊆ Zd to be η-uniformly distributed (modulo qη) if, for some η > 0, its relative upper Banach density on any residue class modulo qη never exceeds (1 + η4) times its density on Zd, namely if δ∗(A | s + (qηZ)d) ≤ (1 + η4) δ∗(A) for all s ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , qη}d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' A straightforward density increment argument allows one to deduce Theorem 1 from the following analogue for η-uniformly distributed subsets of Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let ε > 0, 0 < η ≪ ε2 and k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' If A ⊆ Zd with d ≥ 2k + 3 is η-uniformly distributed, and ∆ = {0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , vk} ⊆ Zd is a non-degenerate k-simplex, then there exist λ0 = λ0(A, ∆, η) such that for any λ1 ≥ λ0 there exists a fixed x ∈ A such that Aλ∆(1A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , 1A)(x) > δ∗(A)k − ε for all λ ∈ [λ0, λ1] ∩ √ N, noting that Aλ∆(1A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , 1A)(x) = |Sλ∆|−1 |{(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk) ∈ Zdk : x + ∆′ ⊆ A with ∆′ = {0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk} ≃ λ∆}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In Proposition 1 above, and throughout this article, we use the notation α ≪ β to denote that α ≤ cβ for some suitably small constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Proposition 1 in fact implies the following stronger optimal formulation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let k ≥ 1, A ⊆ Zd with d ≥ 2k + 3, and ∆ = {0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , vk} ⊆ Zd be a non-degenerate k-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For any ε > 0, there exists an integer q = q(ε, d) and λ0(A, ∆, ε) such that for any λ1 ≥ λ0 there exists a fixed x such that (4) |Sλ∆|−1 |{(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk) ∈ (qZ)dk : x + ∆′ ⊆ A with ∆′ = {0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk} ≃ λq∆}| > δ∗(A)k − ε for all λ ∈ [λ0, λ1] ∩ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 4 NEIL LYALL ´AKOS MAGYAR ALEX NEWMAN PETER WOOLFITT Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' By considering sets A of the form � s∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=',q}d As with each set As a “random” subset of the congruence class s + (qZ)d one can further easily see that conclusion (4) above is in general best possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Proof that Proposition 1 implies Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let 0 < ε ≤ δ ≤ 1 and A ⊆ Zd with d ≥ 2k + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' To prove Corollary 1 it is enough to prove that if δ∗(A) ≥ δ then there exists λ0 = λ0(A, ∆, ε) and q = q(ε, d) such that for any λ1 ≥ λ0 there exists a fixed x ∈ A such that (4) holds for all λ ∈ √ N with λ0 ≤ λ ≤ λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let 0 < η ≪ ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' We prove the above for δm := (1 + η4)−m inductively for all m ≥ 0, using Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For m = 0 the statement is trivial as δ∗(A) = δ0 = 1 and hence A contains arbitrarily large cubic grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Suppose it holds for δ = δm and assume that δ∗(A) ≥ δm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' If A is η-uniformly distributed then the result holds for δ = δm+1 by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In the opposite case there is an s ∈ Zd so that δ∗(A | s+ (qηZ)d) > (1 + η4) δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let φ : s+ (qηZ)d → Zd be defined by φ(x) := q−1 η (x− s) and let A′ := φ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Then δ∗(A′) ≥ δm thus (4) holds for A′ and δ = δm, with some q′ = q′(ε, d) and x′ ∈ A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Note that |{(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk) ∈ (q′Z)dk : x′ + ∆′ ⊆ A′ with ∆′ = {0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk} ≃ q′λ∆}| = |{(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk) ∈ (qηq′Z)dk : qηx′ + ∆′ ⊆ A′ with ∆′ = {0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk} ≃ qηq′λ∆}| which implies that (4) holds for A, δ = δm+1 with q = qηq′ and x = qηx′ + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Before proving proving Proposition 1 we need two preparatory lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' We refer to a subset Q ⊆ Zd as a cube of sidelength ℓ(Q) = N if Q = t0 + QN for some t0 ∈ Zd, where as usual QN = [−N/2, N/2]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Definition (U 1 q,L(Q)-norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For any cube Q ⊆ Zd, integers 1 ≪ q ≪ L ≪ ℓ(Q), and functions f : Q → R we define (5) ∥f∥U1 q,L(Q) = � 1 |Q| � t∈Zd |f ∗ χq,L(t)|2�1/2 where χq,L denotes the normalized characteristic function of the cubes Qq,L := QL ∩ (qZ)d, namely (6) χq,L(x) = �� q L �d if x ∈ (qZ)d ∩ [− L 2 , L 2 ]d 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In (5) above and in the sequel we denote the convolution f ∗ g of two functions f and g by f ∗ g(x) := � y∈Zd f(x − y)g(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' We note that the U 1 q,L(Q)-norm measures the mean square oscillation of a function with respect to cubic grids of size L and gap q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' The first key ingredient in our proof of Proposition 1 is the simple, yet significant, observation from [10] that subsets of Zd with positive upper Banach density that are η-uniformly distributed are also, in a precise sense, uniformly distributed at certain scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Lemma 1 (Consequence of Lemmas 1 and 2 in [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let η > 0 and A ⊆ Zd be η-uniformly distributed with δ := δ∗(A) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' There exists a positive integer L = L(A, η) and cubes Q ⊆ Zd of arbitrarily large sidelength ℓ(Q) with ℓ(Q) ≥ η−4L such that (7) |A ∩ Q| ≥ (δ − O(η))|Q| and (8) ∥(1A − δ)1Q∥U1 qη,L(Q) = O(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' The second key ingredient in our proof of Proposition 1 is the following maximal variant of a so-called generalized von-Neumann-type inequality, which follows in a straightforward manner from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Lemma 2 (Corollary of Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let k ≥ 1, d ≥ 2k + 3, and ∆ = {0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , vk} ⊆ Zd be a non- degenerate k-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For any η > 0, positive integer L, cube Q ⊆ Zd with sidelength N ≥ η−6L, and functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk : Q → [−1, 1] we have 1 |Q| � x∈Zd sup η−3L≤λ≤η3N ��Aλ∆(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)(x) �� ≤ Cd,∆ � min 1≤j≤k ∥fj∥U1 qη,L(Q) + O(η) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' DISCRETE MULTILINEAR MAXIMAL OPERATORS AND PINNED SIMPLICES 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' By Cauchy-Schwarz, it suffices to prove the stronger estimate � 1 |Q| � x∈Zd sup η−3L≤λ≤η3N ��Aλ∆(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)(x) ��2 �1/2 ≤ Cd,∆ � min 1≤j≤k ∥fj∥U1 qη,L(Q) + O(η) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' This follows from Theorem 2 by symmetry and sublinearity after decomposing fk = fk,1 +fk,2 +fk,3 with fk,1 = fk ∗ χqη,L where fk,2 and fk,3 satisfy � fk,2 = �fk 1Ωη,η−1L(1 − � χqη,L) and � fk,3 = �fk 1Ωc η,η−1L(1 − � χqη,L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Indeed, estimate (2) implies that � 1 |Q| � x∈Zd sup λ≥1 ��Aλ∆(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk−1, g)(x) ��2 �1/2 ≤ Cd,∆ � 1 |Q| � x∈Zd |g(x)|2 �1/2 for any g : Zd → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Note that if g = fk,1 then � 1 |Q| � x∈Zd |fk,1(x)|2 �1/2 = ∥fk∥U1 qη,L(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In light of the fact that �χqη,L(ξ) = qd η Ld � x∈[− L 2 , L 2 )d, qη|x e−2πix·ξ it is easy to see that �χq,L(ℓ/q) = 1 for all ℓ ∈ Zd and that there exists some absolute constant C > 0 such that (9) 0 ≤ 1 − �χqη,L(ξ) ≤ C L |ξ − ℓ/qη| for all ξ ∈ Td and ℓ ∈ Zd, and hence that 1 − � χqη,L(ξ) = O(η) for all ξ ∈ Ωη,η−1L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' It thus follows, by Plancherel, that � 1 |Q| � x∈Zd |fk,2(x)|2 �1/2 = O(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Finally, since supp � fk,3 ⊆ Ωc η,η−1L, it follows from estimate (3) that � 1 |Q| � x∈Zd sup η−3L≤λ≤η3N ��Aλ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk−1, fk,3)(x) ��2 �1/2 ≤ Cd,∆ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' □ Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let 0 < ε ≤ δ ≤ 1 and 0 < η ≪ ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Suppose there exists a set A ⊆ Zd with d ≥ 2k + 3 with δ = δ∗(A) > 0 that is η-uniformly distributed but for which the conclusion of Proposition 1 fails, namely that there exists arbitrarily large pairs (λ0, λ1) such that for every x ∈ A one has Aλ∆(1A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , 1A)(x) ≤ δk − ε for some λ ∈ [λ0, λ1] ∩ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Combining this with Lemma 1 we can conclude that there exists a positive integer L and a cube Q ∈ Zd with sidelength N sufficiently large so that in addition to the properties (7) and (8) we also have the property that Aλ∆(1A∩Q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , 1A∩Q)(x) ≤ δk − ε for every x ∈ A for some λ ∈ [η−3L, η3N] ∩ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' We now let A′ := A ∩ Q′, where Q′ denotes the cube of sidelength (1 − η3)N with the same center as Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' It then follows, provided that N was chosen sufficiently large, that Aλ∆(1Q, δ1Q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , δ1Q)(x) = δk 6 NEIL LYALL ´AKOS MAGYAR ALEX NEWMAN PETER WOOLFITT for every x ∈ A′ and hence that for each such x one has k−1 � j=0 Aλ∆(1A∩Q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 1A∩Q � �� � j copies , (1A − δ)1Q, δ1Q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , δ1Q)(x) ≤ −ε for some λ ∈ [η−3L, η3N] ∩ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Consequently, we have that (10) k−1 � j=0 sup η−3L≤λ≤η3N ����Aλ(1A∩Q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 1A∩Q � �� � j copies , (1A − δ)1Q, δ1Q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , δ1Q)(x) ���� ≥ ε for every x ∈ A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Since η ≪ δ and |A′| ≥ |A∩Q|−η3|Q| it follows from (7) that |A′|/|Q| ≥ δ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Combining this observation with (10) we obtain (11) k−1 � j=0 1 |Q| � x∈Zd sup η−3L≤λ≤η3N ����Aλ(1A∩Q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 1A∩Q � �� � j copies , (1A − δ)1Q, δ1Q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , δ1Q)(x) ���� ≥ εδ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' However, Lemma 2 and (8) clearly imply that for each 0 ≤ j ≤ k − 1 one has 1 |Q| � x∈Zd sup η−3L≤λ≤η3N ����Aλ(1A∩Q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 1A∩Q � �� � j copies , (1A − δ)1Q, δ1Q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , δ1Q)(x) ���� = O(η) which leads to a contradiction if η is chosen sufficiently small with respect to ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Proof of Theorem 2 Following the approach in [12] we will deduce Theorem 2 from refined estimates for our maximal operators at a single dyadic scale, namely Proposition 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' We first need to introduce some notation closely related to that in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For any integer j ≥ 0 we let qj = lcm{1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=', 2j} noting that qj ≍ e2j, and for any non-negative integers j and l that satisfy 2j ≤ l , we let (12) Ωj,l := {ξ ∈ Td : ξ ∈ [−2j−l, 2j−l]d + (q−1 j Z)d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' If k ≥ 1, d ≥ 2k + 3, and ∆ = {0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , vk} ⊆ Zd be a non-degenerate k-simplex, then (13) �� sup 2l≤λ≤2l+1 |Aλ∆(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)| �� 2 ≤ Cd,∆ 2−j/2j−1 ∥f1∥2 · · · ∥fk∥2 whenever supp �fi ⊆ Ωc j,l for some 1 ≤ i ≤ k, where Ωc j,l denotes the complement of Ωj,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' It is easy to see that Proposition 2 is equivalent to estimate (3) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Indeed, note that in proving (3) one may restrict the sup to η−2L ≤ λ ≤ 2η−2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Choosing l, j ∈ N such that 2l ≤ η−2L ≤ 2l+1 and 2j ≥ η−2 we have that 2l−j ≤ L and hence Ωj,l ⊆ Ωη,L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Applying Proposition 2 with j and l chosen as above implies estimate (3) of Theorem 2, while applying estimate (3) of Theorem 2 with L = 2l−j and η = 2−j/2 immediately implies Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' We are left with establishing that Proposition 2 implies estimate (2) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Following the approach in [12] we start by introducing a smooth sampling function supported on Ωj,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' A smooth sampling function supported on Ωj,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let ψ ∈ S(Rd) be a Schwartz function satisfying 1Q(ξ) ≤ �ψ(ξ) ≤ 12Q(ξ) where Q = [−1/2, 1/2]d and �ψ(ξ) := ˆ Rd ψ(x)e−2πix·ξdx DISCRETE MULTILINEAR MAXIMAL OPERATORS AND PINNED SIMPLICES 7 denote the Fourier transform of ψ on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For a given q ∈ N and L > q we define ψq,L : Zd → R as ψq,L(x) = �� q L �d ψ � m L � if x ∈ (qZ)d 0 otherwise Writing x = qr + s with r ∈ Zd and s ∈ Zd/qZd, it follows from Poisson summation that �ψq,L(ξ) = � x∈Zd ψ(x)e−2πix·ξ is a q−1-periodic function on Td that satisfies �ψq,L(ξ) = � ℓ∈Zd �ψ(L(ξ − ℓ/q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For a given l ∈ N and 0 ≤ j ≤ Jl := [log2(l)] − 2, we now define the sampling function (14) Ψl,j = ψqj,2l−j and note that supp �Ψl,j ⊆ Ωj,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Finally we define ∆Ψl,j = Ψl,j+1 − Ψl,j and note the important almost orthogonality property they enjoy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Lemma 3 (Lemma 1 in [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' There exists a constant C = CΨ > 0 such that � l≥2j |� ∆Ψl,j(ξ)|2 ≤ C uniformly in j ∈ N and ξ ∈ Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Proof that Proposition 2 implies estimate (2) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let k ≥ 1, d ≥ 2k + 3, and ∆ = {0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , vk} ⊆ Zd be a non-degenerate k-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In [12] the authors gave a direct proof of estimate (2) of Theorem 2 when k = 1, the ℓ2-boundedness of the discrete spherical maximal function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' We may thus, without loss in generality assume that k ≥ 2, supp �fk ⊆ Ωc j,l, and that (15) �� sup λ≥1 |Aλ�∆(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk−1)| �� 2 ≤ Cd,�∆∥f1∥2 · · · ∥fk−1∥2 where �∆ = {0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , vk−1} ⊆ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let (16) Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk) := sup 2l≤λ≤2l+1 |Aλ∆(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Writing fk = fk ∗ Ψl,0 + Jl−1 � j=0 fk ∗ ∆Ψl,j + (fk − fk ∗ Ψl,Jl) it follows by subadditivity that (17) Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk) ≤ Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk ∗ Ψl,0) + Jl−1 � j=0 Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk ∗ ∆Ψl,j) + Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk − fk ∗ Ψl,Jl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Estimate (2) of Theorem 2 will now follow from a few observations and applications of Proposition 2, in light of the fact that sup λ≥1 |Aλ∆(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)| = sup l Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' We first observe that the first term on the right in (17) above satisfies Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk ∗ Ψl,0) ≤ CΨH(fk) sup λ≥1 |Aλ�∆(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk−1)| uniformly in l, where H(f)(x) = sup N>0 1 |QN| ��� � y∈QN f(x − y) ��� 8 NEIL LYALL ´AKOS MAGYAR ALEX NEWMAN PETER WOOLFITT with Q(N) the discrete cube [−N/2, N/2]d ∩ Zd denotes the discrete Hardy-Littlewood maximal operator, which trivially satisfies ∥Hf∥∞ ≤ ∥f∥∞ ≤ ∥f∥2 by the nesting of discrete ℓp spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' It therefore follows from the inductive hypothesis (15) that sup l ∥Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk ∗ Ψl,0)∥2 ≤ C∥f1∥2 · · · ∥fk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For the middle terms in (17) we first note that sup l Jl−1 � j=0 Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk ∗ ∆Ψl,j) ≪ � ∞ � l=0 ��� Jl−1 � j=0 Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk ∗ ∆Ψl,j) ��� 2�1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Taking ℓ2 norms of both sides of the inequality above and applying Minkowski’s inequality, followed by an application of Proposition 2, gives ���sup l � 0≤j≤Jl Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk ∗ ∆Ψl,j) ��� 2 ≤ � j �� l≥2j ∥Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk ∗ ∆Ψl,j)∥2 2 �1/2 ≤ C∥f1∥2 · · · ∥fk−1∥2 � j 2−j/2�� l≥2j ∥fk ∗ ∆Ψl,j∥2 2 �1/2 ≤ C∥f1∥2 · · · ∥fk∥2 where the last inequality above follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' One more application of Proposition 2 with j = [log2 l] − 2 to the last term in (17) gives ���sup l Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk − fk ∗ Ψl,Jl) ��� 2 ≤ � ∞ � l=1 ∥Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk − fk ∗ Ψl,Jl)∥2 2 �1/2 ≤ C � ∞ � l=1 l−1(log2 l)−2�1/2 ∥f1∥2 · · · ∥fk∥2 ≤ C∥f1∥2 · · · ∥fk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Proof of Proposition 2 Given any simplex ∆ = {v0 = 0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , vk} ⊆ Rd, we introduce the associated inner product matrix T = T∆ = (tij)1≤i,j≤k with entries tij := vi ·vj, where “·” stands for the dot product in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Note that T is a positive semi-definite matrix with integer entries and T is positive definite if and only if ∆ is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' It is easy to see that ∆′ ≃ λ∆, with ∆′ = {y0 = 0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk}, if and only if (18) yi · yj = λ2tij for all 1 ≤ i, j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' If we let M ∈ Zd×k be a matrix with column vectors y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk ∈ Zd, then the system of equations above can be written as the matrix equation (19) M tM = λ2T, where M t is the transpose of the matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' It therefore follows that Aλ∆(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)(x) = |Sλ∆|−1 � y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=',yk∈Zd f1(x + y1) · · · fk(x + yk)Sλ2T (M) if we use Sλ2T (M) to denote the indicator function of relation (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let Ik = [0, 2]k(k+1)/2 denote the space of symmetric k × k matrices with entries in the interval [0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Using the fact that tr(XtY ) = tr(Y Xt) = k � i=1 k � j=1 xijyij, for any k × k matrices X = (xij), Y = (yij), one has (20) Sλ2T (M) = 2−k ˆ Ik eπi tr[(MtM−λ2T )X] dX DISCRETE MULTILINEAR MAXIMAL OPERATORS AND PINNED SIMPLICES 9 where dX = � 1≤i≤j≤k dxij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Moreover, if M tM = λ2T then tr(T −1M tM) = tr(MT −1M t) = tr(λ2I) = kλ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Given l ∈ N write Λ = 2l and ε = 2−2l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' We have (21) Sλ2T (M) = 2−kekελ2 ˆ Ik e−πiλ2 tr(T X) eπi tr(M(X+iεT −1)Mt)dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let GX,ε(M) = GX,ε(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk) = eπi tr(M(X+iεT −1)Mt) be the Gaussian function, where y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk ∈ Zd are the column vectors of the matrix M, and define the corresponding multi-linear operator (22) BX,ε(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)(x) := � y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=',yk∈Zd f1(x + y1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' fk(x + yk) GX,ε(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , yk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' It follows that Aλ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)(x) = 2−kekελ2|Sλ∆|−1 ˆ Ik e−πiλ2 tr(T X) BX,ε(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)(x) dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Thus for the maximal function Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk) := sup 2l≤λ≤2l+1 |Aλ∆(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)| we have the pointwise estimate (23) Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)(x) ≤ Cd,∆ Λ−k(d−k−1) ˆ Ik |BX,ε(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)(x)| dX, as ε = Λ−2 = 2−2l and Λ ≤ λ ≤ 2Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Finally, by Minkowski’s inequality (24) ∥Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)∥2 ≤ Cd,∆ Λ−k(d−k−1) ˆ Ik ∥BX,ε(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)(x)∥2 dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Taking the Fourier transform of the expression in (22) we obtain � BX,ε(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)(ξ) equals (25) ˆ Tk−1 �f1(ξ1) · · · �fk−1(ξk−1) �fk(ξ − ξ1 − · · · − ξk−1) � GX,ε(ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , ξk−1, ξ − ξ1 − · · · − ξk−1) dξ1 · · · dξk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Thus by the Cauchy-Schwarz inequality and Plancherel’s indentity, one has (26) ∥BX,ε(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk)∥2 2 ≤ ∥� GX,ε∥2 ∞ k � i=1 ∥fi∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Thus, the ℓ2 × · · · × ℓ2 → ℓ2 boundedness of the dyadic maximal operator Ml(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , fk) follows from the estimate (27) ˆ Ik ∥� GX,ε∥∞ dX ≤ Cd,∆ Λk(d−k−1) with Λ = 2l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For the mollified estimate assume that, supp �fi ⊆ Ωc j,l, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' �fi = 1Ωc j,l �fi for some 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' By symmetry of the expression in (22) we may assume without loss of generality that i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In this case in equal- ity (25) the function � GX,ε(ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , ξk−1, ξ −ξ1 −· · ·−ξk−1) can be replaced by 1Ωc j,l(ξ1) � GX,ε(ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , ξk−1, ξ − ξ1 − · · · − ξk−1), thus to prove Theorem 2, it is enough to show that for j, l ∈ N with 2j+2 ≤ l, one has (28) ˆ Ik ∥1Ωc j,l(ξ1) � GX,ε(ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , ξk)∥∞ dX ≤ Cd,∆ 2−j/2j−1 Λk(d−k−1) with Λ = 2l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 10 NEIL LYALL ´AKOS MAGYAR ALEX NEWMAN PETER WOOLFITT 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Estimates for theta functions on the Siegel upper half space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' To prove estimates (27) and (28) we will follow the approach given in Section 5 of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For the sake of completeness we recall below some of the basic notions and constructs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' If M = [m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , mk] ∈ Zd×k and X = [ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , ξk] ∈ Rd×k are d×k matrices then one has that tr(M tX) = m1 ·ξ1 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='+mk ·ξk where · denotes the usual dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Thus the Fourier transform of a function f(m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , mk) = f(M) may written �f(X) = �f(ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , ξk) = � M∈Zd×k f(M)e−2πi tr(MtX ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' This implies that (29) �GX,ε(X) = � M∈Zd×k eπi tr[(M(X+iεT −1)Mt−2MtX ] = θd,k(X + iεT −1, −X, 0) is the theta-function θd,k : Hk × Rd×k × Rd×k → C defined by (30) θd,k(Z, X, E) = � M∈Zd×k eπi tr[(M−E)Z(M−E)t+2MtX −EtX ] for Z = X + iY ∈ Hk, Hk being the Siegel upper space, see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='1)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='3) in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' We partition the range of integration Ik and estimating the theta function separately on each part by exploiting its transformation properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' This may be viewed as the extension of the classical Farey arcs decomposition to k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Recall the integral symplectic group (31) Γk = � γ = � A B C D � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' ABt = BAt, CDt = DCt, ADt − BCt = Ek, � which acts on the Siegel upper-half space Hk = {Z = X + iY : X ∈ Mk, Y ∈ Pk} as a group of analytic automorphisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' The action being defined by: γ⟨Z⟩ = (AZ + B)(CZ + D)−1 for γ ∈ Γk, Z ∈ Hk, see [15] and also [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let us recall also the subgroup of integral modular substitutions: (32) Γk,∞ = � γ = � A B 0 D � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' ABt = BAt, ADt = Ek � Writing U = At and S = ABt, it is easy to see that D = U −1 and B = SU −1, moreover S is symmetric and U ∈ GL(k, Z), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' det(U) = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' The action of such γ ∈ Γk,∞ on Z ∈ Hk takes the form: (33) γ⟨Z⟩ = Z[U] + S using the notation Z[U] = U tZU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' The general linear group GL(k, Z) acts on the space Pk of positive k × k matrices, via the action: Y → Y [U], Y ∈ Pk, and let Rk denote the corresponding so-called Minkowski domain, see Definition 1 on p12 of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' A matrix Y = (yij) ∈ Rk is called reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' We recall that for a reduced matrix Y (34) Y ≈ YD , y11 ≤ y22 ≤ · · · ≤ ykk where YD = diag(y11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , ykk) denotes the diagonal part of Y , and A ≈ B means that A − ckB > 0, B − ckA > 0 for some constant ck > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For a proof of these facts, see Lemma 2 on p20 in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' A fundamental domain Dk for the action of Γk on Hk, called the Siegel domain, consists of all matrices Z = X + iY , (X = (xij)), satisfying (35) Y ∈ Rk, |xij| ≤ 1/2, | det (CZ + D)| ≥ 1, ∀ γ = � A B C D � ∈ Γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' The second rows of the matrices γ ∈ Γk are parameterized by the so-called coprime symmetric pairs of integral matrices (C, D), which means that CDt is symmetric and the matrices GC and GD with a matrix G of order k are both integral only if G is integral, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='17 in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' It is clear from definition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='6) that if γ2 = γγ1 with second rows (C2, D2) and (C1, D1) for some γ ∈ Γk,∞, then (C2, D2) = (UC1, UD1) for some U ∈ GL(k, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' On the other hand, if both γ1 and γ2 have the same second row (C, D) then γ2γ−1 1 ∈ Γk,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' This gives the parametrization of the group Γk,∞\\Γk by equivalence classes of coprime symmetric pairs (C, D) via the equivalence relation (C2, D2) ∼ (C1, D1) if (C2, D2) = (UC1, UD1) for some U ∈ GL(k, Z), see also p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='54 in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' We will use the notation [γ] = [C, D] ∈ Γk,∞\\Γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' DISCRETE MULTILINEAR MAXIMAL OPERATORS AND PINNED SIMPLICES 11 If one defines the domain: Fk = ∪γ∈Γk,∞γDk, then Hk = � [γ]∈Γk,∞\\Γk γ−1Fk is a non-overlapping cover of the Siegel upper half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Correspondingly, for a given matrix T > 0 of order k, define the Farey arc dissection of level T , as the cover (36) Ik = � [γ]∈Γk,∞\\Γk IT [γ], IT [γ] = {X ∈ Ik : X + iT −1 ∈ γ−1Fk} We recall the basic estimates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='14)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='16) in [15] whose proofs are based on the transformation property |θd,k(Z, X, 0)| = | det (CZ + D)|− d 2 |θd,k(γ⟨Z⟩, XAt − Kγ/2, XCt − Nγ/2)| for some matrices Kγ, Nγ ∈ Zn×k, see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='2 in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Namely, if (C, D) is a coprime symmetric pair, then for Z ∈ IT [C, D] one has (37) |θd,k(Z, X, 0)| ≤ Cd,k | det (CZ + D)|− d 2 uniformly for X ∈ Mk(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Next we describe the “mollified” estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='16) in [15] in slightly different form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For q ∈ N and τ > 0 define the region (38) Ωq,τ = {X ∈ Rd×k : |X − P/2q| ≤ τ for some P ∈ Zd×k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' If [γ] = [C, D] coprime symmetric pair, q := |det(C)| > 0, then for Z ∈ IT [C, D] (39) |θd,k(Z, X, 0)| ≲ | det (CZ + D)|− d 2 � e−c min(Y ) + e−c τ 2µ(CtY C)� uniformly for X ∈ Ωc q,τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Here Y = Imγ⟨Z⟩, min(Y ) = minx∈Zd,x̸=0 |Y x·x| and µ(Y ) = minx∈Rd, |x|=1 |Y x·x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Define, similarly as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='20) in [15] (40) JT [C, D] = ˆ IT [C,D] sup X |θd,k(X + iT −1, −X, 0)| dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' By (37) we have that (41) JT [C, D] ≤ Cd,k J0 T [C, D], where (42) J0 T [C, D] = ˆ X∈IT [C,D] | det(CZ + D)|− d 2 dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' If q := | det(C)| > 0, then for τ > 0 let (43) JT,τ[C, D] := ˆ IT [C,D] sup X 1Ωcτ,q(X) |θd,k(X + iT −1, −X, 0)| dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' By estimate (39) one has (44) JT,τ[C, D] ≤ Cd,k J1 T [C, D] + J2 T,τ[C, D], where (45) J1 T [C, D] = ˆ IT [C,D] | det(CZ + D)|− d 2 e−c min(Y ) dX (46) J2 T,τ[C, D] = ˆ IT [C,D] | det(CZ + D)|− d 2 e−cτ 2 µ(CtY C) dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' where Y = Im γ⟨Z⟩ and γ ∈ Γk such that [γ] = [C, D] ∈ Γk,∞\\Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Then by inequalities (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='24)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='26) given in Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='3-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='4 in [15], we have (47) � St=S JT [C, D + CS] ≤ Cd,k det(T ) d−k−1 2 | det(C)|− d 2 and (48) � St=S JT,τ[C, D + CS] ≤ Cd,k det(T ) d−k−1 2 � | det(C)|−k min(T )− d−2k 4 + | det(C)|− d 2 (τ 2µ(T ))− d−2k 4 � where the summation is over all symmetric integral matrices S ∈ Mk(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 12 NEIL LYALL ´AKOS MAGYAR ALEX NEWMAN PETER WOOLFITT Recall that the map [C, D] → C−1D provides a one-one and onto correspondence between the classes of coprime symmetric pairs [C, D] ∈ Γk,∞\\Γk, with det(C) ̸= 0, and symmetric rational matrices R of order k, and the pairs [C, D +CS] correspond to the matrices R+S with symmetric S ∈ Zk×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let us write Q(1)k×k for the space of modulo 1 incongruent symmetric rational matrices, where Q(1) = Q/Z, Q being the set of rational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' If R = C−1D, for a coprime symmetric pair [C, D] then will write (49) JT [R] := � St=S JT [C, D + CS], (50) JT,τ[R] := � St=S JT,τ[C, D + CS], which is well-defined as it only depends on the equivalence class [R] ∈ Q(1)k×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Finally write d(R) = | det(C)| for R = C−1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Then by (30) and (40), we have with ε = Λ−2 that (51)ˆ Ik sup X |θd,k(X + iεT −1, −X, 0)| dX = � [C,D],det(C)̸=0 JΛ2T [C, D] + � [C,D],det(C)=0 JΛ2T [C, D] =: � 1 + � 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' An estimate for the second sum is given in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='1 in [15], namely it is shown that (52) � 2 ≤ Cd,k |Λ2T |(k−1)(d−k)/2 ≤ Cd,kΛ(d−k)(k−1) where |T | = (� ij t2 ij)1/2 is the Euclidean norm of the matrix T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For the first sum we use estimate (47) for the matrix Λ2T , which implies (53) � 1 = � [R]∈Q(1)k×k JΛ2T [R] ≤ Cd,k Λk(d−k−1) � [R]∈Q(1)k×k d(R)−d/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Recall the following estimate, proved in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='9 in [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' for u ≥ 1 and s > 1 one has (54) u−s � 1≤d(R)≤u d(R)−k + � d(R)≥u d(R)−k−s ≤ C(2 + 1 s − 1) u1−s where the summation is taken over [R] ∈ Q(1)k×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In particular � R d(R)−d/2 ≲ 1 in dimensions d > 2k + 2, thus estimate (27) follows from (29), (51) and estimates (52)-(53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For the mollified estimate (28), we set τ = 2j−l besides Λ = 2l and ε = 2−2l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Again, we note that if q = | det(C)| > 0 and if q | qj i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' if q divides qj then ξ1 ∈ Ωc j,l implies that X ∈ Ωτ,q for X = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , ξd), for the sets Ωj,l and Ωτ,q defined in (12) and (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Using this observation, we have (55) ˆ Ik sup X 1Ωc j,k(ξ1)|θd,k(X + iεT −1, −X, 0)| dX ≲ � d(R)|qj JΛ2T,τ[R] + � d(R)∤qj JΛ2T [R] + � 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' In dimensions d ≥ 2k + 3, using (48) and (54), the first sum on the right side of (55) is crudely estimated by � d(R)|qj JΛ2T,τ[R] ≲ Λk(d−k−1) � 1≤d(R)≤qj � d(R)−kΛ− d−2k 2 + d(R)− d 2 (τΛ)− d−2k 2 � (56) ≲ Λk(d−k−1)� qj2− 3l 2 + 2− 3j 2 � ≲ Λk(d−k−1)2− 3j 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Indeed, qj = lcm{1 ≤ q ≤ 2j} ≈ e2j ≤ 2l as 2j+2 ≤ l by our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' To estimate the second term on the right side of (55), we need the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let j ∈ N and s > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Then (57) � d(R)∤qj d(R)−k−s ≤ C 2j(1−s) j−1 where the constant C may depend on d, k and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' DISCRETE MULTILINEAR MAXIMAL OPERATORS AND PINNED SIMPLICES 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Let (58) Ψ(s) := � [R]∈Q(1)k×k d(R)−k−s = � n≥1 ak(n)n−s, with ak(n) = � d(R)=n d(R)−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' For two Dirichlet series Ψ(s) = � n≥1 a(n)n−s and Φ(s) = � n≥1 b(n)n−s we will write Ψ(s) ⪯ Φ(s) if |a(n)| ≤ b(n) for all n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' It is proved in [7], see (34) Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='9 there, that (59) Ψ(s) ⪯ ζ(s + 1)Kζ(s) =: � n≥1 bK(n)n−s, with K = 2k + k − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Clearly the coefficients of the Dirichlet series ζ(s + 1)Kζ(s) are multiplicative i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' bK(nm) = bK(n)bK(m) if (n, m) = 1, moreover are easy to show that, (60) bK(n) = � m|n dK(m) m , where dK(m) = |{m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' , mk ∈ N : m1m2 · · · mK = m}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Since qj = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' {1 ≤ q ≤ 2j}, if n ∤ qj the either there is a prime p > 2j such that p | n or there is a prime p < 2j such that pγp > 2j but pγp | n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Accordingly, we have the estimate (61) � d(R)∤qj d(R)−k−s = � n∤qj ak(n)n−s ≤ � p>2j � n≥1 bK(pn)p−sn−s + � p<2j � n≥1 bK(pγpn)p−γpsn−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Writing n = prm, the first sum on the right side of (61) is estimated by (62) � p>2j � n≥1 bK(pn)p−sn−s = � p>2j ∞ � r=1 � m≥1,p∤m bK(pr)bK(m)p−rsm−s, using the fact that bK(prm) = bK(pr)bk(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' By (60), we have (63) bK(pr) = 1 + r � s=1 dK(ps) ps ≤ 1 + ∞ � s=1 (s + 1)K 2s ≲ 1, uniformly in r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Thus, for s > 1, (64) � p>2j ∞ � r=1 � m≥1,p∤m bK(pr)bK(m)p−rsm−s ≲ � p>2j p−s ≲ 2j(1−s)j−1, using the fact that the number of primes 2J ≤ p < 2J+1 is bounded by 2J J−1 for all J ≥ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' The second term on the right side of (61) is estimated similarly, except that here we use the fact that pγp > 2j for p < 2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' We have � p<2j � n≥1 bK(pγpn)p−γpsn−s = � p<2j ∞ � r=γp � m≥1,p∤m bK(pr)bK(m)p−rsm−s (65) ≲ � p<2j ∞ � r=γp p−rs ≲ � p<2j p−γps ≲ 2j(1−s)j−1, as the number of primes p < 2j is bounded by 2jj−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Estimate (57) follows immediately from (64)-(65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' □ In dimensions d > 2k + 2, Lemma 4 with s = d/2 − k ≥ 3/2 implies that (66) � d(R)∤qj JΛ2T [R] ≲ Λk(n−k−1)d(R)−d/2 ≲ Λk(n−k−1)2−j/2j−1, with Λ = 2l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Finally, by (52) (55)-(56) and (66) one obtains, in dimensions d > 2k + 2 (67) ˆ Ik sup X 1Ωc j,k(ξ1)|θd,k(X +iεT −1, −X, 0)| dX ≲ Λk(d−k−1)� 2−j/2j−1 +2−3j/2 +2−3l� ≲ Λk(d−k−1)2−j/2j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Estimate (28) follows immediately from (29) and (67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 14 NEIL LYALL ´AKOS MAGYAR ALEX NEWMAN PETER WOOLFITT References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Andrianov, Quadratic Forms and Hecke Operators, Grundlehren der mathematischen Wissenschaften, Springer- Verlag (1987) [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Anderson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Kumchev, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' [11] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Lyall and ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Magyar, Weak hypergraph regularity and applications to geometric Ramsey theory, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=', Series B 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='5 (2022): 160-207 [12] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Lyall, ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Magyar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Newman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Woolfitt, The discrete spherical maximal function: A new proof of ℓ2- boundedness, Proc.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=', v 146/1, (2009) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' 1-34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Raghavan, Modular forms of degree n and representation by quadratic forms, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' (2) 70 (1959), no.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' [18] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Vaughan, The Hardy-Littlewood Method, Second ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=', Cambridge University Press, Cambridge, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content=' Department of Mathematics, The University of Georgia, Athens, GA 30602, USA Email address: lyall@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='uga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='edu Email address: magyar@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='uga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='edu Email address: alxjames@uga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='edu Email address: pwoolfitt@uga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFIT4oBgHgl3EQfyCvK/content/2301.11359v1.pdf'} diff --git a/sdE0T4oBgHgl3EQfbAAV/content/tmp_files/2301.02341v1.pdf.txt b/sdE0T4oBgHgl3EQfbAAV/content/tmp_files/2301.02341v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f35ed790a32e214e0b2c899d2be716b394910722 --- /dev/null +++ b/sdE0T4oBgHgl3EQfbAAV/content/tmp_files/2301.02341v1.pdf.txt @@ -0,0 +1,853 @@ +Alireza Ranjbaran +Ranjbarana@cardiff.ac.uk +Cardiff University School of Biomedical and Life Sciences +Azadeh Nazemi +azadeh1972@gmail.com +Perth, Western Australia +0000-0002-1138-309X +` +A survey on Organoid Image Analysis Platforms +Abstract +An in-vitro cell culture system is used for biological discoveries and hypothesis-driven research on a +particular cell type to understand mechanistic or test pharmaceutical drugs. Conventional in-vitro +cultures have been applied to primary cells and immortalised cell lines plated on 2D surfaces. +However, they are unreliable in complex physiological environments and can not always predict +in-vivo behaviour correctly. Organoids are multicellular spheroids of a primary donor or stem cells +that are replaced in vitro cell culture systems and are widely used in biological, biomedical and +translational studies. Native heterogeneity, microanatomy, and functionality of an organ or diseased +tissue can be represented by three-dimensional in-vitro tissue models such as organoids. Organoids +are essential in in-vitro models for drug discovery and personalised drug screening. Many imaging +artefacts such as organoid occlusion, overlap, out-of-focus spheroids and considerable heterogeneity +in size cause difficulty in conventional image processing. Despite the power of organoid models for +biology, their size and shape have mostly not been considered. Drug responses depend on dynamic +changes in individual organoid morphology, number and size, which means differences in organoid +shape and size, movement through focal planes, and live-cell staining with limited options cause +challenges for drug response and growth analysis. In addition, organoids are imaged on a vast range of +platforms in large structurally complex phenotypes. Hundreds of organoid cultures generate +high-volume images at once, which are challenging to inspect and interpret. Therefore, an automated +coding-free, intuitive and scalable image analysis solution is required. This study primarily introduces +the importance of the role of the organoid culture system in different disciplines of medical science +and various scopes of utilising organoids. Then studies the challenges of operating organoids, +followed by reviewing image analysis systems or platforms applied to organoids to address organoid +utilising challenges. +Keywords: High throughput, Fully automated, Image analysis system, Organoids, Pharmaceutical +drug testing +Introduction +This section briefly introduces utilising organoids in different areas such as organ studies, +investigating physiology, drug testing, cancer biology, precision therapeutics development, antibody +detection, quantifying cellular antigens, responses to chemotherapies and chemoradiation, human +brain development and dysfunction. Organoids are multicellular spheroids of a primary donor or stem +cells. Native heterogeneity, microanatomy, and functionality of an organ or diseased tissue can be +represented by three-dimensional in-vitro tissue models such as organoids. Organoids are required in +in-vitro disease models for drug discovery and personalised drug screening. According to (Clevers, H +et al., 2016), (Schweiger P. J et al.,2016), (Bredenoord, A. L. et al.,2017), and ( Dutta, D. et al.,2017), +In- vitro culture systems have been replaced with using organoid cultures systems. Many researchers +studied organoids of different organs, as Table I indicates: +Table I. Researches and organoid of Organs +Research +Organ + +Cruz-Acuña, R. et al.,2017 Múnera, J. O, et al.,2017; +Gut +Broutier, L. et al.,2016; Grapin‐Botton, A., 2016 and Kim, Y.et al,2016 +Pancreas +Serruya, M. D,2017 +Brain +Skardal, A., et al., 2015 +Liver +Turco, M. Y.,et al,2017 +Endometrium +According to (Shamir, E. R. et al.,2014) and (Skardal, A. et al.,2016), organoids are a suitable model +system for understanding development, investigating physiology, and drug testing.Studying cancer +biology and precision therapeutics development utilises patient-derived tumour organoids (TO) as +high-fidelity models. (Larsen et al., 2021). For more than 20 years, oncology therapy has used +precision medicine or individualised treatment approaches which means identification therapy for the +patient's disease's unique biology, as Table II indicates: +Table II. Researches and organoid of cancer +Research +Cancer type +Gao et al., 2014 +Prostate +Van de Wetering et al., 2015 +Colon +Boj et al., 2015 +Pancreatic +Huang et al., 2015 +Pancreatic +Fujii et al., 2016 +Colon +Broutier et al., 2017 +Hepatobiliary +Turco et al., 2017 +Endometrial +Sachs et al., 2018 +Breast +Tiriac et al., 2018; +Pancreatic +Kijima et al., 2018; Nanki et al., 2018 +Esophagogastric +Romero-Calvo et al., 2019 +Pancreatic +Sachs et al., 2019 +non-small-cell lung cancer (NSCLC) +Boretto et al., 2019 +Endometrial +(Ferguson et al., 2020) hired TOs for drug development and precision medicine studies. They +discovered responses to chemotherapies by observing patient-derived TOs’ mimics. (Ganesh et al., +2019) investigated chemoradiation response. +Brain organoids from Human pluripotent stem cells are self-organised into cytoarchitectural +structures. The three-dimensional models allow studying human brain development and dysfunction +without considering characterisation, spatial information for single-cell, histological analysis, and +whole-tissue analysis(Albanese A. et al., 2020). +Native heterogeneity, microanatomy, and functionality of an organ or diseased tissue can be +represented by three-dimensional in-vitro tissue models such as organoids. Organoids are essential in + +in-vitro models for drug discovery and personalised drug screening. Many imaging artefacts such as +organoid occlusion, overlap, out-of-focus spheroids and considerable heterogeneity in size cause +difficulty in conventional image processing. Despite the power of organoid models for biology, their +size and shape have mostly not been considered. Drug responses depend on dynamic changes in +individual organoid morphology, number and size. It means differences in organoid shape and size, +movement through focal planes, and live-cell staining with limited options cause challenges for drug +response and growth analysis. In addition, organoids are imaged on a vast range of platforms in large +structural complex phenotypic. Hundreds of organoids cultured generate high-volume images at once, +which are challenging to inspect and interpret.About 75% of samples in clinical trials have failed due +to safety and efficacy issues. Hence, a more accurate screening approach is required to improve +sample translation. As a preclinical model, drug discovery pipelines are affected by patient-derived +organoids (PODs). Organoid size in various intra and inter-patient, cellular heterogeneities, temporal +response, and phenotypic evaluation has some drawbacks (Spiller E. R. et al., 2021). Due to a lack of +sufficient knowledge of differential cellular states or genetic backgrounds, studies indicate different +reactions to targeted therapies for tumours. +In 2017 Juan C Caicedo et al. recommended a method for converting the collection of microscopy +images to high-quality image-based (e.g. morphological) profiles to pursue biological discovery, +experimental designs, and laboratory preferences. +In 2017 Andrew M. K. Law et al. designed Andy’s Algorithms to perform batch-processing of +3,3′-diaminobenzidine (DAB) immunohistochemistry, proximity ligation assays (PLAs) and other +standard assays in a quick, accurate and repeatable manner. +In 2018, Borten et al. developed +OrganoSeg to segment, filter and analysis 3D brightfield +images.OrganoSeg quantifies brightfield phenotypes, providing insight into organoid regulation's +/molecular and multicellular mechanisms of organoid regulation. +In 2019, Timothy Kassis et al. implemented OrgaQuant, to locate and quantify the size distribution of +human intestinal organoids in bright field images using a deep convolutional neural network. The +OrgaQuant model does not require any parameters and can analyse thousands of images without +human interaction. +In 2020, Alexandre Albanese et al. presented the SCOUT pipeline for intact cerebral organoids with +an automated multiscale comparative analysis to clear, label, and image intact organoids. +In 2021, Erin R. Spiller et al. proposed an image-based approach and high-content assay to obtain +object-level information on 3D patient-derived tumour organoids without vital dyes requirements. +This method tracks the dynamic response of individual organoids to various drugs in a robust, +nondestructive manner utilising brightfield images at different time points. In addition, they developed +a web-based open-source data visualisation system to simplify the analysis of extensive complex data. +This research proved that imaging, computer vision, and machine learning are practical in biological +research. +In 2021, Brian M. Larsen et al. utilised chemically defined media optimised from over 1,000 patients +to describe the robust pan-cancer TO platform to predict heterogeneity in drug responses for solid +cancers. They used tumour genetic and transcriptomic concordance to accelerate the broad +implementation of organoid models using molecular data, precision medicine research, personalised +therapeutic profiling programs, and defined minimal media for organoid initiation and propagation. +Gritti N. et al. 2021 developed MOrgAna, a python-based platform using machine learning to +segment, quantify and visualise morphological features of high-volume organoid images at once. +In 2022, Jonathan Matthews et al. developed OrganoID, a robust image pixel-by-pixel analysis +platform, in brightfield and phase-contrast microscopy experiments to automatically recognise, label, +and track single organoids. +The next +section +reviews different image analysis platforms or software packages for organoid +image processing and compares the methods and performance. + +Search StrategyMethods: +Several scoping searches were conducted before performing a literature search to provide an +overview of the literature. It was realised during these searches that there were many image +analysis systems used for organoid studies. To ensure no relevant papers were excluded, a +search was conducted to collect as many systems as possible before deciding on the search +terms. This allowed the final search to be broad enough to include all systems from 2003 +until now. +After conducting scoping searches, the research question was proposed to collect a survey of image +organoid image analysis systems addressing drawbacks in using organoids. This was done to ensure +that the research has survey criteria and meets the FINER (Feasible, Interesting, Novel, Ethical, +Relevant) requirement. All relevant repositories were searched for corresponding literature dating +back to 2003. As the research objective is to find a fully automated, high throughput and +general-purpose image analysis system. +Table III. indicates reviewed journals and their relevant +database +Table III. reviewed journals and their relevant database +Journal names +database +Elsevier +PubMed, MEDLINE, EMBASE, and Scopus +CellPress +PubMed, MEDLINE, and the Cochrane +Library +BioRvix +PubMed, +MEDLINE, +EMBASE, +and +Scopus +Scientific Report +PubMed, Web of Science, and Scopus +Frontiers in Oncology +PubMed and Web of Science +Springer +PubMed, Web of Science, and Scopus +The following section briefly explain about the medical databases used by this study. +Pubmed: +PubMed is a free search engine accessing the MEDLINE database of references and abstracts +on life sciences and biomedical topics primarily. The United States National Library of +Medicine (NLM) at the National Institutes of Health maintains the database as part of the +Entrez system of information retrieval. PubMed indexes more than 28 million citations for +biomedical literature from MEDLINE, life science journals, and online books. Citations may +include links to full-text content from PubMed Central and publisher websites. +EMBASE: +EMBASE is a biomedical and pharmacological database that indexes journal articles and +conference proceedings. It is produced by Elsevier and is available via the Elsevier products +ScienceDirect and Scopus. EMBASE covers a wide range of topics, including drug + +discovery, pharmacology, toxicology, epidemiology, and healthcare. It contains over 22 +million records, dating back to 1947. EMBASE is updated weekly and covers over 8,000 +journals, +making +it +an +essential +resource +for +researchers +in +the +biomedical +and +pharmacological sciences. +Web of Science: +Developed by the Institute for Scientific Information, the Web of Science is a platform that +allows users to access research data and citation information. The platform includes the +Science Citation Index, the Social Sciences Citation Index, and the Arts & Humanities +Citation Index. The Web of Science also provides access to the Conference Proceedings +Citation Index and the Book Citation Index. The platform is designed to help users identify, +evaluate, and use scholarly research data. +Scopus: +Scopus is a large abstract and citation database of peer-reviewed literature: scientific journals, +books and conference proceedings. Delivering a comprehensive overview of the world's +research output in the fields of science, technology, medicine, social sciences, and arts and +humanities, Scopus features smart tools to track, analyze and visualize research. +Cochrane Library +The Cochrane Library is a collection of six databases that contain different types of medical +information. The databases are The Cochrane Database of Systematic Reviews (CDSR): This +database contains reviews of healthcare interventions, such as drugs, medical devices, and +surgeries. The Cochrane Central Register of Controlled Trials (CENTRAL): This database +contains information on clinical trials. The Cochrane Methodology Register (CMR): This +database contains information on methods used in healthcare research. The Health +Technology Assessment Database (HTA): This database contains information on the +effectiveness and safety of health technologies, such as drugs, medical devices, and +diagnostic tests. The NHS Economic Evaluation Database (NHS EED): This database +contains information on the cost-effectiveness of healthcare interventions. The Joanna Briggs +Institute EBP Database (JBI): This database contains information on evidence-based practice. +Result +This section presents a summary review of some image analysis pipelines for organoids. Table IV +denotes these platforms, scopes, techniques and properties in terms of highlighted keywords for this +research +Table IV. Image analysis platform for organoids +Platform +Applied to/purposes +Technique +Batch +processing/High +throughput +Fully +automated +Metamorph +Time-lapse, +multi-dimensional +acquisition and 3D reconstruction on +microscopic image analysis of live +cells +Image processing +Not tell +Manual user interface +Yes + +Caicedo +Biological systems on a large scale +by +using +chemical +and +genetic +perturbations. +Data-analysis strategies for +image-based cell profiling +Yes +Yes +CALYPSO +Intrinsic +heterogeneity +of +cancer +tissues +Image processing, +Yes +Not tell +Andy’s Algorithm +Oncology drug development +Image processing +Yes +Yes +OrganoSeg +Brightfield 3D organoid populations +Morphometry +Standalone +Not say +Yes +OrgaQuant +Human +Intestinal +Organoid +Localization and Quantification +Faster R-CNN +Not tell +Yes +SCOUT +3D phenotyping of human cerebral +organoids +UNet +Yes +Yes +Larsen +Remove vital dye requirements. +Pix2Pix GAN +Yes +Yes +OBIA +Drug +treatment +responses +on +Patient-Derived Organoids (PDOs) +Supervised linear classifier +to separate live and dead +Yes +Yes +MOrgAna +Segmentation +and +quantifying +organoid brightfield images. +Gaussian /Laplacian filters, +Logistic regression +Yes +Yes +OrganoID +Pancreatic +ductal +adenocarcinoma +(PDAC) +modified UNet +Yes +Yes +Literature review +Utilising organoids assists scientists in different areas such as organ studies, investigating physiology, +cancer biology, precision therapeutics development, responses to chemotherapies and chemoradiation. +Organoids can be imaged on platforms such as benchtop stereoscopes or high-content confocal-based. +Investigation, interpretation and inspection of high-volume organoids are problematic. Data collection +from various time points and conditions among thousands of regions of interest is still challenging for +biologists. Using organoids still faces some drawbacks that need to be investigated. Some problems +regard ignorance of organoid size, shape, and morphological status. Furthermore, images have been +provided in a high volume in quantity requiring automatic batch processing. Therefore, +Fully +automated high throughput image analysis pipelines for organoids are required. Image analysis is a +powerful tool in many medical disciplines. Today, researchers use machine learning to improve image +processing results. The software using machine learning are MetaMorph, Imaris, Harmony, ZEN, FIJI, +CellProfiler and ilastik (Berg et al., 2019; Schindelin et al., 2012; Carpenter et al., 2006). Some +identify thousands of cells in vast fields and extract biologically related features (McQuin et al., +2018). +Metamorph(2003) +MetaMorph is a microscopic image analysis tool to manipulate various image processing tasks such as +intensity logging, advanced morphology, colocalisation, image stack alignment, montage generation, +movie making, 3D rendering, image colour combination, light equalisation, topographical surface +map generation, convolve and deconvolve images, arithmetic operations, orthogonal planes +visualisation, image stitching and 3D calculation. + +The graphical interface offers colour combination, integrated morphometry analysis, region +measurements and counting cells. Metamorph has many optional modules such as counting nuclei, +neurite outgrowth analysis, tracking motile cells or assessing cell cycle phases. +MetaMorph software is a quick, practical and robust system. It performs time-lapse and +multi-dimensional acquisition and 3D reconstruction. +Furthermore, Methamorph has brightening operations for biological experiments applying to live cell +imaging. +Caicedo research +In 2017, Juan C Caicedo et al. recommended a method for converting the collection of microscopy +images +to +high-quality +image-based +(e.g. +morphological) +profiles +to biological discovery, +experimental designs, and laboratory preferences. +Image-based cell profiling is a high-throughput microscopy system for quantifying phenotypic +differences among various cell populations. It uses chemical and genetic perturbations and provides +an approach to studying biological systems on a large scale (Caicedo, J.C et al., 2017). +Their image analysis system performs two main steps: +1-Illumination detection and correction using retrospective multi-image. In the centre of the field of +view, pixels are brighter than those on the edges. +2-Pixel foreground or background classification using image segmentation. +Figure. 1 illustrates the main Caicedo steps +Figure.1 Caicedo overview +Comprehensive image analysis procedure for structurally complex organotypic cultures +(CALYPSO) +In 2017, Anne-Laure Bulin et al. proposed a comprehensive high-volume image analysis system. +They aimed to study complex heterotypic organoids to discover the therapeutic efficacy +architecturally. They extracted fluorescence intensity-based parameters for tumour treatment. +CALYPSO measures the organoid area without considering the shape and volumetric estimations. For +binarization, adaptive thresholding was used instead of a constant threshold for accurate tumour +nodules detection. To eliminate out-of-focus objects, filtering has been done by adding fluorescence +channels and binarising Otsu’s threshold method. The obtained binary images were multiplied with +initial masks. +Organoids are indexed based on final masks. These indices are unique for each organoid. CALYPSO +calculates tumour nodules viability factor using fluorescence intensity. This factor is varied between +zero and one and indicates organoids' health. +CALYPSO performed the following steps for image analysis: masking, background subtraction, +threshold calculation and finally, live area thresholding. Viability and live area fraction are used for +correlated analysis. Live area fraction is obtained from the rate of the fraction of live area to total area. +Viability is obtained from the rate of live intensity to the total value of live intensity and dead +intensity. +Figure.2 illustrates the main steps of CALYPSO + +Image illumination detection and correction-->Pixel foreground or background classification -->lmage segmentationAdaptive threshod-->Filtering-->Adding fluorescence channels-->Otsu thresholdFigure.2 CALYPSO overview +Andy’s Algorithm +Image analysis based on antibody detection faces many challenges in quantifying cellular antigens +and obtaining accurate high-throughput quantitation, specifically for users with insufficient image +processing skills. Cognitive bias and scientific reproducibility are emerging problems in oncology +drug development. +In 2017 Andrew M. K. Law et al. designed Andy’s Algorithms to address the stated issues. This +algorithm performs batch-processing of 3,3′-diaminobenzidine (DAB) immunohistochemistry, +proximity ligation assays (PLAs) and other standard assays in a quick, accurate and repeatable +manner. +Andy’s algorithm performs optimisation before image processing to determine optimal parameters. +These parameters are used to address the problems caused by variation of cell types, tissue type, +fixation method, antigen detection method, staining pattern, colour discrimination, magnification, +illumination settings, and resolution. +Andy’s algorithm provides interactive environments for users with basic knowledge. It was designed +for compassionate pre-clinical oncology purposes. Andy’s algorithm is a high throughput and iterative +image analysis platform for biology, this platform does not require advanced image processing skills. +OrganoSeg +In 2018, Borten et al. developed OrganoSeg as an open source and parameters based on conventional +image processing techniques to segment, filter and analyse 3D brightfield images. OrganoSeg +quantifies brightfield phenotypes, providing insight into the molecular and multicellular mechanisms +of organoid regulation. Its high performance has been approved by classifying 5167 breast-cancer +spheroids and 5743 colon cancer cells. +OrganoSeg works on jpg, png or tiff format, grayscale images, performs an open-close morphological +function, smooths bulk components preserving sharp spheroid boundaries, and then binarises the +smoothed image by local adaptive thresholding or Otsu’s method. Users are enabled to find minimum +threshold size and separate poor growth organoids by segmentation. OrganoSeg extracts up to 23 +morphometry standard measures for downstream. It works with 3D-cultured breast images or movies, +brightfield, phase-contrast, and differential interference contrast images. OrganoSeg, regardless of +confocal or high contrast fluorescence micrographs, translates obtained images to measurable +datasets. +High +volume +organoid +shape +analysis +generates a collection of non-continuous +morphological references. Combining OrganoSeg and RNA sequencing indicated a relationship in +colorectal cancer. +OrganoSeg is standalone with a graphical user interface to quantify organoid +cultures and 3D spheroid transmitted-light images. +Figure.3 illustrates the main OrganoSeg steps. +Figure.3 OrganoSeg overview +OrgaQuant + +Grayscale-->Open/close morphology-->Adaptive Threshols-->Superimpose-->Binarise-->Filter noise-->Fill holes-->Smooth-->ldentify Rol-->Extract metricsIn 2019, Timothy Kassis et al. described OrgaQuant implementation, a deep convolutional neural +network to locate and quantify bright field human intestinal organoids images distribution size. To +train OrgaQuant, they manually annotated human intestinal organoid images and generated a dataset. +They conducted this research to address the lack of technique to localise and quantify organoids in a +3D environment and the presence of many imaging artefacts (Kassis T. et al., 2019). They used +extracted features from a neural network (NN) for Clustering or subtle changes detection in organoid +morphology. +Object detection is a complicated process in computer vision, particularly in cases with speed +concerns. Some developed detection algorithms have been designed to be quick and accurate such as +single Shot Multibox Detector (SSD) and You Only Look Once (YOLO). OrgaQuant was designed +based on the Region Convolutional Neural Network (R-CNN) and or faster R-CNN. Faster R-CNN +uses ResNet and Inception architecture. +OrgaQuant for pre-training utilises COCO box annotations and needs an extensive dataset provided +by augmentation. OrgaQuant automatically localises an organoid within a brightfield image and +annotates the bounding box. Downstream image processing uses cropped organoid images. This +pipeline measures the intestinal organoid size. +OrgaQuant facilitates tracking organoid growth kinetics in droplets during the time. OrgaQuant is a +basic, intelligent and open-source organoid quantification technique. an open source and parameters +based on conventional image processing techniques. +The OrgaQuant model does not require any parameters and can analyse thousands of images without +human interaction. Figure.4 illustrates the main OrgaQuant steps +Figure.4 illustrates the main OrgaQuant steps +Single-cell and Cytoarchitecture analysis of Organoids using Unbiased Techniques (SCOUT) +In 2020, Alexandre Albanese et al. presented the SCOUT pipeline for intact cerebral organoids with +an automated multiscale comparative analysis to clear, label and image organoids. From the +microscopic fluorescence dataset, SCOUT CNN extracts hundreds of features characterising +molecular, cellular, spatial, cytoarchitectural, and organoid-wide properties. SCOUT performed a +comprehensive analysis of 46 intact organoids and ~100 million cells and revealed quantitative +multiscale "phenotypes" for organoid development and Zika virus infection. +Imaging each organoid takes about 15 minutes for reliable, accurate single-cell analysis and 300 +multiscale features takes about six hours. +SCOUT has performed quantifying spatial features such as cellular context, ventricle morphology, +cytoarchitecture distribution, and uncommon events detection. Zika virus infection decreases cell +population, and this pipeline quantifies this reduction. +SCOUT has three main modules: single cell analysis, regional architecture analysis, whole organoid +analysis, and finding correlation. +UNet was slightly modified, and two layers were added before and after the UNet bottleneck to +improve the model. Then modified UNet was trained using weighted binary class entropy (WBCE). +WBCE support model to converge to accurate ventricle segment. This model was used to segment +ventricles automatically. Experimental results denote 97.2% Dice score similarity for UNet. +Quantifying multiscale feature correlation, maturation-related changes, protocol comparisons, and + +Brightfield image--> Faster R-CNN -->Localised organoidZika virus pathology demonstrated SCOUT's power. SCOUT is a versatile platform for automatic +single-cell analysis. +Figure.5 illustrates the main steps of SCOUT +Figure5. SCOUT overview +Larsen +In +2021, +Larsen +et +al. +proposed +a +pan-cancer +precision +medicine +organoid +platform. +Genomic/transcriptomic fidelity has been extracted from over 1000 patients' tumour organoid(TOs) +cultures. They determined solid tumour chemical minimal media. They studied growth factors to +initiate and propagate organoids. Their results indicated that Epidermal Growth Factor (EGF) and +Noggin are sufficient for most organoid cultures. They developed a convolutional neural +network(CNN) based on pix2pix Generative Adversarial Network (GAN) and predicted label-free +light microscopy drug response. CNN contains a fluorescence generator and a viability discriminator. +Pix2pix GAN accuracy has been approved in cell biology by Isola et al. 2017; Tsuda and Hotta, 2019. +They adopted +70x70 patch GAN as a viability discriminator (Isola et al., 2017). +Concatenated +brightfield and fluorescent are inputs of the discriminator. The generator gets a brightfield image as +input and generates fluorescence. +Realistic image prediction and image translation from the black and white domain to the colourful +domain is possible by +Pix2Pix (Isola et al., 2017), conditional generative adversarial networks +(GANs), and CycleGAN (Zhu et al., 2017). +This technique was developed as a light-microscopy-based assay to omit costly vital dye +requirements. High content image analysis was performed by inverting confocal microscopy images, +followed by ability measurements. Then binary classification for TO viability was executed for drug +screening optimisation. Figure.6 illustrates the main Larsen steps +Figure.6 Larsen steps overview +Object-Based Image Analysis (OBIA) +In 2021, Erin R. Spiller et al. developed an object-based image analysis (OBIA) system to convert +cells to population-level analysis and investigate disturbances in drug treatment responses of +heterogeneous object-based Patient-Derived Organoids (PDOs). It observes phenotypic changes. +OBIA considered the textural and morphological features from brightfield images as discriminators +between dead and live organoids. They used a supervised linear classifier to separate live and dead +organoids for drug response recognition. The OBIA ML image analysis was initially performed in a +small sample set. They designed a tool to visualise the differences of all collected features over time. +The OBIA team developed a web-based Application Programming Interface (API) to upload machine +learning output analysis and download feature metrics and survival curves. +They proposed a label-free high content screening (HCS) robust, on-time approach working with +colorectal cancer (CRC) organoid live-cell image. It is an automated dynamic cellular visualisation +and multiparametric extraction data system. + +Gaussian Blur -->Convert to 8bit-->Threshold-->Watershed-->Fill holes-->Particle anlysise-->SavaBrightfield image -->pix2pix GAN generator--> fluorescenceMoreover, this method can be scaled to apply to high-volume drug screen data of different cancer +types PDOs. OBIA has done computational intensive image analysis automatically and iteratively. +Figure.7 illustrates the main OBIA steps +Figure.7 OBIA overview +MOrgAna +Gritti N. et al. 2021 developed MOrgAna, a python-based platform using machine learning to +segment, quantify and visualise morphological features of high-volume organoid images at once. +MOrgAna is an appropriate package for users with basic to advanced skills. +So, this technology assists in atlas-free 3D whole-organoid analysis to gain quantitative comparative +analysis between experimental conditions. Brightfield image availability, in contrast, fluorescence is +the reason behind utilising them. MOrgAna monitors the organoid in bright field mode for biomedical +purposes. The training dataset includes morphological parameters, Gaussian and Laplacian filters, +Logistic regression, improved flexibility and feasibility. This pipeline is designed on a multilayer +perceptron for organoid recognition. +Figure.8 illustrates the main MOrgAna steps +Figure.8 MOrgAna overview +OrganoID +In 2022, Jonathan Matthews et al. developed OrganoID, a robust image pixel-by-pixel analysis +platform, in brightfield and phase-contrast microscopy experiments to automatically recognise, label, +and track single organoids. This model was trained based on pancreatic cancer organoid images and +tested on images of pancreatic, lung, colon, and adenoid cystic carcinoma organoids. OrganoID is an +open-source image analysis platform which determines organoid morphology pixel-wised. This +platform does not require fluorescence or transgenic labelling. It can analyse different organoid types +in microscopy experiments promptly. It is designed based on a modified UNet and has minimum +layers for quick performance. The training dataset was pancreas cancer samples. Testing validating +images come from pancreatic, lung, colon, and adenoid cystic carcinoma with more than 89% +tracking accuracy for four days of the validation period. OrganoID performs two highlighted tasks: +individual organoid tracking over time and Drug screening experiments. General image processing +steps with OrganoID are contour detection, single and bulk organoid separation and determination of +response over time by time-lapse image sequence. CNN has been designed to indicate the presence of +organoid probability in each pixel. UNet architecture includes layers of convolutional, maximum +filtering, concatenation, extraction and localisation features. +The training dataset was obtained from 66 manually organoid microscopy images of pancreatic ductal +adenocarcinoma (PDAC), then collected samples divided into 80% training and 20% validating and +augmentation applied to the training dataset. The OrganoID platform effectively and accurately uses +organoid images in high-volume data physiological research. + +Figure.9 is an illustration of the main steps of OrganoID +Figure.9 OrganoID overview +Critical Analysis +Reviewing image analysis pipelines for organoids of 11 researches indicates that since 2019 four +studies have used deep learning for organoid analysis. Generally using deep learning, in contrast, +traditional methods require larger training datasets, which is expensive, time-consuming, and requires +expert annotation. Although such training dataset preparation needs more effort, the result of models +will be more robust and efficient. +Almost all pipelines are automated, and more than 75% of them are high throughput and can process a +high number of images at once. +As Table IV indicates, these pipelines have been designed for different targets. Since five pipelines +specifically have been designed for oncology studies; thus it denotes organoid significance in cancer +studies. Almost all projects consider organoids’ morphology for different purposes, such as drug +response monitoring. They address the issues regarding ignorance of organoid shape and size. +Conclusion and future development +In this study, we have introduced the vast range of organoids usage in biology and medical research, +such as organ studies, investigating physiology, drug testing, cancer biology, precision therapeutics +development, +antibody +detection, +quantifying +cellular +antigens, +responses +to +chemotherapies/chemoradiation, and human brain development /dysfunction. +Then briefly describe the drawbacks of using organoids and finally review eleven designed image +analysis systems to solve the drawbacks. Since 2014, more than 23 researches have been conducted to +observe different organs' functions and dysfunction using organoids. +Organoids can be imaged on platforms such as benchtop stereoscopes or high-content confocal-based +microscopes. Investigation, interpretation and inspection of high-volume organoids are problematic. +Data collection from various time points and conditions among thousands of regions of interest is still +challenging for biologists. Some problems regard ignorance of organoid size, shape, and +morphological status. Hence, the essential properties of an organoid image analysis pipeline are fully +automated, high throughput, and considering morphology features. We reviewed the development of +image analysis systems for Organoids in biological disciplines. Starting from 2003 with MetaMorph +development which was able to perform some simple image processing tasks such as montage, +rendering, alignment, colour combination and arithmetic operation robustly and quickly. Proceeding +to 2017, Caicedo et al. developed another system to encourage biological observation and +experimental laboratory preference, followed by the development of CALYPSO to measure the +viability of an organoid to discover its therapeutic efficacy. At the same time, Andy’s algorithm came +as a high throughput, iterative and user-friendly image analysis biology platform for compassionate +pre-clinical oncology purposes. In 2018, OrganoSeg was released as a standalone graphical +high-volume platform to extract metrics from organoids. OrganoQuant in 2019 was designed based on +faster RCNN to address the issues for organoid quantification regarding image artefacts. In 2020, +SCOUT was revealed as a versatile platform based on UNet and came in 3 modules: single cell, +regional and whole organoid to automatically segment ventricles. In 2021, Larcen’s pan-cancer +organoid platform was designed for precision medicine based on pix2pix GAN network image +translation. This platform was developed to remove costly vital dye requirements and optimise drug +screening. OBIA was released for drug response recognition levels by classifying dead and live + +Neural Network predication map-->Canny edge detection--> Watershedorganoids. The OBIA team also developed a visualisation tool and web-based API to address the +biologist's +challenges +regarding +high-volume +data. MOrgAna was developed for organoid +segmentation, quantification and visualisation. OrganoID was designed for organoid tracking and drug +screening based on UNet. +Eleven projects have been studied. These projects aimed to provide organoid image analysis systems. +Regardless of the area these pipelines are designed for, they all meet the general criteria for organoid +image analysis systems. These criteria are being automated, high throughput and considering organoid +morphology. +Four out of eleven pipelines used Deep Neural Network(DNN). DNN, despite being more expensive, +expert-required annotation, time-consuming preparation generates reliable results. For instance, +Figure 10 illustrates the results of computed tomography (CT) of abdominal multi organs image +segmentation +using UNet (we developed the UNet model prior to this research to evaluate the +segmentation accuracy of this model). This figure shows spleen, right kidney and left kidney +segmentation from top to bottom. As observed, segmentation accuracy decreased from top to bottom +due to reducing the number of images in the training dataset. +Figure. 10 UNet abdominal CT scan organs segmentation +As these pipelines followed different targets, comparing them in terms of performance can not be +correct. We can only compare them based on image processing techniques or can compare pipelines +with common targets/scops, as Table V indicates. +The summary of nine reviewed pipelines shows in Table V. +Table V. pipelines overview +Pipelines +Initial Image processing steps + +Caicedo +Image illumination detection and correction →pixel background/ foreground +classification +CALYPSO +Adaptive threshold→ Filtering →Adding fluorescence channel→Otsu +threshold +OrganoSeg +Grayscale→open/close Morphology→Adaptive +threshold→Superimpose→Filter noise and holes→Smooth→ROI +OrgaQuant +Brightfield images→Faster RCNN→Localised organoid +SCOUT +Gaussian →8 bits→Threshold→Watershed→Fill holes→Particle analysis +Larsen +Brightfield images→Pix2pix GAN→generator→Fluorescence +OBIA +Brightfield images→ Morphological features→Linear classifier +MOrgAna +Brightfield images→ Morphological features→Laplacian and Gaussian +filters→Logistic Regression +OrganoID +NN prediction map→Canny edge→ Watershed +Future work and further development +Since during this study, despite a limited training dataset, we successfully developed UNet to be +familiar with the NN model design. We could get reasonable cross-validation results; thus, further +development will be conducted to design a trustworthy system to classify live and dead cells using +Pix2Pix GAN in the oncology area. This system includes two networks, a UNet as a generator and +ResNet as a discriminator to classify gold standard and live/dead masks generated by the generator. +The generator accepts brightfield images and generates live/dead masks. Two models will be trained +by this method: one model generates masks of live cells, and another one generates masks of dead +cells. For providing training datasets for two models, we can use CALYPSO or OBIA. These +pipelines support us in saving time and budget. Then +brightfield images and dead/live organoid +masks will be concatenated as input and output of the generator. 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(2022). +OrganoID: a versatile deep learning platform for organoid image analysis. bioRxiv. + diff --git a/sdE0T4oBgHgl3EQfbAAV/content/tmp_files/load_file.txt b/sdE0T4oBgHgl3EQfbAAV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b57d2778fa27ed4d43257ba90fc50f03293f45d9 --- /dev/null +++ b/sdE0T4oBgHgl3EQfbAAV/content/tmp_files/load_file.txt @@ -0,0 +1,1036 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf,len=1035 +page_content='Alireza Ranjbaran Ranjbarana@cardiff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='uk Cardiff University School of Biomedical and Life Sciences Azadeh Nazemi azadeh1972@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='com Perth, Western Australia 0000-0002-1138-309X ` A survey on Organoid Image Analysis Platforms Abstract An in-vitro cell culture system is used for biological discoveries and hypothesis-driven research on a particular cell type to understand mechanistic or test pharmaceutical drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Conventional in-vitro cultures have been applied to primary cells and immortalised cell lines plated on 2D surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' However, they are unreliable in complex physiological environments and can not always predict in-vivo behaviour correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Organoids are multicellular spheroids of a primary donor or stem cells that are replaced in vitro cell culture systems and are widely used in biological, biomedical and translational studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Native heterogeneity, microanatomy, and functionality of an organ or diseased tissue can be represented by three-dimensional in-vitro tissue models such as organoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Organoids are essential in in-vitro models for drug discovery and personalised drug screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Many imaging artefacts such as organoid occlusion, overlap, out-of-focus spheroids and considerable heterogeneity in size cause difficulty in conventional image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Despite the power of organoid models for biology, their size and shape have mostly not been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Drug responses depend on dynamic changes in individual organoid morphology, number and size, which means differences in organoid shape and size, movement through focal planes, and live-cell staining with limited options cause challenges for drug response and growth analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' In addition, organoids are imaged on a vast range of platforms in large structurally complex phenotypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Hundreds of organoid cultures generate high-volume images at once, which are challenging to inspect and interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Therefore, an automated coding-free, intuitive and scalable image analysis solution is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' This study primarily introduces the importance of the role of the organoid culture system in different disciplines of medical science and various scopes of utilising organoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Then studies the challenges of operating organoids, followed by reviewing image analysis systems or platforms applied to organoids to address organoid utilising challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Keywords: High throughput, Fully automated, Image analysis system, Organoids, Pharmaceutical drug testing Introduction This section briefly introduces utilising organoids in different areas such as organ studies, investigating physiology, drug testing, cancer biology, precision therapeutics development, antibody detection, quantifying cellular antigens, responses to chemotherapies and chemoradiation, human brain development and dysfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Organoids are multicellular spheroids of a primary donor or stem cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Native heterogeneity, microanatomy, and functionality of an organ or diseased tissue can be represented by three-dimensional in-vitro tissue models such as organoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Organoids are required in in-vitro disease models for drug discovery and personalised drug screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' According to (Clevers, H et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2016), (Schweiger P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' J et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=',2016), (Bredenoord, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=',2017), and ( Dutta, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=',2017), In- vitro culture systems have been replaced with using organoid cultures systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Many researchers studied organoids of different organs, as Table I indicates: Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Researches and organoid of Organs Research Organ Cruz-Acuña, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=',2017 Múnera, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' O, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=',2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Gut Broutier, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=',2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Grapin‐Botton, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2016 and Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='et al,2016 Pancreas Serruya, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' D,2017 Brain Skardal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2015 Liver Turco, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=',et al,2017 Endometrium According to (Shamir, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=',2014) and (Skardal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=',2016), organoids are a suitable model system for understanding development, investigating physiology, and drug testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='Studying cancer biology and precision therapeutics development utilises patient-derived tumour organoids (TO) as high-fidelity models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' (Larsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=" For more than 20 years, oncology therapy has used precision medicine or individualised treatment approaches which means identification therapy for the patient's disease's unique biology, as Table II indicates: Table II." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Researches and organoid of cancer Research Cancer type Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2014 Prostate Van de Wetering et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2015 Colon Boj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2015 Pancreatic Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2015 Pancreatic Fujii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2016 Colon Broutier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2017 Hepatobiliary Turco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2017 Endometrial Sachs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2018 Breast Tiriac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Pancreatic Kijima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Nanki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2018 Esophagogastric Romero-Calvo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2019 Pancreatic Sachs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2019 non-small-cell lung cancer (NSCLC) Boretto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2019 Endometrial (Ferguson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2020) hired TOs for drug development and precision medicine studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' They discovered responses to chemotherapies by observing patient-derived TOs’ mimics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' (Ganesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2019) investigated chemoradiation response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Brain organoids from Human pluripotent stem cells are self-organised into cytoarchitectural structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The three-dimensional models allow studying human brain development and dysfunction without considering characterisation, spatial information for single-cell, histological analysis, and whole-tissue analysis(Albanese A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Native heterogeneity, microanatomy, and functionality of an organ or diseased tissue can be represented by three-dimensional in-vitro tissue models such as organoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Organoids are essential in in-vitro models for drug discovery and personalised drug screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Many imaging artefacts such as organoid occlusion, overlap, out-of-focus spheroids and considerable heterogeneity in size cause difficulty in conventional image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Despite the power of organoid models for biology, their size and shape have mostly not been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Drug responses depend on dynamic changes in individual organoid morphology, number and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' It means differences in organoid shape and size, movement through focal planes, and live-cell staining with limited options cause challenges for drug response and growth analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' In addition, organoids are imaged on a vast range of platforms in large structural complex phenotypic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Hundreds of organoids cultured generate high-volume images at once, which are challenging to inspect and interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='About 75% of samples in clinical trials have failed due to safety and efficacy issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Hence, a more accurate screening approach is required to improve sample translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' As a preclinical model, drug discovery pipelines are affected by patient-derived organoids (PODs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Organoid size in various intra and inter-patient, cellular heterogeneities, temporal response, and phenotypic evaluation has some drawbacks (Spiller E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Due to a lack of sufficient knowledge of differential cellular states or genetic backgrounds, studies indicate different reactions to targeted therapies for tumours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' In 2017 Juan C Caicedo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' recommended a method for converting the collection of microscopy images to high-quality image-based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' morphological) profiles to pursue biological discovery, experimental designs, and laboratory preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' In 2017 Andrew M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' designed Andy’s Algorithms to perform batch-processing of 3,3′-diaminobenzidine (DAB) immunohistochemistry, proximity ligation assays (PLAs) and other standard assays in a quick, accurate and repeatable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' In 2018, Borten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' developed OrganoSeg to segment, filter and analysis 3D brightfield images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content="OrganoSeg quantifies brightfield phenotypes, providing insight into organoid regulation's /molecular and multicellular mechanisms of organoid regulation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' In 2019, Timothy Kassis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' implemented OrgaQuant, to locate and quantify the size distribution of human intestinal organoids in bright field images using a deep convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The OrgaQuant model does not require any parameters and can analyse thousands of images without human interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' In 2020, Alexandre Albanese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' presented the SCOUT pipeline for intact cerebral organoids with an automated multiscale comparative analysis to clear, label, and image intact organoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' In 2021, Erin R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Spiller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' proposed an image-based approach and high-content assay to obtain object-level information on 3D patient-derived tumour organoids without vital dyes requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' This method tracks the dynamic response of individual organoids to various drugs in a robust, nondestructive manner utilising brightfield images at different time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' In addition, they developed a web-based open-source data visualisation system to simplify the analysis of extensive complex data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' This research proved that imaging, computer vision, and machine learning are practical in biological research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' In 2021, Brian M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Larsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' utilised chemically defined media optimised from over 1,000 patients to describe the robust pan-cancer TO platform to predict heterogeneity in drug responses for solid cancers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' They used tumour genetic and transcriptomic concordance to accelerate the broad implementation of organoid models using molecular data, precision medicine research, personalised therapeutic profiling programs, and defined minimal media for organoid initiation and propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Gritti N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' 2021 developed MOrgAna, a python-based platform using machine learning to segment, quantify and visualise morphological features of high-volume organoid images at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' In 2022, Jonathan Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' developed OrganoID, a robust image pixel-by-pixel analysis platform, in brightfield and phase-contrast microscopy experiments to automatically recognise, label, and track single organoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The next section reviews different image analysis platforms or software packages for organoid image processing and compares the methods and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Search StrategyMethods: Several scoping searches were conducted before performing a literature search to provide an overview of the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' It was realised during these searches that there were many image analysis systems used for organoid studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' To ensure no relevant papers were excluded, a search was conducted to collect as many systems as possible before deciding on the search terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' This allowed the final search to be broad enough to include all systems from 2003 until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' After conducting scoping searches, the research question was proposed to collect a survey of image organoid image analysis systems addressing drawbacks in using organoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' This was done to ensure that the research has survey criteria and meets the FINER (Feasible, Interesting, Novel, Ethical, Relevant) requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' All relevant repositories were searched for corresponding literature dating back to 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' As the research objective is to find a fully automated, high throughput and general-purpose image analysis system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' indicates reviewed journals and their relevant database Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' reviewed journals and their relevant database Journal names database Elsevier PubMed, MEDLINE, EMBASE, and Scopus CellPress PubMed, MEDLINE, and the Cochrane Library BioRvix PubMed, MEDLINE, EMBASE, and Scopus Scientific Report PubMed, Web of Science, and Scopus Frontiers in Oncology PubMed and Web of Science Springer PubMed, Web of Science, and Scopus The following section briefly explain about the medical databases used by this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Pubmed: PubMed is a free search engine accessing the MEDLINE database of references and abstracts on life sciences and biomedical topics primarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The United States National Library of Medicine (NLM) at the National Institutes of Health maintains the database as part of the Entrez system of information retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' PubMed indexes more than 28 million citations for biomedical literature from MEDLINE, life science journals, and online books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Citations may include links to full-text content from PubMed Central and publisher websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' EMBASE: EMBASE is a biomedical and pharmacological database that indexes journal articles and conference proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' It is produced by Elsevier and is available via the Elsevier products ScienceDirect and Scopus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' EMBASE covers a wide range of topics, including drug discovery, pharmacology, toxicology, epidemiology, and healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' It contains over 22 million records, dating back to 1947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' EMBASE is updated weekly and covers over 8,000 journals, making it an essential resource for researchers in the biomedical and pharmacological sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Web of Science: Developed by the Institute for Scientific Information, the Web of Science is a platform that allows users to access research data and citation information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The platform includes the Science Citation Index, the Social Sciences Citation Index, and the Arts & Humanities Citation Index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The Web of Science also provides access to the Conference Proceedings Citation Index and the Book Citation Index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The platform is designed to help users identify, evaluate, and use scholarly research data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Scopus: Scopus is a large abstract and citation database of peer-reviewed literature: scientific journals, books and conference proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=" Delivering a comprehensive overview of the world's research output in the fields of science, technology, medicine, social sciences, and arts and humanities, Scopus features smart tools to track, analyze and visualize research." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Cochrane Library The Cochrane Library is a collection of six databases that contain different types of medical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The databases are The Cochrane Database of Systematic Reviews (CDSR): This database contains reviews of healthcare interventions, such as drugs, medical devices, and surgeries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The Cochrane Central Register of Controlled Trials (CENTRAL): This database contains information on clinical trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The Cochrane Methodology Register (CMR): This database contains information on methods used in healthcare research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The Health Technology Assessment Database (HTA): This database contains information on the effectiveness and safety of health technologies, such as drugs, medical devices, and diagnostic tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The NHS Economic Evaluation Database (NHS EED): This database contains information on the cost-effectiveness of healthcare interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The Joanna Briggs Institute EBP Database (JBI): This database contains information on evidence-based practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Result This section presents a summary review of some image analysis pipelines for organoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Table IV denotes these platforms, scopes, techniques and properties in terms of highlighted keywords for this research Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Image analysis platform for organoids Platform Applied to/purposes Technique Batch processing/High throughput Fully automated Metamorph Time-lapse, multi-dimensional acquisition and 3D reconstruction on microscopic image analysis of live cells Image processing Not tell Manual user interface Yes Caicedo Biological systems on a large scale by using chemical and genetic perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Data-analysis strategies for image-based cell profiling Yes Yes CALYPSO Intrinsic heterogeneity of cancer tissues Image processing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Yes Not tell Andy’s Algorithm Oncology drug development Image processing Yes Yes OrganoSeg Brightfield 3D organoid populations Morphometry Standalone Not say Yes OrgaQuant Human Intestinal Organoid Localization and Quantification Faster R-CNN Not tell Yes SCOUT 3D phenotyping of human cerebral organoids UNet Yes Yes Larsen Remove vital dye requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Pix2Pix GAN Yes Yes OBIA Drug treatment responses on Patient-Derived Organoids (PDOs) Supervised linear classifier to separate live and dead Yes Yes MOrgAna Segmentation and quantifying organoid brightfield images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Gaussian /Laplacian filters, Logistic regression Yes Yes OrganoID Pancreatic ductal adenocarcinoma (PDAC) modified UNet Yes Yes Literature review Utilising organoids assists scientists in different areas such as organ studies, investigating physiology, cancer biology, precision therapeutics development, responses to chemotherapies and chemoradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Organoids can be imaged on platforms such as benchtop stereoscopes or high-content confocal-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Investigation, interpretation and inspection of high-volume organoids are problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Data collection from various time points and conditions among thousands of regions of interest is still challenging for biologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Using organoids still faces some drawbacks that need to be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Some problems regard ignorance of organoid size, shape, and morphological status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Furthermore, images have been provided in a high volume in quantity requiring automatic batch processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Therefore, Fully automated high throughput image analysis pipelines for organoids are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Image analysis is a powerful tool in many medical disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Today, researchers use machine learning to improve image processing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The software using machine learning are MetaMorph, Imaris, Harmony, ZEN, FIJI, CellProfiler and ilastik (Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Schindelin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Carpenter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Some identify thousands of cells in vast fields and extract biologically related features (McQuin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Metamorph(2003) MetaMorph is a microscopic image analysis tool to manipulate various image processing tasks such as intensity logging, advanced morphology, colocalisation, image stack alignment, montage generation, movie making, 3D rendering, image colour combination, light equalisation, topographical surface map generation, convolve and deconvolve images, arithmetic operations, orthogonal planes visualisation, image stitching and 3D calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The graphical interface offers colour combination, integrated morphometry analysis, region measurements and counting cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Metamorph has many optional modules such as counting nuclei, neurite outgrowth analysis, tracking motile cells or assessing cell cycle phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' MetaMorph software is a quick, practical and robust system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' It performs time-lapse and multi-dimensional acquisition and 3D reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Furthermore, Methamorph has brightening operations for biological experiments applying to live cell imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Caicedo research In 2017, Juan C Caicedo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' recommended a method for converting the collection of microscopy images to high-quality image-based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' morphological) profiles to biological discovery, experimental designs, and laboratory preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Image-based cell profiling is a high-throughput microscopy system for quantifying phenotypic differences among various cell populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' It uses chemical and genetic perturbations and provides an approach to studying biological systems on a large scale (Caicedo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='C et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Their image analysis system performs two main steps: 1-Illumination detection and correction using retrospective multi-image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' In the centre of the field of view, pixels are brighter than those on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' 2-Pixel foreground or background classification using image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' 1 illustrates the main Caicedo steps Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='1 Caicedo overview Comprehensive image analysis procedure for structurally complex organotypic cultures (CALYPSO) In 2017, Anne-Laure Bulin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' proposed a comprehensive high-volume image analysis system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' They aimed to study complex heterotypic organoids to discover the therapeutic efficacy architecturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' They extracted fluorescence intensity-based parameters for tumour treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' CALYPSO measures the organoid area without considering the shape and volumetric estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' For binarization, adaptive thresholding was used instead of a constant threshold for accurate tumour nodules detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' To eliminate out-of-focus objects, filtering has been done by adding fluorescence channels and binarising Otsu’s threshold method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The obtained binary images were multiplied with initial masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Organoids are indexed based on final masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' These indices are unique for each organoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' CALYPSO calculates tumour nodules viability factor using fluorescence intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=" This factor is varied between zero and one and indicates organoids' health." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' CALYPSO performed the following steps for image analysis: masking, background subtraction, threshold calculation and finally, live area thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Viability and live area fraction are used for correlated analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Live area fraction is obtained from the rate of the fraction of live area to total area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Viability is obtained from the rate of live intensity to the total value of live intensity and dead intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='2 illustrates the main steps of CALYPSO Image illumination detection and correction-->Pixel foreground or background classification -->lmage segmentationAdaptive threshod-->Filtering-->Adding fluorescence channels-->Otsu thresholdFigure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='2 CALYPSO overview Andy’s Algorithm Image analysis based on antibody detection faces many challenges in quantifying cellular antigens and obtaining accurate high-throughput quantitation, specifically for users with insufficient image processing skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Cognitive bias and scientific reproducibility are emerging problems in oncology drug development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' In 2017 Andrew M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' designed Andy’s Algorithms to address the stated issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' This algorithm performs batch-processing of 3,3′-diaminobenzidine (DAB) immunohistochemistry, proximity ligation assays (PLAs) and other standard assays in a quick, accurate and repeatable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Andy’s algorithm performs optimisation before image processing to determine optimal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' These parameters are used to address the problems caused by variation of cell types, tissue type, fixation method, antigen detection method, staining pattern, colour discrimination, magnification, illumination settings, and resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Andy’s algorithm provides interactive environments for users with basic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' It was designed for compassionate pre-clinical oncology purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Andy’s algorithm is a high throughput and iterative image analysis platform for biology, this platform does not require advanced image processing skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OrganoSeg In 2018, Borten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' developed OrganoSeg as an open source and parameters based on conventional image processing techniques to segment, filter and analyse 3D brightfield images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OrganoSeg quantifies brightfield phenotypes, providing insight into the molecular and multicellular mechanisms of organoid regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Its high performance has been approved by classifying 5167 breast-cancer spheroids and 5743 colon cancer cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OrganoSeg works on jpg, png or tiff format, grayscale images, performs an open-close morphological function, smooths bulk components preserving sharp spheroid boundaries, and then binarises the smoothed image by local adaptive thresholding or Otsu’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Users are enabled to find minimum threshold size and separate poor growth organoids by segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OrganoSeg extracts up to 23 morphometry standard measures for downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' It works with 3D-cultured breast images or movies, brightfield, phase-contrast, and differential interference contrast images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OrganoSeg, regardless of confocal or high contrast fluorescence micrographs, translates obtained images to measurable datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' High volume organoid shape analysis generates a collection of non-continuous morphological references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Combining OrganoSeg and RNA sequencing indicated a relationship in colorectal cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OrganoSeg is standalone with a graphical user interface to quantify organoid cultures and 3D spheroid transmitted-light images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='3 illustrates the main OrganoSeg steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='3 OrganoSeg overview OrgaQuant Grayscale-->Open/close morphology-->Adaptive Threshols-->Superimpose-->Binarise-->Filter noise-->Fill holes-->Smooth-->ldentify Rol-->Extract metricsIn 2019, Timothy Kassis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' described OrgaQuant implementation, a deep convolutional neural network to locate and quantify bright field human intestinal organoids images distribution size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' To train OrgaQuant, they manually annotated human intestinal organoid images and generated a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' They conducted this research to address the lack of technique to localise and quantify organoids in a 3D environment and the presence of many imaging artefacts (Kassis T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' They used extracted features from a neural network (NN) for Clustering or subtle changes detection in organoid morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Object detection is a complicated process in computer vision, particularly in cases with speed concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Some developed detection algorithms have been designed to be quick and accurate such as single Shot Multibox Detector (SSD) and You Only Look Once (YOLO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OrgaQuant was designed based on the Region Convolutional Neural Network (R-CNN) and or faster R-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Faster R-CNN uses ResNet and Inception architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OrgaQuant for pre-training utilises COCO box annotations and needs an extensive dataset provided by augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OrgaQuant automatically localises an organoid within a brightfield image and annotates the bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Downstream image processing uses cropped organoid images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' This pipeline measures the intestinal organoid size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OrgaQuant facilitates tracking organoid growth kinetics in droplets during the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OrgaQuant is a basic, intelligent and open-source organoid quantification technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' an open source and parameters based on conventional image processing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The OrgaQuant model does not require any parameters and can analyse thousands of images without human interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='4 illustrates the main OrgaQuant steps Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='4 illustrates the main OrgaQuant steps Single-cell and Cytoarchitecture analysis of Organoids using Unbiased Techniques (SCOUT) In 2020, Alexandre Albanese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' presented the SCOUT pipeline for intact cerebral organoids with an automated multiscale comparative analysis to clear, label and image organoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' From the microscopic fluorescence dataset, SCOUT CNN extracts hundreds of features characterising molecular, cellular, spatial, cytoarchitectural, and organoid-wide properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' SCOUT performed a comprehensive analysis of 46 intact organoids and ~100 million cells and revealed quantitative multiscale "phenotypes" for organoid development and Zika virus infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Imaging each organoid takes about 15 minutes for reliable, accurate single-cell analysis and 300 multiscale features takes about six hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' SCOUT has performed quantifying spatial features such as cellular context, ventricle morphology, cytoarchitecture distribution, and uncommon events detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Zika virus infection decreases cell population, and this pipeline quantifies this reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' SCOUT has three main modules: single cell analysis, regional architecture analysis, whole organoid analysis, and finding correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' UNet was slightly modified, and two layers were added before and after the UNet bottleneck to improve the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Then modified UNet was trained using weighted binary class entropy (WBCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' WBCE support model to converge to accurate ventricle segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' This model was used to segment ventricles automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Experimental results denote 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='2% Dice score similarity for UNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=" Quantifying multiscale feature correlation, maturation-related changes, protocol comparisons, and Brightfield image--> Faster R-CNN -->Localised organoidZika virus pathology demonstrated SCOUT's power." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' SCOUT is a versatile platform for automatic single-cell analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='5 illustrates the main steps of SCOUT Figure5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' SCOUT overview Larsen In 2021, Larsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' proposed a pan-cancer precision medicine organoid platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=" Genomic/transcriptomic fidelity has been extracted from over 1000 patients' tumour organoid(TOs) cultures." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' They determined solid tumour chemical minimal media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' They studied growth factors to initiate and propagate organoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Their results indicated that Epidermal Growth Factor (EGF) and Noggin are sufficient for most organoid cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' They developed a convolutional neural network(CNN) based on pix2pix Generative Adversarial Network (GAN) and predicted label-free light microscopy drug response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' CNN contains a fluorescence generator and a viability discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Pix2pix GAN accuracy has been approved in cell biology by Isola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Tsuda and Hotta, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' They adopted 70x70 patch GAN as a viability discriminator (Isola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Concatenated brightfield and fluorescent are inputs of the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The generator gets a brightfield image as input and generates fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Realistic image prediction and image translation from the black and white domain to the colourful domain is possible by Pix2Pix (Isola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2017), conditional generative adversarial networks (GANs), and CycleGAN (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' This technique was developed as a light-microscopy-based assay to omit costly vital dye requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' High content image analysis was performed by inverting confocal microscopy images, followed by ability measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Then binary classification for TO viability was executed for drug screening optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='6 illustrates the main Larsen steps Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='6 Larsen steps overview Object-Based Image Analysis (OBIA) In 2021, Erin R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Spiller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' developed an object-based image analysis (OBIA) system to convert cells to population-level analysis and investigate disturbances in drug treatment responses of heterogeneous object-based Patient-Derived Organoids (PDOs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' It observes phenotypic changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OBIA considered the textural and morphological features from brightfield images as discriminators between dead and live organoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' They used a supervised linear classifier to separate live and dead organoids for drug response recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The OBIA ML image analysis was initially performed in a small sample set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' They designed a tool to visualise the differences of all collected features over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The OBIA team developed a web-based Application Programming Interface (API) to upload machine learning output analysis and download feature metrics and survival curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' They proposed a label-free high content screening (HCS) robust, on-time approach working with colorectal cancer (CRC) organoid live-cell image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' It is an automated dynamic cellular visualisation and multiparametric extraction data system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Gaussian Blur -->Convert to 8bit-->Threshold-->Watershed-->Fill holes-->Particle anlysise-->SavaBrightfield image -->pix2pix GAN generator--> fluorescenceMoreover, this method can be scaled to apply to high-volume drug screen data of different cancer types PDOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OBIA has done computational intensive image analysis automatically and iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='7 illustrates the main OBIA steps Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='7 OBIA overview MOrgAna Gritti N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' 2021 developed MOrgAna, a python-based platform using machine learning to segment, quantify and visualise morphological features of high-volume organoid images at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' MOrgAna is an appropriate package for users with basic to advanced skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' So, this technology assists in atlas-free 3D whole-organoid analysis to gain quantitative comparative analysis between experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Brightfield image availability, in contrast, fluorescence is the reason behind utilising them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' MOrgAna monitors the organoid in bright field mode for biomedical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The training dataset includes morphological parameters, Gaussian and Laplacian filters, Logistic regression, improved flexibility and feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' This pipeline is designed on a multilayer perceptron for organoid recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='8 illustrates the main MOrgAna steps Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='8 MOrgAna overview OrganoID In 2022, Jonathan Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' developed OrganoID, a robust image pixel-by-pixel analysis platform, in brightfield and phase-contrast microscopy experiments to automatically recognise, label, and track single organoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' This model was trained based on pancreatic cancer organoid images and tested on images of pancreatic, lung, colon, and adenoid cystic carcinoma organoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OrganoID is an open-source image analysis platform which determines organoid morphology pixel-wised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' This platform does not require fluorescence or transgenic labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' It can analyse different organoid types in microscopy experiments promptly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' It is designed based on a modified UNet and has minimum layers for quick performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The training dataset was pancreas cancer samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Testing validating images come from pancreatic, lung, colon, and adenoid cystic carcinoma with more than 89% tracking accuracy for four days of the validation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OrganoID performs two highlighted tasks: individual organoid tracking over time and Drug screening experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' General image processing steps with OrganoID are contour detection, single and bulk organoid separation and determination of response over time by time-lapse image sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' CNN has been designed to indicate the presence of organoid probability in each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' UNet architecture includes layers of convolutional, maximum filtering, concatenation, extraction and localisation features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The training dataset was obtained from 66 manually organoid microscopy images of pancreatic ductal adenocarcinoma (PDAC), then collected samples divided into 80% training and 20% validating and augmentation applied to the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The OrganoID platform effectively and accurately uses organoid images in high-volume data physiological research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='9 is an illustration of the main steps of OrganoID Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='9 OrganoID overview Critical Analysis Reviewing image analysis pipelines for organoids of 11 researches indicates that since 2019 four studies have used deep learning for organoid analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Generally using deep learning, in contrast, traditional methods require larger training datasets, which is expensive, time-consuming, and requires expert annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Although such training dataset preparation needs more effort, the result of models will be more robust and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Almost all pipelines are automated, and more than 75% of them are high throughput and can process a high number of images at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' As Table IV indicates, these pipelines have been designed for different targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Since five pipelines specifically have been designed for oncology studies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' thus it denotes organoid significance in cancer studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Almost all projects consider organoids’ morphology for different purposes, such as drug response monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' They address the issues regarding ignorance of organoid shape and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Conclusion and future development In this study, we have introduced the vast range of organoids usage in biology and medical research, such as organ studies, investigating physiology, drug testing, cancer biology, precision therapeutics development, antibody detection, quantifying cellular antigens, responses to chemotherapies/chemoradiation, and human brain development /dysfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Then briefly describe the drawbacks of using organoids and finally review eleven designed image analysis systems to solve the drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=" Since 2014, more than 23 researches have been conducted to observe different organs' functions and dysfunction using organoids." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Organoids can be imaged on platforms such as benchtop stereoscopes or high-content confocal-based microscopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Investigation, interpretation and inspection of high-volume organoids are problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Data collection from various time points and conditions among thousands of regions of interest is still challenging for biologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Some problems regard ignorance of organoid size, shape, and morphological status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Hence, the essential properties of an organoid image analysis pipeline are fully automated, high throughput, and considering morphology features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' We reviewed the development of image analysis systems for Organoids in biological disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Starting from 2003 with MetaMorph development which was able to perform some simple image processing tasks such as montage, rendering, alignment, colour combination and arithmetic operation robustly and quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Proceeding to 2017, Caicedo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' developed another system to encourage biological observation and experimental laboratory preference, followed by the development of CALYPSO to measure the viability of an organoid to discover its therapeutic efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' At the same time, Andy’s algorithm came as a high throughput, iterative and user-friendly image analysis biology platform for compassionate pre-clinical oncology purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' In 2018, OrganoSeg was released as a standalone graphical high-volume platform to extract metrics from organoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OrganoQuant in 2019 was designed based on faster RCNN to address the issues for organoid quantification regarding image artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' In 2020, SCOUT was revealed as a versatile platform based on UNet and came in 3 modules: single cell, regional and whole organoid to automatically segment ventricles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' In 2021, Larcen’s pan-cancer organoid platform was designed for precision medicine based on pix2pix GAN network image translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' This platform was developed to remove costly vital dye requirements and optimise drug screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OBIA was released for drug response recognition levels by classifying dead and live Neural Network predication map-->Canny edge detection--> Watershedorganoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=" The OBIA team also developed a visualisation tool and web-based API to address the biologist's challenges regarding high-volume data." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' MOrgAna was developed for organoid segmentation, quantification and visualisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' OrganoID was designed for organoid tracking and drug screening based on UNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Eleven projects have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' These projects aimed to provide organoid image analysis systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Regardless of the area these pipelines are designed for, they all meet the general criteria for organoid image analysis systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' These criteria are being automated, high throughput and considering organoid morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Four out of eleven pipelines used Deep Neural Network(DNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' DNN, despite being more expensive, expert-required annotation, time-consuming preparation generates reliable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' For instance, Figure 10 illustrates the results of computed tomography (CT) of abdominal multi organs image segmentation using UNet (we developed the UNet model prior to this research to evaluate the segmentation accuracy of this model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' This figure shows spleen, right kidney and left kidney segmentation from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' As observed, segmentation accuracy decreased from top to bottom due to reducing the number of images in the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' 10 UNet abdominal CT scan organs segmentation As these pipelines followed different targets, comparing them in terms of performance can not be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' We can only compare them based on image processing techniques or can compare pipelines with common targets/scops, as Table V indicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The summary of nine reviewed pipelines shows in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' pipelines overview ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='Pipelines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='Initial Image processing steps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='Caicedo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='Image illumination detection and correction →pixel background/ foreground ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='classification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='CALYPSO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='Adaptive threshold→ Filtering →Adding fluorescence channel→Otsu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='threshold ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='OrganoSeg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='Grayscale→open/close Morphology→Adaptive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='threshold→Superimpose→Filter noise and holes→Smooth→ROI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='OrgaQuant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='Brightfield images→Faster RCNN→Localised organoid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='SCOUT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='Gaussian →8 bits→Threshold→Watershed→Fill holes→Particle analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='Larsen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='Brightfield images→Pix2pix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='GAN→generator→Fluorescence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='OBIA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='Brightfield images→ Morphological features→Linear classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='MOrgAna ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='Brightfield images→ Morphological features→Laplacian and Gaussian ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='filters→Logistic Regression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='OrganoID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='NN prediction map→Canny edge→ Watershed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='Future work and further development ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content='Since during this study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' despite a limited training dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' we successfully developed UNet to be familiar with the NN model design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' We could get reasonable cross-validation results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' thus, further development will be conducted to design a trustworthy system to classify live and dead cells using Pix2Pix GAN in the oncology area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' This system includes two networks, a UNet as a generator and ResNet as a discriminator to classify gold standard and live/dead masks generated by the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' The generator accepts brightfield images and generates live/dead masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Two models will be trained by this method: one model generates masks of live cells, and another one generates masks of dead cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' For providing training datasets for two models, we can use CALYPSO or OBIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' These pipelines support us in saving time and budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Then brightfield images and dead/live organoid masks will be concatenated as input and output of the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' To have samples for testing or cross-validation training, datasets will be divided into 80% train and 20% test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Reviewing table V indicates the ideal pipeline for organoid image analysis is a fully automated, high-throughput and general-purpose system designed based on Deep Neural Network, covering brightfield and fluorescence images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Bibliography Clevers, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Modelling development and disease with organoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Cell, 165(7), 1586-1597.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', Toshimitsu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', Nakazato, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', Kawasaki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' A Colorectal Tumour Organoid Library Demonstrates Progressive Loss of Niche Factor Requirements during Tumorigenesis.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', & Trivedi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' MOrgAna: accessible quantitative analysis of organoids with machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Development, 148(18), dev199611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=' Berg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfbAAV/content/2301.02341v1.pdf'} +page_content=', Kutra, D.' metadata={'source': 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Yang2 +1State Key Laboratory for Mechanical Behavior of Materials, +Xi’an Jiaotong University, Shaanxi 710049, China +2Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore +3Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE), +Beijing Institute of Technology, Beijing 100081, China +4College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China +5School of Physics, Northwest University, Shaanxi 710127, China +Recent experiments have demonstrated interesting physics in a family of two-dimensional (2D) +composition-tunable materials Nb2n+1SinTe4n+2. +Here, we show that owing to its intrinsic low +symmetry, metallic nature, tunable composition, and ambient stability, these materials offer a good +platform for studying Berry curvature dipole (BCD) and nonlinear Hall effect. Using first-principles +calculations, we find that BCD exhibits pronounced peaks in monolayer Nb3SiTe6 (n = 1 case). +Its magnitude decreases monotonically with n and completely vanishes in the n → ∞ limit. This +variation manifests a special hidden dimensional crossover of the low-energy electronic states in this +system. The resulting nonlinear Hall response from BCD in these materials is discussed. Our work +reveals pronounced geometric quantities and nonlinear transport physics in Nb2n+1SinTe4n+2 family +materials, which should be readily detected in experiment. +I. +INTRODUCTION +The Hall effects, in which a transverse current jH is +induced by a longitudinal driving E field, are of funda- +mental importance in condensed matter physics [1–3]. At +linear order, i.e., with jH ∼ E, the Hall effect requires the +broken time reversal symmetry T , which can be achieved +either by an applied magnetic field or by intrinsic mag- +netism. This constraint is loosened when considering Hall +responses at nonlinear order, as the non-equilibrium elec- +tron distribution driven by E field already breaks T at +its first order. Focusing on the second-order response, +in nonmagnetic materials and in the absence of mag- +netic field, Sodemann and Fu proposed a Berry curvature +dipole (BCD) contribution to the nonlinear Hall current +jH ∼ E2 within the semiclassical theory framework [4]. +Their work attracted great interest in the past few years, +and the effect has been successfully detected in several +material systems [5–24]. It was suggested that this effect +offers a new mechanism for nonlinear applications, such +as frequency-doubling and rectification [25–27]. +For experimental study, two-dimensional (2D) mate- +rials have advantages in their great tunability. For ex- +ample, the Fermi level in 2D materials can be readily +tuned via gating technique to a large extent not possible +in 3D bulk materials [5, 28]. However, regarding BCD +and its induced nonlinear Hall effect, the constraint from +crystalline symmetry in 2D is rather stringent. It was +shown that the largest symmetry in 2D that allows for a +nonzero BCD is a single in-plane mirror line [4]. Hence, +to realize the effect, one has to choose crystals with very +low symmetry, which are rather limited, or takes extra +∗ caojin.phy@gmail.com +† zzy@mail.buct.edu.cn +‡ mafei@mail.xjtu.edu.cn +effort to exert strain or twist on the crystal to lower the +symmetry. This severely hinders the experimental study. +Recently, the family of composition-tunable materials +Nb2n+1SinTe4n+2 have attracted interest from both the- +ory and experiment [29–34]. +In the bulk form, these +materials are van der Waals layered materials. +Their +high-quality 2D layers can be obtained by mechanical +exfoliation method [35]. The special feature of this fam- +ily is the tunable composition embodied by the integer +n [36–39]. +For each n, the system is a stoichiometric +crystal, and the physical properties have an interesting +dependence on n. For example, it was shown that in a +2D monolayer, for finite n, the material is nonsymmor- +phic nodal-line semimetal [29]; whereas the n → ∞ limit, +i.e., the material Nb2SiTe4, is a narrow-gap semiconduc- +tor [33, 40]. With increasing n, the low-energy states at +Fermi level exhibits a dimensional change from 2D-like +states to 1D-like states [32]. +We note that 2D Nb2n+1SinTe4n+2 materials actually +offer a good platform to explore BCD related physics. +First, except for the n → ∞ limit, all members of the +family have a sufficiently low symmetry to allow an in- +trinsic BCD, without the need of applied strain. +Sec- +ond, they offer an opportunity of systematic investigation +of the evolution of BCD with the tunable composition. +Third, these 2D materials are stable at ambient condi- +tions [32], which facilitates experimental study as well as +possible applications. +Motivated by the above considerations, in this work, +we theoretically study BCD and nonlinear Hall effect +in monolayer Nb2n+1SinTe4n+2 materials. +With first- +principles calculations, we show that the n = 1 case, +i.e., Nb3SiTe6, possesses a pronounced BCD. The magni- +tude can reach 1.54 ˚A in the hole doped case, larger than +previously reported values in 2D Td-WTe2 [18], strained +NbS2 [21] and WSe2 [18]. With increasing n, the BCD +peaks in the spectrum show a monotonic decrease and +arXiv:2301.00946v1 [cond-mat.mtrl-sci] 3 Jan 2023 + +2 +eventually vanish in the n → ∞ limit. This behavior can +be understood from two perspectives. One is from the +symmetry perspective, and the other is from the dimen- +sional evolution of the electronic states. The latter view +manifests that although structurally, these materials are +strongly bonded in both directions in 2D, electronically, +the states exhibit a dimensional crossover from 2D to +1D. This hidden crossover dictates the change in BCD. +The key features of the results are further captured by +our constructed tight-binding models for this family of +materials. To guide experiment, we discuss properties of +the nonlinear Hall response arising from BCD. Our work +reveals interesting properties of Nb2n+1SinTe4n+2 family +materials and suggests them as a suitable platform to +explore BCD and nonlinear Hall physics. +II. +COMPUTATION METHOD +Our first-principle calculations were based on the den- +sity functional theory (DFT), performed by using the +Vienna ab initio simulation package [41–43]. The ionic +potentials were treated by using the projector aug- +mented wave method [44]. +The exchange-correlation +functional was treated by the generalized gradient ap- +proximation [45] in the scheme by Perdew, Burke, and +Ernzerhof [46]. The plane-wave cutoff energy was set to +be 400 eV, and a 10 × 4 × 1 Γ-centered k-point mesh was +used for the Brillouin zone (BZ) sampling. The conver- +gence criteria for the total energy and the force were set +to be 10−6 eV and 0.01 eV/˚A, respectively. +To avoid +artificial interaction between periodic images, a vacuum +space of 15 ˚A thickness was added. +Spin-orbital cou- +pling (SOC) was included in all calculations. Based on +the band structure calculation, an ab initio tight-binding +model was constructed by using the Wannier90 pack- +age [47]. The d orbitals of Nb atoms and p orbitals of +Te atoms were used as the initial guess of the local basis. +The BCD was calculated based on this ab initio tight- +binding model. In evaluating BCD, we set T = 100 K in +the Fermi distribution function. +III. +CRYSTAL AND ELECTRONIC +STRUCTURES +The Nb2n+1SinTe4n+2 family materials were first syn- +thesized in the 1990s by chemical vapour transport +method [36]. The lattice structures of their 2D mono- +layers are illustrated in Fig. 1. +Here, each monolayer +consists of three atomic layers: the middle layer con- +taining Nb and Si atoms is sandwiched by two Te lay- +ers [Fig. 1(a)]. From the top view [see Figs. 1(b)-1(d)], +these materials can be viewed as composed of three build- +ing blocks, which are conventionally called the a, b, c +chains. As shown in Fig. 1(b), a and b chains contain Si +atoms and share the same composition of NbSi1/2Te2, +whereas the c chain does not contain Si and has the +a +b +c +(ab) +c +c +(ab)×n +c +c +... +... +(a) +(b) +(c) +(d) +Nb +Si +Te +x +y +y +z +x +FIG. 1. (a) Lattice structure of monolayer Nb3SiTe6. (b) The +three building blocks of of Nb2n+1SinTe4n+2 family materials: +a, b and c chains. (c) Top view of n = 1 case (Nb3SiTe6). The +dashed box marks the unit cell. (d) Nb2n+1SinTe4n+2 can be +constructed by n copies of (ab) chains and one c chain in a +unit cell (the dashed box). +composition of NbTe2. Assuming these chains are along +the x direction [as in Fig. 1(c)], then a and b are con- +nected by a glide mirror operation ˜ +My = {My| 1 +20}, and +in these materials they always appear together. Mem- +bers of this family are formed by assembling these chains +along the lateral direction (y) in a periodic manner, such +that Nb2n+1SinTe4n+2 corresponds to the arrangement +of (ab)nc. Namely, in a period, we have one c chain and +n copies of (ab) chains, as illustrated in Fig. 1(d). In the +n = ∞ limit, there is no c chain in the structure any +more, and we reach the composition of Nb2SiTe4. +Our optimized lattice parameters for n = 1, 2, 3, ∞ are +listed in Table I. These values are in good agreement with +experiment and previous calculations [29, 30, 32]. +We +also note that for members with finite n, they all have +the space group symmetry Pmc21, with C2v point group. +In comparison, Nb2SiTe4 with n = ∞ has a larger space +group Pbam and a point group D2h. The main difference +is the extra glide mirror ˜ +Mx = {Mx|0 1 +2} for n = ∞ case +but not for any finite n. From Fig. 1(c), one can see that +it is the c chains that break the +˜ +Mx symmetry which +holds for (ab) chains. +In Fig. 2, we plot the calculated electronic band struc- +tures for the four representative members in Table I. One +can see that the band structures for n = 1, 2, 3 show sim- +ilar features. +Previous works have shown that in the + +3 +TABLE I. Optimized lattice parameters and the correspond- +ing symmetries of representative monolayer Nb2n+1SinTe4n+2 +materials. +n +a (˚A) +b (˚A) +Thickness (˚A) Space group Point Group +1 +6.408 11.633 +3.649 +Pmc21 +C2v +2 +6.405 19.590 +3.770 +Pmc21 +C2v +3 +6.404 27.552 +3.651 +Pmc21 +C2v +∞ 6.401 +7.962 +3.783 +Pbam +D2h +absence of SOC, these materials are nodal-line semimet- +als [29, 32]. The nodal line on the X-M path around +Fermi level is enforced by the nonsymmorphic T ˜ +My sym- +metry. The detailed analysis was given in our previous +works [48], so we will not repeat it here. It should be +noted that in Fig. 2, the band structures include the SOC +effects. Under SOC, the T ˜ +My symmetry protection is no +longer exact, so the original nodal line degeneracy will be +lifted. In the enlarged view in Fig. 2(b), one can clearly +see the splitting of the nodal line. Nevertheless, there +is still a degenerate nodal point at X (and also at M). +This point is a fourfold degenerate Dirac point enforced +by nonsymmorphic symmetries of the system. +Its for- +mation mechanism has been discussed in Ref. [29]. The +SOC induced change to the band structure is weak, so for +many properties, SOC may just be neglected. However, +band geometric properties like Berry curvature and BCD +are very sensitive to small-gap regions in band structures, +such as those due to SOC splitting. Therefore, to study +BCD and its nonlinear Hall effect, we have to include +SOC in the calculation. +The low-energy states around Fermi level are mostly +distributed on the c chains. Previous scanning tunnel- +ing spectroscopy (STS) experiments also verified this fea- +ture [32, 33]. With increasing n, the distance between +two c chains will increase and hence the coupling between +them will decrease. As a result, the band dispersion will +become flatter along the y direction, as can be seen in +Figs. 2(c)-2(e) along the Γ-Y and X-M paths. +For Nb2SiTe4 with n = ∞, Fig. 2(f) shows that it is +a narrow-gap semiconductor. The band gap is ∼ 0.51 +eV, which is slightly larger than the band gap of lay- +ered Nb2SiTe4 (∼ 0.39 eV) [40]. This different character +can now be understood from the discussion above. One +can view the c chains as metals, whereas the (ab) chains +are insulating. Since Nb2SiTe4 is entirely made of (ab) +chains, its spectrum would naturally be gapped. +The features discussed above, particularly the evolu- +tion of band structure with n, will have important im- +plications on BCD and nonlinear Hall response in these +materials. +IV. +BERRY CURVATURE DIPOLE +Berry curvature is an intrinsic band geometric quan- +tity. It plays an important role in many physical prop- +E (eV) +0.2 +-0.2 +0 +Y +Y +M +X +E (eV) +0.2 +-0.2 +0 +0 +-80 +80 +E (meV) +E (eV) +0.2 +-0.2 +0 +Y +M +Γ +X +（a） +（b） +（c） +（d） +（e） +（f） +Γ +E (eV) +0.2 +-0.2 +0 +Y +Y +M +X +Γ +Y +Y +M +X +Γ +Y +Y +M +X +Γ +M +X +k x +k y +FIG. 2. (a) Brillouin zone for monolayer Nb2n+1SinTe4n+2. +(b-f) Band structures for monolayer Nb2n+1SinTe4n+2: (b, c) +n = 1, where (b) is an enlarged figure around the path X-M +in (c); (d) n = 2; (e) n = 3; and (f) n = ∞. +erties, especially anomalous transport properties [49]. In +nonmagnetic materials, nonzero Berry curvature requires +the breaking of inversion symmetry. This condition is ful- +filled in monolayer Nb2n+1SinTe4n+2 with finite n. For +Nb2SiTe4 with n = ∞, inversion symmetry is respected +and hence Berry curvature vanishes identically. +For a 2D system, Berry curvature only has a single +component, which can be expressed as (we set e = ℏ = 1 +in the formulas) +Ωz(nk) = −2 Im +� +n′̸=n +⟨unk|vx|un′k⟩⟨un′k|vy|unk⟩ +(εnk − εn′k)2 +, +(1) +for a state |unk⟩, where vx and vy are the velocity oper- +ators, and εnk is the energy of |unk⟩. +Consider Nb3SiTe6 (n = 1). In Fig. 3(a), we plot the +distribution of its Berry curvature in BZ for occupied +states, i.e., the quantity +Ω(k) = +� +n +f0Ωz(nk), +(2) +where f0 is the Fermi distribution function. One observes +that the Berry curvature is odd in ky and even in kx, as +required by T and ˜ +My, and its value is quite pronounced +along the Γ-Y path. +BCD is the first moment of Berry curvature in BZ. It +is a pseudovector in 2D, defined as [4] +Da = +� +n +� +BZ +d2k +(2π)2 f0∂aΩz(nk) += − +� +n +� +BZ +d2k +(2π)2 f ′ +0va(nk)Ωz(nk), +(3) + +4 +(a) +(b) +(c) +(d) +(e) +(f) +FIG. 3. Berry curvature and its dipole (BCD) in Nb3SiTe6. +(a) Distribution of Berry curvature for the occupied states. +(b) The BCD Dy versus chemical potential µ. (c-f) The k- +resolved BCD as defined in Eq. (4). Here, (c, d) are plotted +for µ = 0.064 eV (the upper peak in (b)), and (e, f) are for +µ = −0.180 eV (the lower peak in (b)). The Fermi contours +at these energies are indicated by the black curves. +where a ∈ {x, y}, ∂a ≡ ∂ka, and in the second line, we +write it as a Fermi surface integral. +For finite n, Nb2n+1SinTe4n+2 only has a single mirror +line along x, which allows a nonzero BCD. Since D is +a pseudovector, it must be along the y direction, i.e., +D = Dyˆy. In Fig. 3(b), we plot the calculated Dy versus +the chemical potential µ for n = 1. One observes two +peaks in the figure: one is at 0.064 eV with a value of +0.399 ˚A, and the other is at −0.180 eV with a value of +−1.540 ˚A. The two peaks are of opposite signs. We note +that the magnitude of -1.540 ˚A is quite large. This is +comparable or larger than those found in monolayer Td- +WTe2 (0.1 ∼ 0.7 ˚A) [18], strained NbS2 (0.2 ˚A) [21] and +strained WSe2 (0.02 ˚A) [18]. +To understand the origin of the large BCD in mono- +layer Nb3SiTe6, in Figs. 3(c)-3(f), we plot the k-resolved +BCD on Fermi surface, namely the quantity +Da(k) = − +� +n +f ′ +0va(nk)Ωz(nk), +(4) +for µ = 0.064 eV (upper peak) and −0.180 eV (lower +peak). First of all, one observes that Dx(k) is an odd +function in ky whereas Dy(k) is an even function, as re- +quired by the ˜ +My symmetry. Hence, after integral over +BZ, BCD only has the y component left. From Figs. 3(c)- +3(f), one can see that the nodal line region along X-M +does not make a sizable contribution to BCD. For the +upper peak [Fig. 3(d)], large contribution to Dy is from +the Γ-Y path, which corresponds to the SOC splitting +µ (eV) +-0.8 +0 +0 +0.2 +-0.2 +0 +0.2 +-0.2 +µ (eV) +0.8 +n = 2 +n = 3 +(a) +(b) +-0.8 +0 +0.8 +Dy (Å) +Dy (Å) +FIG. 4. +The BCD Dy versus chemical potential µ for (a) +n = 2, and (b) n = 3 cases. +gap indicated in Fig. 2(c). +The spin splitting gap on +the outer Fermi surface [marked by the green arrow in +Fig. 3(d)] also gives a non-negligible contribution. +As +for the lower peak, Figs. 3(e)-3(f) show that the Fermi +surface has two separate pieces. By examining the band +structure around the hot spots in Fig. 3(f), we find that +the large negative contribution is also from SOC splitting +of the band structure. +Next, we consider the cases with n = 2 and 3. From +the results in Fig. 4, one can see that the magnitude of +BCD decreases with increasing n. For n = 3, the BCD +value above µ = 0 (which is also the energy of nodal +line) is already negligibly small. As for the lower peak, +the value is about 0.663 ˚A for n = 2 and 0.396 ˚A for +n = 3. +This trend of decreasing BCD with increasing n +in +monolayer +Nb2n+1SinTe4n+2 +can +be +understood +from two perspectives. +First, in terms of symmetry, +Nb2n+1SinTe4n+2 with finite n supports BCD because +of its low symmetry. +The presence of c chains is cru- +cial because they break the ˜ +Mx symmetry of (ab) chains +(Fig. 5). Without c chains, ˜ +Mx becomes an exact sym- +metry and it suppresses BCD (given the other mirrors in +the system) as in the n = ∞ limit. Hence, the density +of c chains in the system can be viewed as a measure of +the extent of symmetry breaking. It is strongest in n = 1 +case, and gradually decreases as n increases, determining +the trend in BCD. +Meanwhile, the trend is also connected with the di- +mensional crossover in this system [34]. As discussed, the +low-energy states are mostly distributed on the c chains. +One may view the c chains as metallic 1D subsystems put +in an insulating matrix formed by the (ab) chains. For +small n, the system retains a 2D character, because the +c chains are not far from each other and the inter-chain +coupling is sizable. However, with increasing n, the inter- +chain coupling will decrease, and the system approaches +the quasi-1D character. Berry curvature is a differential +2-form, which vanishes in the 1D limit [as can also be seen +from Eq. (1)]. Thus, BCD must decrease and approach +zero during this dimensional crossover. +It must be emphasized that the dimensional crossover +here is referring to the low-energy electronic states. +Structurally, Nb2n+1SinTe4n+2 materials always main- +tain a 2D material character: the lattices are strongly + +5 +x +y +y +FIG. 5. (a) (ab) chains preserve the ˜ +Mx symmetry, whereas c +chains break it. Hence, the density of c chains represents the +extend of ˜ +Mx symmetry breaking. (b) The Nb2n+1SinTe4n+2 +system may be schematically viewed as 1D metallic chains +(c chains) embedded in a 2D insulator matrix (made of (ab) +chains). +bonded in both x and y directions. Thus, the crossover +is a hidden feature that occurs only for the electronic +sector. This is a very interesting piece of physics for 2D +Nb2n+1SinTe4n+2 materials. Now, we revealed its man- +ifestation in BCD, which can be detected via nonlinear +Hall measurement. +V. +A MODEL STUDY +To understand the features in band structure and in +BCD, we construct an effective lattice model to describe +the low-energy bands in monolayer Nb2n+1SinTe4n+2 +with finite n. The model may also serve as a good start- +ing point for other theoretical studies. +In Refs. [34, 48], we have proposed a 2D Dirac Su- +Schrieffer-Heeger (SSH) model, which is spinless (i.e., +without SOC) and captures the nonsymmorphic nodal +line feature in monolayer Nb2n+1SinTe4n+2. +However, +to study BCD, as we noted, the consideration of SOC +is necessary. Therefore, we need to extend the previous +spinless Dirac SSH model to include SOC effects. +The Dirac SSH model is defined on a rectangular lat- +tice, as shown in Fig. 6. It consists of an array of zigzag +chains running in the x direction. Physically, each chain +corresponds to a c chain in Nb2n+1SinTe4n+2. In a unit +cell, there are two sites A and B. Assigning one orbital +at each site and considering the nearest intra-chain and +inter-chain hoppings, one obtains the following model +constrained by T , ˜ +My, and Mz symmetries: +H0 = t +� +0 +1 + e−ikx +1 + eikx +0 +� +σ0 ++t′ +� +0 +e−iky � +1 + e−ikx� +eiky � +1 + eikx� +0 +� +σ0, (5) +where the momenta are measured in unit of the inverses +of lattice constants, and the Pauli matrices σ denote the +x +y +My +t +t’ +A +B +E /t +0.2 +-0.2 +0 +Y +Y +M +X +Γ +Dy(Å) +0 +0.2 +0.1 +0 +0.8 +0.4 +t’/t +~ +x +y +A +B +λ1 +λ3 +λ2 +0 +0.2 +-0.2 +0 +0.3 +-0.3 +µ (eV) +Dy(Å) +(a) +(b) +(c) +(d) +(e) +FIG. 6. (a) Schematic figure showing the tight-binding model. +The model consists of zigzag chains. A primitive cell contains +two sites A and B. t and t′ are the amplitudes for intrachain +and interchain hoppings. +(b) illustrates the three hopping +processes corresponding to the SOC terms in Eq. (6). +(c) +Band structure of the tight-binding model. (d) Corresponding +BCD Dy versus chemical potential. (e) Variation of BCD peak +value as a function of the interchain coupling. The solid curve +is a guide to the eye. In (c, d), we set t = 0.2 eV, t′ = 0.16 eV, +λ1 = 1, and λ2 = λ3 = 0.1. The same values of t and λ’s are +taken in (e). +spin degree of freedom. +Next, we add SOC to the model. The above mentioned +symmetries resulted in the following SOC terms up to +second neighbor hopping processes: +HSOC = t +� +2λ1 sin kx +0 +0 +−2λ1 sin kx +� +σz ++t′ +� +2λ3 sin ky +iλ2eiky � +1 + e−ikx� +−iλ2e−iky � +1 + eikx� +2λ3 sin ky +� +σz. +(6) +Here, the first term is from intrachain hopping process, +whereas the second term is from interchain process, as +indicated in Fig. 6(b). +Therefore, our spin-orbit-coupled Dirac SSH model is +obtained as +H = H0 + HSOC. +(7) +In Fig. 6(c), we plot a typical band structure of this +model. Namely, there is an approximate nodal line on +the X-M path (split by SOC); the SOC splitting is ob- +served on X-M and Y -Γ paths, but not on the Γ-X and +M-Y paths. The double degeneracy on X-M and Y -Γ + +6 +4 +0 +-4 0 +π +2π +χ⊥ (10-4 nm · V-1 Ω-1) +(b) +(a) +x +y +ɵ +E +jH +ɵ +FIG. 7. (a) Illustration of the nonlinear Hall current induced +by E field in Nb3SiTe6. The in-plane E field makes an angle +θ from the mirror line. The induced Hall current is perpen- +dicular to the E field, as indicated by the green arrow. (b) +Nonlinear Hall conductivity χH versus the angle θ. +is due to the anti-commutation between ˜ +My and Mz on +these two paths. +One can see that it indeed captures +the main features of DFT band structures in Fig. 2(c). +In Fig. 6(c), we plot the BCD calculated for this model. +The two BCD peaks in Fig. 3(b) are reproduced in this +simple model. One peak is above the nodal-line energy +and the other one is below, and they have opposite signs. +Finally, we plot the BCD peak magnitude as a function of +interchain coupling t′. One can see that the value mono- +tonically increases with the interchain coupling. +Since +t′ decreases with n in monolayer Nb2n+1SinTe4n+2, the +behavior in Fig. 6(d) agrees with our result from DFT +calculations. +VI. +NONLINEAR HALL EFFECT +It was shown that BCD leads to a second-order non- +linear Hall current. For a 2D system, the current can be +expressed as +jH = −1 +2τ ˆz × E(D · E), +(8) +where E is the applied in-plane E field, and τ is the +relaxation time. Consider monolayer Nb2n+1SinTe4n+2 +with the coordinate setup in Fig. 7(a). +Assuming ap- +plied E field is in the direction specified by the polar +angle θ (with respect to the mirror line), i.e., (Ex, Ey) = +E(cos θ, sin θ), then the Hall current will be in the direc- +tion of (jx, jy) = jH(− sin θ, cos θ), with the Hall current +magnitude +jH = χH(θ)E2, +(9) +and the nonlinear Hall conductivity +χH(θ) = −1 +2τDy sin θ. +(10) +Experimentally, a 2D material sample can be etched +into a disk shape and attached with multiple pairs of +leads [6, 50], such that the sin θ angular dependence in +the nonlinear Hall response can be verified in experiment. +To measure the second-order nonlinear response, one typ- +ically modulates the driving source with a low frequency +and detects the signal at doubled frequency using the +lock-in technique [5, 6]. The Fermi level of 2D materials +can be readily tuned by using electric gating technique. +Here, consider monolayer Nb3SiTe6 (i.e., n = 1). With +our calculated Dy ∼ 1.54 ˚A at the lower peak, assuming +τ = 10 ps which is typical for 2D materials, the magni- +tude of χH can reach 2.9 × 10−4 nm·S/V and its angular +dependence is shown in Fig. 7(b). Under a driving field +of E ∼ 104 V/m, the resulting nonlinear Hall current +density can reach ∼ 0.6 µA/cm . +For n = 2 (3), the +signal is expected to be smaller by a factor ∼ 2 (∼ 4), +which is still detectable in experiment. +VII. +CONCLUSION +We have revealed monolayer Nb2n+1SinTe4n+2 materi- +als as a suitable platform for studying BCD and nonlinear +Hall effect. These materials have the adequate symmetry +to support the effect without extra strain, enjoy stability +at ambient conditions, and exhibit composition tunabil- +ity. We show that BCD is most pronounced for the n = 1 +case, where its magnitude can reach 1.54 ˚A. The BCD +value decreases with increasing n. This can be under- +stood from degree of symmetry breaking and also from a +dimensional crossover. It is interesting that this crossover +occurs only for the low-energy electronic states, whereas +structurally, the system is always strongly bonded in 2D. +The evolution of BCD with n can be regarded as a man- +ifestation of this hidden transition. +We construct the +spin-orbit-coupled Dirac SSH model, which captures the +main features of DFT results. The nonlinear Hall con- +ductivity and its angular dependence are analyzed. Our +work uncovers interesting geometric quantities and non- +linear physics in Nb2n+1SinTe4n+2 family materials. It +provides useful guidance for subsequent experiments on +these systems. +ACKNOWLEDGMENTS +The authors thank D. L. Deng for helpful discus- +sions. +This work is supported by Singapore MOE +AcRF Tier 2 (T2EP50220-0026), +National Natural +Science Foundation of China (Grant Nos. 52271136, +11704304, 12204378), and Natural Science Foundation of +Shaanxi Province (Grant Nos. 2019TD-020, 2019JLM-30, +2017JZ015, 2018JQ1028). 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Yang, and W.-b. +Gao, Nature Nanotechnology 16, 869 (2021). + diff --git a/t9AzT4oBgHgl3EQfBvof/content/tmp_files/load_file.txt b/t9AzT4oBgHgl3EQfBvof/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..68ae0639ad929c146a02a121dc3ed8499b473dea --- /dev/null +++ b/t9AzT4oBgHgl3EQfBvof/content/tmp_files/load_file.txt @@ -0,0 +1,948 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf,len=947 +page_content='Berry curvature dipole and nonlinear Hall effect in two-dimensional Nb2n+1SinTe4n+2 Yiwei Zhao,1, 2 Jin Cao,3, 2, ∗ Zeying Zhang,4, 2, † Si Li,5 Yan Li,1 Fei Ma,1, ‡ and Shengyuan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Yang2 1State Key Laboratory for Mechanical Behavior of Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Xi’an Jiaotong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Shaanxi 710049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' China 2Research Laboratory for Quantum Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Singapore University of Technology and Design,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Singapore 487372,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Singapore 3Centre for Quantum Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Beijing Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Beijing 100081,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' China 4College of Mathematics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Beijing University of Chemical Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Beijing 100029,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' China 5School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Northwest University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Shaanxi 710127,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' China Recent experiments have demonstrated interesting physics in a family of two-dimensional (2D) composition-tunable materials Nb2n+1SinTe4n+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Here, we show that owing to its intrinsic low symmetry, metallic nature, tunable composition, and ambient stability, these materials offer a good platform for studying Berry curvature dipole (BCD) and nonlinear Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Using first-principles calculations, we find that BCD exhibits pronounced peaks in monolayer Nb3SiTe6 (n = 1 case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Its magnitude decreases monotonically with n and completely vanishes in the n → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' This variation manifests a special hidden dimensional crossover of the low-energy electronic states in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The resulting nonlinear Hall response from BCD in these materials is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Our work reveals pronounced geometric quantities and nonlinear transport physics in Nb2n+1SinTe4n+2 family materials, which should be readily detected in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' INTRODUCTION The Hall effects, in which a transverse current jH is induced by a longitudinal driving E field, are of funda- mental importance in condensed matter physics [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' At linear order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=', with jH ∼ E, the Hall effect requires the broken time reversal symmetry T , which can be achieved either by an applied magnetic field or by intrinsic mag- netism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' This constraint is loosened when considering Hall responses at nonlinear order, as the non-equilibrium elec- tron distribution driven by E field already breaks T at its first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Focusing on the second-order response, in nonmagnetic materials and in the absence of mag- netic field, Sodemann and Fu proposed a Berry curvature dipole (BCD) contribution to the nonlinear Hall current jH ∼ E2 within the semiclassical theory framework [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Their work attracted great interest in the past few years, and the effect has been successfully detected in several material systems [5–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' It was suggested that this effect offers a new mechanism for nonlinear applications, such as frequency-doubling and rectification [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' For experimental study, two-dimensional (2D) mate- rials have advantages in their great tunability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' For ex- ample, the Fermi level in 2D materials can be readily tuned via gating technique to a large extent not possible in 3D bulk materials [5, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' However, regarding BCD and its induced nonlinear Hall effect, the constraint from crystalline symmetry in 2D is rather stringent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' It was shown that the largest symmetry in 2D that allows for a nonzero BCD is a single in-plane mirror line [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Hence, to realize the effect, one has to choose crystals with very low symmetry, which are rather limited, or takes extra ∗ caojin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='phy@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='com † zzy@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='buct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='cn ‡ mafei@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='cn effort to exert strain or twist on the crystal to lower the symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' This severely hinders the experimental study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Recently, the family of composition-tunable materials Nb2n+1SinTe4n+2 have attracted interest from both the- ory and experiment [29–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' In the bulk form, these materials are van der Waals layered materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Their high-quality 2D layers can be obtained by mechanical exfoliation method [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The special feature of this fam- ily is the tunable composition embodied by the integer n [36–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' For each n, the system is a stoichiometric crystal, and the physical properties have an interesting dependence on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' For example, it was shown that in a 2D monolayer, for finite n, the material is nonsymmor- phic nodal-line semimetal [29];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' whereas the n → ∞ limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=', the material Nb2SiTe4, is a narrow-gap semiconduc- tor [33, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' With increasing n, the low-energy states at Fermi level exhibits a dimensional change from 2D-like states to 1D-like states [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' We note that 2D Nb2n+1SinTe4n+2 materials actually offer a good platform to explore BCD related physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' First, except for the n → ∞ limit, all members of the family have a sufficiently low symmetry to allow an in- trinsic BCD, without the need of applied strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Sec- ond, they offer an opportunity of systematic investigation of the evolution of BCD with the tunable composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Third, these 2D materials are stable at ambient condi- tions [32], which facilitates experimental study as well as possible applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Motivated by the above considerations, in this work, we theoretically study BCD and nonlinear Hall effect in monolayer Nb2n+1SinTe4n+2 materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' With first- principles calculations, we show that the n = 1 case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=', Nb3SiTe6, possesses a pronounced BCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The magni- tude can reach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='54 ˚A in the hole doped case, larger than previously reported values in 2D Td-WTe2 [18], strained NbS2 [21] and WSe2 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' With increasing n, the BCD peaks in the spectrum show a monotonic decrease and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='00946v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='mtrl-sci] 3 Jan 2023 2 eventually vanish in the n → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' This behavior can be understood from two perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' One is from the symmetry perspective, and the other is from the dimen- sional evolution of the electronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The latter view manifests that although structurally, these materials are strongly bonded in both directions in 2D, electronically, the states exhibit a dimensional crossover from 2D to 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' This hidden crossover dictates the change in BCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The key features of the results are further captured by our constructed tight-binding models for this family of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' To guide experiment, we discuss properties of the nonlinear Hall response arising from BCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Our work reveals interesting properties of Nb2n+1SinTe4n+2 family materials and suggests them as a suitable platform to explore BCD and nonlinear Hall physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' COMPUTATION METHOD Our first-principle calculations were based on the den- sity functional theory (DFT), performed by using the Vienna ab initio simulation package [41–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The ionic potentials were treated by using the projector aug- mented wave method [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The exchange-correlation functional was treated by the generalized gradient ap- proximation [45] in the scheme by Perdew, Burke, and Ernzerhof [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The plane-wave cutoff energy was set to be 400 eV, and a 10 × 4 × 1 Γ-centered k-point mesh was used for the Brillouin zone (BZ) sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The conver- gence criteria for the total energy and the force were set to be 10−6 eV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='01 eV/˚A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' To avoid artificial interaction between periodic images, a vacuum space of 15 ˚A thickness was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Spin-orbital cou- pling (SOC) was included in all calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Based on the band structure calculation, an ab initio tight-binding model was constructed by using the Wannier90 pack- age [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The d orbitals of Nb atoms and p orbitals of Te atoms were used as the initial guess of the local basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The BCD was calculated based on this ab initio tight- binding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' In evaluating BCD, we set T = 100 K in the Fermi distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' CRYSTAL AND ELECTRONIC STRUCTURES The Nb2n+1SinTe4n+2 family materials were first syn- thesized in the 1990s by chemical vapour transport method [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The lattice structures of their 2D mono- layers are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Here, each monolayer consists of three atomic layers: the middle layer con- taining Nb and Si atoms is sandwiched by two Te lay- ers [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' From the top view [see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 1(b)-1(d)], these materials can be viewed as composed of three build- ing blocks, which are conventionally called the a, b, c chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 1(b), a and b chains contain Si atoms and share the same composition of NbSi1/2Te2, whereas the c chain does not contain Si and has the a b c (ab) c c (ab)×n c c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (a) (b) (c) (d) Nb Si Te x y y z x FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (a) Lattice structure of monolayer Nb3SiTe6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (b) The three building blocks of of Nb2n+1SinTe4n+2 family materials: a, b and c chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (c) Top view of n = 1 case (Nb3SiTe6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The dashed box marks the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (d) Nb2n+1SinTe4n+2 can be constructed by n copies of (ab) chains and one c chain in a unit cell (the dashed box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' composition of NbTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Assuming these chains are along the x direction [as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 1(c)], then a and b are con- nected by a glide mirror operation ˜ My = {My| 1 20}, and in these materials they always appear together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Mem- bers of this family are formed by assembling these chains along the lateral direction (y) in a periodic manner, such that Nb2n+1SinTe4n+2 corresponds to the arrangement of (ab)nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Namely, in a period, we have one c chain and n copies of (ab) chains, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' In the n = ∞ limit, there is no c chain in the structure any more, and we reach the composition of Nb2SiTe4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Our optimized lattice parameters for n = 1, 2, 3, ∞ are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' These values are in good agreement with experiment and previous calculations [29, 30, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' We also note that for members with finite n, they all have the space group symmetry Pmc21, with C2v point group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' In comparison, Nb2SiTe4 with n = ∞ has a larger space group Pbam and a point group D2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The main difference is the extra glide mirror ˜ Mx = {Mx|0 1 2} for n = ∞ case but not for any finite n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 1(c), one can see that it is the c chains that break the ˜ Mx symmetry which holds for (ab) chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 2, we plot the calculated electronic band struc- tures for the four representative members in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' One can see that the band structures for n = 1, 2, 3 show sim- ilar features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Previous works have shown that in the 3 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Optimized lattice parameters and the correspond- ing symmetries of representative monolayer Nb2n+1SinTe4n+2 materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' n a (˚A) b (˚A) Thickness (˚A) Space group Point Group 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='408 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='633 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='649 Pmc21 C2v 2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='405 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='590 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='770 Pmc21 C2v 3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='404 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='552 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='651 Pmc21 C2v ∞ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='401 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='962 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='783 Pbam D2h absence of SOC, these materials are nodal-line semimet- als [29, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The nodal line on the X-M path around Fermi level is enforced by the nonsymmorphic T ˜ My sym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The detailed analysis was given in our previous works [48], so we will not repeat it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' It should be noted that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 2, the band structures include the SOC effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Under SOC, the T ˜ My symmetry protection is no longer exact, so the original nodal line degeneracy will be lifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' In the enlarged view in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 2(b), one can clearly see the splitting of the nodal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Nevertheless, there is still a degenerate nodal point at X (and also at M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' This point is a fourfold degenerate Dirac point enforced by nonsymmorphic symmetries of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Its for- mation mechanism has been discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The SOC induced change to the band structure is weak, so for many properties, SOC may just be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' However, band geometric properties like Berry curvature and BCD are very sensitive to small-gap regions in band structures, such as those due to SOC splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Therefore, to study BCD and its nonlinear Hall effect, we have to include SOC in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The low-energy states around Fermi level are mostly distributed on the c chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Previous scanning tunnel- ing spectroscopy (STS) experiments also verified this fea- ture [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' With increasing n, the distance between two c chains will increase and hence the coupling between them will decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' As a result, the band dispersion will become flatter along the y direction, as can be seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 2(c)-2(e) along the Γ-Y and X-M paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' For Nb2SiTe4 with n = ∞, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 2(f) shows that it is a narrow-gap semiconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The band gap is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='51 eV, which is slightly larger than the band gap of lay- ered Nb2SiTe4 (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='39 eV) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' This different character can now be understood from the discussion above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' One can view the c chains as metals, whereas the (ab) chains are insulating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Since Nb2SiTe4 is entirely made of (ab) chains, its spectrum would naturally be gapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The features discussed above, particularly the evolu- tion of band structure with n, will have important im- plications on BCD and nonlinear Hall response in these materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' BERRY CURVATURE DIPOLE Berry curvature is an intrinsic band geometric quan- tity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' It plays an important role in many physical prop- E (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 0 Y Y M X E (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 0 0 80 80 E (meV) E (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 0 Y M Γ X （a） （b） （c） （d） （e） （f） Γ E (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 0 Y Y M X Γ Y Y M X Γ Y Y M X Γ M X k x k y FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (a) Brillouin zone for monolayer Nb2n+1SinTe4n+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (b-f) Band structures for monolayer Nb2n+1SinTe4n+2: (b, c) n = 1, where (b) is an enlarged figure around the path X-M in (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (d) n = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (e) n = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' and (f) n = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' erties, especially anomalous transport properties [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' In nonmagnetic materials, nonzero Berry curvature requires the breaking of inversion symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' This condition is ful- filled in monolayer Nb2n+1SinTe4n+2 with finite n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' For Nb2SiTe4 with n = ∞, inversion symmetry is respected and hence Berry curvature vanishes identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' For a 2D system, Berry curvature only has a single component, which can be expressed as (we set e = ℏ = 1 in the formulas) Ωz(nk) = −2 Im � n′̸=n ⟨unk|vx|un′k⟩⟨un′k|vy|unk⟩ (εnk − εn′k)2 , (1) for a state |unk⟩, where vx and vy are the velocity oper- ators, and εnk is the energy of |unk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Consider Nb3SiTe6 (n = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 3(a), we plot the distribution of its Berry curvature in BZ for occupied states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=', the quantity Ω(k) = � n f0Ωz(nk), (2) where f0 is the Fermi distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' One observes that the Berry curvature is odd in ky and even in kx, as required by T and ˜ My, and its value is quite pronounced along the Γ-Y path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' BCD is the first moment of Berry curvature in BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' It is a pseudovector in 2D, defined as [4] Da = � n � BZ d2k (2π)2 f0∂aΩz(nk) = − � n � BZ d2k (2π)2 f ′ 0va(nk)Ωz(nk), (3) 4 (a) (b) (c) (d) (e) (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Berry curvature and its dipole (BCD) in Nb3SiTe6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (a) Distribution of Berry curvature for the occupied states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (b) The BCD Dy versus chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (c-f) The k- resolved BCD as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Here, (c, d) are plotted for µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='064 eV (the upper peak in (b)), and (e, f) are for µ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='180 eV (the lower peak in (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The Fermi contours at these energies are indicated by the black curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' where a ∈ {x, y}, ∂a ≡ ∂ka, and in the second line, we write it as a Fermi surface integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' For finite n, Nb2n+1SinTe4n+2 only has a single mirror line along x, which allows a nonzero BCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Since D is a pseudovector, it must be along the y direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=', D = Dyˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 3(b), we plot the calculated Dy versus the chemical potential µ for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' One observes two peaks in the figure: one is at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='064 eV with a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='399 ˚A, and the other is at −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='180 eV with a value of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='540 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The two peaks are of opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' We note that the magnitude of -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='540 ˚A is quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' This is comparable or larger than those found in monolayer Td- WTe2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='1 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='7 ˚A) [18], strained NbS2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 ˚A) [21] and strained WSe2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='02 ˚A) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' To understand the origin of the large BCD in mono- layer Nb3SiTe6, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 3(c)-3(f), we plot the k-resolved BCD on Fermi surface, namely the quantity Da(k) = − � n f ′ 0va(nk)Ωz(nk), (4) for µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='064 eV (upper peak) and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='180 eV (lower peak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' First of all, one observes that Dx(k) is an odd function in ky whereas Dy(k) is an even function, as re- quired by the ˜ My symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Hence, after integral over BZ, BCD only has the y component left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' From Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 3(c)- 3(f), one can see that the nodal line region along X-M does not make a sizable contribution to BCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' For the upper peak [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 3(d)], large contribution to Dy is from the Γ-Y path, which corresponds to the SOC splitting µ (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='8 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 µ (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='8 n = 2 n = 3 (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='8 Dy (Å) Dy (Å) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The BCD Dy versus chemical potential µ for (a) n = 2, and (b) n = 3 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' gap indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The spin splitting gap on the outer Fermi surface [marked by the green arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 3(d)] also gives a non-negligible contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' As for the lower peak, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 3(e)-3(f) show that the Fermi surface has two separate pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' By examining the band structure around the hot spots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 3(f), we find that the large negative contribution is also from SOC splitting of the band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Next, we consider the cases with n = 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' From the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 4, one can see that the magnitude of BCD decreases with increasing n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' For n = 3, the BCD value above µ = 0 (which is also the energy of nodal line) is already negligibly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' As for the lower peak, the value is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='663 ˚A for n = 2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='396 ˚A for n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' This trend of decreasing BCD with increasing n in monolayer Nb2n+1SinTe4n+2 can be understood from two perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' First, in terms of symmetry, Nb2n+1SinTe4n+2 with finite n supports BCD because of its low symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The presence of c chains is cru- cial because they break the ˜ Mx symmetry of (ab) chains (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Without c chains, ˜ Mx becomes an exact sym- metry and it suppresses BCD (given the other mirrors in the system) as in the n = ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Hence, the density of c chains in the system can be viewed as a measure of the extent of symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' It is strongest in n = 1 case, and gradually decreases as n increases, determining the trend in BCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Meanwhile, the trend is also connected with the di- mensional crossover in this system [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' As discussed, the low-energy states are mostly distributed on the c chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' One may view the c chains as metallic 1D subsystems put in an insulating matrix formed by the (ab) chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' For small n, the system retains a 2D character, because the c chains are not far from each other and the inter-chain coupling is sizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' However, with increasing n, the inter- chain coupling will decrease, and the system approaches the quasi-1D character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Berry curvature is a differential 2-form, which vanishes in the 1D limit [as can also be seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Thus, BCD must decrease and approach zero during this dimensional crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' It must be emphasized that the dimensional crossover here is referring to the low-energy electronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Structurally, Nb2n+1SinTe4n+2 materials always main- tain a 2D material character: the lattices are strongly 5 x y y FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (a) (ab) chains preserve the ˜ Mx symmetry, whereas c chains break it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Hence, the density of c chains represents the extend of ˜ Mx symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (b) The Nb2n+1SinTe4n+2 system may be schematically viewed as 1D metallic chains (c chains) embedded in a 2D insulator matrix (made of (ab) chains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' bonded in both x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Thus, the crossover is a hidden feature that occurs only for the electronic sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' This is a very interesting piece of physics for 2D Nb2n+1SinTe4n+2 materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Now, we revealed its man- ifestation in BCD, which can be detected via nonlinear Hall measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' A MODEL STUDY To understand the features in band structure and in BCD, we construct an effective lattice model to describe the low-energy bands in monolayer Nb2n+1SinTe4n+2 with finite n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The model may also serve as a good start- ing point for other theoretical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' [34, 48], we have proposed a 2D Dirac Su- Schrieffer-Heeger (SSH) model, which is spinless (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=', without SOC) and captures the nonsymmorphic nodal line feature in monolayer Nb2n+1SinTe4n+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' However, to study BCD, as we noted, the consideration of SOC is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Therefore, we need to extend the previous spinless Dirac SSH model to include SOC effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The Dirac SSH model is defined on a rectangular lat- tice, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' It consists of an array of zigzag chains running in the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Physically, each chain corresponds to a c chain in Nb2n+1SinTe4n+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' In a unit cell, there are two sites A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Assigning one orbital at each site and considering the nearest intra-chain and inter-chain hoppings, one obtains the following model constrained by T , ˜ My, and Mz symmetries: H0 = t � 0 1 + e−ikx 1 + eikx 0 � σ0 +t′ � 0 e−iky � 1 + e−ikx� eiky � 1 + eikx� 0 � σ0, (5) where the momenta are measured in unit of the inverses of lattice constants, and the Pauli matrices σ denote the x y My t t’ A B E /t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 0 Y Y M X Γ Dy(Å) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='4 t’/t ~ x y A B λ1 λ3 λ2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='3 µ (eV) Dy(Å) (a) (b) (c) (d) (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (a) Schematic figure showing the tight-binding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The model consists of zigzag chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' A primitive cell contains two sites A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' t and t′ are the amplitudes for intrachain and interchain hoppings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (b) illustrates the three hopping processes corresponding to the SOC terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (c) Band structure of the tight-binding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (d) Corresponding BCD Dy versus chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (e) Variation of BCD peak value as a function of the interchain coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The solid curve is a guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' In (c, d), we set t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='2 eV, t′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='16 eV, λ1 = 1, and λ2 = λ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The same values of t and λ’s are taken in (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' spin degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Next, we add SOC to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The above mentioned symmetries resulted in the following SOC terms up to second neighbor hopping processes: HSOC = t � 2λ1 sin kx 0 0 −2λ1 sin kx � σz +t′ � 2λ3 sin ky iλ2eiky � 1 + e−ikx� −iλ2e−iky � 1 + eikx� 2λ3 sin ky � σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (6) Here, the first term is from intrachain hopping process, whereas the second term is from interchain process, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Therefore, our spin-orbit-coupled Dirac SSH model is obtained as H = H0 + HSOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (7) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 6(c), we plot a typical band structure of this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Namely, there is an approximate nodal line on the X-M path (split by SOC);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' the SOC splitting is ob- served on X-M and Y -Γ paths, but not on the Γ-X and M-Y paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The double degeneracy on X-M and Y -Γ 6 4 0 4 0 π 2π χ⊥ (10-4 nm · V-1 Ω-1) (b) (a) x y ɵ E jH ɵ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (a) Illustration of the nonlinear Hall current induced by E field in Nb3SiTe6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The in-plane E field makes an angle θ from the mirror line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The induced Hall current is perpen- dicular to the E field, as indicated by the green arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (b) Nonlinear Hall conductivity χH versus the angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' is due to the anti-commutation between ˜ My and Mz on these two paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' One can see that it indeed captures the main features of DFT band structures in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 6(c), we plot the BCD calculated for this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The two BCD peaks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 3(b) are reproduced in this simple model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' One peak is above the nodal-line energy and the other one is below, and they have opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Finally, we plot the BCD peak magnitude as a function of interchain coupling t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' One can see that the value mono- tonically increases with the interchain coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Since t′ decreases with n in monolayer Nb2n+1SinTe4n+2, the behavior in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 6(d) agrees with our result from DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' NONLINEAR HALL EFFECT It was shown that BCD leads to a second-order non- linear Hall current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' For a 2D system, the current can be expressed as jH = −1 2τ ˆz × E(D · E), (8) where E is the applied in-plane E field, and τ is the relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Consider monolayer Nb2n+1SinTe4n+2 with the coordinate setup in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Assuming ap- plied E field is in the direction specified by the polar angle θ (with respect to the mirror line), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=', (Ex, Ey) = E(cos θ, sin θ), then the Hall current will be in the direc- tion of (jx, jy) = jH(− sin θ, cos θ), with the Hall current magnitude jH = χH(θ)E2, (9) and the nonlinear Hall conductivity χH(θ) = −1 2τDy sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' (10) Experimentally, a 2D material sample can be etched into a disk shape and attached with multiple pairs of leads [6, 50], such that the sin θ angular dependence in the nonlinear Hall response can be verified in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' To measure the second-order nonlinear response, one typ- ically modulates the driving source with a low frequency and detects the signal at doubled frequency using the lock-in technique [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The Fermi level of 2D materials can be readily tuned by using electric gating technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Here, consider monolayer Nb3SiTe6 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=', n = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' With our calculated Dy ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='54 ˚A at the lower peak, assuming τ = 10 ps which is typical for 2D materials, the magni- tude of χH can reach 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='9 × 10−4 nm·S/V and its angular dependence is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Under a driving field of E ∼ 104 V/m, the resulting nonlinear Hall current density can reach ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='6 µA/cm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' For n = 2 (3), the signal is expected to be smaller by a factor ∼ 2 (∼ 4), which is still detectable in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' CONCLUSION We have revealed monolayer Nb2n+1SinTe4n+2 materi- als as a suitable platform for studying BCD and nonlinear Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' These materials have the adequate symmetry to support the effect without extra strain, enjoy stability at ambient conditions, and exhibit composition tunabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' We show that BCD is most pronounced for the n = 1 case, where its magnitude can reach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='54 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The BCD value decreases with increasing n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' This can be under- stood from degree of symmetry breaking and also from a dimensional crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' It is interesting that this crossover occurs only for the low-energy electronic states, whereas structurally, the system is always strongly bonded in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The evolution of BCD with n can be regarded as a man- ifestation of this hidden transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' We construct the spin-orbit-coupled Dirac SSH model, which captures the main features of DFT results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' The nonlinear Hall con- ductivity and its angular dependence are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Our work uncovers interesting geometric quantities and non- linear physics in Nb2n+1SinTe4n+2 family materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' It provides useful guidance for subsequent experiments on these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Deng for helpful discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' This work is supported by Singapore MOE AcRF Tier 2 (T2EP50220-0026), National Natural Science Foundation of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 52271136, 11704304, 12204378), and Natural Science Foundation of Shaanxi Province (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' 2019TD-020, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Yu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Guan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Sheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Yao, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Yang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' B 97, 045131 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Nie, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfBvof/content/2301.00946v1.pdf'} +page_content=' Yang, D.' 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We prove conjecturally sharp upper bounds for the Dirichlet character +moments +1 +r−1 +� +χ mod r | � +n≤x χ(n)|2q, where r is a large prime, 1 ≤ x ≤ r, and +0 ≤ q ≤ 1 is real. +In particular, if both x and r/x tend to infinity with r then +1 +r−1 +� +χ mod r | � +n≤x χ(n)| = o(√x), and so the sums � +n≤x χ(n) typically exhibit +“better than squareroot cancellation”. We prove analogous better than squareroot +bounds for the moments +1 +T +� T +0 | � +n≤x nit|2qdt of zeta sums; of Dirichlet theta func- +tions θ(1, χ); and of the sums � +n≤x h(n)χ(n), where h(n) is any suitably bounded +multiplicative function (for example the M¨obius function µ(n)). +The proofs depend on similar better than squareroot cancellation phenomena for +low moments of random multiplicative functions. An important ingredient is a reor- +ganisation of the conditioning arguments from the random case, so that one only needs +to “condition” on a small collection of fairly short prime number sums. The condi- +tioned quantities arising can then be well approximated by twisted second moments, +whose behaviour is the same for character and zeta sums as in the random case. +1. Introduction +In this paper we are interested in the size of sums such as +� +n≤x +nit +and +� +n≤x +χ(n), +where t ∈ R and χ(n) is a non-principal Dirichlet character modulo a large prime r. +These zeta sums and character sums are among the most studied objects in analytic +number theory. We would like to show, on the widest possible range of x, that we have +substantial cancellation amongst the terms in the sums. Furthermore, we would like to +understand the extent of the cancellation. +By periodicity, we can confine our study of character sums to the range x ≤ r. And +if t is large and x ≥ t, then standard Fourier analysis (see Lemma 1.2 of Ivi´c [13], for +example) shows that � +x 0, +which follows from Theorem 1 of Montgomery and Vaughan [22] (who actually proved +the bound with the fixed sum � +n≤x χ(n) replaced by M(χ) := maxx | � +n≤x χ(n)|). +See also Cochrane and Zheng [4], Granville and Soundararajan [7], and Kerr [14], for +a selection of stronger upper bounds on high moments when x is small. Notice that +it is important to exclude the principal character χ0 in Montgomery and Vaughan’s +result [22] when q and x are large (it would give a large contribution ≍ x2q/r), but in +the first statement its contribution ⌊x⌋2/(r − 1) is not overwhelming (compared with +⌊x⌋) provided x ≤ 0.99r, say. +It is natural to ask whether one typically (e.g. for a positive proportion of characters χ +mod r) has squareroot behaviour � +n≤x χ(n) ≍ √x, and thus whether the low moments +are really ≍ xq or not1. Combining the Cauchy–Schwarz inequality with the preceding +estimates, for any x ≤ 0.99r we get +1 +r − 2 +� +χ̸=χ0 mod r +| +� +n≤x +χ(n)|2q ≥ +( +1 +r−2 +� +χ̸=χ0 mod r | � +n≤x χ(n)|2)2 +1 +r−2 +� +χ̸=χ0 mod r | � +n≤x χ(n)|2(2−q) ≫q +x2 +r2−q +∀ 0 ≤ q ≤ 1. +In particular, for any fixed small α > 0 and all αr ≤ x ≤ 0.99r, we now find that indeed +1 +r−2 +� +χ̸=χ0 mod r | � +n≤x χ(n)|2q ≍α,q xq for all 0 ≤ q ≤ 1. Since the order of the second +1See the MathOverflow post http://mathoverflow.net/questions/129264/short-character-sums-averaged-on-the-character +for explicit discussion of this question. + +TYPICAL CHARACTER AND ZETA SUMS +3 +moment is the square of the first moment, another standard Cauchy–Schwarz argument +(often called the Paley–Zygmund inequality) implies that | � +n≤x χ(n)| ≫α +√x for a +positive proportion of characters mod r, when αr ≤ x ≤ 0.99r. But when x = o(r), the +lower bound +x2 +r2−q does not match xq, and the typical size of � +n≤x χ(n) remains unclear. +(Note that depending on the size of x, one could substantially improve the “simple” +lower bound +x2 +r2−q by suitably applying H¨older’s inequality rather than the Cauchy– +Schwarz inequality, and using the high moment bounds of e.g. [4, 7, 14] rather than +Montgomery and Vaughan’s result [22]. See also section 1.5 of La Bret`eche, Munsch and +Tenenbaum [3]. But this would still not deliver a matching lower bound xq in general.) +For zeta sums � +n≤x nit, it is not too difficult to show that | � +n≤x nit| ≪ +√ +t log t for +all large x ≤ t, although this estimate seems much less celebrated than the analogous +P´olya–Vinogradov inequality for character sums. See chapter 7.6 of Montgomery [20], +and the paper of Fujii, Gallagher and Montgomery [5]. +The Vinogradov–Korobov +method yields non-trivial estimates o(x) provided (log x)/ log2/3 |t| → ∞ as |t| → ∞, +see e.g. chapter 6 of Ivi´c [13]. Obtaining bounds on a wide range of x is particularly +important when bounding the Riemann zeta function ζ(s) for ℜ(s) close to 1, with conse- +quences for the distribution of primes. When ℜ(s) = 1/2, obtaining a larger saving on a +more limited range of x is important. The state of the art is recent work of Bourgain [2]. +Somewhat surprisingly, the analogues of the precise conditional character sum results +of Montgomery and Vaughan [21] and of Granville and Soundararajan [7] do not seem +to have been worked out explicitly for zeta sums. However, one could deduce various +such results by adapting their methods, using the approximate functional equation for +ζ(s) (which implies in particular that | � +n≤x nit| ≍ +√ +t| � +t/(2πx) 0, which is a partial ana- +logue (with x fixed rather than taking an inner maximum over x) of Montgomery and +Vaughan’s moment bound [22] for character sums. We remark that although the restric- +tion to T/2 ≤ t ≤ T could be significantly relaxed here, we would not have such a bound +when integrating over all 0 ≤ t ≤ T with q and x large, due to large contributions from +small t (analogously to the contribution from the principal character χ0 that should be +excluded from high moments of character sums). +Now if x ≤ cT for a suitable small constant c > 0, then Montgomery and Vaughan’s +mean value theorem also immediately implies that 2 +T +� T +T/2 | � +n≤x nit|2dt = ⌊x⌋+O(x2/T) ≫ +x (and with more work one could show this for all 1 ≤ x ≤ T). So, similarly as for charac- +ter sums, the Cauchy–Schwarz inequality implies that 2 +T +� T +T/2 | � +n≤x nit|2qdt ≫q x2/T 2−q +for all 1 ≤ x ≤ T and 0 ≤ q ≤ 1. In particular, if x ≍ T then these moments are ≍ xq, +but when x = o(T) the true order of the low moments remains unclear. See also section +1.5 of La Bret`eche, Munsch and Tenenbaum [3] for discussion of lower bounds for the +moments. +One way of exploring the behaviour of � +n≤x χ(n) or � +n≤x nit is to consider an +appropriate random model. +Let (f(p))p prime be a sequence of independent random +variables, each distributed uniformly on the complex unit circle. +Then we define a +Steinhaus random multiplicative function f(n), by setting f(n) := � +pa||n f(p)a for all +n ∈ N. Steinhaus random multiplicative functions have been used quite extensively +to model a randomly chosen Dirichlet character χ(n) or “continuous character” nit: +see the papers of Granville and Soundararajan [7] and Lamzouri [15], for example. +Helson [12] conjectured, by a rough analogy with the first moment of the Dirichlet kernel +in classical Fourier analysis, that one should have E| � +n≤x f(n)| = o(√x) as x → ∞. +This conjecture was somewhat surprising, given the general philosophy of squareroot +(and not more than squareroot) cancellation for oscillating number theoretic sums, and +various counter-conjectures were made by other authors. See the introduction to [11] for +discussion and references. However, the author [11] recently proved Helson’s conjecture, +in fact showing that +E| +� +n≤x +f(n)|2q ≍ +� +x +1 + (1 − q)√log log x +�q +(1.1) +uniformly for all large x and all 0 ≤ q ≤ 1. This raises the question whether one should +now expect better than squareroot cancellation for character and zeta sums, on some +range of x rather smaller than the conductor, or whether the random multiplicative +model simply fails to capture the arithmetic truth in these problems. + +TYPICAL CHARACTER AND ZETA SUMS +5 +1.1. Statement of results. Our main results establish, for character and zeta sums, +the natural analogues of the upper bound part of (1.1). +Theorem 1. Let r be a large prime. Then uniformly for any 1 ≤ x ≤ r and any +0 ≤ q ≤ 1, we have +1 +r − 1 +� +χ mod r +| +� +n≤x +χ(n)|2q ≪ +� +x +1 + (1 − q) +� +log log(10L) +�q +, +where L = Lr := min{x, r/x}. +Theorem 2. Let T be a large real number. Then uniformly for any 1 ≤ x ≤ T and any +0 ≤ q ≤ 1, we have +1 +T +� T +0 +| +� +n≤x +nit|2qdt ≪ +� +x +1 + (1 − q) +� +log log(10LT) +�q +, +where LT := min{x, T/x}. +In particular, for any fixed 0 < q < 1 and any x = x(r) such that x and r/x both +tend to infinity with r, we have +1 +r−1 +� +χ mod r | � +n≤x χ(n)|2q ≪ +xq +(log log(10L))q/2 = o(xq). +We shall discuss the proofs in detail in section 1.2, below. We note here that there is +a well known “symmetry” in the behaviour of character sums and zeta sums, whereby +e.g. +1 +√x| � +n≤x χ(n)| ≈ �x +r | � +n≤r/x χ(n)| (very roughly speaking). See section 10 of +Granville and Soundararajan [7]. This symmetry is sometimes called the “Fourier flip”, +and manifests itself in the (approximate) functional equations of the corresponding L- +functions, and in the very structured nature of long sums � +n≤x χ(n), � +n≤x nit where x +is of the same order as the conductor (i.e. as r or |t|, respectively). Given the symmetry +between character sums of lengths x and r/x, and between zeta sums of lengths x and +|t|/x, the quantities log log(10Lr) and log log(10LT) appearing in Theorems 1 and 2 are +natural substitutes for the log log x saving factor in (1.1). +The shape of the bounds in Theorems 1 and 2 might initially seem peculiar, and +perhaps open to improvement. In most number theoretic settings, if one obtains a saving +one expects to save at least a power of a logarithm. But in fact it seems reasonable to +conjecture that Theorems 1 and 2 are sharp (provided in Theorem 1 that x ≤ 0.99r, +say, so we are away from the point where periodicity trivially induces substantial extra +cancellation2). Note that in the probabilistic setting of (1.1) we already have an order +of magnitude result, rather than just an upper bound. +It is possible that the methods leading to Theorems 1 and 2 would produce matching +lower bounds when x ≤ elogc r and x ≤ elogc T, say, for a certain small c > 0, although the +2For non-principal χ, if 0.99r < x ≤ r we can observe that � +n≤x χ(n) = − � +x 0, we +define +θ(s, χ) := +� �∞ +n=1 χ(n)e−πn2s/r +if χ is even, +�∞ +n=1 nχ(n)e−πn2s/r +if χ is odd. +(These differ by a factor of 2 from the definitions sometimes given, but that will be +unimportant for our purposes here.) Theta functions arise in the theory of automor- +phic forms, and are an important tool in classical proofs of the functional equation for +Dirichlet L-functions. See e.g. chapter 10.1 of Montgomery and Vaughan [23]. + +TYPICAL CHARACTER AND ZETA SUMS +7 +In the last few years, a sequence of papers have investigated the moments of θ(1, χ), +as χ varies over even (non-principal) characters or over odd characters. For example, for +each fixed q ∈ N Munsch and Shparlinski [25] proved conjecturally sharp lower bounds +for the 2q-th moments, namely +1 +r − 1 +� +χ mod r, +χ even, +χ̸=χ0 +|θ(1, χ)|2q ≫q rq/2 log(q−1)2 r, +1 +r − 1 +� +χ mod r, +χ odd +|θ(1, χ)|2q ≫q r3q/2 log(q−1)2 r. +Munsch [24] proved almost sharp upper bounds assuming the Generalised Riemann Hy- +pothesis for Dirichlet L-functions (losing a factor logǫ r compared with the presumed +truth), again when q ∈ N. One application of such estimates is deducing non-vanishing +results for θ(1, χ), the subject of a conjecture of Louboutin [17]. Thus Louboutin [16], +and Louboutin and Munsch [18], proved that θ(1, χ) ̸= 0 for ≫ r/ log r odd characters +and ≫ r/ log r even characters modulo prime r, by computing and comparing second +and fourth moments. Most recently, La Bret`eche, Munsch and Tenenbaum [3] intro- +duced weights coming from G´al sums into such arguments, and proved that θ(1, χ) ̸= 0 +for ≫ r/ logδ+o(1) r even characters modulo prime r, for a certain explicit constant +δ ≈ 0.086. See also the work of Bengoechea [1] and of Guo and Peng [8], who use Galois +theoretic techniques to deduce that θ(1, χ) ̸= 0 for almost all χ, but only for moduli r +from certain sparse subsets of primes. +Using our methods, we can prove: +Corollary 2. Let r be a large prime. Uniformly for any 0 ≤ q ≤ 1, we have +1 +r − 1 +� +χ mod r, +χ even +|θ(1, χ)|2q ≪ +� +√r +1 + (1 − q)√log log r +�q +, +and +1 +r − 1 +� +χ mod r, +χ odd +|θ(1, χ)|2q ≪ +� +r3/2 +1 + (1 − q)√log log r +�q +. +Since we have θ(1, χ) = �∞ +n=1 χ(n)e−πn2/r ≈ � +n≤√r χ(n) for even characters, and +θ(1, χ) ≈ � +n≤√r nχ(n) ≈ √r � +n≤√r χ(n) for odd characters, one sees that Corollary +2 should follow fairly directly from Theorem 1. Again, see section 4 below for the full +proof, which is just a partial summation argument. +Similarly as for Theorems 1 and 2, it is reasonable to conjecture that one should have +matching lower bounds for the moments in Corollary 2. If one could prove this for any +fixed 0 < q < 1, then (since the low moments scale proportionally with the exponent q, +unlike the higher moments where there is quadratic dependence in the exponent of log r) +another standard Paley–Zygmund type argument with H¨older’s inequality, comparing + +8 +ADAM J HARPER +upper and lower bounds for two low moments, would immediately yield that θ(1, χ) ̸= 0 +for a positive proportion of χ. +We are also highly interested in variants of Theorems 1 and 2, where the sums are +more exotic. +In the case of the second moment of such sums, one has exactly the +same mean value estimates +1 +r−1 +� +χ mod r | � +n≤x anχ(n)|2 = ⌊x⌋ (where x < r) and +1 +T +� T +0 | � +n≤x annit|2dt = ⌊x⌋+ O( x2 +T ) for any complex coefficients an with absolute value +1. We cannot seek analogues of Theorems 1 and 2 at this level of generality, since for +a generic sequence of unimodular coefficients an the moments will have order xq (e.g. +for independent, uniformly random an, this is implied by standard moment results like +Khintchine’s inequalities). But if we tweak the sums in ways that don’t disrupt the +multiplicative structure too much, then it turns out that better than squareroot bounds +like Theorems 1 and 2 do endure. Thus we have: +Theorem 3. Let r be a large prime. Then uniformly for any 1 ≤ x ≤ r, any 0 ≤ q ≤ 1, +and any multiplicative function h(n) that has absolute value 1 on primes and absolute +value at most 1 on prime powers, we have +1 +r − 1 +� +χ mod r +| +� +n≤x +h(n)χ(n)|2q ≪ +� +x +1 + (1 − q) +� +log log(10L) +�q +, +where L = Lr := min{x, r/x}. +The size conditions on h(n) could be adjusted a little here, but the current formu- +lation already permits important examples such as the M¨obius function µ(n). The key +point is that the multiplicative structure of h(n) allows one to adapt the proof of Theo- +rem 1, to show that the moments of � +n≤x h(n)χ(n) behave like the moments of twisted +random multiplicative sums � +n≤x h(n)f(n). And since, on the primes, h(p)f(p) are +again independent and uniform on the complex unit circle (here we use the unimodu- +larity of h(p)), these are essentially the same as the untwisted moments. See section 4 +for further details. One could also prove the analogous result for � +n≤x h(n)nit. +Unlike the sums � +n≤x χ(n) and � +n≤x nit, there is no symmetry or “Fourier flip” in +the presence of a general multiplicative twist h(n). So whilst the appearance of Lr, LT +in Theorems 1 and 2 was very natural, and one would conjecture those bounds to be +sharp, for many h(n) it seems likely that Theorem 3 should hold in a stronger form with +log log(10L) replaced by log log(10x). Similarly, the restriction to x ≤ r or to x ≤ T no +longer seems natural for � +n≤x h(n)χ(n) and � +n≤x h(n)nit. Indeed, if we take h(n) to +be a Steinhaus random multiplicative function, then (1.1) implies that for any large x + +TYPICAL CHARACTER AND ZETA SUMS +9 +and large prime r we have +E +1 +r − 1 +� +χ mod r +| +� +n≤x +h(n)χ(n)|2q = +1 +r − 1 +� +χ mod r +E| +� +n≤x +h(n)χ(n)|2q ≍ +� +x +1 + (1 − q)√log log x +�q +, +since h(n)χ(n) will also be a Steinhaus random multiplicative function3 for any given +Dirichlet character χ. Consequently, for most realisations of h(n) we must have the +stronger bound +1 +r−1 +� +χ mod r | � +n≤x h(n)χ(n)|2q ≪ ( +x +1+(1−q)√log log x)q. +In particular, we can return to the case of the M¨obius function. One tends to think +of µ(n) as being “random looking” in various senses, and in a recent paper Gorodet- +sky [6] explored this in the context of character sums. Based on function field consid- +erations, he ultimately conjectured that for all natural number exponents q < log r, +the moments +1 +r−1 +� +χ mod r | � +n≤x µ(n)χ(n)|2q should be asymptotic to the correspond- +ing random multiplicative moments as x and r become large. Likewise, it now seems +reasonable to conjecture that for all 0 ≤ q ≤ 1 and any fixed A > 0, we should have +1 +r − 1 +� +χ mod r +| +� +n≤x +µ(n)χ(n)|2q ≪ ( +x +1 + (1 − q)√log log x)q +∀ x ≤ rA, +and +1 +2T +� T +−T +| +� +n≤x +µ(n)nit|2qdt ≪ ( +x +1 + (1 − q)√log log x)q +∀ x ≤ T A. +(1.2) +This conjecture seems worthy of further investigation, since if true it would have signif- +icant arithmetic consequences. Standard arguments with Perron’s formula imply that +����� +� +x√x f(n), the subsum over +numbers whose largest prime factor P(n) is greater than √x. +By multiplicativity we can write � +n≤x,P (n)>√x f(n) = � +√x√x +remaining random, so the conditional expectation of any quantity is a function of the + +TYPICAL CHARACTER AND ZETA SUMS +11 +values (f(p))p≤√x), we get +˜E +����� +� +n≤x,P (n)>√x +f(n) +����� +2 += +� +√x√x remain independent with mean zero. But by the Tower Property +of conditional expectation (which in this case is simply Fubini’s theorem, breaking up +the multiple “integration” E into separate integrations corresponding to the (f(p))p≤√x +and the (f(p))p>√x), we can write E| � +n≤x,P (n)>√x f(n)|2q = E˜E| � +n≤x,P (n)>√x f(n)|2q. +So applying H¨older’s inequality to the conditional expectation ˜E, we get +E +����� +� +n≤x,P (n)>√x +f(n) +����� +2q +≤ E +� +˜E +����� +� +n≤x,P (n)>√x +f(n) +����� +2�q += E +� � +√x√x f(n)|2q +by writing E| � +n≤x,P (n)>√x f(n)|2q = E˜E| � +n≤x,P (n)>√x f(n)|2q will be (roughly) to write +Echar| +� +n≤x +χ(n)|2q = +� +j +EcharGj(χ)| +� +n≤x +χ(n)|2q, +where Gj(χ) are functions satisfying � +j Gj(χ) ≡ 1 that approximately pick out all +characters χ for which sums like ℜ � +p≤√x +χ(p) +p1/2+it have a given collection of values. We +can apply H¨older’s inequality to each inner average EcharGj(χ)| � +n≤x χ(n)|2q here, as we +did with ˜E| � +n≤x,P (n)>√x f(n)|2q previously, and end up needing to work with quantities +like EcharGj(χ)| � +n≤x χ(n)|2. Furthermore, if the functions Gj are sufficiently nice we +could hope to approximate Gj(χ) by a polynomial in the χ(p), at which point we might +be able to invoke (1.4) and replace EcharGj(χ)| � +n≤x χ(n)|2 by EGj(f)| � +n≤x f(n)|2. +This is a fairly good high level description of how the proof of Theorem 1 proceeds, +and so a key tool will be a collection of nice functions forming a smooth partition of +unity, from which we can ultimately form our multipliers Gj by taking suitable products +(corresponding to all the different t values in our net). See Approximation Result 1, in +section 2.1, for the technical statement we use. +But there is an important issue that must be addressed. Notice that the conductor +r doesn’t appear in the preceding paragraph, and one doesn’t yet see how the quantity +log log(10Lr) will arise in Theorem 1 in place of log log x. The point is that in (1.4) we + +TYPICAL CHARACTER AND ZETA SUMS +13 +must ensure, when everything is expanded out, that our products of primes are smaller +than r. See Propositions 1 and 2, in section 2.2, for the precise statements that we will +use to compare EcharGj(χ)| � +n≤x χ(n)|2 with EGj(f)| � +n≤x f(n)|2, where it is crucial +that the prime sums ℜ � +p≤P +χ(p) +p1/2+it involved run up to P for some P that is suitably +small compared with r/x. Indeed, for prime number sums of length P we should work +with a net of t values with spacing roughly 1/ log P, so with roughly log P different sums. +And for each of those, when we apply Taylor’s theorem to expand its contribution to +Gj(χ) we must expand as far as degree logO(1) P to achieve a suitable level of precision. +Since the square | � +n≤x χ(n)|2 also contributes terms χ(n), χ(m) with n, m up to x in +size, we must have xP logO(1) P < r to match up character sum and random multiplicative +function expressions. +In particular, we cannot work with P = √x here unless x is rather small compared +with r. The largest permissible choice of P will rather be something like elogc(r/x), for +a suitable constant c. Since we also want P < x (it would not make sense to include +sums over primes > x, which are not involved in � +n≤x χ(n), in the conditioning), one +reasonable choice turns out to be P ≈ exp{log1/6 Lr}. Fortunately, for an upper bound +(but not for a lower bound) we are free to select the range of primes involved in our +“conditioning” as we wish. (This observation would actually allow some simplification +of the upper bound arguments in the purely random setting [11] as well. Rather than +breaking up � +n≤x f(n) into various subsums according to the size of the largest prime +factor P(n), as there, and then conditioning on all somewhat smaller primes, we can +simply condition the full sum on all primes up to one suitably chosen point.) As P +becomes smaller, the possible saving ( +1 +1+(1−q)√log log P )q that we can ultimately achieve +using multiplicative chaos results will diminish, but since log log P varies so slowly we +can vary P quite a lot without visibly changing the final bounds. In particular, note +that although our choice P ≈ exp{log1/6 Lr} is (necessarily) significantly smaller than +Lr, we still have log log P ≍ log log Lr and so we get the desired factor +1 +1+(1−q)√ +log log(10L) +in Theorem 1. +In sections 3.1–3.3, we implement an argument along the above lines and succeed +in replacing all terms of the form EcharGj(χ)| � +n≤x χ(n)|2 by EGj(f)| � +n≤x f(n)|2, +thus passing to the purely random setting. +However, because this is done in the +setup of coarser conditioning, more work remains to confirm that the resulting ran- +dom multiplicative function expression can really still be bounded by something like +E( +1 +log P +� 1/2 +−1/2 |F rand +P +(1/2 + it)|2dt)q. Initially one can apply a similar kind of smoothing +and Parseval procedure as was originally done in [11], see section 2.3 (where we recall +a version of Parseval’s identity for Dirichlet series) and section 3.4 below. This brings +us to random Euler products, but with a (weighted) sum over j on the outside of the +q-th power in place of a “genuine” expectation E, and with some additional averaging + +14 +ADAM J HARPER +(weighted by Gj(f)) on the inside of the q-th power. Since each term Gj(f) only contains +information about ℜ � +p≤P +f(p) +p1/2+it at a net of points t, as opposed to all −1/2 ≤ t ≤ 1/2 +or all t ∈ R, we want to restrict to working with Euler products at those t. However, +provided the net of t are sufficiently close together (slightly closer than 1/ log P), this +can be achieved with simple mean square arguments. +Finally, we must check that the weighting by Gj(f) fixes enough information about +all the sums ℜ � +p≤P +f(p) +p1/2+it that, even with this extra averaging on the inside of the q-th +power, we still end up with something roughly like E( +1 +log P +� 1/2 +−1/2 |F rand +P +(1/2+it)|2dt)q (or +actually a version of this with the integral replaced by a sum over our discrete points +t). Provided the parameters were selected properly in terms of P when constructing +the smooth functions underlying Gj(·), (which is ultimately why one must Taylor ex- +pand as far as degree logO(1) P in the above discussion), it turns out that averaging +against Gj(f) does essentially restrict all the sums ℜ � +p≤P +f(p) +p1/2+it to their intended +boxes depending on j. Thus the weighted outer sum over j does perform essentially +the same role as a genuine expectation, and we do arrive at (a discretised version of) +E( +1 +log P +� 1/2 +−1/2 |F rand +P +(1/2+it)|2dt)q. This argument is completed in sections 3.5–3.6, using +properties of the functions from Approximation Result 1 and using multiplicative chaos +results presented in section 2.4. +2. Preparations +2.1. A smooth partition of unity. As explained in the introduction, one important +tool for us will be a collection of fairly well behaved functions that we can use to +approximately detect the values of various sums involving χ(p), and therefore simulate a +conditioning process in our main proofs. These sorts of constructions are fairly standard +in modern analysis and number theory, and it will not be too difficult to prove the +following. +Approximation Result 1. Let N ∈ N be large, and δ > 0 be small. There exist +functions g : R → R (depending on δ) and gN+1 : R → R (depending on δ and N) +such that, if we define gj(x) = g(x − j) for all integers |j| ≤ N, we have the following +properties: +(i) � +|j|≤N gj(x) + gN+1(x) = 1 for all x ∈ R; +(ii) g(x) ≥ 0 for all x ∈ R, and g(x) ≤ δ whenever |x| > 1; +(iii) gN+1(x) ≥ 0 for all x ∈ R, and gN+1(x) ≤ δ whenever |x| ≤ N; +(iv) for all l ∈ N and all x ∈ R, we have the derivative estimate | dl +dxlg(x)| ≤ +1 +π(l+1)( 2π +δ )l+1. + +TYPICAL CHARACTER AND ZETA SUMS +15 +Proof of Approximation Result 1. Our construction will be a minor variant of the proof +of Approximation Result 1 of Harper [9] (with R = 0 there). +As such, we content +ourselves with outlining the main steps. +Let b(x) be a Beurling–Selberg function majorising the indicator function 1|x|≤1/2, +with Fourier transform supported on [−1/δ, 1/δ]. See e.g. Vaaler’s paper [26] for back- +ground on such majorants. Thus we have b(x) ≥ 1|x|≤1/2 for all x ∈ R; and +� ∞ +−∞ b(x)dx = +1+δ; and b(x) = +� 1/δ +−1/δ ˆb(t)e2πixtdt for all x ∈ R, where |ˆb(t)| = | +� +b(x)e−2πixtdx| ≤ 1+δ. +We define g(x) as a convolution of b, namely +g(x) = +� ∞ +−∞ +1|u|≤1/2 +b(x − u) +1 + δ +du = +� ∞ +−∞ +1|x−u|≤1/2 +b(u) +1 + δdu. +Then it is clear that g(x) is non-negative, since b(x) is non-negative. The other claims +about g(x) in (ii) and (iv) follow identically as in Harper [9]. +Now by definition of g and gj, the sum � +|j|≤N gj(x) is += +� ∞ +−∞ +� +|j|≤N +1|x−j−u|≤1/2 +b(u) +1 + δdu = +� ∞ +−∞ +1|x−u|≤N+1/2 +b(u) +1 + δdu +for all real x. Thus we always have � +|j|≤N gj(x) ≤ +� ∞ +−∞ +b(u) +1+δdu = 1, and furthermore +1 − � +|j|≤N gj(x) is equal to +� ∞ +−∞ +1|x−u|>N+1/2 +b(u) +1 + δdu = +� ∞ +−∞ +1|x−u|>N+1/2 +b(u) − 1|u|≤1/2 +1 + δ +du+ +� ∞ +−∞ +1|x−u|>N+1/2 +1|u|≤1/2 +1 + δ du. +When |x| ≤ N the second integral here vanishes, and so +1 − +� +|j|≤N +gj(x) ≤ +� ∞ +−∞ +|b(u) − 1|u|≤1/2| +1 + δ +du = +� ∞ +−∞ +b(u) − 1|u|≤1/2 +1 + δ +du = +δ +1 + δ ≤ δ. +So the first and third statements in Approximation Result 1 follow if we simply set +gN+1(x) := 1 − � +|j|≤N gj(x) for all x ∈ R. +□ +2.2. Mean value estimates. Having introduced approximating functions as in Ap- +proximation Result 1, our arguments will require us to evaluate various character aver- +ages involving these functions. A basic tool for this will be the following bound, which +is a fairly standard even moment estimate for character sums of appropriate lengths. +Lemma 1 (Even moment estimate). Let x ≥ 1, and let (c(n))n≤x be any complex +numbers. Let P be any finite set of primes, let Q be any (non-empty) set consisting +of some elements of P and squares of elements of P, and write U := max{q ∈ Q} . +Finally, let Q(χ) := � +q∈Q +a(q)χ(q) +√q +, where the a(q) are any complex numbers. + +16 +ADAM J HARPER +Then for any natural number k such that xUk < r, we have +Echar +����� +� +n≤x +c(n)χ(n) +����� +2 +|Q(χ)|2k ≪ +�� +n≤x +˜d(n)|c(n)|2 +� +· (k!) +� +2 +� +q∈Q +vq|a(q)|2 +q +�k +, +where ˜d(n) := � +d|n 1p|d⇒p∈P, and vq is 1 if q is a prime and 6 if q is the square of a +prime. +Proof of Lemma 1. This is a character sum version of Lemma 2 of Harper [9], which +dealt with t-averages of � +n≤x c(n)n−it and � +q∈Q +a(q) +q1/2+it . It may be proved in the same +way as that result (in fact slightly more easily, since for Dirichlet characters one has +perfect rather than approximate orthogonality). +□ +Using Taylor expansion and Lemma 1 we can deduce the following crucial Propo- +sition, which guarantees that (provided we keep sufficient control on the sizes of the +various parameters) character averages involving the functions gj behave in the same +way as the corresponding averages involving random multiplicative functions. +Proposition 1 (Characters behave like random model). Let the functions gj, with +associated parameters N, δ, be as in Approximation Result 1. Suppose that x ≥ 1, and +let (c(n))n≤x be any complex numbers having absolute values ≤ 1. Furthermore, let P be +large, and let Y ∈ N be such that xP 400(Y/δ)2 log(N log P ) < r. Let f(n) denote a Steinhaus +random multiplicative function. +Then for any indices −N ≤ j(1), j(2), ..., j(Y ) ≤ N + 1, and for any sequences +(a1(p))p≤P, (a1(p2))p≤P, ..., (aY (p))p≤P, (aY (p2))p≤P of complex numbers having absolute +values ≤ 1, we have +Echar +Y� +i=1 +gj(i)(ℜ( +� +p≤P +ai(p)χ(p) +√p ++ ai(p2)χ(p2) +p +)) +����� +� +n≤x +c(n)χ(n) +����� +2 += +E +Y� +i=1 +gj(i)(ℜ( +� +p≤P +ai(p)f(p) +√p ++ ai(p2)f(p2) +p +)) +����� +� +n≤x +c(n)f(n) +����� +2 ++ O +� +x +(N log P)Y (1/δ)2 +� +. +Proof of Proposition 1. Using property (iv) from Approximation Result 1, for any thresh- +old 2S ∈ 2N we can write gj(x) = ˜gj(x) + rj(x), where ˜gj(·) is a polynomial of degree +2S − 1 (namely the degree 2S − 1 Taylor polynomial of gj(x) about zero), and where +|rj(x)| ≤ |x|2S +(2S)! sup|y|≤|x| | d2S +dy2S gj(y)| ≪ N|2πx/δ|2S +δS(2S)! . (Note that the factor N here is to ac- +count for the case where gj(y) = gN+1(y) = 1 − � +|i|≤N gi(y).) First we examine the +contribution from the “main terms” ˜gj(·) to the left hand side in the Proposition. Pro- +vided that xP 4SY < r, we can expand all the polynomials and the square out and find + +TYPICAL CHARACTER AND ZETA SUMS +17 +that +Echar +Y� +i=1 +˜gj(i)(ℜ( +� +p≤P +ai(p)χ(p) +√p ++ ai(p2)χ(p2) +p +)) +����� +� +n≤x +c(n)χ(n) +����� +2 += +E +Y� +i=1 +˜gj(i)(ℜ( +� +p≤P +ai(p)f(p) +√p ++ ai(p2)f(p2) +p +)) +����� +� +n≤x +c(n)f(n) +����� +2 +, +since if 1 ≤ U, V < r then we have the equality Echarχ(U)χ(V ) = 1U=V = Ef(U)f(V ). +(Note that Ef(U)f(V ) = 1U=V for all natural numbers U, V on the random multiplica- +tive function side, but on the character side one needs the restriction that 1 ≤ U, V < r +to boost a congruence mod r to an equality.) +Next, dividing up according to the smallest index i at which we get a remainder, we +see the contribution from all of the remainders rj(i)(·) to the left hand side in Proposition +1 is +≪ +Echar +Y +� +i=1 +N|2π/δ|2S +δS(2S)! | +� +p≤P +ai(p)χ(p) +√p ++ ai(p2)χ(p2) +p +|2S · +· +i−1 +� +l=1 +� +1 + O(N|2π/δ|2S +δS(2S)! | +� +p≤P +al(p)χ(p) +√p ++ al(p2)χ(p2) +p +|2S) +������ +� +n≤x +c(n)χ(n) +����� +2 +. +Here we used the fact that |˜gj(l)(x)| ≤ |gj(l)(x)| + |rj(l)(x)| ≤ 1 + O( N|2πx/δ|2S +δS(2S)! ). Us- +ing Lemma 1 and the condition xP 4SY < r again, along with the bounds (jS)! ≪ +√jS(jS/e)jS and (2S)! ≥ (2S/e)2S to handle the factorials produced by that lemma, +we find this is all +≪ +�� +n≤x +˜d(n)|c(n)|2 +� +· +Y +� +i=1 +N|2π/δ|2S +δS(2S)! +� +√ +iS(iS +e )S(2 log log P + O(1))S +� +· +· +i−1 +� +l=1 +� +1 + O +� +N|2π/δ|2S +δS(2S)! (iS +e )S(2 log log P + O(1))S +�� +≪ +x log P · +Y +� +i=1 +N +δ +� +i +S (e(π/δ)2i(2 log log P + O(1)) +S +)S · +· +� +1 + O +� +N +δS (e(π/δ)2i(2 log log P + O(1)) +S +)S +��i−1 +. +Here we also used our assumptions that |c(n)| ≤ 1 and |ai(p)|, |ai(p2)| ≤ 1, which in par- +ticular imply that 2 � +p≤P( |ai(p)|2 +p ++ 6|ai(p2)|2 +p2 +) ≤ 2 � +p≤P +1 +p+O(1) = 2 log log P +O(1). One +has the same overall bound for the contribution from the remainders rj(i)(·) to the right +hand side in Proposition 1, since one has the same bound for E| � +n≤x c(n)f(n)|2|Q(f)|2k + +18 +ADAM J HARPER +as for the character average in Lemma 1 (indeed this quantity is again exactly equal to +Echar| � +n≤x c(n)χ(n)|2|Q(χ)|2k, under the size conditions in the lemma). +Now if we set S = 100Y ⌊(1/δ)2 log(N log P)⌋, then the condition xP 4SY < r is +satisfied in view of our assumption that xP 400(Y/δ)2 log(N log P ) < r. And with this choice, +the error term produced by all of the remainders is +≪ x log P · +Y +� +i=1 +N +δ +� +i +S 0.6S +� +1 + O( N +δS 0.6S) +�i−1 +≪ x log P ·NY ·0.6S ≪ +x +(N log P)Y (1/δ)2 , +as desired. +□ +Taking x = 1 and c(1) = 1 in Proposition 1, we obtain the following important +special case. +Proposition 2. Let the functions gj, with associated parameters N, δ, be as in Approxi- +mation Result 1. Suppose P is large, and let Y ∈ N be such that P 400(Y/δ)2 log(N log P ) < r. +Let f(n) denote a Steinhaus random multiplicative function. +Then for any indices −N ≤ j(1), j(2), ..., j(Y ) ≤ N + 1, and for any sequences +(a1(p))p≤P, (a1(p2))p≤P, ..., (aY (p))p≤P, (aY (p2))p≤P of complex numbers having absolute +values ≤ 1, we have +Echar +Y� +i=1 +gj(i)(ℜ( +� +p≤P +ai(p)χ(p) +√p ++ ai(p2)χ(p2) +p +)) += +E +Y� +i=1 +gj(i)(ℜ( +� +p≤P +ai(p)f(p) +√p ++ ai(p2)f(p2) +p +)) + ++O +� +1 +(N log P)Y (1/δ)2 +� +. +2.3. Parseval’s identity for Dirichlet series. As in the proof of the random analogue +of Theorem 1, we shall need the following version of Parseval’s identity for Dirichlet +series. +Harmonic Analysis Result 1 (See (5.26) in sec. 5.1 of Montgomery and Vaughan [23]). +Let (an)∞ +n=1 be any sequence of complex numbers, and let A(s) := �∞ +n=1 +an +ns denote the +corresponding Dirichlet series, and σc denote its abscissa of convergence. Then for any +σ > max{0, σc}, we have +� ∞ +0 +| � +n≤x an|2 +x1+2σ +dx = 1 +2π +� ∞ +−∞ +���� +A(σ + it) +σ + it +���� +2 +dt. +We shall deploy Harmonic Analysis Result 1 at a point in our argument where we +have already used Propositions 1 and 2 to move from studying character averages to +studying random multiplicative functions f(n). As such, we will be able to take an as +the values f(n) restricted to P-smooth numbers n (for a certain parameter P), and with +A(s) being the partial Euler product corresponding to f(n) on all P-smooth numbers, +similarly as in the random case in [11]. Note that we could not proceed in this way + +TYPICAL CHARACTER AND ZETA SUMS +19 +working with Dirichlet characters mod r directly, because we would not retain sufficient +control on terms in the Euler product corresponding to those n > r. +2.4. Random Euler products. The final key tool we require, and the ultimate source +of the better than squareroot cancellation that we look to establish in our theorems, is +an upper bound for the small moments of a short integral of a random Euler product. +This builds on ideas from the probabilistic theory of critical multiplicative chaos. Recall +that (f(p))p prime is a sequence of independent random variables distributed uniformly on +the complex unit circle, and for any large quantity P and any complex s with ℜ(s) > 0, +let F rand +P +(s) := � +p≤P(1 − f(p) +ps )−1. +Multiplicative Chaos Result 1 (See section 4 of Harper [11]). Uniformly for all large +P and 2/3 ≤ q ≤ 1, we have +E( +� 1/2 +−1/2 +|F rand +P +(1/2 + it)|2dt)q ≪ +� +log P +1 + (1 − q)√log log P +�q +. +This is proved in section 4 of [11] (see the proof of the Theorem 1 upper bound +there), in slightly different notation: the product F0(1/2 + it) from [11] corresponds to +F rand +P +(1/2 + it) with P replaced by x1/e. +Although the reader is welcome to treat Multiplicative Chaos Result 1 entirely as a +black box for our purposes here, it seems worthwhile to note that the bound is rather +subtle. An upper bound ≪ (log P)q would follow immediately by using H¨older’s inequal- +ity to compare with the q = 1 case, and applying standard Euler product calculations. +Obtaining the crucial saving 1 + (1 −q)√log log P in the denominator requires a careful +analysis of the behaviour of various subproducts of F rand +P +(1/2 + it). It is also shown +in section 5.2 of Harper [11] that the upper bound in Multiplicative Chaos Result 1 is +sharp, but we won’t need (or be able) to exploit that here. +When dealing with character sums we cannot perform such a precise and complete +“conditioning” procedure as in the genuine random multiplicative case, so it will be +more straightforward (although not absolutely essential) to work with discrete sums +rather than integral averages +� 1/2 +−1/2 in the argument. We now deduce a discrete version +of Multiplicative Chaos Result 1 to fit with this proof structure. +Lemma 2. For any large P, we have +E +� +|k|≤ log1.01 P +2 +� +1 +2 log1.01 P +− +1 +2 log1.01 P +|F rand +P +(1/2+i +k +log1.01 P +it)−F rand +P +(1/2+i +k +log1.01 P )|2dt ≪ log0.99 P. +Proof of Lemma 2. Since we have F rand +P +(s) = �∞ +n=1, +n is P smooth +f(n) +ns , with f(n) a Steinhaus +random multiplicative function, a mean square calculation shows that the left hand side + +20 +ADAM J HARPER +in Lemma 2 is += +� +|k|≤ log1.01 P +2 +� +1 +2 log1.01 P +− +1 +2 log1.01 P +E| +∞ +� +n=1, +n is P smooth +f(n) +n +1/2+i +k +log1.01 P +(n−it − 1)|2dt += +� +|k|≤ log1.01 P +2 +� +1 +2 log1.01 P +− +1 +2 log1.01 P +∞ +� +n=1, +n is P smooth +|n−it − 1|2 +n +dt. +Here we have |n−it − 1| ≪ min{|t| log n, 1} ≤ min{ +log n +log1.01 P , 1}. +Thus the contribu- +tion to the series from those n ≤ P log log P is ≪ +(log log P )2 +log0.02 P +�∞ +n=1, +n is P smooth +1 +n, and since +�∞ +n=1, +n is P smooth +1 +n = � +p≤P(1 − 1 +p)−1 ≪ log P this is all ≪ (log log P)2 log0.98 P. +The +contribution to the series from those n > P log log P (where we use the trivial bound +|n−it − 1| ≪ 1) is also acceptably small, namely +≪ e− log log P +∞ +� +n=1, +n is P smooth +1 +n1−1/ log P = e− log log P � +p≤P +(1− +1 +p1−1/ log P )−1 ≪ e− log log P log P = 1. +□ +Multiplicative Chaos Result 2. Uniformly for all large P and 2/3 ≤ q ≤ 1, we have +E( +1 +log1.01 P +� +|k|≤(log1.01 P )/2 +|F rand +P +(1/2 + i +k +log1.01 P )|2)q ≪ +� +log P +1 + (1 − q)√log log P +�q +. +Proof of Multiplicative Chaos Result 2. To deduce this from Multiplicative Chaos Re- +sult 1, it will suffice to prove a suitable upper bound for +E( +� +|k|≤(log1.01 P )/2 +� 1/(2 log1.01 P ) +−1/(2 log1.01 P ) +|F rand +P +(1/2 + i +k +log1.01 P + it) −F rand +P +(1/2 + i +k +log1.01 P )|2dt)q. +But using H¨older’s inequality to compare the q-th moment with the first moment, we +see this quantity is at most +� +E +� +|k|≤ log1.01 P +2 +� +1 +2 log1.01 P +− +1 +2 log1.01 P +|F rand +P +(1/2 + i +k +log1.01 P + it) − F rand +P +(1/2 + i +k +log1.01 P )|2dt +�q +, +which we can bound acceptably using Lemma 2. +□ +3. Proof of Theorem 1 +3.1. Notation and set-up. We may restrict attention to the range 2/3 ≤ q ≤ 1, since +if q is smaller we can use H¨older’s inequality to upper bound Echar| � +n≤x χ(n)|2q by +(Echar| � +n≤x χ(n)|4/3)3q/2, and invoke the q = 2/3 case. + +TYPICAL CHARACTER AND ZETA SUMS +21 +Recall that L := min{x, r/x}, which we may assume to be large since otherwise the +Theorem is trivial. We have a parameter P at our disposal, which we must choose to +be comparable to L on a doubly logarithmic scale, but (it turns out) somewhat smaller +than L on a logarithmic scale. In fact it will suffice if P is around exp{log1/6 L}, and +(for very minor technical reasons) we shall actually choose P to be the largest number +below exp{log1/6 L} such that log0.01 P is an integer. Thus we will have log P ≍ log1/6 L +and log log P ≍ log log L, and to prove Theorem 1 it will suffice to show that +Echar| +� +n≤x +χ(n)|2q ≪ +� +x +1 + (1 − q)√log log P +�q +. +(3.1) +We set M := 2 log1.02 P, say (note this is an integer with our choice of P), and +for each integer k satisfying |k| ≤ M and each character χ mod r we set Sk(χ) := +ℜ � +p≤P( +χ(p) +p1/2+ik/ log1.01 P + +χ(p)2 +2p1+2ik/ log1.01 P ). These are the prime number sums on which we +shall “condition” in the next subsection. +Finally, recall that in Approximation Result 1 we have two parameters N, δ to set, +for our construction of functions gj(·) forming a smooth partition of unity. We shall +make the final choices of these at the end of the proof, but it will turn out that the +“precision” parameter δ may be chosen as a suitable negative power of log P, and the +“range” parameter N (for which we have much flexibility) as a suitable multiple of +log log P, say. +3.2. The conditioning argument. Firstly, let P(n) denote the largest prime factor +of n, and as usual set Ψ(x, y) := #{n ≤ x : n is y smooth} = #{n ≤ x : P(n) ≤ y}. +Then we have +Echar| +� +n≤x,P (n)≤x1/ log log x +χ(n)|2q ≤ +� +Echar| +� +n≤x,P (n)≤x1/ log log x +χ(n)|2 +�q += Ψ(x, x1/ log log x)q, +by H¨older’s inequality and orthogonality of Dirichlet characters. Using standard smooth +number estimates (see Theorem 7.6 of Montgomery and Vaughan [23], for example) this +is ≪ (x(log x)−c log log log x)q, which is a negligible contribution in Theorem 1. +So it will suffice to bound Echar| � +n≤x,P (n)>x1/ log log x χ(n)|2q. In the random version +of the argument, one proceeds by conditioning on the values of f(p) on all “small” +primes p. +We now look to emulate this procedure for character sums, breaking up +the average Echar according to the values of all of the sums Sk(χ). +Thus, recalling +that the gj from Approximation Result 1 form a partition of unity, we may rewrite + +22 +ADAM J HARPER +Echar| � +n≤x,P (n)>x1/ log log x χ(n)|2q as +Echar +M +� +i=−M +( +N+1 +� +j=−N +gj(Si(χ))) +����� +� +n≤x, +P (n)>x1/ log log x +χ(n) +����� +2q += +� +−N≤j(−M),...,j(0),...,j(M)≤N+1 +Echar +M +� +i=−M +gj(i)(Si(χ)) +����� +� +n≤x, +P (n)>x1/ log log x +χ(n) +����� +2q += +� +−N≤j(−M),...,j(0),...,j(M)≤N+1 +σ(j)Ej +����� +� +n≤x, +P (n)>x1/ log log x +χ(n) +����� +2q +, +where for any (2M+1)-vector j from the outer sum we set σ(j) := Echar �M +i=−M gj(i)(Si(χ)), +and EjW := σ(j)−1EcharW �M +i=−M gj(i)(Si(χ)) for all functions W(χ). This Ej is our +character sum version of a conditional expectation. In particular, for the constant func- +tion 1, by the definitions we have Ej1 = 1 for all choices of the vector j. So applying +H¨older’s inequality to Ej, we conclude overall that +Echar +����� +� +n≤x, +P (n)>x1/ log log x +χ(n) +����� +2q +≤ +� +−N≤j(−M),...,j(0),...,j(M)≤N+1 +σ(j) +� +Ej +����� +� +n≤x, +P (n)>x1/ log log x +χ(n) +����� +2�q +. +3.3. Passing to the random case. At this point, we can use Propositions 1 and 2 +to move from working with σ(j) and Ej +����� +� +n≤x, +P (n)>x1/ log log x χ(n) +����� +2 +to working with their +analogues involving random multiplicative functions. Actually it isn’t essential to do +this at such an early stage in the argument, but doing it early will simplify various steps +of the analysis, and will ultimately allow us to establish (3.1) once we reach a point +where we can invoke our Multiplicative Chaos Results. +Using Proposition 1 with Y = 2M + 1, provided that our choices of N, δ ultimately +satisfy xP 400((2M+1)/δ)2 log(N log P ) < r we will have +Ej +����� +� +n≤x, +P (n)>x1/ log log x +χ(n) +����� +2 += +1 +σ(j) +� +E +M +� +i=−M +gj(i)(Si(f)) +����� +� +n≤x, +P (n)>x1/ log log x +f(n) +����� +2 ++O +� +x +(N log P)(2M+1)(1/δ)2 +�� +. +Now observe that � +j σ(j) = Echar �M +i=−M(�N+1 +j=−N gj(Si(χ))) = Echar1 = 1. Thus, apply- +ing H¨older’s inequality to � +j, we see the total contribution to Echar| � +n≤x,P (n)>x1/ log log x χ(n)|2q +from all of the “big Oh” terms from Proposition 1 is +≪ +� +� +−N≤j(−M),...,j(0),...,j(M)≤N+1 +σ(j)· 1 +σ(j) +x +(N log P)(2M+1)(1/δ)2 +�q += +� +x( +(2N + 2) +(N log P)(1/δ)2 )2M+1 +�q +. + +TYPICAL CHARACTER AND ZETA SUMS +23 +This is negligible for (3.1). +Next, if we define σrand(j) := E �M +i=−M gj(i)(Si(f)) for all (2M+1)-vectors j, where f is +a Steinhaus random multiplicative function, then using Proposition 2 we get σ(j)1−q ≪ +σrand(j)1−q + ( +1 +(N log P )(2M+1)(1/δ)2 )1−q. Therefore we have +� +−N≤j(−M),...,j(0),...,j(M)≤N+1 +σ(j) +� +1 +σ(j)E +M +� +i=−M +gj(i)(Si(f)) +����� +� +n≤x, +P (n)>x1/ log log x +f(n) +����� +2�q +≪ +� +j +σrand(j) +� +1 +σrand(j)E +M +� +i=−M +gj(i)(Si(f)) +����� +� +n≤x, +P (n)>x1/ log log x +f(n) +����� +2�q ++ ++( +1 +(N log P)(2M+1)(1/δ)2 )1−q � +j +� +E +M +� +i=−M +gj(i)(Si(f)) +����� +� +n≤x, +P (n)>x1/ log log x +f(n) +����� +2�q +. +Yet another application of H¨older’s inequality to the sum over j, and recalling again +that the gj form a partition of unity, implies that the final line here is +≪ +( +1 +(N log P)(2M+1)(1/δ)2 )1−q · ((2N + 2)2M+1)1−q · +�� +j +E +M +� +i=−M +gj(i)(Si(f)) +����� +� +n≤x, +P (n)>x1/ log log x +f(n) +����� +2�q += +� +( +(2N + 2) +(N log P)(1/δ)2 )2M+1 +�1−q� +E +����� +� +n≤x, +P (n)>x1/ log log x +f(n) +����� +2�q +. +This is certainly ≤ e−(1−q) log log Pxq, say, which is acceptable for (3.1). +In summary, if we now define Ej,randW := σrand(j)−1EW �M +i=−M gj(i)(Si(f)) for all +random variables W, then to prove Theorem 1 it remains for us to show that +� +−N≤j(−M),...,j(0),...,j(M)≤N+1 +σrand(j) +� +Ej,rand +����� +� +n≤x, +P (n)>x1/ log log x +f(n) +����� +2�q +≪ +� +x +1 + (1 − q)√log log P +�q +. +(3.2) +Note that we have now removed all mention of Dirichlet characters, and (3.2) is purely +a statement about random multiplicative functions. +3.4. Passing to Euler products. Our next step is to move from the left hand side of +(3.2), which still involves sums of f(n), to an expression featuring Euler products. This +part of the argument will be very similar to the corresponding work in section 2.4 of +Harper [11], although not exactly the same because the way we set up our “conditioning” +(using a smooth partition of unity) was necessarily different here. + +24 +ADAM J HARPER +Firstly (and crucially), since the f(p) are independent mean zero random variables, +and the various expressions �M +i=−M gj(i)(Si(f)) in the definition of Ej,rand only involve +the f(p) for p ≤ P (not those for p > P), we find by expanding the square that +� +j +σrand(j) +� +Ej,rand +����� +� +n≤x, +P (n)>x1/ log log x +f(n) +����� +2�q += +� +j +σrand(j) +� +Ej,rand +� +m≤x, +P (m)>x1/ log log x, +p|m⇒p>P +����� +� +n≤x/m, +n is P smooth +f(n) +����� +2�q +. +Here we implicitly used the fact that P < x1/ log log x. +Setting X = e +√log x, and replacing the condition that P(m) > x1/ log log x by the +weaker condition that m > x1/ log log x (for an upper bound), and introducing an integral +to smooth out on a scale of 1/X, we find the bracketed term is +≪ +� +Ej,rand +� +x1/ log log xP +X +m +� m(1+1/X) +m +| +� +n≤x/t, +n is P smooth +f(n)|2dt +�q ++ +� +Ej,rand +� +x1/ log log xP +X +m +� m(1+1/X) +m +| +� +x/tP +X +m +� m(1+1/X) +m +| +� +x/tP +X +m +� m(1+1/X) +m +| +� +x/tP +X +m +� m(1+1/X) +m +| +� +x/tP +(1 + +x +mX ) +�q +. +Applying a standard sieve bound (e.g. Theorem 3.6 of Montgomery and Vaughan [23]) +to detect the condition that p|m ⇒ p > P implies this quantity is ≪ ( +x +log P )q, which is +more than good enough for us. + +TYPICAL CHARACTER AND ZETA SUMS +25 +Meanwhile, the total contribution to (3.2) from the first bracket in (3.3) is +≪ +� +j +σrand(j) +� +Ej,rand +� x +x1/ log log x | +� +n≤x/t, +n is P smooth +f(n)|2 +� +t/(1+1/X)P +X +mdt +�q +≪ +� +j +σrand(j)( +1 +log P )q +� +Ej,rand +� x +x1/ log log x | +� +n≤x/t, +n is P smooth +f(n)|2dt +�q += +� +j +σrand(j)( +x +log P )q +� +Ej,rand +� x1−1/ log log x +1 +| +� +n≤z, +n is P smooth +f(n)|2dz +z2 +�q +. +Here the second line follows by applying a standard sieve bound again to the sum over m, +and the final equality follows from the substitution z = x/t. Using Harmonic Analysis +Result 1, we deduce that this is all +≪ ( +x +log P )q � +j +σrand(j)(Ej,rand +� ∞ +−∞ +|F rand +P +(1/2 + it)|2 +|1/2 + it|2 +dt)q, +(3.4) +where F rand +P +(s) := � +p≤P(1 − f(p) +ps )−1 = �∞ +n=1, +n is P smooth +f(n) +ns +is the Euler product corre- +sponding to the random multiplicative function f(n) on P-smooth numbers. +Since we have restricted to the range 2/3 ≤ q ≤ 1, we can break up the integral in +(3.4) into sub-intervals of length 1 and obtain a bound +≤ +( +x +log P )q � +j +σrand(j) +∞ +� +v=−∞ +(Ej,rand +� v+1/2 +v−1/2 +|F rand +P +(1/2 + it)|2 +|1/2 + it|2 +dt)q +≪ +( +x +log P )q +∞ +� +v=−∞ +1 +(|v| + 1)4/3 +� +j +σrand(j)(Ej,rand +� v+1/2 +v−1/2 +|F rand +P +(1/2 + it)|2dt)q. +Those terms with |v| > log0.01 P, say, trivially make a negligible contribution here. +Indeed, using H¨older’s inequality again their contribution is +≤ +( +x +log P )q +� +|v|>log0.01 P +1 +(|v| + 1)4/3( +� +j +σrand(j)Ej,rand +� v+1/2 +v−1/2 +|F rand +P +(1/2 + it)|2dt)q += +( +x +log P )q +� +|v|>log0.01 P +1 +(|v| + 1)4/3(E +� v+1/2 +v−1/2 +|F rand +P +(1/2 + it)|2dt)q, +and the orthogonality of random multiplicative functions implies this is += ( +x +log P )q +� +|v|>log0.01 P +1 +(|v| + 1)4/3( +∞ +� +n=1, +n is P smooth +1 +n)q ≪ ( +x +log P )q +1 +log1/300 P +logq P, +which is acceptable for Theorem 1. + +26 +ADAM J HARPER +To complete the proof of the theorem, in view of (3.2) and (3.4) it will now suffice +to show that uniformly for all |v| ≤ log0.01 P, we have +� +j +σrand(j)(Ej,rand +� v+1/2 +v−1/2 +|F rand +P +(1/2 + it)|2dt)q ≪ +� +log P +1 + (1 − q)√log log P +�q +. +(3.5) +3.5. Strengthening the conditioning. We now embark on establishing (3.5). For +notational simplicity, we will write out the details in the case where v = 0. +The +treatment of all other |v| ≤ log0.01 P is exactly similar4. +Firstly, since our “conditional expectations” Ej,rand contain information about all the +sums Sk(f), corresponding to the discrete points t = +k +log1.01 P , we need to move from the +integral in (3.5) to a discretised version. Thus we have +� +j +σrand(j)(Ej,rand +� 1/2 +−1/2 +|F rand +P +(1/2 + it)|2dt)q +≪ +� +j +σrand(j)(Ej,rand +� +|k|≤ log1.01 P +2 +� +1 +2 log1.01 P +− +1 +2 log1.01 P +|F rand +P +(1 +2 + i +k +log1.01 P + it) − F rand +P +(1 +2 + i +k +log1.01 P )|2dt)q ++ +� +j +σrand(j)(Ej,rand +1 +log1.01 P +� +|k|≤(log1.01 P )/2 +|F rand +P +(1/2 + i +k +log1.01 P )|2)q. +Applying H¨older’s inequality to the sum over j once again, we see the first line here is +≤ +� +E +� +|k|≤ log1.01 P +2 +� +1 +2 log1.01 P +− +1 +2 log1.01 P +|F rand +P +(1/2 + i +k +log1.01 P + it) − F rand +P +(1/2 + i +k +log1.01 P )|2dt +�q +, +which is ≪ log0.99q P in view of Lemma 2 from section 2.4. This is more than acceptable +for (3.5), so it remains to handle the discrete sum on the second line. +It will also shortly be helpful to have some size restrictions on the |F rand +P +(1/2 + +i +k +log1.01 P )|, to aid us with setting the “range” parameter N from Approximation Result +1. So let us define the (random) “bad set” +T := {k ∈ Z : |F rand +P +(1/2+i +k +log1.01 P )| ≥ log1.1 P +or +|F rand +P +(1/2+i +k +log1.01 P )| ≤ +1 +log1.1 P }, +4Observe that our “conditioning” in Ej,rand is on the Sk(f) for all |k| ≤ M = 2 log1.02 P, corresponding +to imaginary parts ≤ +M +log1.01 P = 2 log0.01 P, which includes (with a little room to spare) the full range +|v| ≤ log0.01 P that we must handle in (3.5). Notice also that since we assume that log0.01 P is an +integer, the points i(v + +k′ +log1.01 P ) for v, k′ ∈ Z are of the desired form i +k +log1.01 P with k ∈ Z, which +appear inside Sk(f). + +TYPICAL CHARACTER AND ZETA SUMS +27 +say. Splitting up the sum over |k| ≤ (log1.01 P)/2 according to whether k ∈ T or not, +and using H¨older’s inequality as (many times) before, we find that +� +j +σrand(j)(Ej,rand +1 +log1.01 P +� +|k|≤(log1.01 P )/2 +|F rand +P +(1/2 + i +k +log1.01 P )|2)q +≤ +� +j +σrand(j)(Ej,rand +1 +log1.01 P +� +|k|≤(log1.01 P )/2, +k /∈T +|F rand +P +(1/2 + i +k +log1.01 P )|2)q + ++ +� +1 +log1.01 P +� +|k|≤(log1.01 P )/2 +E1k∈T |F rand +P +(1/2 + i +k +log1.01 P )|2 +�q +. +On the final line, E1k∈T |F rand +P +(1/2 + i +k +log1.01 P )|2 is at most (log1.1 P)−0.2E|F rand +P +(1/2 + +i +k +log1.01 P )|2.2 + ( +1 +log1.1 P )2 (say). Standard results on the moments of random Euler prod- +ucts (see e.g. Euler Product Result 1 of [10]) imply that (log1.1 P)−0.2E|F rand +P +(1/2 + +i +k +log1.01 P )|2.2 ≪ (log1.1 P)−0.2 log1.21 P = log0.99 P, so we get another more than accept- +able contribution ≪ log0.99q P to (3.5). +Now the purpose of introducing the functions gj(i) (in the definitions of Ej,rand and +σrand(j)) was to approximately localise the values of the sums Si(f). At this stage, we +will replace this approximate localisation by an exact version, with a view to ultimately +producing a connection with Multiplicative Chaos Result 2. Thus the contribution to +(3.5) from the sum over k /∈ T is +≤ +� +j +σrand(j)(Ej,rand +1 +log1.01 P +� +|k|≤(log1.01 P )/2, +k /∈T +1|Sk(f)−j(k)|≤1|F rand +P +(1/2 + i +k +log1.01 P )|2)q + ++ +� +j +σrand(j)(Ej,rand +1 +log1.01 P +� +|k|≤(log1.01 P )/2 +1|Sk(f)−j(k)|>11k /∈T |F rand +P +(1/2 + i +k +log1.01 P )|2)q. +Applying H¨older’s inequality yet again to the sum over j on the second line, and recalling +the definitions of Ej,rand and σrand(j), we can bound that line by +( +1 +log1.01 P +� +|k|≤ log1.01 P +2 +� +j +E +M +� +i=−M +gj(i)(Si(f))·1|Sk(f)−j(k)|>11k /∈T |F rand +P +(1/2+i +k +log1.01 P )|2)q. +Now when −N ≤ j(k) ≤ N, by property (ii) from Approximation Result 1 we have +gj(k)(Sk(f)) · 1|Sk(f)−j(k)|>1 ≤ δ. Furthermore, if k /∈ T then |Sk(f)| = | log |F rand +P +(1/2 + +i +k +log1.01 P )|| + O(1) ≤ 1.1 log log P + O(1). +Then provided we take N ≥ 1.2 log log P +(say), when j(k) = N + 1 we have gj(k)(Sk(f)) · 1k /∈T ≤ δ, by property (iii) from +Approximation Result 1. So in any case we have �M +i=−M gj(i)(Si(f))·1|Sk(f)−j(k)|>11k /∈T ≤ +δ � +i̸=k gj(i)(Si(f)), and then if we perform the sum over all possible values of j(i) for +i ̸= k we get δ � +i̸=k +� +−N≤j(i)≤N+1 gj(i)(Si(f)) = δ. Thus the second line above is at + +28 +ADAM J HARPER +most +( +δ +log1.01 P +� +|k|≤ log1.01 P +2 +� +−N≤j(k)≤N+1 +E|F rand +P +(1/2 + i +k +log1.01 P )|2)q +≪ +(δN +∞ +� +n=1, +n is P smooth +1 +n)q +≪ +(δN log P)q. +This bound will be acceptable for (3.5) provided that δ ≤ +1 +N√log log P . +3.6. Conclusion. Finally, to establish (3.5) (in the case v = 0) it remains to prove that +� +j +σrand(j) +� +Ej,rand +1 +log1.01 P +� +|k|≤(log1.01 P )/2, +k /∈T +1|Sk(f)−j(k)|≤1|F rand +P +(1/2 + i +k +log1.01 P )|2 +�q +≪ +� +log P +1 + (1 − q)√log log P +�q +. +Recall now that |F rand +P +(1/2 + i +k +log1.01 P )| = exp{−ℜ � +p≤P log(1 − +f(p) +p +1/2+i +k +log1.01 P )} ≍ +exp{Sk(f)} for all k and all realisations of the random multiplicative function f(n). +And the point of our manipulations has been that the only k values that now contribute +to the sum over k are those for which Sk(f) ∈ [j(k) − 1, j(k) + 1]. Noting also that we +must have |Sk(f)| ≤ 1.1 log log P + O(1) if k /∈ T , we see the left hand side is +≪ +� +j +σrand(j)(Ej,rand +1 +log1.01 P +� +|k|≤(log1.01 P )/2, +|j(k)|≤1.1log log P +O(1) +e2j(k))q += +� +j +σrand(j)( +1 +log1.01 P +� +|k|≤(log1.01 P )/2, +|j(k)|≤1.1 log log P +O(1) +e2j(k))q. +Here the quantity inside the bracket is a deterministic function of j, with the only +remaining appearance of any randomness coming inside the multipliers σrand(j) in the +outer “averaging” over j. Thus we are very close to the structure of the quantity bounded +in Multiplicative Chaos Result 2, where all of the averaging E occurs on the outside of +the q-th power. + +TYPICAL CHARACTER AND ZETA SUMS +29 +Indeed, recalling the definition of σrand(j) we can rewrite the sum here as +E +� +j +M +� +i=−M +gj(i)(Si(f))( +1 +log1.01 P +� +|k|≤(log1.01 P )/2, +|j(k)|≤1.1 log log P +O(1) +e2j(k))q +≪ +E +� +j +M +� +i=−M +gj(i)(Si(f))( +1 +log1.01 P +� +|k|≤(log1.01 P )/2, +|j(k)|≤1.1 log log P +O(1) +|F rand +P +(1/2 + i +k +log1.01 P )|2)q + ++E +� +j +M +� +i=−M +gj(i)(Si(f))( +1 +log1.01 P +� +|k|≤(log1.01 P )/2, +|j(k)|≤1.1 log log P +O(1) +1|Sk(f)−j(k)|>1 log2.2 P)q. +And since � +j +�M +i=−M gj(i)(Si(f)) ≡ 1, the first line on the right hand side is at most +E( +1 +log1.01 P +� +|k|≤(log1.01 P )/2 |F rand +P +(1/2 + i +k +log1.01 P )|2)q, which is ≪ +� +log P +1+(1−q)√log log P +�q +by +Multiplicative Chaos Result 2. +Meanwhile, one last application of H¨older’s inequality (to the expectation and sum +over j simultaneously), and using properties (ii) and (iii) from Approximation Result +1 (as in section 3.5) to deduce that gj(k)(Sk(f))1|j(k)|≤1.1 log log P +O(1)1|Sk(f)−j(k)|>1 ≤ δ, +reveals the second line is +≤ +� +E +� +j +M +� +i=−M +gj(i)(Si(f)) +1 +log1.01 P +� +|k|≤(log1.01 P )/2, +|j(k)|≤1.1 log log P +O(1) +1|Sk(f)−j(k)|>1 log2.2 P +�q +≤ + + +1 +log1.01 P +� +|k|≤(log1.01 P )/2 +log2.2 P · E +� +j +δ +� +i̸=k +gj(i)(Si(f)) + + +q +. +Since the functions gj form a partition of unity, this is all +≤ + + +1 +log1.01 P +� +|k|≤(log1.01 P )/2 +log2.2 P · δ +� +−N≤j(k)≤N+1 +1 + + +q +≪ (δN log2.2 P)q, +which will be acceptable for (3.5) provided that δ ≤ +1 +N log1.2 P √log log P . +To conclude, we need only check that we can make an acceptable choice of the +parameters N, δ. In section 3.3, we needed to have xP 400((2M+1)/δ)2 log(N log P ) < r so +that Proposition 1 could be applied. Recalling that M = 2 log1.02 P, we see this will be +satisfied provided that P 6499(log2.04 P )(1/δ)2 log(N log P ) < r/x, say. In section 3.5 we needed +N ≥ 1.2 log log P and δ ≤ +1 +N√log log P , and now we have the more stringent condition +δ ≤ +1 +N log1.2 P √log log P . + +30 +ADAM J HARPER +So taking N = ⌈1.2 log log P⌉ and δ = +1 +log1.3 P , say, all of our conditions will be +satisfied provided that P 6500(log4.64 P ) log log P < r/x. This indeed holds with our choice of +P, slightly smaller than exp{log1/6 L}. +□ +4. Proofs of the other Theorems and Corollaries +4.1. Proof of Theorem 2. The neatest way to proceed is to let Φ : R → R be a fixed +non-negative function that is ≥ 1[0,1], and whose Fourier transform �Φ is supported in +[−1/2π, 1/2π]. For example, we can take Φ(x) to be a Beurling–Selberg function, as +described in Vaaler’s paper [26]. Then if we write EcontW(t) to denote the continuous +average +1 +T �Φ(0) +� ∞ +−∞ Φ(t/T)W(t)dt, we see +1 +T +� T +0 +| +� +n≤x +nit|2qdt ≤ �Φ(0)Econt| +� +n≤x +nit|2q ≪ Econt| +� +n≤x +nit|2q, +and so it will suffice to prove the claimed Theorem 2 bound for Econt| � +n≤x nit|2q. +The point of introducing Econt is that we have Econt1 = +1 +T �Φ(0) +� ∞ +−∞ Φ(t/T)dt = +1 +�Φ(0) +� ∞ +−∞ Φ(u)du = 1, and crucially EcontUitV it = +1 +T �Φ(0) +� ∞ +−∞ Φ(t/T)e−it log(V/U)dt = +�Φ((T/2π) log(V/U)) +�Φ(0) += 1U=V for all natural numbers 1 ≤ U, V < T. (If we worked with +1 +T +� T +0 directly, there would be error terms rather than an exact equality here.) These +are the same properties that we had for our character average Echar in the proof of +Theorem 1. In particular, the analogues of Lemma 1 and of Propositions 1 and 2 hold +with Echar replaced by Econt, with χ(n) replaced by nit, and with the various conditions +xUk, xP 400(Y/δ)2 log(N log P ), P 400(Y/δ)2 log(N log P ) < r replaced by their obvious substitutes +xUk, xP 400(Y/δ)2 log(N log P ), P 400(Y/δ)2 log(N log P ) < T. This means that we can repeat the +argument in sections 3.1–3.3, simply replacing L = min{x, r/x} by LT := min{x, T/x} +and replacing Sk(χ) by ℜ � +p≤P( +pit +p1/2+ik/ log1.01 P + +p2it +2p1+2ik/ log1.01 P ). At the end of section +3.3, the proof of Theorem 2 then reduces to establishing exactly the same bound (3.2) +for random multiplicative functions that we already proved in sections 3.4–3.6. +□ +4.2. Proof of Corollary 1. This is a simple consequence of Theorem 1 together with +Markov’s inequality. +Thus for any 0 ≤ q ≤ 1, the left hand side in Corollary 1 is +≤ Echar| � +n≤x χ(n)|2q +(λ +√x +(log log(10L))1/4 )2q . +If λ ≥ e +√ +log log(10L), then simply taking q = 1 and using the fact that Echar| � +n≤x χ(n)|2 ≤ +x yields the desired bound. + +TYPICAL CHARACTER AND ZETA SUMS +31 +For smaller λ, if we set q = 1 − δ with 0 < δ ≤ 1, and apply Theorem 1, we obtain +that +Echar| � +n≤x χ(n)|2q +(λ +√x +(log log(10L))1/4 )2q ≪ 1 +λ2 +λ2δ(log log(10L))q/2 +(1 + δ +� +log log(10L))q ≤ 1 +λ2 +λ2δ +δ += log λ +λ2 +e2δ log λ +δ log λ . +Choosing δ = +1 +2 log λ yields the claimed upper bound. +□ +4.3. Proof of Corollary 2. As usual, it will suffice to prove Corollary 2 when 2/3 ≤ +q ≤ 1, because when q is smaller we can use H¨older’s inequality to compare with the +q = 2/3 case. +If χ is an even Dirichlet character mod r, then using the definition of the theta +function and partial summation we have +θ(1, χ) = +∞ +� +n=1 +χ(n)e−πn2/r += +� +n≤√r log r +χ(n)e−πn2/r + O(1) += +� +n≤√r log r +χ(n) +� √r log r +n +2πu +r e−πu2/rdu + O(1) += +� √r log r +1 +2πu +r e−πu2/r � +n≤u +χ(n)du + O(1). +By noting that +� √r log r +1 +2πu +r e−πu2/rdu ≍ 1, and applying H¨older’s inequality to the in- +tegral over u (here we use the fact that 2q ≥ 1), we deduce that for each even χ we +have +|θ(1, χ)|2q ≪ +� √r log r +1 +2πu +r e−πu2/r| +� +n≤u +χ(n)|2qdu + 1. +The contribution to this integral from 1 ≤ u ≤ r1/4 is trivially ≪ +� r1/4 +1 +u +r u2qdu ≪ 1, +which is negligible. On the remaining range r1/4 ≤ u ≤ √r log r, Theorem 1 implies +that Echar| � +n≤u χ(n)|2q ≪ ( +u +1+(1−q)√log log r)q, giving an acceptable contribution +≪ ( +1 +1 + (1 − q)√log log r)q +� √r log r +r1/4 +u +r e−πu2/ruqdu ≪ ( +√r +1 + (1 − q)√log log r)q +for Corollary 2. +If χ is an odd character, then by definition we have θ(1, χ) = �∞ +n=1 nχ(n)e−πn2/r, +and a similar partial summation as before yields that +θ(1, χ) = +� √r log r +1 +(2πu2 +r +− 1)e−πu2/r � +n≤u +χ(n)du + O(1). + +32 +ADAM J HARPER +Since we now have +� √r log r +1 +| 2πu2 +r +−1|e−πu2/rdu ≍ √r, when we apply H¨older’s inequality +and Theorem 1 (as in the even character case) we now have an extra factor (√r)2q = rq in +our bounds, producing the claimed upper bound for the average over odd characters. +□ +4.4. Proof of Theorem 3. Once we introduce the multiplicative twist h(n), the Euler +product factor corresponding to a prime p changes from being (1 − +χ(p) +p1/2+it)−1 (or (1 − +f(p) +p1/2+it)−1 on the random multiplicative side), to +1 + h(p)χ(p) +p1/2+it ++ +∞ +� +k=2 +h(pk)χ(p)k +pk(1/2+it) +or +1 + h(p)f(p) +p1/2+it ++ +∞ +� +k=2 +h(pk)f(p)k +pk(1/2+it) . +Here we have | h(p)χ(p) +p1/2+it + �∞ +k=2 +h(pk)χ(p)k +pk(1/2+it) | ≤ �∞ +k=1 +1 +pk/2 = +1 +√p−1. Provided that p ≥ 5, this +is all ≤ +1 +√ +5−1 < 1, so we can still apply Taylor expansion to analyse the logarithms of +the Euler factors. This is not the case for the primes 2 and 3, but for those we still have +an upper bound ≪ 1 for the Euler factors. +With the above observations, the proof of Theorem 3 is a fairly obvious adjustment of +the proof of Theorem 1. Rather than Sk(χ) := ℜ � +p≤P( +χ(p) +p1/2+ik/ log1.01 P + +χ(p)2 +2p1+2ik/ log1.01 P ), +in section 3 we work with Sk,h(χ) := ℜ � +5≤p≤P( +h(p)χ(p) +p1/2+ik/ log1.01 P + (h(p2)−(1/2)h(p)2)χ(p)2 +p1+2ik/ log1.01 P +) +(coming from the first and second order terms in the Taylor expansions of the logarithms +of the Euler factors). Note that the primes 2 and 3 are omitted here. The calculations +then proceed essentially without change, until in (3.4) we end up with | � +p≤P(1 + +h(p)f(p) +p1/2+it + �∞ +k=2 +h(pk)f(p)k +pk(1/2+it) )| in place of |F rand +P +(1/2 + it)|. +And this is ≪ |F rand +P,h (1/2 + +it)|, where F rand +P,h (s) := � +5≤p≤P(1 + h(p)f(p) +ps ++ �∞ +k=2 +h(pk)f(p)k +pks +) (with the primes 2 and 3 +omitted). +Continuing through sections 3.5 and 3.6, we only need to verify that we have the same +upper bound E|F rand +P,h (1/2 + i +k +log1.01 P )|2.2 ≪ log1.21 P as for E|F rand +P +(1/2 + i +k +log1.01 P )|2.2, +and (most importantly) that Multiplicative Chaos Result 1 continues to hold with +F rand +P +(1/2 + it) replaced by F rand +P,h (1/2 + it). The former is an easy modification of Euler +Product Result 1 of [10]. The latter is also quite straightforward to check by working +through the proofs in sections 3.1–3.2 and 4.1–4.3 of [11], noting that Lemma 1 from +that paper holds without change for the Euler product F rand +P,h (s) (here it is important +that |h(p)| = 1 on primes p, which produce the main terms there), and all subsequent +results flow from Lemma 1. +□ +References +[1] P. Bengoechea. Galois action on special theta values. J. Th´eor. Nombres Bordeaux, 28, no. 2, pp +347-360. 2016 +[2] J. Bourgain. Decoupling, exponential sums and the Riemann zeta function. J. Amer. Math. Soc., +30, no. 1, pp 205-224. 2017 + +TYPICAL CHARACTER AND ZETA SUMS +33 +[3] R. de la Bret`eche, M. Munsch, G. Tenenbaum. Small G´al sums and applications. J. Lond. Math. +Soc. (2), 103, no. 1, pp 336-352. 2021 +[4] T. Cochrane, Z. Zheng. High order moments of character sums. Proc. Amer. Math. Soc., 126, no. +4, pp 951-956. 1998 +[5] A. Fujii, P. X. Gallagher, H. L. Montgomery. 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(N.S.), 12, no. 2, +pp 183-216. 1985 +Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 +7AL, England +Email address: A.Harper@warwick.ac.uk + diff --git a/vtE3T4oBgHgl3EQfOQkJ/content/tmp_files/load_file.txt b/vtE3T4oBgHgl3EQfOQkJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1da0227704f8e7c7f5096919167ded9ffee0c2df --- /dev/null +++ b/vtE3T4oBgHgl3EQfOQkJ/content/tmp_files/load_file.txt @@ -0,0 +1,1028 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf,len=1027 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content='04390v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content='NT] 11 Jan 2023 THE TYPICAL SIZE OF CHARACTER AND ZETA SUMS IS o(√x) ADAM J HARPER Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content=' We prove conjecturally sharp upper bounds for the Dirichlet character moments 1 r−1 � χ mod r | � n≤x χ(n)|2q, where r is a large prime, 1 ≤ x ≤ r, and 0 ≤ q ≤ 1 is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content=' In particular, if both x and r/x tend to infinity with r then 1 r−1 � χ mod r | � n≤x χ(n)| = o(√x), and so the sums � n≤x χ(n) typically exhibit “better than squareroot cancellation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content=' We prove analogous better than squareroot bounds for the moments 1 T � T 0 | � n≤x nit|2qdt of zeta sums;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content=' of Dirichlet theta func- tions θ(1, χ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content=' and of the sums � n≤x h(n)χ(n), where h(n) is any suitably bounded multiplicative function (for example the M¨obius function µ(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content=' The proofs depend on similar better than squareroot cancellation phenomena for low moments of random multiplicative functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content=' An important ingredient is a reor- ganisation of the conditioning arguments from the random case, so that one only needs to “condition” on a small collection of fairly short prime number sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content=' The condi- tioned quantities arising can then be well approximated by twisted second moments, whose behaviour is the same for character and zeta sums as in the random case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content=' Introduction In this paper we are interested in the size of sums such as � n≤x nit and � n≤x χ(n), where t ∈ R and χ(n) is a non-principal Dirichlet character modulo a large prime r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content=' These zeta sums and character sums are among the most studied objects in analytic number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content=' We would like to show, on the widest possible range of x, that we have substantial cancellation amongst the terms in the sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content=' Furthermore, we would like to understand the extent of the cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content=' By periodicity, we can confine our study of character sums to the range x ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content=' And if t is large and x ≥ t, then standard Fourier analysis (see Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQfOQkJ/content/2301.04390v1.pdf'} +page_content='2 of Ivi´c [13], for example) shows that � x 500) to be +comparable to MVP. With only ≤ 50 pruning iterations, +MVP can reach a reasonable accuracy, while conventional +pruning methods perform poorly. +These imply that the +initialized sub-network obtained by majority voting already +contains helpful knowledge from its similar tasks to speed up +the training of the pruned model. +Effect of Similarities between Tasks MVP consistently +achieves better performance when applied to nearest tasks +with the highest similarities. +In plot (b) of Fig. 4, we +compare the LEEP score with the Wordnet similarity and +study the effect of applying MVP to neighbour tasks with +different similarities. From similarity group 1 to group 5, +the similarities between tasks decrease. We find that for both +the two similarity metrics, the accuracy of MVP improves +significantly when the similarities between tasks increase. +When the pruning iterations are small(= 20), where the +initialization of the sub-network is more important, the +accuracy of tasks from similarity group 1 leads to similarity +group 5 by 15%. Despite the accuracy of similarity group 5 +improving when the pruning iterations increase to 100, there +is still a gap of 7%. This result indicates that neighbour tasks +with high similarities share more knowledge with the target +task. In this plot, we also find that tasks in different similarity +groups classified by LEEP score show larger differences than +Wordnet similarity, implying that LEEP score can better eval- +uate similarities between tasks. This result is consistent with +our observation in the empirical study. The performance of +Wordnet similarity is also good and can still be an alternative +when the time and computational resources are limited. +Comparison between Pruned Models Extracted by +Different Pruning Method In this part, we apply MVP to + +20 +50 +100 +200 +500 +Number of pruning iterations +40 +50 +60 +70 +80 +90 +Accuracy% of MVP +MVP on ResNet-18 +MVP on ViT +IFP on ResNet-18 +AHNP on ViT +(a) Effect of Pruning Iterations +1 +2 +3 +4 +5 +Similarity groups +60 +62 +64 +66 +68 +70 +72 +74 +Accuracy% of MVP +LEEP, Iters=20 +Wordnet, Iters=20 +1 +2 +3 +4 +5 +Similarity groups +83 +84 +85 +86 +87 +88 +89 +LEEP, Iters=100 +Wordnet, Iters=100 +(b) Effect of Similarities between Tasks +Figure 4: (a) Comparison between MVP and conventional pruning methods with different pruning iterations on different architectures. For +both ResNet-18 and ViT, MVP converges much faster in a small number of iterations than conventional pruning methods. (b) Comparison +between LEEP score and Wordnet similarity for MVP with different pruning iterations. From similarity groups 1 to 5, the similarities between +tasks decrease. For both similarity metrics, more similar tasks get better performance. LEEP score has a better ability to measure similarities +between tasks than Wordnet similarity. +0 1 2 3 4 5 6 7 8 9 101112131415161718 +Layer index +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +0.225 +IoU of filters between different tasks +similarity group 1 +similarity group 2 +similarity group 3 +similarity group 4 +similarity group 5 +(a) IoU of Layers for IFP(Taylor Pruning) +1 +2 +3 +4 +5 +Number of neighbours +84 +85 +86 +87 +88 +89 +Accuracy% of MVP +IFP(Feature Pruning) +IFP(Taylor Pruning) +(b) Effect of Neighbour Numbers +Figure 5: (a) IoU of layers in ResNet-18 between tasks whose pruned models are extracted by IFP (Taylor Pruning) and more similar tasks +also share more filters, especially in deeper layers. (b) Results of applying MVP to pruned models from Activation Pruning and Taylor +Pruning over different number of neighbours. MVP(neighbour number ≥ 2) can improve the performance of transfer learning(neighbour +number = 1) by a large margin when applied to pruned models extracted by different pruning methods. +pruned models extracted by Taylor Pruning [Molchanov et +al., 2019] on ResNet-18 for CIFAR-100 tasks, to prove that +MVP works well on pruned models extracted by various +pruning methods. Taylor Pruning measures the importance +of each filter by the effect of removing this filter on final +loss. In plot (a) of Figure 5, we show the IoU of each layer +for pairs of tasks with different task similarities, of which the +pruned models are extracted by Taylor Pruning. Consistent +with our observation in the empirical study, pruned models +with higher similarities share more filters. +Effect of Number of Neighbours In plot (b) of Figure 5, +we investigate the effect of the number of neighbours for +MVP. When the number = 1, MVP reduces to transfer learn- +ing which learns from the pruned model of a single selected +similar task. +In the plot, when the number of neighbours +increase from 1 to 2, the performance improves sharply. +This result implies the effectiveness of meta knowledge from +different neighbours. When the number of neighbours ≥ 3, +for both Activation Pruning and Taylor Pruning, the accuracy +improves little, which indicates that 3 neighbours are enough +for MVP to produce a high-quality initialization. +6 +Conclusion +In this paper, we study “non-parametric meta-pruning” +problem that aims to reduce the memory and computational +costs of single-task pruning, via reusing a pre-trained model +and similar tasks’ pruned models to find an initialization +sub-network for a new task. We conduct an empirical study +to investigate the relationship between task similarity and the +pruned models of two tasks for different datasets and deep +neural networks. +The empirical study motivates a simple +yet strong baseline for meta-pruning, called “meta-vote +pruning (MVP)” (Alg. 1). +By extensive experiments on +multiple tasks drawn from several datasets under different +training settings, we demonstrate the advantages of MVP +over other SOTA pruning methods in the region of limited +computation and show its potential to reduce the carbon +footprint of pruning/fine-tuning large networks for billions +of edge devices and tasks. + +References +[Bai et al., 2022] Yue Bai, Huan Wang, Zhiqiang Tao, Kun- +peng Li, and Yun Fu. Dual lottery ticket hypothesis. arXiv +preprint arXiv:2203.04248, 2022. +[Chen and He, 2020] Xinlei Chen and Kaiming He. 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In +NeurIPS, 2020. + +7 +Appendix +Algorithm 2 ITERATIVE FILTER PRUNING (IFP) +Input +: Pre-trained network F(·; θ), Task T and training +set DT , Hyperparameters J, h, r, p +Initialize: Ωℓ ← [nℓ], the set of filters preserved in layer-ℓ +9 for j ← 1 to J do +10 +if j%h = 0 and |Ωℓ| > (1 − r)nℓ then +11 +for ℓ ← 1 to L − 1 do +12 +Prune p% of filters in Ωℓ with the smallest impor- +tance score over DT ; +13 +end +14 +end +15 +Apply one SGD step on a mini-batch of DT to fine-tune +the remained filters {θℓ,i : ℓ ∈ [L − 1], i ∈ Ωℓ} and θL; +16 end +7.1 +Iterative Filter Pruning +The detailed procedure of IFP is described in Algorithm 2. +Given a pre-trained network F(·; θ) of L layers (layer-L +is fully-connected) with parameter θ = {θℓ}ℓ=1:L and a +training set DT of a target task T, let θℓ = {θℓ,i}i=1:nℓ +denote all parameters in layer-ℓ composed of θℓ,i for every +filter-i. IFP fine-tunes the model for total J iterations. It +prunes p% of the filters remained in each layer every h +iterations according to their activation values fℓ,i(x). It stops +to prune layer-ℓ if reaching the targeted pruning ratio r. +7.2 +Automatic Head&Node Pruning +The detailed procedure of AHNP is described in Algorithm 3. +Given a pre-trained network F(·; θ) of L layers (layer-L +is fully-connected) with parameter θ = {θℓ}ℓ=1:L and a +training set DT of a target task T, let θℓ = {θℓ,i}i=1:nℓ +denote all parameters in layer-ℓ composed of θℓ,i for ev- +ery head/node-i. +Sℓ,i denote the score for each prunable +head/node-i in layer-ℓ. +AHNP fine-tunes the model and +scores for total J iterations. It prunes the heads/nodes if their +scores are smaller than the threshold τ. +It stops to prune +layer-ℓ if reaching the targeted pruning ratio r. Then, AHNP +fine-tunes the pruned model for K iterations. +7.3 +Results of MVP on Oxford Flowers-102 and +Oxford-IIIT Pets +Table 5: Accuracy of MVP on fine-grained tasks. +Methods +Oxford Flowers-102 +Oxford-IIIT Pets +Iters +Acc +FLOPs +Iters +Acc +FLOPs +IFP +800 95.20±1.39 47.98(T) 1000 77.38±0.96 59.33(T) +IFP +60 54.20±6.61 3.04(T) +100 55.76±3.86 4.57(T) +MVP(ours) 60 95.40±1.34 0.95(T) +100 78.29±0.77 1.58(T) +Similar to previous observations, MVP outperforms IFP +with much more training iterations when applied to target +tasks sampled from fine-grained datasets Oxford Flowers-102 +and Oxford-IIIT Pets. +The results indicate that MVP can +Algorithm +3 +AUTOMATIC +HEAD&NODE +PRUNING +(AHNP) +Input +: Pre-trained network F(·; θ), Task T and training +set DT , Hyperparameters J, K, r, τ +Initialize: Ωℓ ← [nℓ], the set of heads/nodes preserved in +layer-ℓ. Sℓ,i ← 1, the score for each prunable +head/node in layer-ℓ. +17 for j ← 1 to J do +18 +for ℓ ← 1 to L − 1 do +19 +for i ∈ Ωℓ do +20 +Prune the head/node if its score Sℓ,i < τ; +21 +end +22 +end +23 +Stop pruning if reaching the target pruning ratio r. +24 +Apply one optimization step on a mini-batch of DT +to fine-tune the remained heads/nodes and scores +{θℓ,i, Sℓ,i : ℓ ∈ [L − 1], i ∈ Ωℓ} and θL; +25 end +26 Remove S, fine-tune the pruned model for K iterations on Di. +build high-quality initialization, which contains fine-grained +pattern information for each class in hard-to-distinguish tar- +get tasks. +7.4 +Discussion about the cost of creating a pruned +model-zoo by training on hundreds of tasks +In the experimental setting, there exists a one-time cost for +preparing the model-zoo. However, like training any meta- +learning model, this can facilitate many future tasks by sig- +nificantly reducing their computation and required samples. +Moreover, we can keep adding MVP pruned models of new +tasks into this model-zoo and keep improving it in a life-long +learning manner with no extra cost. +Meta-pruning moves required computation from new-task +adaptation to pre-training (preparing the model-zoo). In prac- +tice, this is an even more important advantage over single-task +pruning because the meta-pruning cost is offline on the server +side so it is tolerable. Practitioners care more about the de- +ployment cost of new tasks, e.g., on edge devices with limited +data and computation. Because of the model-zoo, our method +makes the practical deployment of model pruning more af- +fordable. +7.5 +The difference of IoU between different +similarity groups and similarity metrics +In Fig.6, we draw the difference of IoU between tasks of sim- +ilarity group 1 and similarity group 5 for ResNet-50. As the +layer gets deeper, the difference increases. +In Fig.2 of the paper, the average difference of IoU be- +tween similarity group 1 and similarity group 5 over all layers +is 0.195 and 0.150 respectively for LEEP and Wordnet simi- +larity, which has a large gap. In Fig.4 of the paper, the LEEP +score performs a little better than Wordnet similarity in MVP +which indicates that models with the larger IoU share more +relevant parameters and LEEP has a good ability to find the +nearest neighbours. + +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Layer index +0.00 +0.02 +0.04 +0.06 +0.08 +Difference of IoU +Figure 6: The difference of IoU between tasks of similarity group 1 +and similarity group 5 for ResNet-50. +0 +2 +4 +6 +8 +10 +Attention block index +0.0 +0.5 +1.0 +IoU of heads +0 +2 +4 +6 +8 +10 +Feedforward block index +0.0 +0.5 +IoU of nodes +similarity group 1 +similarity group 2 +similarity group 3 +similarity group 4 +similarity group 5 +random pruning +Figure 7: Comparison of IoU between MVP and random pruning. +7.6 +Comparison of IoU between MVP and random +pruning +In Fig.7, besides MVP, we also show the IoU of pruned mod- +els extracted by random pruning. When the similarity be- +tween tasks is large, the IoU of MVP is much larger than ran- +dom pruning, implying that these tasks contain lots of rele- +vant information. When the similarity between tasks is small, +in the last few blocks, the IoU of nodes is similar to that of +random pruning, which indicates that tasks of low similarity +share little high-level information with the target task. + diff --git a/wtFJT4oBgHgl3EQfgSwn/content/tmp_files/load_file.txt b/wtFJT4oBgHgl3EQfgSwn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b34912d38f9f2a3335d438df34bbbf0ee554938f --- /dev/null +++ b/wtFJT4oBgHgl3EQfgSwn/content/tmp_files/load_file.txt @@ -0,0 +1,868 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf,len=867 +page_content='Voting from Nearest Tasks: Meta-Vote Pruning of Pre-trained Models for Downstream Tasks Haiyan Zhao1 , Tianyi Zhou2 , Guodong Long1 , Jing Jiang1 , Chengqi Zhang1 1University of Technology Sydney 2University of Maryland Haiyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='Zhao-2@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='au, zhou@umiacs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='edu, {guodong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='long, jing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='jiang, Chengqi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='Zhang}@uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='au Abstract As a few large-scale pre-trained models become the major choices of various applications, new challenges arise for model pruning, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', can we avoid pruning the same model from scratch for every downstream task?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' How to reuse the pruning results of previous tasks to accelerate the pruning for a new task?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' To address these challenges, we create a small model for a new task from the pruned models of similar tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We show that a few fine-tuning steps on this model suffice to produce a promising pruned-model for the new task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We study this “meta-pruning” from nearest tasks on two major classes of pre-trained models, convolutional neural network (CNN) and vision transformer (ViT), under a limited budget of prun- ing iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Our study begins by investigating the overlap of pruned models for similar tasks and how the overlap changes over different layers and blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Inspired by these discoveries, we develop a simple but effective “Meta-Vote Pruning (MVP)” method that significantly reduces the pruning iter- ations for a new task by initializing a sub-network from the pruned models of its nearest tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In experiments, we demonstrate MVP’s advantages in accuracy, efficiency, and generalization through extensive empirical studies and comparisons with popular pruning methods over several datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 1 Introduction Large-scale pre-trained models usually contain tens of millions or even billions of parameters for promising gen- eralization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The computation and memory of modern GPUs or clusters can support to train such models, but directly deploying them to edge devices can easily violate the hardware limits on memory and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Network pruning [Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Chin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2020] has been widely studied to compress neural nets by removing redundant connections and nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Numer- ous empirical results have verified that pruning can compress the original network into much smaller sub-networks that still enjoy the comparable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Instead of reducing the network to the target size by one-time pruning, iterative prun- ing that alternates between pruning and fine-tuning for iter- ations usually achieves better performance [Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Theoretically, a line of recent works [Frankle and Carbin, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Savarese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Malach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2020] attempts to prove the lottery ticket hypothesis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', the existence of such sub-networks, for different pruning settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In a variety of practical applications, a large-scale pre- trained network like ResNet-50 [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2016] or Vision Transformer (ViT) [Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2021] usually needs to be pruned for a wide variety of devices and adapted to different downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Running an iterative pruning algorithm for every device or task from the same pre-trained network can create enormous carbon footprint overload in our biosphere and waste a lot of computational power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' On the other hand, the wide applications of a few pre-trained models have already created thousands of pruned models for different downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Can we reuse these pruned models as prior knowledge to save the pruning computation on new tasks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We call this problem “meta-pruning”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In this paper, we mainly focus on a special case of it, which initializes a sub-network for a given new task based on the pruned models of similar tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Meta-pruning is non- parametric if no parametric model is trained to produce the initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' It is analogous to MAML [Finn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2017] in that the meta-objective optimizes the initialization of a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' It differs from MAML in that (1) both the sub-network’s architecture and weights are initialized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' and (2) the initialization is not universal but task-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Since meta-pruning aims to find better sub-network ini- tialization for new tasks, we limit the iterations during meta- pruning to strengthen the impact of initialization on the final pruned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' This also controls the computational cost and carbon footprint of meta-pruning much less than conventional pruning that requires many iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Under this constraint, a well-performed pre-trained model is critical to the meta- pruning performance because (1) it needs to provide initial- ized sub-networks for different tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' and (2) a few iterations of fine-tuning to the sub-networks should suffice to produce high-quality pruned models for targeted tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Meta-pruning follows a practical setting where one single pre-trained model is tailored for different tasks using limited iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We study two classes of the most widely used pre-trained arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='11560v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='LG] 27 Jan 2023 models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', convolutional neural networks (CNN) and ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The primary contribution of this paper is two folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In the first part, we conduct a thorough empirical study that applies different pruning methods to CNN and ViT and compare their produced sub-networks for hundreds of downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' No meta-pruning is studied in this part and its primary purpose is to (1) find the nearest tasks for a new task using different similarity metrics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' and (2) compare the pruned models for different but similar tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' To this end, we build a dataset of tasks and their sub-networks pruned from the same pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Statistics and evaluations on this dataset indicate similar tasks with high similarity tend to share more nodes/filters/heads preserved in their pruned models, especially in deeper layers that notably capture high-level task-specific features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Motivated by the empirical study, the second part of this paper proposes a simple yet strong meta-pruning method called “meta-vote pruning (MVP)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' It can significantly reduce the pruning cost and memory required by previous pruning approaches yet still produce pruned models with promising performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Given a pre-trained model, MVP finds a sub-network for a new task by selecting nodes/filters/heads through majority voting among its nearest tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', a filter will be sampled with a higher chance if it is selected into more sub-networks of similar tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' To keep the method simple, we sample the same proportion of nodes/filters/heads as the targeted pruning ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Then we apply a few iterations of fine- tuning to the initialized sub-network using training data of the new task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Although a more sophisticated procedure can be developed, the proposed method already saves a substantial amount of computation and memory while maintaining a high test accuracy of pruned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We demonstrate these via ex- periments over tasks from CIFAR-100 [Krizhevsky and Hin- ton, 2009], ImageNet [Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2009], Caltech-256 [Grif- fin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2007] and several fine-grained datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The pruned models extracted from an ImageNet pre-trained model can also vote for tasks drawn from the unseen datasets with great performance, which shows the generalization of MVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 2 Related works Network pruning Network pruning has been widely stud- ied to compress network and accelerate its inference for a single task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We mainly summarize structure pruning below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In CNN, to encourage the sparsity of the pruned network, L0 [Louizos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2018], L1 [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2017] or L2 [Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2015] regularization have been used, and polarization regularization [Zhuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2020] shrinks some nodes to- wards 0 and meanwhile strengthen the others to keep impor- tant nodes intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Different criteria have been proposed to evaluate the importance of nodes/filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' prunes filters with the smallest sum of parameters’ absolute val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' prune filters according to the second-order Taylor expansion of the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Methods [Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Frankle and Carbin, 2018] based on lottery ticket hypothesis try to find a well-performed sparse initialization for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' ViT has been widely used in computer vision and achieved SOTA performance in many tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The input patches for each block can be pruned to save computation for ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' propose a metric for the importance of each patch and dynam- ically prune patches in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' PatchSlimming [Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2022] retains patches critical to preserve the original final output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' HVT [Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2021] is a CNN-like method which shortens the patch sequence by max-pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Another line of works [Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2022a] au- tomatically prunes the unimportant heads, nodes and blocks in ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' These methods excel on single-task pruning but their cost linearly increases for multiple tasks (and thus more ex- pensive than meta-pruning) because: (1) a large model needs to be trained for every task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' (2) every task requires to prune its own large pre-trained model from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' For both CNN and ViT, it is time-consuming for these pruning methods to build a pruned model for each unseen target task from a large pre- trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Our proposed method can borrow the knowl- edge of the existing pruned models extracted by these pruning methods and use them to generate a well-performed pruned model for the unseen task with a few fine-tuning iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Meta-pruning To the best of our knowledge, the non- parametric meta-pruning problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', how to prune a model for a target task using the pruned models of other tasks, has not been specifically studied in previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' However, several recent researches aim at learning meta(prior) knowledge that can improve pruning in other scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' MetaPruning [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2019] trains a weight- generation meta-network to prune the same network for the same task under different constraints, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', user/hardware defined pruning ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' DHP [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2020] addresses the same problem but does not rely on any reinforcement learning or evolutionary algorithm since it makes the pruning procedure differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Meta-learning has been studied to find better weight-initialization for pruning on different tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' applies Reptile [Nichol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2018] for overfitting reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Meta-learning has also been studied to select the best pruning criterion for different tasks [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In [Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2020], a shared sparse backbone network is trained for multi-task learning but it cannot be adapted to new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Our method is the first one to use meta-learning to extract a pruned model for a new task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The main differences of our approach to them are: (1) we do not train a parametric meta-learner but instead use majority voting from similar tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' and (2) our meta-voting generates a pruned small sub-network to initialize the target task training, which significantly reduces the pruning cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 3 Empirical Study: Pruning a Pre-trained model for Different Tasks In this section, we conduct an empirical study that applies different methods to prune a CNN or ViT pre-trained model for over hundreds of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Our study focuses on the overlap between the pruned models for different tasks and whether/how it relates to their similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' To this end, we introduce different task similarities and compare the overlap associated with different similarity groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The results show that more similar tasks tend to share more nodes/filters/heads in their pruned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' And this holds across different pruning methods, datasets and pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' No meta-pruning is used in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='1 A Dataset of Pruned Models While the number of possible downstream tasks and users can be huge in practice, the current progress on foundation models show that one or a few large-scale pre-trained models with light fine-tuning usually achieve the SOTA performance on most of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' To simulate this scenario on a standard dataset, our empirical study creates a dataset of pruned models for hundreds of tasks from the same pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We choose CIFAR-100 and ImageNet for the study due to many classes in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' For each dataset, we randomly draw 1000 classification tasks, each defined on 5 classes sampled without replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We adopt ResNet-18 [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2016] pre-trained on CIFAR-100, ResNet-50 and a small version of DeiT [Touvron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2021a] pre-trained on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' For ResNet-18 and ResNet-50, we prune two types of pre-trained models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', the supervised training following [Devries and Taylor, 2017] and the self-supervised training following SimSiam [Chen and He, 2020] (only the encoder is used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' For ViT, the training of its pre-trained model follows [Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Iterative Pruning We apply iterative filter-pruning (IFP) to ResNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Unlike magnitude-based pruning [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2017] with one-time selection of nodes/weights, iterative pruning alternates between network pruning and fine-tuning of model weights for multiple iterations, each of which prunes p% of the remaining nodes/weights so it progressively prunes a large network to the targeted size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' It usually performs better than other pruning methods and has also been mainly studied in theoretical works about Lottery Ticket Hypothe- sis [Frankle and Carbin, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We take the activation values of filters averaged over all training samples to measure the importance of filters [Molchanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2016], referred as Activation Pruning, in which filters with smaller activation values contain less information of input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The detailed procedure of IFP is described in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Automatic Pruning Inspired by the SOTA ViT structured pruning method [Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2022b], we prune ViT by auto- matic head&node pruning (AHNP) for a given task, which parameterizes the sub-network as the pre-trained model with a learnable score multiplied to each prunable head and node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' To encourage sparsity, the differentiable scores of all prunable heads and nodes are optimized with an additional L1 regularization loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' After each optimization step, we apply a simple thresholding to these scores to remove heads and nodes with small scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The optimization stops if the pruned model reaches the targeted size and the model will be fine-tuned for a few iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The detailed procedure of AHNP can be found in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' For tasks of CIFAR-100, we run IFP for all 1000 tasks on ResNet-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' And we apply IFP and AHNP to tasks of ImageNet on ResNet-50 and ViT respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Finally, we create a dataset of pruned models for thousands of tasks over different pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' For each task i, we record its la- bels Ci, the set of preserved nodes/filters/heads {Ωℓ}ℓ=1:L−1 and the pruned model θT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We use the same hyper-parameters for different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' For IFP on ResNet, we use a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='005, pruning iterations of 1000 and batch-size of 128 for both the tasks of CIFAR-100 and ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' When applying AHNP to ViT, we follow the ViT training in [Touvron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2021b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We reduce the pruning iterations to 1000 and use a small learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='00005 for parameters inherited from the pre-trained ViT (to preserve its knowledge) and a large learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='05 for the learnable scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The pruning ratio is 90% for all pruned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Refer to the Appendix for the detailed discussion about the computational cost of the model zoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='2 Do similar tasks share more nodes/filters/heads on each layer of their pruned models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The representations learned for a task can still be helpful to its similar tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' This motivates transfer/multi-task/meta learn- ing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' But do similar tasks also share more structures in their pruned sub-networks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We apply two metrics to mea- sure the similarity between classification tasks in our dataset and study whether/how the similarity relate to their shared nodes/filters/heads in different layers of their pruned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Similarity Metrics We apply two metrics to compute the similarity between tasks and find the nearest tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', the Log Expected Empirical Prediction (LEEP) [Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2020] and the Wordnet wup similarity [Pedersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Wu and Palmer, 1994].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' LEEP score is widely used in transfer learning to estimate the knowledge transferability from a source task to a target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In our study, for each target task, we can rank the other tasks by their LEEP similarity score from each of them to the target one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Computing the LEEP score only requires a single forward pass of the pruned model on the target task’s data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Wordnet wup similarity only requires the semantic labels of classes in each task and it is based on the depths of the their corresponding synsets in the Wordnet [Miller, 1995] taxonomies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' It does not depend on the pruned model so it is more efficient to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Overlap Between Tasks Let Ωi ℓ and Ωj ℓ denote the sets of filters/nodes/heads remained in layer-ℓ after running IFP or AHNP for task i and j (when using the same pre- trained model), we measure the overlap of the two sets by intersection over union (IoU) ratio [Jaccard, 1901], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', IoU = |Ωi ℓ∩Ωj ℓ|/|Ωi ℓ∪Ωj ℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 1 (ResNet) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 2 (ViT) report the IoU of each layer/block for pairs of tasks with different similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' For each target task, the tasks in the dataset are partitioned into 5 similarity groups according to their LEEP scores or Wordnet similarities to the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The similarity decreases from group 1 to group 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Specifically, for each test task, we compute its similarity scores with its neighbours in the model zoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We sort these similarity scores and partition them into five groups of equal intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Neighbours whose similarity scores fall into a certain interval will be assigned to the corresponding group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' For all the datasets and architectures, more similar tasks tend to share more filters/nodes/heads (larger IoU) between their pruned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Therefore, for a new task, the pruned models of its nearest tasks preserve many important filters for it and combining them might result in a better and much smaller sub-network to initialize the new task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Moreover, for deeper layers/blocks in both ResNet and ViT, the gap between different similarity groups on the IoU increases because the features are more task-specific in deeper layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Due to the same reason, for every similarity group, IoU decreases with (a) CIFAR-100, Supervised model (b) CIFAR-100, Self-supervised model (c) ImageNet, LEEP Similarity (d) ImageNet, Wordnet Similarity Figure 1: IoU of layers in ResNet between tasks with different similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 0 2 4 6 8 10 Attention block index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='0 IoU of heads 0 2 4 6 8 10 Feedforward block index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='5 IoU of nodes similarity group 1 similarity group 2 similarity group 3 similarity group 4 similarity group 5 (a) ImageNet, LEEP Similarity 0 2 4 6 8 10 Attention block index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='5 IoU of heads 0 2 4 6 8 10 Feedforward block index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='5 IoU of nodes similarity group 1 similarity group 2 similarity group 3 similarity group 4 similarity group 5 (b) ImageNet, Wordnet Similarity Figure 2: IoU of blocks in ViT between tasks with different similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' depth in the overall trend (though fluctuating locally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Fur- thermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 2 shows that the IoU gap between similarity groups defined by the LEEP score is larger than that obtained by Wordnet similarity (refer to Appendix for detailed results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' This indicates that the semantic similarity between class la- bels might not be as accurate as the LEEP score that takes the pruned model and its learned representations into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Algorithm 1 META-VOTE PRUNING (MVP) Input : Target task i and its training set Di, pruning ratio r, J, N, a dataset of pruned models for different tasks Output : A pruned model for target task-i Initialize: Ωℓ ← ∅, the set of filters in layer-ℓ 1 Sample/find N similar tasks N i to task i according to LEEP score or Wordnet similarity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 2 for ℓ ← 1 to L − 1 do 3 Sample (1 − r)nℓ filters with probability p(k) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' (1)) and add them to Ωℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 4 for k ∈ Ωℓ do 5 Initialize filter-k by averaging its parameters of tasks in {j ∈ N i : k ∈ Ωj ℓ};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 6 end 7 end 8 Fine-tune the pruned model for J iterations on Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 4 Meta-Vote Pruning (MVP) Inspired by the empirical study above, we propose a simple yet strong baseline “meta-vote pruning (MVP)” (Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 1) for non-parametric meta-pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The procedure of MVP majority voting is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Given a target task i, MVP draws a sub-network of a pre-trained network by sampling filters/nodes/heads in each layer using majority voting from its nearest tasks N i and their pruned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In particular, for each filter-k ∈ [nℓ] from layer-ℓ of the pre-trained model, we apply softmax (with temperature τ) to the times of each filter being selected by tasks in N i, which yields a probability distribution over all the filters [nℓ], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', ∀k ∈ [nℓ], p(k) = exp(|{j ∈ N i : k ∈ Ωj ℓ}|/τ) � h∈[nℓ] exp(|{j ∈ N i : h ∈ Ωj ℓ}|/τ) (1) To initialize layer-ℓ of the sub-network, MVP samples filters from this distribution (without replacement) according to the targeted pruning ratio r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We further initialize the parameters of each filter-k by averaging its parameters in the pruned models of the similar tasks which preserve filter-k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' MVP then fine-tunes the initialized sub-network for a few iterations on the training set of the target task since MVP targets to keep the computational cost low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 5 Experiments In this section, we conduct extensive experiments on CIFAR- 100 and ImageNet over different pre-trained models, which evaluate MVP (Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 1) and compare it with SOTA pruning methods under different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We validate the strong generalization of MVP by applying it to unseen tasks from Caltech-256 and other fine-grained datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We further study the effect of different pruning iterations, neighbour numbers, task sizes and similarity metrics for MVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' All the results show that MVP can outperform other methods with better performance and higher efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='1 Implementation Details The experiments of MVP are mainly based on the tasks from the dataset introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' For each setting of exper- iments, we randomly draw 100 test tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', the target task Task N-1 Task 2 Task 3 Task 4 2 0 4 0 0 2 0 1 0 1 1 1 Softmax (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' (1)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='147 Votes for filters in layer- ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Probabilities of being selected Target task .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Task 1 Task N Similarity score = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='25 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='68 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='11 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='57 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='78 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='46 Target task selected filters Find nearest tasks 1 12 2 3 4 5 6 7 8 9 10 11 1 3 6 Task2 selected filters 3 6 10 Task3 selected filters 1 3 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' TaskN-1 selected filters 1 3 11 Task4 selected filters 3 6 12 Figure 3: The example of majority voting in MVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Each similar neighbour task of the target task vote for filters which are reserved by its pruned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Then, softmax is applied to the votes of all filters in layer-ℓ and filters with more votes have higher probability to be selected by the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 1) from the dataset and treat the rest tasks as training tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' To evaluate MVP on CNN, we run MVP on the pruned models of ResNet-18 and ResNet-50 for CIFAR-100 and Im- ageNet respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' For both these two experiments, we use the meta-pruning iterations of 100, batch size of 128, learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='01 and optimizer of SGD with the cosine-annealing learning rate schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' For experiments of ViT, MVP is applied to the pruned models of ViT for ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The meta- pruning iterations and batch size are also set as 100 and 128 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Following the setting of training ViT in [Tou- vron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2021b], we apply a small learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='0002 and optimizer of AdamW with the cosine-annealing learning rate schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The small number of meta-pruning iterations demonstrates the efficiency of MVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The target pruning ratio of MVP for all tasks is 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' All the results of accuracy shown in this section are averaged over the 100 test tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='2 Baseline Methods We compare MVP with several baselines and SOTA pruning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We first implement two baselines to show the advantages of MVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' (1) Conventional pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We apply a larger number of pruning iterations to extract pruned models for each target task by IFP or AHNP introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' This baseline can be regarded as the upper bound performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' (2) Random pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' To validate whether the initialization of MVP makes sense, for each target task, we initialize its sub-network by randomly sampling the same number of nodes/filters/heads as MVP from the pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We take this baseline as the lower bound performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We also include other SOTA pruning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' For MVP on CNN, we compare MVP with IHT-based Reptile [Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2020], a meta-pruning method that uses Reptile [Nichol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2018] and iterative pruning to find better weight- initialization for a pruned meta-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Given a new task, it fine-tunes the pruned meta-model for a limited number of it- erations to obtain the final pruned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' MEST [Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2021] is the SOTA method in sparse training community, which trains a model from a sparse sub-network so that less computation is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' DLTH [Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2022] is based on a variant of the Lottery Ticket Hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' It transforms random tickets into winning tickets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We compare MVP with UVC [Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2022b] and PoWER [Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2020] on ViT pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Unlike AHNP, which prunes heads and nodes, UVC also skips the unimportant layers and blocks in ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Unlike parameter pruning, PoWER adopts a dynamic method pruning the input patches of each block for each input sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' For a fair comparison, except for the upper bound baseline, the pruning iterations of all other methods and MVP are set to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' And the pruning ratios of all methods are set to 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Table 1: Comparison between MVP and baseline methods on CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The ’-SSL’ behind each method means applying this method to pruned models extracted from self-supervised pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Bold and Bold gray mark the best and second best accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Methods Pruning Iterations ResNet-18 ResNet-50 Acc FLOPs Acc FLOPs IFP 1000 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='99±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='47 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='88(T) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='68 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='06(T) IFP-SSL 1000 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='22±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='52 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='88(T) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='75 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='06(T) Random Pruning 100 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='12±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='43(T) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='42±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='92 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='16(T) IHT-based Reptile[Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2020] 100 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='23±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='43(T) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='40±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='16(T) MEST[Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2021] 100 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='28±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='47(T) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='25±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='33 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='48(T) DLTH[Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2022] 100 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='46±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='28(T) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='33±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='56 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='64(T) MVP(ours) 100 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='98±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='43(T) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='80±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='16(T) MVP-SSL(ours) 100 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='43(T) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='16(T) Table 2: Comparison between MVP and baseline methods on ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Bold and Bold gray mark the best and second best accuracy Methods Pruning Iterations ViT Acc FLOPs AHNP 1000 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='48±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='62 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='50(T) Random Pruning 100 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='71±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='25(T) UVC[Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2022b] 100 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='57 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='73(T) PoWER[Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2020] 100 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='76±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='18 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='86(T) MVP(ours) 100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='23±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='49 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='25(T) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='3 Main Results The results of applying MVP to tasks from CIFAR- 100(ImageNet) on ResNet-18(ResNet-50) supervised and self-supervised pre-trained model, and the baseline methods are reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' On both datasets and pre-trained models, MVP outperforms IFP which spends 10× iterations of MVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Hence, MVP can produce a higher-quality pruned model when using fewer iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The results demonstrate that MVP can work well on tasks from both supervised and self-supervised pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The random pruning performs much poorer than MVP, which indicates the impor- tance of majority voting from nearest tasks in selecting filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We also compare MVP with SOTA pruning methods for CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' IHT-based Reptile [Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2020] trains a universal sparse sub-network for all target tasks by applying meta- learning on training tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' MVP achieves higher accuracy than IHT-based Reptile under the same training iterations, implying that MVP can find an accurate sub-network for each target task as its initialization and improve its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' MEST [Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2021] can speed up pruning by starting training from a well-designed sub-network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' As a variant of Lottery Ticket Hypothesis, DLTH [Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2022] proposes a method to transform any random ticket into the winning ticket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' MVP outperforms MEST and DLTH by a large margin because MVP is trained on a sub-network selected using meta knowledge from similar tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In contrast, the initial sub-network for MEST or the winning ticket of DLTH does not leverage any prior knowledge about the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 2 shows the comparison between MVP and baseline methods on ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Similar to the results on pruning CNN, the performance of MVP on ViT is comparable to AHNP that ap- plies much more pruning iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The accuracy of random pruning is still much worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' MVP also outperforms SOTA pruning methods developed for ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Hence, on ViT, MVP can efficiently produce a small yet high-quality sub-network for each new task by exploiting the nearest tasks’ models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The baselines are slower and require more iterations than MVP because they need to re-train the model to achieve a small loss when some parameters or patches are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Both UVC and PoWER cannot recover the accuracy under this strong constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In contrast, the majority voting in MVP directly produces a small sub-network from similar tasks’ models so only a few iterations suffice to reach a downstream task performance comparable to AHNP with 10x iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Table 3: Accuracy of MVP on unseen tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Methods Caltech-256 CUB200-2011 Iters Acc FLOPs Iters Acc FLOPs IFP 800 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='28±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='64 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='06(T) 800 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='51 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='99(T) IFP 60 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='90±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='79 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='73(T) 80 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='85±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='17(T) MVP(ours) 60 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='90(T) 80 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='54±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='63(T) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='4 Performance on Unseen Dataset In this section, to validate the generalization of MVP, we apply MVP to produce pruned models for target tasks from unseen dataset Caltech-256 [Griffin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2007] and fine- grained dataset CUB200-2011 [Wah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2011], Oxford Flowers-102 [Nilsback and Zisserman, 2008] and Oxford- IIIT Pets [Parkhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2012], using the pruned models of tasks from ResNet-50 training on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The data of these datasets are never seen by the pre-trained model and tasks in the pruned model dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Each target task is defined on 5 classes sampled without replacement from each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The performance of MVP on Caltech-256 is shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 3, which is still comparable to the IFP using 10x prun- ing iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' When the number of pruning iterations of IFP decreases, its performance becomes much worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Besides Caltech-256, we also validate the effectiveness of MVP on more difficult fine-grained datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', CUB200-2011, Ox- ford Flowers-102 and Oxford-IIIT Pets where images in dif- ferent classes are from various species of birds, flowers and animals, which are hard to distinguish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' On target tasks from fine-grained datasets, MVP also works better than IFP which needs much more computation costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Results of Oxford Flowers-102 and Oxford-IIIT Pets can be found in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The results show that MVP can still produce a high-quality initialization for the task from unseen datasets by majority voting of similar tasks so that the pruned model can converge quickly with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' MVP’s great performance on fine-grained datasets implies that MVP can learn from different objects to facilitate the classification of hard-to- distinguish target tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' This experiment demonstrates that MVP can be applied to various datasets and generalizes well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Table 4: Results on sub-tasks of different sizes for CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Methods 10-classification 3-classification Iters Acc FLOPs Iters Acc FLOPs IFP 1500 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='26 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='48(T) 500 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='71 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='59(T) MVP(ours) 190 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='53±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='21(T) 60 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='12(T) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='5 Results of MVP on sub-tasks of different sizes To evaluate the performance of MVP on sub-tasks of different sizes, we build a dataset of pruned models for 10- classification and 3-classification sub-tasks from CIFAR-100, of which the pruning ratio is set to 85% and 95% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The results are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' From the results we can find that When changing the size of the sub-tasks, MVP can consistently achieve comparable or better performance than SoTA methods by spending much less computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' MVP is applicable to a variety of tasks of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='6 Ablation Study Effect of Iteration Numbers Given a new target task and a pre-trained model, MVP can build a well-performed small model in a few iterations, demonstrating its capability in reducing adaptation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In plot (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 4, we compare MVP with conventional pruning methods using different numbers of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' On different architectures of pre- trained models, MVP converges to a high accuracy after nearly 100 iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' On the contrary, the conventional pruning methods need much more iterations(> 500) to be comparable to MVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' With only ≤ 50 pruning iterations, MVP can reach a reasonable accuracy, while conventional pruning methods perform poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' These imply that the initialized sub-network obtained by majority voting already contains helpful knowledge from its similar tasks to speed up the training of the pruned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Effect of Similarities between Tasks MVP consistently achieves better performance when applied to nearest tasks with the highest similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In plot (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 4, we compare the LEEP score with the Wordnet similarity and study the effect of applying MVP to neighbour tasks with different similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' From similarity group 1 to group 5, the similarities between tasks decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We find that for both the two similarity metrics, the accuracy of MVP improves significantly when the similarities between tasks increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' When the pruning iterations are small(= 20), where the initialization of the sub-network is more important, the accuracy of tasks from similarity group 1 leads to similarity group 5 by 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Despite the accuracy of similarity group 5 improving when the pruning iterations increase to 100, there is still a gap of 7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' This result indicates that neighbour tasks with high similarities share more knowledge with the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In this plot, we also find that tasks in different similarity groups classified by LEEP score show larger differences than Wordnet similarity, implying that LEEP score can better eval- uate similarities between tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' This result is consistent with our observation in the empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The performance of Wordnet similarity is also good and can still be an alternative when the time and computational resources are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Comparison between Pruned Models Extracted by Different Pruning Method In this part,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' we apply MVP to 20 50 100 200 500 Number of pruning iterations 40 50 60 70 80 90 Accuracy% of MVP MVP on ResNet-18 MVP on ViT IFP on ResNet-18 AHNP on ViT (a) Effect of Pruning Iterations 1 2 3 4 5 Similarity groups 60 62 64 66 68 70 72 74 Accuracy% of MVP LEEP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Iters=20 Wordnet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Iters=20 1 2 3 4 5 Similarity groups 83 84 85 86 87 88 89 LEEP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Iters=100 Wordnet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Iters=100 (b) Effect of Similarities between Tasks Figure 4: (a) Comparison between MVP and conventional pruning methods with different pruning iterations on different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' For both ResNet-18 and ViT, MVP converges much faster in a small number of iterations than conventional pruning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' (b) Comparison between LEEP score and Wordnet similarity for MVP with different pruning iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' From similarity groups 1 to 5, the similarities between tasks decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' For both similarity metrics, more similar tasks get better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' LEEP score has a better ability to measure similarities between tasks than Wordnet similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 8 9 101112131415161718 Layer index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='225 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='IoU of filters between different tasks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='similarity group 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='similarity group 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='similarity group 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='similarity group 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='similarity group 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='(a) IoU of Layers for IFP(Taylor Pruning) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='Number of neighbours ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='84 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='86 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='87 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='88 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='89 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='Accuracy% of MVP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='IFP(Feature Pruning) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='IFP(Taylor Pruning) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='(b) Effect of Neighbour Numbers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='Figure 5: (a) IoU of layers in ResNet-18 between tasks whose pruned models are extracted by IFP (Taylor Pruning) and more similar tasks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='also share more filters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' especially in deeper layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' (b) Results of applying MVP to pruned models from Activation Pruning and Taylor Pruning over different number of neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' MVP(neighbour number ≥ 2) can improve the performance of transfer learning(neighbour number = 1) by a large margin when applied to pruned models extracted by different pruning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' pruned models extracted by Taylor Pruning [Molchanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2019] on ResNet-18 for CIFAR-100 tasks, to prove that MVP works well on pruned models extracted by various pruning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Taylor Pruning measures the importance of each filter by the effect of removing this filter on final loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In plot (a) of Figure 5, we show the IoU of each layer for pairs of tasks with different task similarities, of which the pruned models are extracted by Taylor Pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Consistent with our observation in the empirical study, pruned models with higher similarities share more filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Effect of Number of Neighbours In plot (b) of Figure 5, we investigate the effect of the number of neighbours for MVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' When the number = 1, MVP reduces to transfer learn- ing which learns from the pruned model of a single selected similar task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In the plot, when the number of neighbours increase from 1 to 2, the performance improves sharply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' This result implies the effectiveness of meta knowledge from different neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' When the number of neighbours ≥ 3, for both Activation Pruning and Taylor Pruning, the accuracy improves little, which indicates that 3 neighbours are enough for MVP to produce a high-quality initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 6 Conclusion In this paper, we study “non-parametric meta-pruning” problem that aims to reduce the memory and computational costs of single-task pruning, via reusing a pre-trained model and similar tasks’ pruned models to find an initialization sub-network for a new task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' We conduct an empirical study to investigate the relationship between task similarity and the pruned models of two tasks for different datasets and deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The empirical study motivates a simple yet strong baseline for meta-pruning, called “meta-vote pruning (MVP)” (Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' By extensive experiments on multiple tasks drawn from several datasets under different training settings, we demonstrate the advantages of MVP over other SOTA pruning methods in the region of limited computation and show its potential to reduce the carbon footprint of pruning/fine-tuning large networks for billions of edge devices and tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' References [Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2022] Yue Bai, Huan Wang, Zhiqiang Tao, Kun- peng Li, and Yun Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Dual lottery ticket hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='04248, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' [Chen and He, 2020] Xinlei Chen and Kaiming He.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2021] Mingjian Zhu, Yehui Tang, and Kai Han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Vision transformer pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='08500, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' [Zhuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', 2020] Tao Zhuang, Zhixuan Zhang, Yuheng Huang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Zeng, Kai Shuang, and Xiang Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Neuron- level structured pruning using polarization regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In NeurIPS, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 7 Appendix Algorithm 2 ITERATIVE FILTER PRUNING (IFP) Input : Pre-trained network F(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' θ), Task T and training set DT , Hyperparameters J, h, r, p Initialize: Ωℓ ← [nℓ], the set of filters preserved in layer-ℓ 9 for j ← 1 to J do 10 if j%h = 0 and |Ωℓ| > (1 − r)nℓ then 11 for ℓ ← 1 to L − 1 do 12 Prune p% of filters in Ωℓ with the smallest impor- tance score over DT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 13 end 14 end 15 Apply one SGD step on a mini-batch of DT to fine-tune the remained filters {θℓ,i : ℓ ∈ [L − 1], i ∈ Ωℓ} and θL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 16 end 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='1 Iterative Filter Pruning The detailed procedure of IFP is described in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Given a pre-trained network F(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' θ) of L layers (layer-L is fully-connected) with parameter θ = {θℓ}ℓ=1:L and a training set DT of a target task T, let θℓ = {θℓ,i}i=1:nℓ denote all parameters in layer-ℓ composed of θℓ,i for every filter-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' IFP fine-tunes the model for total J iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' It prunes p% of the filters remained in each layer every h iterations according to their activation values fℓ,i(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' It stops to prune layer-ℓ if reaching the targeted pruning ratio r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='2 Automatic Head&Node Pruning The detailed procedure of AHNP is described in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Given a pre-trained network F(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' θ) of L layers (layer-L is fully-connected) with parameter θ = {θℓ}ℓ=1:L and a training set DT of a target task T, let θℓ = {θℓ,i}i=1:nℓ denote all parameters in layer-ℓ composed of θℓ,i for ev- ery head/node-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Sℓ,i denote the score for each prunable head/node-i in layer-ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' AHNP fine-tunes the model and scores for total J iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' It prunes the heads/nodes if their scores are smaller than the threshold τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' It stops to prune layer-ℓ if reaching the targeted pruning ratio r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Then, AHNP fine-tunes the pruned model for K iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='3 Results of MVP on Oxford Flowers-102 and Oxford-IIIT Pets Table 5: Accuracy of MVP on fine-grained tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Methods Oxford Flowers-102 Oxford-IIIT Pets Iters Acc FLOPs Iters Acc FLOPs IFP 800 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='20±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='39 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='98(T) 1000 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='96 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='33(T) IFP 60 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='20±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='61 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='04(T) 100 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='76±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='86 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='57(T) MVP(ours) 60 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='40±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='95(T) 100 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='77 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='58(T) Similar to previous observations, MVP outperforms IFP with much more training iterations when applied to target tasks sampled from fine-grained datasets Oxford Flowers-102 and Oxford-IIIT Pets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' The results indicate that MVP can Algorithm 3 AUTOMATIC HEAD&NODE PRUNING (AHNP) Input : Pre-trained network F(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' θ), Task T and training set DT , Hyperparameters J, K, r, τ Initialize: Ωℓ ← [nℓ], the set of heads/nodes preserved in layer-ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Sℓ,i ← 1, the score for each prunable head/node in layer-ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 17 for j ← 1 to J do 18 for ℓ ← 1 to L − 1 do 19 for i ∈ Ωℓ do 20 Prune the head/node if its score Sℓ,i < τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 21 end 22 end 23 Stop pruning if reaching the target pruning ratio r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 24 Apply one optimization step on a mini-batch of DT to fine-tune the remained heads/nodes and scores {θℓ,i, Sℓ,i : ℓ ∈ [L − 1], i ∈ Ωℓ} and θL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 25 end 26 Remove S, fine-tune the pruned model for K iterations on Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' build high-quality initialization, which contains fine-grained pattern information for each class in hard-to-distinguish tar- get tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='4 Discussion about the cost of creating a pruned model-zoo by training on hundreds of tasks In the experimental setting, there exists a one-time cost for preparing the model-zoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' However, like training any meta- learning model, this can facilitate many future tasks by sig- nificantly reducing their computation and required samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Moreover, we can keep adding MVP pruned models of new tasks into this model-zoo and keep improving it in a life-long learning manner with no extra cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Meta-pruning moves required computation from new-task adaptation to pre-training (preparing the model-zoo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In prac- tice, this is an even more important advantage over single-task pruning because the meta-pruning cost is offline on the server side so it is tolerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Practitioners care more about the de- ployment cost of new tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=', on edge devices with limited data and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' Because of the model-zoo, our method makes the practical deployment of model pruning more af- fordable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='5 The difference of IoU between different similarity groups and similarity metrics In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='6, we draw the difference of IoU between tasks of sim- ilarity group 1 and similarity group 5 for ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' As the layer gets deeper, the difference increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='2 of the paper, the average difference of IoU be- tween similarity group 1 and similarity group 5 over all layers is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='195 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='150 respectively for LEEP and Wordnet simi- larity, which has a large gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='4 of the paper, the LEEP score performs a little better than Wordnet similarity in MVP which indicates that models with the larger IoU share more relevant parameters and LEEP has a good ability to find the nearest neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 0 5 10 15 20 25 30 35 40 45 50 Layer index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='08 Difference of IoU Figure 6: The difference of IoU between tasks of similarity group 1 and similarity group 5 for ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 0 2 4 6 8 10 Attention block index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='0 IoU of heads 0 2 4 6 8 10 Feedforward block index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='5 IoU of nodes similarity group 1 similarity group 2 similarity group 3 similarity group 4 similarity group 5 random pruning Figure 7: Comparison of IoU between MVP and random pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='6 Comparison of IoU between MVP and random pruning In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content='7, besides MVP, we also show the IoU of pruned mod- els extracted by random pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' When the similarity be- tween tasks is large, the IoU of MVP is much larger than ran- dom pruning, implying that these tasks contain lots of rele- vant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} +page_content=' When the similarity between tasks is small, in the last few blocks, the IoU of nodes is similar to that of random pruning, which indicates that tasks of low similarity share little high-level information with the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFJT4oBgHgl3EQfgSwn/content/2301.11560v1.pdf'} diff --git a/z9AyT4oBgHgl3EQfbfcG/content/tmp_files/2301.00261v1.pdf.txt b/z9AyT4oBgHgl3EQfbfcG/content/tmp_files/2301.00261v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9f4072fa3dd76d0057fb6bfa690e9fca0549450a --- /dev/null +++ b/z9AyT4oBgHgl3EQfbfcG/content/tmp_files/2301.00261v1.pdf.txt @@ -0,0 +1,1771 @@ +arXiv:2301.00261v1 [nucl-th] 31 Dec 2022 +Cluster radioactivity in trans-lead region: A systematic study with +modified empirical formulas +A. Jaina,b,c, P. K. Sharmad, S. K. Jaina, J. K. Deegwale, G. Saxenac,f +aDepartment of Physics, School of Basic Sciences, Manipal University Jaipur, Jaipur-303007, India +bDepartment of Physics, S. S. Jain Subodh P.G.(Autonomous) College, Jaipur-302004, India +cDepartment of Physics (H&S), Govt. Women Engineering College, Ajmer-305002, India +dGovt. Polytechnic College, Rajsamand-313324, India +eGovt. Women Engineering College, Ajmer-305002, India +fDepartment of Physics, Faculty of Science, University of Zagreb, Bijeni˘cka c. 32, 10000 Zagreb, Croatia. +Abstract +The possibility of cluster emission from trans-lead (86≤Z≤96) region of periodic chart has been +explored comprehensively by employing few empirical formulas which are modified by adding an- +gular momentum (l) or isospin-dependent (I = (N − Z)/A) or both terms for the calculation of +cluster decay half-lives. These modified versions of the formulas are found with lesser χ2 per degree +of freedom and root mean-square error, in addition to the smaller values of some other statistical +parameters, while compared to their corresponding old versions on available 61 experimental data +of cluster radioactivity. By applying the modified version of the formula given by Balasubramaniam +et al. [PRC 70 (2004) 017301], the most accurate formula among these, half-lives of several clusters +i.e. isotopes of Be, B, C, N, O, F, Ne, Na, Mg, and Si are predicted systematically for the several +isotopes in the trans-lead region. The contest of cluster emission with α-decay has been investigated +in form of branching ratio which brings several potential cluster emissions into the probable decay +modes of these nuclei. The accurate prediction of half-lives of such clusters is expected to be crucial +for the future experimental observations where α-decay is observed dominantly. +Keywords: +Cluster decay, Trans-lead Nuclei, Empirical formulas, α-decay. +1. Introduction +In 1980, Sandulescu et al. [1] firstly predicted a new type of radioactivity: cluster radioactivity, +which was based on fragmentation theory, where fusion and fission reaction valleys were generated +by the shell closure effect [2]. Later in 1984, Rose and Jones experimentally proved the existence +of this new type of exotic decay [3], in which 14C decays from actinide parent nucleus 223Ra and +forms a stable doubly magic (Z=82, N=126) nucleus 208Pb. Till now, many clusters decays from +light to heavy clusters (14C to 32Si) have been observed from various trans-lead nuclei (Fr, Ra, +Ac, Pa, Th, U, Pu, etc.) resulting the corresponding daughter nuclei as magic nuclei (Z=82) or +neighboring ones (Z=80, 81, and 83), which indicate the importance of shell and pairing effects +in cluster radioactivity [4–6]. These clusters are observed with long half-lives (T1/2) in the range +1011-1030 sec. [7]. +Theoretically, the half-lives of cluster emissions are predicted using various models such as uni- +fied fission model (UFM) [8], generalised liquid drop model (GLDM) [9], super-asymmetric fission +Preprint submitted to Nuclear Physics A +January 3, 2023 + +model (SAFM) [10], preformation cluster model (PCM) [11], etc. Cluster decay half-lives are also +calculated by using various semi-empirical formulas such as (i) the empirical relation suggested +by Balasubramaniam et al. (BKAG formula) for cluster decay half-lives with only three parame- +ters [12], (ii) the empirical relation suggested by Ren et al. (RenA formula) using a microscopic +density-dependent cluster model with the re-normalized M3Y nucleon-nucleon interaction [13]. Con- +comitantly, based on experimental observations about the characteristics of exotic cluster decays, +scaling law proposed by Horoi [14] in which logarithmic half-life is proportional to scaling variable +(ZcZd)0.6/√Q and also proportional to √µ, where µ is the reduced mass of cluster and daughter +nuclei which was followed by another semi-empirical formula (NRDX), proposed by Ni et al. [15] +considering WKB barrier penetration probability with some approximations. In 2009, Qi et al. +introduced universal decay law (UDL) [16] that originates from the mechanism of charged particle +decay and R-matrix for all sort of decays of clusters, which includes monopole radioactive decays +as well. Poenaru et al. [17] plotted a universal curve (UNIV) which is found to be a straight line +for cluster decay and α-decay. +All the above-mentioned formulas have been fitted to the available experimental data without +considering the dependence of half-lives on angular momentum taken away by the cluster: expected +to be crucial alike to the α-decay [18] to delineate all sets of experimental data. The importance +of angular momentum on the α-decay half-lives has already been established in a few of our recent +works [19, 20] which has invoked us to probe similar dependence on the cluster decay half-lives. In +addition to this, isospin (I = (N − Z)/A) of parent nucleus is found to be pivotal for the case of +α-decay in heavy and superheavy nuclei [20–25] pointing towards its significance in terms of cluster +decay as well. Considering these two effects together, modified UDL formula (new UDL) by Soylu +and Qi [26], and improved NRDX formula (named as improved unified formula (IUF)) by Ismail et +al. [27] have explained recently that angular momentum and isospin are indeed crucial quantities +in determining the cluster decay half-lives. Importance of isospin effect is also probed by improving +semi-empirical formula (ISEM) for the cluster radioactivity in Ref. [28]. +In this article, we have modified the BKAG [12], RenA [13], Horoi [14], NRDX [15], UDL [16], +and UNIV [17] formulas by investigating the effect of centrifugal barrier and isospin terms. These +six modified formulas are fitted by using 61 experimental cluster decay data [7, 9, 26, 29]. The +comparison of RMSE (root mean square error) between the older and modified version manifestly +shows the significance of inclusion of angular momentum and isospin-dependent terms in cluster +emission. Furthermore, one of the modified formulas i.e. MBKAG formula (emerged with least +RMSE) is employed to calculate the cluster decay half-lives for various cluster emissions like isotopes +of Be, B, C, N, O, F, Ne, Na, Mg, and Si in trans-lead region (86≤Z≤96). For these theoretical +estimates, the requirement of disintegration energy (Q-value) is tested by 121 available experimental +Q-values [7, 9, 26, 29] from various mass models [30–33]. Consequently, various potential clusters +are proposed from trans-lead region along with their accurate estimation of half-lives. +2. Formalism +In 2004, Balasubramaniam et al. fitted a formula (BKAG) [12] for cluster decay. In the course +of that year, Ren et al. +established a formula [13] that can be treated as a natural extension +of the Geiger-Nuttall law [34] as well as the Viola-Seaborg formula [35] from simple α-decay to +complex cluster radioactivity. In the same year, Horoi also suggested an independent model for +α-decay which was generalized for cluster emission [14]. +In 2008, Ni et al. +established NRDX +semi-empirical formula for the calculation of half-lives of α and cluster decays [15]. Afterwards, Qi +2 + +et al. has introduced universal decay law (UDL) [16] which is widely used by many authors for the +estimation of half-lives of cluster radioactivity. In 2011, Poenaru et al. fitted UNIV formula [17] +and represented a single line of the universal curve on the graph for α-decay and cluster decay. The +original versions of these formulas are mentioned below: +log10T BKAG +1/2 +(sec.) = [aAc(Ad − Ac)/A + bZc(Zd − Zc)/Z]Q−1/2 + c +(1) +log10T RenA +1/2 +(sec.) = aZdZcQ−1/2 + bZdZc + c +(2) +log10T Horoi +1/2 +(sec.) = (a√µ + b)[(ZcZd)0.607Q−1/2 − 7] + (c√µ + d) +(3) +log10T NRDX +1/2 +(sec.) = aZcZd +� µ +Q + b√µ(ZcZd)1/2 + c +(4) +log10T UDL +1/2 +(sec.) += +aZcZd +� µ +Q + b[µZcZd(Ac +1/3 + Ad +1/3)]1/2 + c +(5) +log10T UNIV +1/2 +(sec.) += +−logP + log10S − [log10(ln2) − log10υ] +(6) +In the above-mentioned formulas Ad, Ac and Zd, Zc denote the mass numbers and atomic numbers +of the daughter nucleus and cluster, respectively. Q (in MeV) is the energy released in cluster +decay, and µ = AdAc/(Ad + Ac) is the reduced mass. +In Eqn. +(6), −logP is determined by +a(µZcZdRb)1/2[arccos√r − +� +r(1 − r)], r = Ra/Rb with Ra = 1.2249(Ac +1/3 + Ad +1/3) fm, Rb = +1.43998ZdZc/Q fm, and the logarithmic form of preformation factor is given by log10S = −b(Ac−1) +along with [log10(ln2) − log10υ] = d is the additive constant. The values of fitting coefficients a, b, +c, and d of the above mentioned formulas can be found in their respective Refs. [12–17]. +On account of the importance of angular momentum (l) as mentioned above, in the present +work, as the first step we have modified these formulas by adding only l dependent term (l(l + 1)), +where l is the minimum angular momentum of cluster particle, which is obtained by following +selection rules: +l = + + + + + + + +△j +for even △j and πi = πf +△j + 1 +for even △j and πi ̸= πf +△j +for odd △j and πi ̸= πf +△j + 1 +for odd △j and πi = πf +(7) +here, △j = |jp−jd−jc| with jp, πi, are the spin and parity values of the parent nucleus, respectively. +jd is the spin of the daughter nucleus. πf = (πd)(πc), in which, πd and πc are the parities of the +daughter nucleus and cluster, respectively. For the purpose of fitting, the data of spin and parity are +taken from NUBASE2020 [36]. In the next step, the formulas are also modified by adding isospin +I(= (N − Z)/A) dependent term (I(I + 1)). The accuracy and need of addition of different terms +belong to the modified formulas are checked by χ2 per degree of freedom (χ2) and RMSE values +for various versions, which are listed in Table 1 and calculated by using the following relations: +χ2 = +1 +Nnucl − Np +Nnucl +� +i=1 +� +log T i +T h. +T i +Exp. +�2 +(8) +3 + +RMSE = +� +� +� +� +1 +Nnucl +Nnucl +� +i=1 +� +log T i +T h. +T i +Exp. +�2 +(9) +where, Nnucl is the total number of nuclei (data) and Np is the number of degree of freedom (or +no. of coefficients). T i +Exp. and T i +T h. are the experimental and theoretical values of half-lives for ith +data point, respectively. +Table 1: The χ2 and RMSE of various versions of BKAG, RenA, Horoi, NRDX, UDL, and UNIV formulas for 61 +cluster decay data. +Formula +BKAG +RenA +Horoi +NRDX +UDL +UNIV +χ2 +RMSE +χ2 +RMSE +χ2 +RMSE +χ2 +RMSE +χ2 +RMSE +χ2 +RMSE +Original +1.01 +0.98 +1.10 +0.95 +1.45 +1.16 +0.85 +0.90 +1.88 +1.34 +0.87 +0.91 +With l term only +0.66 +0.78 +0.92 +0.93 +0.76 +0.84 +0.66 +0.78 +0.51 +0.69 +0.65 +0.78 +With l and I terms +0.44 +0.63 +0.68 +0.79 +0.77 +0.83 +0.66 +0.77 +0.49 +0.67 +0.67 +0.77 +The investigation of addition of different terms leads to the following conclusion from Table +1: (i) the addition of l-dependent term which reflects the hindrance effect of centrifugal barrier, +significantly reduces χ2 and RMSE for all the considered six formulas, (ii) whereas, the addition of +I-dependent term minimises χ2 and RMSE values only for BKAG and RenA formulas. As a result, +the final versions of these modified formulas adopted in the present article are given by: +log10T MBKAG +1/2 +(sec.) = [aAc(Ad − Ac)/A + bZc(Zd − Zc)/Z]Q−1/2 + cl(l + 1) + dI(I + 1) + e (10) +log10T MRenA +1/2 +(sec.) = aZdZcQ−1/2 + bZdZc + cl(l + 1) + dI(I + 1) + e +(11) +log10T MHoroi +1/2 +(sec.) = (a√µ + b)[(ZcZd)0.607Q−1/2 − 7] + (c√µ + d) + el(l + 1) +(12) +log10T MNRDX +1/2 +(sec.) += +aZcZd +� µ +Q + b√µ(ZcZd)1/2 + cl(l + 1) + d +(13) +log10T MUDL +1/2 +(sec.) += +aZcZd +� µ +Q + b[µZcZd(Ac +1/3 + Ad +1/3)]1/2 + cl(l + 1) + d +(14) +log10T MUNIV +1/2 +(sec.) += +−logP − log10S + cl(l + 1) + d +(15) +The coefficients a, b, c, d, and e of these modified formulas are mentioned in Table 2. +3. Results and discussions +To ascertain the impact on accuracy for the estimation of half-lives of cluster decay by the +addition of the above mentioned terms, we have plotted the ratio of decay widths WExp./WT h. = +log10T T h. +1/2 /log10T Exp. +1/2 +as a function of A for our six modified formulas (MBKAG, MRenA, MHoroi, +4 + +Table 2: The coefficients of MBKAG, MRenA, MHoroi, MNRDX, MUDL, and MUNIV formulas proposed in the +present work. +Formula +a +b +c +d +e +MBKAG +6.5279 +89.2684 +0.0798 +70.0439 +-100.4122 +MRenA +1.2947 +-0.0423 +0.0771 +89.9255 +-101.5076 +MHoroi +10.1451 +-23.1954 +4.4835 +-10.9094 +0.0567 +MNRDX +0.3590 +-1.0063 +0.0634 +-18.8444 +- +MUDL +0.3564 +-0.3199 +0.0737 +-24.8301 +- +MUNIV +0.2369 +0.6104 +0.0648 +-23.7267 +- +MNRDX, MUDL, and MUNIV) along with their original versions in Fig. 1. Most of the points +corresponding to our modified formulas (red diamonds) are between half order of magnitude while +the points corresponding to the original formulas (blue triangles) are somewhat widely scattered, +which indicate the improvement for the estimation of half-lives of cluster decay after the addition +of angular momentum (l) or isospin-dependent (I = (N − Z)/A) or both terms. +0.8 +0.9 +1.0 +1.1 +0.8 +0.9 +1.0 +1.1 +220 +225 +230 +235 +240 +0.8 +0.9 +1.0 +1.1 +220 +225 +230 +235 +240 + BKAG (RMSE: 0.98) + MBKAG (RMSE: 0.63) + + RenA (RMSE: 0.95) + MRenA (RMSE: 0.79) + + + Horoi (RMSE: 1.16) + MHoroi +(RMSE: 0.84 +) +W +Exp. +/W +Th. +=log +10 +T +Th. +1/2 +/log +10 +T +Exp. +1/2 + + + NRDX (RMSE: 0.90) + MNRDX (RMSE: 0.79) + + + UDL (RMSE: 1.34) + MUDL (RMSE: 0.69) + + + UNIV (RMSE: 0.91) + MUNIV (RMSE: 0.78) +A +Figure 1: (Colour online) Ratio of experimental to theoretical decay widths WExp./WT h. = log10T T h. +1/2 /log10T Exp. +1/2 +for the comparison of our six modified formulas with their respective original versions by using 61 cluster emission +data. The RMSE values are also indicated in front of the name of the respective formula. +For the comparison among our modified formulas with a few of latest fitted/modified formulas +[26–28] for cluster decay half-lives, we have calculated some other statistical parameters such as +standard deviation (σ), uncertainty (u), average deviation factor (x), and mean deviation δ for +61 experimentally known cluster decay half-lives [7, 9, 26, 29]. All these statistical parameters for +5 + +these formulas are mentioned in Table 3. These statistical parameters are defined as: +σ = +� +� +� +� +1 +Nnucl − 1 +Nnucl +� +i=1 +� +log T i +T h. +T i +Exp. +�2 +(16) +u = +� +� +� +� +1 +Nnucl(Nnucl − 1) +Nnucl +� +i=1 +� +log T i +T h. +T i +Exp. +− µ +�2 +(17) +x = +1 +Nnucl +Nnucl +� +i=1 +� +|logT i +Exp. − logT i +T h.| +logT i +Exp. +� +(18) +δ = +1 +Nnucl +Nnucl +� +i=1 +�����log T i +T h. +T i +Exp. +����� +(19) +The terms in above equations are already defined in Eqns. (8) and (9). µ in Eqn. (17) refers to +the mean of full data set. +Table 3: Comparison of MBKAG, MRenA, MHoroi, MNRDX, MUDL, and MUNIV formulas with few others for- +mulas. +Formula +σ +u +x +δ +MBKAG (Present Work) +0.64 +0.08 +0.02 +0.51 +MRenA(Present Work) +0.80 +0.10 +0.02 +0.62 +MHoroi (Present Work) +0.84 +0.11 +0.03 +0.66 +MNRDX (Present Work) +0.79 +0.10 +0.02 +0.60 +MUDL (Present Work) +0.70 +0.09 +0.03 +0.53 +MUNIV (Present Work) +0.79 +0.10 +0.03 +0.59 +New UDL [26] +0.81 +0.10 +0.03 +0.68 +IUF [27] +0.84 +0.11 +0.03 +0.64 +ISEF [28] +0.93 +0.12 +0.04 +0.76 +It is clear from Table 3 that the isospin (only for BKAG and RenA) and angular momentum +play a crucial role to improve the cluster decay formulas and result in lesser statistical parameters +σ, u, x, and δ for the modified formulas introduced in the present work, as compared with a few of +the latest fitted/modified formulas (new UDL, IUF, and ISEF formulas) for the cluster decay. It is +to be noted that among all the modified formulas, MBKAG formula renders more accurate half-life +while compared through all the statistical parameters. Hence, MBKAG formula can be employed +to predict the more precise half-lives of cluster decay and the probable decay emission. With this +in view, the possibility of cluster emission from the experimentally known trans-lead (86≤Z≤96) +isotopes is probed by considering the daughter nuclei near the proton shell closure i.e., the emission +of a cluster is chosen in such a way that the proton number of daughter nucleus Zd is close to 82 +(Pb). +Before predicting possibilities of new cluster decays in trans-lead regions, we first calculate +the half-lives of experimentally known cluster decay using the MBKAG formula which are listed +in Table 4. We have taken only one parent-cluster combination out of 61 experimental data of +cluster decay, to compare with α-decay half-lives. For the α-decay half-lives, we have used the +NMHF (new modified Horoi formula) whose accuracy in determining the half-lives has already +6 + +Table 4: The calculated logarithmic half-lives using MBKAG formula together with experimental values [7, 9, 26, 29] +for cluster decay. The α-decay half-lives are calculated by using NMHF formula [20]. BR refers for branching ratios +calculated by using Eqn. (20). Q and Qα are the disintegration energies for cluster decay and α-decay, taken from +Refs.[7, 9, 26, 29] and AME2020 [37], respectively. For the l values, spin and parity of parent, daughter, and cluster +nuclei are used from NUBASE2020 [36]. +Parent +Daughter +Emitted +Q +Qα +l +log10T1/2(sec.) +BRExp. +BR +nucleus +nucleus +cluster +(MeV) +(MeV) +Exp. +MBKAG +NMHF +(Cluster) +(α) +221Fr +207Tl +14C +31.28 +6.46 +3 +14.52 +15.44 +2.96 +-11.56 +-12.48 +221Ra +207Pb +14C +32.39 +6.88 +3 +13.39 +13.01 +1.74 +-11.65 +-11.27 +222Ra +208Pb +14C +33.05 +6.68 +0 +11.22 +11.46 +2.32 +-8.90 +-9.14 +223Ra +209Pb +14C +31.85 +5.98 +4 +15.25 +15.18 +5.17 +-10.08 +-10.01 +223Ac +209Bi +14C +33.06 +6.78 +2 +12.60 +11.54 +2.38 +-10.22 +-9.16 +223Ac +208Pb +15N +39.47 +6.78 +2 +14.76 +14.36 +2.38 +-12.38 +-11.98 +224Ra +210Pb +14C +30.54 +5.79 +0 +15.90 +15.99 +5.87 +-10.03 +-10.12 +225Ac +211Bi +14C +30.48 +5.94 +4 +17.16 +17.30 +5.70 +-11.46 +-11.60 +226Ra +212Pb +14C +28.21 +4.87 +0 +21.19 +20.68 +10.52 +-10.67 +-10.16 +226Th +212Po +14C +30.67 +6.45 +0 +15.30 +15.02 +3.79 +-11.51 +-11.24 +228Th +208Pb +20O +44.72 +5.52 +0 +20.72 +21.34 +7.82 +-12.90 +-13.52 +230Th +206Hg +24Ne +57.78 +4.77 +0 +24.64 +25.78 +11.91 +-12.73 +-13.87 +230U +208Pb +22Ne +61.40 +5.99 +0 +19.57 +20.38 +6.32 +-13.25 +-14.06 +231Pa +207Tl +24Ne +60.42 +5.15 +1 +23.23 +23.33 +10.11 +-13.12 +-13.22 +232Th +208Hg +24Ne +55.62 +4.08 +0 +29.20 +28.56 +16.63 +-12.57 +-11.94 +232Th +206Hg +26Ne +55.97 +4.08 +0 +29.20 +29.21 +16.63 +-12.57 +-12.59 +232U +208Pb +24Ne +62.31 +5.41 +0 +21.06 +21.32 +9.08 +-11.98 +-12.24 +232U +204Hg +28Mg +74.32 +5.41 +0 +22.26 +25.01 +9.08 +-13.18 +-15.93 +233U +209Pb +24Ne +60.50 +4.91 +2 +24.82 +23.71 +11.86 +-12.96 +-11.85 +233U +208Pb +25Ne +60.75 +4.91 +2 +24.82 +23.97 +11.86 +-12.96 +-12.12 +233U +205Hg +28Mg +74.24 +4.91 +3 +27.59 +26.38 +11.86 +-15.73 +-14.53 +234U +210Pb +24Ne +58.84 +4.86 +0 +25.88 +25.06 +12.19 +-13.69 +-12.87 +234U +208Pb +26Ne +59.47 +4.86 +0 +25.88 +25.46 +12.19 +-13.69 +-13.27 +234U +206Hg +28Mg +74.13 +4.86 +0 +25.14 +25.86 +12.19 +-12.95 +-13.67 +235U +211Pb +24Ne +57.36 +4.68 +1 +27.42 +26.95 +13.37 +-14.05 +-13.58 +235U +210Pb +25Ne +57.83 +4.68 +3 +27.42 +27.81 +13.37 +-14.05 +-14.43 +235U +207Hg +28Mg +72.20 +4.68 +1 +28.09 +27.81 +13.37 +-14.72 +-14.44 +235U +206Hg +29Mg +72.61 +4.68 +3 +28.09 +28.70 +13.37 +-14.72 +-15.32 +236U +212Pb +24Ne +55.96 +4.57 +0 +25.90 +28.50 +14.04 +-11.86 +-14.46 +236U +210Pb +26Ne +56.75 +4.57 +0 +25.90 +28.73 +14.04 +-11.86 +-14.69 +236U +208Hg +28Mg +71.69 +4.57 +0 +27.58 +28.40 +14.04 +-13.54 +-14.36 +236U +206Hg +30Mg +72.51 +4.57 +0 +27.58 +28.56 +14.04 +-13.54 +-14.52 +236Pu +208Pb +28Mg +79.67 +5.87 +0 +21.52 +21.72 +7.63 +-13.89 +-14.09 +237Np +207Tl +30Mg +75.02 +4.96 +2 +26.93 +27.03 +12.09 +-14.84 +-14.94 +238Pu +210Pb +28Mg +75.93 +5.59 +0 +25.70 +24.98 +8.98 +-16.72 +-16.00 +238Pu +208Pb +30Mg +77.03 +5.59 +0 +25.70 +24.97 +8.98 +-16.72 +-15.99 +238Pu +206Hg +32Si +91.21 +5.59 +0 +25.30 +25.27 +8.98 +-16.32 +-16.28 +240Pu +206Hg +34Si +90.95 +5.26 +0 +25.62 +26.74 +10.78 +-14.84 +-15.96 +241Am +207Tl +34Si +93.84 +5.64 +3 +25.26 +25.94 +9.21 +-16.05 +-16.73 +242Cm +208Pb +34Si +96.53 +6.22 +0 +23.15 +23.39 +6.84 +-16.31 +-16.55 +7 + +been demonstrated in Ref. [20]. The first, second, and third columns of Table 4 show the parent, +daughter, and cluster nuclei, respectively. Next two columns represent the disintegration energies +of cluster decay and α-decay taken from Refs. [7, 9, 26, 29] and from AME2020 [37], respectively. +The sixth column lists angular momentum taken away by cluster particle after emission which is +calculated by using selection rules explained in the Eqn. (7). We have calculated logarithmic half- +lives of cluster decay (using Eqn. (10)), tabulated them in the eighth column, and compared these +results with the experimental results (presented in the seventh column). It is clear from the Table 4 +that calculated half-lives of cluster emission by using the MBKAG formula (present work) are very +close to experimental results. Branching ratio (BR) which quantifies comparison between cluster +decay to the α-decay and is defined as the ratio of α-decay half-life (listed in the ninth column) to +the cluster decay half-life as below: +BR = log10bc = log10(λc/λα) = log10(Tα/Tc) +(20) +where, λα and λc are referred as the decay constants of α-decay and cluster emission, respectively. +The calculated branching ratios are shown in the last column which are indeed close to experimental +branching ratios [7, 9, 26, 29] (presented in the second last column). In fact, an excellent match of +half-lives of almost all mentioned clusters in Table 4 validates the pertinence of MBKAG formula. +Furthermore, one can note that the experimental cluster decay half-life goes maximum nearly upto +1030 sec., therefore, it can be reasoned out that the clusters with a half-life less than 1030 sec. +seemingly be of experimental interest. +In the next step of our study, we have utilized the degree of accuracy of MBKAG formula, +as exhibited in Table 4, to predict the logarithmic half-lives of unknown cluster emissions in the +trans-lead region. For this estimation, the Q-values are calculated by the following relation: +Q(MeV ) = B.E.(d) + B.E.(c) − B.E.(p) + k[Zǫ +p − Zǫ +d] +(21) +where, the term k[Zǫ +p − Zǫ +d] indicates screening effect caused by the surrounding electrons around +the nuclei [38] with k=8.7 eV [8.7 × 10−6MeV] and ǫ=2.517 for Z (proton number) ≥ 60, and +k=13.6 eV [13.6 × 10−6MeV] and ǫ =2.408 for Z < 60 have been deducted from the data shown by +Huang et al. [39]. For accurate prediction of theoretical Q-values, we have selected an effective and +reliable possible treatment among various theoretical approaches viz. relativistic mean-field theory +(RMF) [32, 40–44], Finite Range Droplet Model (FRDM) [31], nonrelativistic Skyrme Hartree-Fock- +Bogoliubov (HFB) [33], and Weizsacker-Skyrme mass model (WS4) [30]. From these approaches, +we have calculated RMSE, listed in Table 5, for the known 121 Q-values related to cluster emissions +[7, 9, 26, 29]. Table 5 establishes that WS4 mass model provides an excellent agreement with the +minimum RMSE compared to all other considered theoretical approaches and hence justifies the +calculation of Q-values for cluster emission by taking binding energies (for daughter(d), cluster(c), +and parent(p) nuclei) from this mass model [30]. +Table 5: RMSE of various mass models for Q-value data for cluster emission. +Theory +RMSE +WS4 +0.43 +FRDM +0.78 +HFB +1.17 +RMF +3.61 +8 + +120 +125 +130 +135 +140 +0 +20 +40 +60 +80 +100 +120 +125 +130 +135 +20 +40 +60 +80 +100 +120 +115 +120 +125 +130 +135 +140 +20 +40 +60 +80 +110 +115 +120 +125 +130 +135 +140 +20 +40 +60 +80 +100 +115 +120 +125 +130 +135 +20 +40 +60 +80 +100 +115 +120 +125 +130 +135 +140 +20 +40 +60 +110 +115 +120 +125 +130 +135 +20 +40 +60 +115 +120 +125 +130 +135 +140 +20 +40 +60 +80 +100 +115 +120 +125 +130 +135 +140 +20 +40 +60 +80 +110 +115 +120 +125 +130 +135 +140 +20 +40 +60 +80 +100 +115 +120 +125 +130 +135 +140 +145 +20 +40 +60 +80 +100 + +8 +Be + +9 +Be + +10 +Be +Rn-Isotopes +N=126 +N=126 +N=126 +N=126 + +10 +B +Fr-Isotopes + +11 +B + +12 +B + +13 +B + +14 +B + +13 +C + 14 +C + +15 +C + +16 +C +Ra-Isotopes +log +10 +T +1/2 + (sec.) + +15 +O + +16 +O + +17 +O + +18 +O + +19 +O + +20 +O + +21 +O + +22 +O + +23 +O +Th-Isotopes +N=126 +N=126 +N=126 +N=126 +N=126 +N=126 +N=126 + +13 +N + +14 +N + +15 +N +Ac-Isotopes + +16 +N + +17 +N + +18 +N + +19 +F + +20 +F + +21 +F + +22 +F + +23 +F +Pa-Isotopes + +22 +Ne + +23 +Ne +U-Isotopes + +24 +Ne + +25 +Ne + +26 +Ne + +27 +Ne + +21 +Na + +22 +Na + +23 +Na + +24 +Na + +25 +Na + +26 +Na + +27 +Na + +28 +Na +Np-Isotopes + +23 +Mg + +24 +Mg + +25 +Mg + +26 +Mg + +27 +Mg + +28 +Mg + +29 +Mg +Pu-Isotopes + +24 +Al + +25 +Al + +26 +Al + +27 +Al + +28 +Al + +29 +Al + +30 +Al + +31 +Al + +32 +Al +Am-Isotopes +N +d + +26 +Si + +27 +Si + +28 +Si + +29 +Si + +30 +Si + +31 +Si + +32 +Si + +33 +Si +Cm-Isotopes +Figure 2: (Colour online) Variation of half-lives of various cluster emissions from experimentally known isotopes of +trans-lead nuclei (86≤Z≤96) as a function of neutron number of daughter nuclei (considering proton number Zd=82). +These half-lives are calculated by using MBKAG formula and the Q-values are taken from the WS4 mass model[30]. +9 + +After the selection of efficacious empirical formula as well as the theoretical Q-values, we have +chosen all the parent-cluster combinations for this extensive study to find the possible clusters emit- +ted from 211−231Rn, 213−226Fr, 214−235Ra, 215−233Ac, 216−237Th, 218−241Pa, 228−243U, 226−245Np, +226−245Pu, 227−248Am, and 231−252Cm isotopes leading to 208Pb daughter (doubly magic) and neigh- +bouring nuclei. We have plotted our results (up to T=10100 sec.) in Fig. 2 where the minima of +log10T1/2 in several panels (Ra-isotopes to U-isotopes) correspond to 208Pb daughter i.e., doubly +magic (Z=82, N=126) or near to it. These minima provide us the most probable clusters emitted +from the respective isotopes. However, the probability of cluster emission always competes with +α-decay which is quantified by branching ratio as we have discussed in Eqn. (20). The limit of +experimental branching ratio related to α-decay is around BR = −17 as can be seen in Table 4 +and also explained by Poenaru et al. [45]. Accordingly, cluster emission emerges more probable +if BR ≥ −17: the criteria for the listed probable clusters in Table 6. These clusters are selected +from the Fig. 2 for the particular isotopic chain of parent trans-lead nuclei 211−231Rn, 213−226Fr, +214−235Ra, 215−233Ac, 216−237Th, 218−241Pa, and 228−243U. Most of our results are within the ex- +perimental reach and also in close match with the recent predictions of Refs. [46–48]. +Table 6: The calculated logarithmic half-lives and branching ratios of probable clusters emitted from various isotopes +of trans-lead nuclei (86≤Z≤96). Cluster decay and α-decay half-lives are calculated by using MBKAG formula (Eqn. +10) and NMHF formula [20], respectively. Disintegration energies (Q-values) for the cluster decay and α-decay are +taken from WS4 mass model [30] and AME2020 [37], respectively. +For the l values, spin and parity of parent, +daughter, and cluster nuclei are used from NUBASE2020 [36]. +Parent +Daughter +Emitted +Q +Qα +l +log10T1/2(sec.) +BR +nucleus +nucleus +cluster +(MeV) +(MeV) +MBKAG +NMHF +(Cluster) +(α) +216Rn +208Pb +8Be +17.13 +8.20 +0 +6.65 +-2.84 +-9.49 +222Fr +207Pb +14B +21.56 +5.85 +0 +20.23 +5.24 +-14.99 +221Ra +208Pb +13C +31.70 +6.88 +3 +13.13 +1.74 +-11.39 +223Ra +208Pb +15C +29.22 +5.98 +2 +19.15 +5.17 +-13.98 +222Ac +208Pb +14N +35.64 +7.14 +1 +17.93 +1.03 +-16.90 +222Ac +207Pb +15N +39.10 +7.14 +1 +14.09 +1.03 +-13.06 +224Ac +208Pb +16N +36.43 +6.33 +2 +19.44 +3.99 +-15.45 +225Ac +208Pb +17N +35.64 +5.94 +2 +21.68 +5.70 +-15.98 +224Th +208Pb +16O +46.63 +7.30 +0 +15.11 +0.81 +-14.30 +225Th +208Pb +17O +45.02 +6.92 +2 +18.39 +2.22 +-16.17 +226Th +208Pb +18O +45.88 +6.45 +0 +17.98 +3.79 +-14.19 +227Th +208Pb +19O +44.36 +6.15 +2 +21.19 +5.16 +-16.03 +228Th +208Pb +20O +44.87 +5.52 +0 +21.12 +7.96 +-13.16 +229Th +208Pb +21O +43.41 +5.17 +0 +23.84 +9.77 +-14.37 +230Th +208Pb +22O +43.48 +4.77 +0 +24.73 +11.91 +-12.82 +231Th +208Pb +23O +41.08 +4.21 +2 +29.26 +15.75 +-13.51 +228Pa +208Pb +20F +50.90 +6.26 +2 +22.42 +5.13 +-17.29 +229Pa +208Pb +21F +51.83 +5.84 +0 +21.94 +6.74 +-15.20 +231Pa +208Pb +23F +52.01 +5.15 +1 +23.75 +10.11 +-13.64 +231U +208Pb +23Ne +60.99 +5.58 +0 +21.55 +8.53 +-13.02 +231U +206Pb +25Ne +59.91 +5.58 +2 +23.95 +8.53 +-15.42 +On the other side, in the panels from Np-isotopes to Cm-isotopes in Fig. 2, in-spite of a clear +minima, there is incessantly some probability of emission of clusters since many of the clusters own +half-lives less than 1030 sec. (experimental limit of half-lives of cluster emissions). For examples, +10 + +21Na from 226−229Np, 22Na from 226−230Np, 23Na from 226−233Np, 24Na from 226−234Np, 25,27Na +from 226−237Np, 26Na from 226−236Np and 28Na from 224−236Np. Similarly, some possible clusters +(Mg-isotopes) emitted from various Pu-isotopes (Zp=94) are 23Mg from 226−231Pu, 24,25Mg from +226−235Np, 26Mg from 226−238Np, 27Mg from 226−239Np, and 28,29Mg from 226−241Np. Among Am- +isotopes the potential clusters are 24Al from 227−230Am, 25Al from 227−233Am, 26Al from 227−236Am, +27Al from 227−239Am, 28Al from 227−240Am, 29Al from 227−241Am, and 30−32Al from 227−242Am as +well as 26−33Si from the 231−252Cm isotopes. In the emission of odd mass clusters, the odd-even +staggering is noticeable in Fig. 2 which is usually attributed to the existence of nucleonic pairing +correlations [49]. The above-mentioned detailed study about favorable clusters having T1/2 < 1030 +sec. is expected to be certainly useful for future experimental inputs. +4. Conclusions +Several empirical formulas are investigated by adding angular momentum and isospin depen- +dence. Their modified versions are turned into MBKAG, MRenA, MHoroi, MNRDX, MUDL, and +MUNIV formulas. Experimental data of a total of 61 nuclei have been utilized for fitting which of- +fers improved results of all the modified formulas while compared to their earlier versions. Among +these six modified formulas, after comparison of several statistical parameters the MBKAG for- +mula is found most precise which is used to examine cluster decay half-lives for trans-lead region: +211−231Rn, 213−226Fr, 214−235Ra, 215−233Ac, 216−237Th, 218−241Pa, 228−243U, 226−245Np, 226−245Pu, +227−248Am, and 231−252Cm isotopes leading to 208Pb daughter (doubly magic) and neighbouring +nuclei. We have found the considerable probability of emission of various isotopes of Be, B, C, N, +O, F, Ne, Na, Mg, and Si from above mentioned trans-lead nuclei, respectively, and many of them +are found to be favorable for the measurement (T1/2 < 1030 sec.). This study reveals that doubly +magic daughter nuclei play a crucial role in the cluster decay process and could serve as a stimulus +to the experiments eyeing on cluster radioactivity. +5. Acknowledgement +AJ and GS acknowledge the support provided by SERB (DST), Govt. of India under CRG/2019/001851 +and SIR/2022/000566, respectively. +References +[1] A. Sandulescu, D.N. Poenaru, W. Greiner, Sov. J. Part. Nucl. II 11 (1980) 528. +[2] R.K. Gupta, W. Greiner, Int. J. Mod. Phys. E 03 (1994) 335. +[3] H.J. Rose, G.A. Jones, Nature (London) 307 (1984) 245. +[4] S. Kumar, et al., J. Phys. G: Nucl. Part. Phys. 29 (2003) 625. +[5] S. Kumar, R. Rani, R. Kumar, J. Phys. G: Nucl. Part. Phys. 36 (2009) 015110. +[6] R.K. Gupta, et al., Phys. Rev. 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A 41 (2009) 109 . +13 + diff --git a/z9AyT4oBgHgl3EQfbfcG/content/tmp_files/load_file.txt b/z9AyT4oBgHgl3EQfbfcG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..65deb9eef97f387224ba6783fa5918ec234951ae --- /dev/null +++ b/z9AyT4oBgHgl3EQfbfcG/content/tmp_files/load_file.txt @@ -0,0 +1,1416 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf,len=1415 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='00261v1 [nucl-th] 31 Dec 2022 Cluster radioactivity in trans-lead region: A systematic study with modified empirical formulas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Jaina,b,c, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Sharmad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Jaina, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Deegwale, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Saxenac,f aDepartment of Physics, School of Basic Sciences, Manipal University Jaipur, Jaipur-303007, India bDepartment of Physics, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Jain Subodh P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' (Autonomous) College, Jaipur-302004, India cDepartment of Physics (H&S), Govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Women Engineering College, Ajmer-305002, India dGovt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Polytechnic College, Rajsamand-313324, India eGovt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Women Engineering College, Ajmer-305002, India fDepartment of Physics, Faculty of Science, University of Zagreb, Bijeni˘cka c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 32, 10000 Zagreb, Croatia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Abstract The possibility of cluster emission from trans-lead (86≤Z≤96) region of periodic chart has been explored comprehensively by employing few empirical formulas which are modified by adding an- gular momentum (l) or isospin-dependent (I = (N − Z)/A) or both terms for the calculation of cluster decay half-lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' These modified versions of the formulas are found with lesser χ2 per degree of freedom and root mean-square error, in addition to the smaller values of some other statistical parameters, while compared to their corresponding old versions on available 61 experimental data of cluster radioactivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' By applying the modified version of the formula given by Balasubramaniam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [PRC 70 (2004) 017301], the most accurate formula among these, half-lives of several clusters i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' isotopes of Be, B, C, N, O, F, Ne, Na, Mg, and Si are predicted systematically for the several isotopes in the trans-lead region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' The contest of cluster emission with α-decay has been investigated in form of branching ratio which brings several potential cluster emissions into the probable decay modes of these nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' The accurate prediction of half-lives of such clusters is expected to be crucial for the future experimental observations where α-decay is observed dominantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Keywords: Cluster decay, Trans-lead Nuclei, Empirical formulas, α-decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Introduction In 1980, Sandulescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [1] firstly predicted a new type of radioactivity: cluster radioactivity, which was based on fragmentation theory, where fusion and fission reaction valleys were generated by the shell closure effect [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Later in 1984, Rose and Jones experimentally proved the existence of this new type of exotic decay [3], in which 14C decays from actinide parent nucleus 223Ra and forms a stable doubly magic (Z=82, N=126) nucleus 208Pb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Till now, many clusters decays from light to heavy clusters (14C to 32Si) have been observed from various trans-lead nuclei (Fr, Ra, Ac, Pa, Th, U, Pu, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=') resulting the corresponding daughter nuclei as magic nuclei (Z=82) or neighboring ones (Z=80, 81, and 83), which indicate the importance of shell and pairing effects in cluster radioactivity [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' These clusters are observed with long half-lives (T1/2) in the range 1011-1030 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Theoretically, the half-lives of cluster emissions are predicted using various models such as uni- fied fission model (UFM) [8], generalised liquid drop model (GLDM) [9], super-asymmetric fission Preprint submitted to Nuclear Physics A January 3, 2023 model (SAFM) [10], preformation cluster model (PCM) [11], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Cluster decay half-lives are also calculated by using various semi-empirical formulas such as (i) the empirical relation suggested by Balasubramaniam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' (BKAG formula) for cluster decay half-lives with only three parame- ters [12], (ii) the empirical relation suggested by Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' (RenA formula) using a microscopic density-dependent cluster model with the re-normalized M3Y nucleon-nucleon interaction [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Con- comitantly, based on experimental observations about the characteristics of exotic cluster decays, scaling law proposed by Horoi [14] in which logarithmic half-life is proportional to scaling variable (ZcZd)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='6/√Q and also proportional to √µ, where µ is the reduced mass of cluster and daughter nuclei which was followed by another semi-empirical formula (NRDX), proposed by Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [15] considering WKB barrier penetration probability with some approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' In 2009, Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' introduced universal decay law (UDL) [16] that originates from the mechanism of charged particle decay and R-matrix for all sort of decays of clusters, which includes monopole radioactive decays as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Poenaru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [17] plotted a universal curve (UNIV) which is found to be a straight line for cluster decay and α-decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' All the above-mentioned formulas have been fitted to the available experimental data without considering the dependence of half-lives on angular momentum taken away by the cluster: expected to be crucial alike to the α-decay [18] to delineate all sets of experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' The importance of angular momentum on the α-decay half-lives has already been established in a few of our recent works [19, 20] which has invoked us to probe similar dependence on the cluster decay half-lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' In addition to this, isospin (I = (N − Z)/A) of parent nucleus is found to be pivotal for the case of α-decay in heavy and superheavy nuclei [20–25] pointing towards its significance in terms of cluster decay as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Considering these two effects together, modified UDL formula (new UDL) by Soylu and Qi [26], and improved NRDX formula (named as improved unified formula (IUF)) by Ismail et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [27] have explained recently that angular momentum and isospin are indeed crucial quantities in determining the cluster decay half-lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Importance of isospin effect is also probed by improving semi-empirical formula (ISEM) for the cluster radioactivity in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' In this article, we have modified the BKAG [12], RenA [13], Horoi [14], NRDX [15], UDL [16], and UNIV [17] formulas by investigating the effect of centrifugal barrier and isospin terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' These six modified formulas are fitted by using 61 experimental cluster decay data [7, 9, 26, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' The comparison of RMSE (root mean square error) between the older and modified version manifestly shows the significance of inclusion of angular momentum and isospin-dependent terms in cluster emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Furthermore, one of the modified formulas i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' MBKAG formula (emerged with least RMSE) is employed to calculate the cluster decay half-lives for various cluster emissions like isotopes of Be, B, C, N, O, F, Ne, Na, Mg, and Si in trans-lead region (86≤Z≤96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' For these theoretical estimates, the requirement of disintegration energy (Q-value) is tested by 121 available experimental Q-values [7, 9, 26, 29] from various mass models [30–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Consequently, various potential clusters are proposed from trans-lead region along with their accurate estimation of half-lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Formalism In 2004, Balasubramaniam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' fitted a formula (BKAG) [12] for cluster decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' In the course of that year, Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' established a formula [13] that can be treated as a natural extension of the Geiger-Nuttall law [34] as well as the Viola-Seaborg formula [35] from simple α-decay to complex cluster radioactivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' In the same year, Horoi also suggested an independent model for α-decay which was generalized for cluster emission [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' In 2008, Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' established NRDX semi-empirical formula for the calculation of half-lives of α and cluster decays [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Afterwards, Qi 2 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' has introduced universal decay law (UDL) [16] which is widely used by many authors for the estimation of half-lives of cluster radioactivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' In 2011, Poenaru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' fitted UNIV formula [17] and represented a single line of the universal curve on the graph for α-decay and cluster decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' The original versions of these formulas are mentioned below: log10T BKAG 1/2 (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=') = [aAc(Ad − Ac)/A + bZc(Zd − Zc)/Z]Q−1/2 + c (1) log10T RenA 1/2 (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=') = aZdZcQ−1/2 + bZdZc + c (2) log10T Horoi 1/2 (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=') = (a√µ + b)[(ZcZd)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='607Q−1/2 − 7] + (c√µ + d) (3) log10T NRDX 1/2 (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=') = aZcZd � µ Q + b√µ(ZcZd)1/2 + c (4) log10T UDL 1/2 (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=') = aZcZd � µ Q + b[µZcZd(Ac 1/3 + Ad 1/3)]1/2 + c (5) log10T UNIV 1/2 (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=') = −logP + log10S − [log10(ln2) − log10υ] (6) In the above-mentioned formulas Ad, Ac and Zd, Zc denote the mass numbers and atomic numbers of the daughter nucleus and cluster, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Q (in MeV) is the energy released in cluster decay, and µ = AdAc/(Ad + Ac) is the reduced mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' In Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' (6), −logP is determined by a(µZcZdRb)1/2[arccos√r − � r(1 − r)], r = Ra/Rb with Ra = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='2249(Ac 1/3 + Ad 1/3) fm, Rb = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='43998ZdZc/Q fm, and the logarithmic form of preformation factor is given by log10S = −b(Ac−1) along with [log10(ln2) − log10υ] = d is the additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' The values of fitting coefficients a, b, c, and d of the above mentioned formulas can be found in their respective Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [12–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' On account of the importance of angular momentum (l) as mentioned above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' in the present work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' as the first step we have modified these formulas by adding only l dependent term (l(l + 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' where l is the minimum angular momentum of cluster particle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' which is obtained by following selection rules: l = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 △j for even △j and πi = πf △j + 1 for even △j and πi ̸= πf △j for odd △j and πi ̸= πf △j + 1 for odd △j and πi = πf (7) here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' △j = |jp−jd−jc| with jp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' are the spin and parity values of the parent nucleus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' jd is the spin of the daughter nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' πf = (πd)(πc), in which, πd and πc are the parities of the daughter nucleus and cluster, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' For the purpose of fitting, the data of spin and parity are taken from NUBASE2020 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' In the next step, the formulas are also modified by adding isospin I(= (N − Z)/A) dependent term (I(I + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' The accuracy and need of addition of different terms belong to the modified formulas are checked by χ2 per degree of freedom (χ2) and RMSE values for various versions, which are listed in Table 1 and calculated by using the following relations: χ2 = 1 Nnucl − Np Nnucl � i=1 � log T i T h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' T i Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' �2 (8) 3 RMSE = � � � � 1 Nnucl Nnucl � i=1 � log T i T h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' T i Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' �2 (9) where, Nnucl is the total number of nuclei (data) and Np is the number of degree of freedom (or no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' of coefficients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' T i Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' and T i T h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' are the experimental and theoretical values of half-lives for ith data point, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Table 1: The χ2 and RMSE of various versions of BKAG, RenA, Horoi, NRDX, UDL, and UNIV formulas for 61 cluster decay data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Formula BKAG RenA Horoi NRDX UDL UNIV χ2 RMSE χ2 RMSE χ2 RMSE χ2 RMSE χ2 RMSE χ2 RMSE Original 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='91 With l term only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='78 With l and I terms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='77 The investigation of addition of different terms leads to the following conclusion from Table 1: (i) the addition of l-dependent term which reflects the hindrance effect of centrifugal barrier, significantly reduces χ2 and RMSE for all the considered six formulas, (ii) whereas, the addition of I-dependent term minimises χ2 and RMSE values only for BKAG and RenA formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' As a result, the final versions of these modified formulas adopted in the present article are given by: log10T MBKAG 1/2 (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=') = [aAc(Ad − Ac)/A + bZc(Zd − Zc)/Z]Q−1/2 + cl(l + 1) + dI(I + 1) + e (10) log10T MRenA 1/2 (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=') = aZdZcQ−1/2 + bZdZc + cl(l + 1) + dI(I + 1) + e (11) log10T MHoroi 1/2 (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=') = (a√µ + b)[(ZcZd)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='607Q−1/2 − 7] + (c√µ + d) + el(l + 1) (12) log10T MNRDX 1/2 (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=') = aZcZd � µ Q + b√µ(ZcZd)1/2 + cl(l + 1) + d (13) log10T MUDL 1/2 (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=') = aZcZd � µ Q + b[µZcZd(Ac 1/3 + Ad 1/3)]1/2 + cl(l + 1) + d (14) log10T MUNIV 1/2 (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=') = −logP − log10S + cl(l + 1) + d (15) The coefficients a, b, c, d, and e of these modified formulas are mentioned in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Results and discussions To ascertain the impact on accuracy for the estimation of half-lives of cluster decay by the addition of the above mentioned terms, we have plotted the ratio of decay widths WExp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='/WT h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' = log10T T h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 1/2 /log10T Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 1/2 as a function of A for our six modified formulas (MBKAG, MRenA, MHoroi, 4 Table 2: The coefficients of MBKAG, MRenA, MHoroi, MNRDX, MUDL, and MUNIV formulas proposed in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Formula a b c d e MBKAG 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='5279 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='2684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='0798 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='0439 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='4122 MRenA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='2947 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='0423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='0771 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='9255 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='5076 MHoroi 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='1451 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='1954 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='4835 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='9094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='0567 MNRDX 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='3590 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='0063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='0634 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='8444 MUDL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='3564 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='3199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='0737 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='8301 MUNIV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='2369 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='6104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='0648 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='7267 MNRDX, MUDL, and MUNIV) along with their original versions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Most of the points corresponding to our modified formulas (red diamonds) are between half order of magnitude while the points corresponding to the original formulas (blue triangles) are somewhat widely scattered, which indicate the improvement for the estimation of half-lives of cluster decay after the addition of angular momentum (l) or isospin-dependent (I = (N − Z)/A) or both terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='1 220 225 230 235 240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='1 220 225 230 235 240 BKAG (RMSE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='98) MBKAG (RMSE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='63) RenA (RMSE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='95) MRenA (RMSE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='79) Horoi (RMSE: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='16) MHoroi (RMSE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='84 ) W Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' /W Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' =log 10 T Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 1/2 /log 10 T Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 1/2 NRDX (RMSE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='90) MNRDX (RMSE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='79) UDL (RMSE: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='34) MUDL (RMSE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='69) UNIV (RMSE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='91) MUNIV (RMSE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='78) A Figure 1: (Colour online) Ratio of experimental to theoretical decay widths WExp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='/WT h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' = log10T T h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 1/2 /log10T Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 1/2 for the comparison of our six modified formulas with their respective original versions by using 61 cluster emission data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' The RMSE values are also indicated in front of the name of the respective formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' For the comparison among our modified formulas with a few of latest fitted/modified formulas [26–28] for cluster decay half-lives, we have calculated some other statistical parameters such as standard deviation (σ), uncertainty (u), average deviation factor (x), and mean deviation δ for 61 experimentally known cluster decay half-lives [7, 9, 26, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' All these statistical parameters for 5 these formulas are mentioned in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' These statistical parameters are defined as: σ = � � � � 1 Nnucl − 1 Nnucl � i=1 � log T i T h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' T i Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' �2 (16) u = � � � � 1 Nnucl(Nnucl − 1) Nnucl � i=1 � log T i T h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' T i Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' − µ �2 (17) x = 1 Nnucl Nnucl � i=1 � |logT i Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' − logT i T h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='| logT i Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' � (18) δ = 1 Nnucl Nnucl � i=1 �����log T i T h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' T i Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' ����� (19) The terms in above equations are already defined in Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' (8) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' µ in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' (17) refers to the mean of full data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Table 3: Comparison of MBKAG, MRenA, MHoroi, MNRDX, MUDL, and MUNIV formulas with few others for- mulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Formula σ u x δ MBKAG (Present Work) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='51 MRenA(Present Work) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='62 MHoroi (Present Work) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='66 MNRDX (Present Work) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='60 MUDL (Present Work) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='53 MUNIV (Present Work) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='59 New UDL [26] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='68 IUF [27] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='64 ISEF [28] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='76 It is clear from Table 3 that the isospin (only for BKAG and RenA) and angular momentum play a crucial role to improve the cluster decay formulas and result in lesser statistical parameters σ, u, x, and δ for the modified formulas introduced in the present work, as compared with a few of the latest fitted/modified formulas (new UDL, IUF, and ISEF formulas) for the cluster decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' It is to be noted that among all the modified formulas, MBKAG formula renders more accurate half-life while compared through all the statistical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Hence, MBKAG formula can be employed to predict the more precise half-lives of cluster decay and the probable decay emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' With this in view, the possibility of cluster emission from the experimentally known trans-lead (86≤Z≤96) isotopes is probed by considering the daughter nuclei near the proton shell closure i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=', the emission of a cluster is chosen in such a way that the proton number of daughter nucleus Zd is close to 82 (Pb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Before predicting possibilities of new cluster decays in trans-lead regions, we first calculate the half-lives of experimentally known cluster decay using the MBKAG formula which are listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' We have taken only one parent-cluster combination out of 61 experimental data of cluster decay, to compare with α-decay half-lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' For the α-decay half-lives, we have used the NMHF (new modified Horoi formula) whose accuracy in determining the half-lives has already 6 Table 4: The calculated logarithmic half-lives using MBKAG formula together with experimental values [7, 9, 26, 29] for cluster decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' The α-decay half-lives are calculated by using NMHF formula [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' BR refers for branching ratios calculated by using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Q and Qα are the disintegration energies for cluster decay and α-decay, taken from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [7, 9, 26, 29] and AME2020 [37], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' For the l values, spin and parity of parent, daughter, and cluster nuclei are used from NUBASE2020 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Parent Daughter Emitted Q Qα l log10T1/2(sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=') BRExp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' BR nucleus nucleus cluster (MeV) (MeV) Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' MBKAG NMHF (Cluster) (α) 221Fr 207Tl 14C 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='28 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='46 3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='52 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='96 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='56 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='48 221Ra 207Pb 14C 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='39 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='88 3 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='39 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='74 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='65 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='27 222Ra 208Pb 14C 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='68 0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='22 11.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='39 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='84 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='31 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='55 7 been demonstrated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' The first, second, and third columns of Table 4 show the parent, daughter, and cluster nuclei, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Next two columns represent the disintegration energies of cluster decay and α-decay taken from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [7, 9, 26, 29] and from AME2020 [37], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' The sixth column lists angular momentum taken away by cluster particle after emission which is calculated by using selection rules explained in the Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' We have calculated logarithmic half- lives of cluster decay (using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' (10)), tabulated them in the eighth column, and compared these results with the experimental results (presented in the seventh column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' It is clear from the Table 4 that calculated half-lives of cluster emission by using the MBKAG formula (present work) are very close to experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Branching ratio (BR) which quantifies comparison between cluster decay to the α-decay and is defined as the ratio of α-decay half-life (listed in the ninth column) to the cluster decay half-life as below: BR = log10bc = log10(λc/λα) = log10(Tα/Tc) (20) where, λα and λc are referred as the decay constants of α-decay and cluster emission, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' The calculated branching ratios are shown in the last column which are indeed close to experimental branching ratios [7, 9, 26, 29] (presented in the second last column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' In fact, an excellent match of half-lives of almost all mentioned clusters in Table 4 validates the pertinence of MBKAG formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Furthermore, one can note that the experimental cluster decay half-life goes maximum nearly upto 1030 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=', therefore, it can be reasoned out that the clusters with a half-life less than 1030 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' seemingly be of experimental interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' In the next step of our study, we have utilized the degree of accuracy of MBKAG formula, as exhibited in Table 4, to predict the logarithmic half-lives of unknown cluster emissions in the trans-lead region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' For this estimation, the Q-values are calculated by the following relation: Q(MeV ) = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' (d) + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' (c) − B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' (p) + k[Zǫ p − Zǫ d] (21) where, the term k[Zǫ p − Zǫ d] indicates screening effect caused by the surrounding electrons around the nuclei [38] with k=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='7 eV [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='7 × 10−6MeV] and ǫ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='517 for Z (proton number) ≥ 60, and k=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='6 eV [13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='6 × 10−6MeV] and ǫ =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='408 for Z < 60 have been deducted from the data shown by Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' For accurate prediction of theoretical Q-values, we have selected an effective and reliable possible treatment among various theoretical approaches viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' relativistic mean-field theory (RMF) [32, 40–44], Finite Range Droplet Model (FRDM) [31], nonrelativistic Skyrme Hartree-Fock- Bogoliubov (HFB) [33], and Weizsacker-Skyrme mass model (WS4) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' From these approaches, we have calculated RMSE, listed in Table 5, for the known 121 Q-values related to cluster emissions [7, 9, 26, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Table 5 establishes that WS4 mass model provides an excellent agreement with the minimum RMSE compared to all other considered theoretical approaches and hence justifies the calculation of Q-values for cluster emission by taking binding energies (for daughter(d), cluster(c), and parent(p) nuclei) from this mass model [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Table 5: RMSE of various mass models for Q-value data for cluster emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Theory RMSE WS4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='43 FRDM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='78 HFB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='17 RMF 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='61 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='130 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='Si ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='Cm-Isotopes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='Figure 2: (Colour online) Variation of half-lives of various cluster emissions from experimentally known isotopes of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='trans-lead nuclei (86≤Z≤96) as a function of neutron number of daughter nuclei (considering proton number Zd=82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' These half-lives are calculated by using MBKAG formula and the Q-values are taken from the WS4 mass model[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 9 After the selection of efficacious empirical formula as well as the theoretical Q-values, we have chosen all the parent-cluster combinations for this extensive study to find the possible clusters emit- ted from 211−231Rn, 213−226Fr, 214−235Ra, 215−233Ac, 216−237Th, 218−241Pa, 228−243U, 226−245Np, 226−245Pu, 227−248Am, and 231−252Cm isotopes leading to 208Pb daughter (doubly magic) and neigh- bouring nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' We have plotted our results (up to T=10100 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=') in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 2 where the minima of log10T1/2 in several panels (Ra-isotopes to U-isotopes) correspond to 208Pb daughter i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=', doubly magic (Z=82, N=126) or near to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' These minima provide us the most probable clusters emitted from the respective isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' However, the probability of cluster emission always competes with α-decay which is quantified by branching ratio as we have discussed in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' The limit of experimental branching ratio related to α-decay is around BR = −17 as can be seen in Table 4 and also explained by Poenaru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Accordingly, cluster emission emerges more probable if BR ≥ −17: the criteria for the listed probable clusters in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' These clusters are selected from the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 2 for the particular isotopic chain of parent trans-lead nuclei 211−231Rn, 213−226Fr, 214−235Ra, 215−233Ac, 216−237Th, 218−241Pa, and 228−243U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Most of our results are within the ex- perimental reach and also in close match with the recent predictions of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [46–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Table 6: The calculated logarithmic half-lives and branching ratios of probable clusters emitted from various isotopes of trans-lead nuclei (86≤Z≤96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Cluster decay and α-decay half-lives are calculated by using MBKAG formula (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 10) and NMHF formula [20], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Disintegration energies (Q-values) for the cluster decay and α-decay are taken from WS4 mass model [30] and AME2020 [37], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' For the l values, spin and parity of parent, daughter, and cluster nuclei are used from NUBASE2020 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Parent Daughter Emitted Q Qα l log10T1/2(sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=') BR nucleus nucleus cluster (MeV) (MeV) MBKAG NMHF (Cluster) (α) 216Rn 208Pb 8Be 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='13 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='20 0 6.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='75 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='51 228Pa 208Pb 20F 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='90 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='26 2 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='42 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='13 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='29 229Pa 208Pb 21F 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='83 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='84 0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='94 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='74 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='20 231Pa 208Pb 23F 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='15 1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='75 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='11 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='64 231U 208Pb 23Ne 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='99 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='58 0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='55 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='53 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='02 231U 206Pb 25Ne 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='91 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='58 2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='95 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='53 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='42 On the other side, in the panels from Np-isotopes to Cm-isotopes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 2, in-spite of a clear minima, there is incessantly some probability of emission of clusters since many of the clusters own half-lives less than 1030 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' (experimental limit of half-lives of cluster emissions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' For examples, 10 21Na from 226−229Np, 22Na from 226−230Np, 23Na from 226−233Np, 24Na from 226−234Np, 25,27Na from 226−237Np, 26Na from 226−236Np and 28Na from 224−236Np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Similarly, some possible clusters (Mg-isotopes) emitted from various Pu-isotopes (Zp=94) are 23Mg from 226−231Pu, 24,25Mg from 226−235Np, 26Mg from 226−238Np, 27Mg from 226−239Np, and 28,29Mg from 226−241Np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Among Am- isotopes the potential clusters are 24Al from 227−230Am, 25Al from 227−233Am, 26Al from 227−236Am, 27Al from 227−239Am, 28Al from 227−240Am, 29Al from 227−241Am, and 30−32Al from 227−242Am as well as 26−33Si from the 231−252Cm isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' In the emission of odd mass clusters, the odd-even staggering is noticeable in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 2 which is usually attributed to the existence of nucleonic pairing correlations [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' The above-mentioned detailed study about favorable clusters having T1/2 < 1030 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' is expected to be certainly useful for future experimental inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Conclusions Several empirical formulas are investigated by adding angular momentum and isospin depen- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Their modified versions are turned into MBKAG, MRenA, MHoroi, MNRDX, MUDL, and MUNIV formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Experimental data of a total of 61 nuclei have been utilized for fitting which of- fers improved results of all the modified formulas while compared to their earlier versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Among these six modified formulas, after comparison of several statistical parameters the MBKAG for- mula is found most precise which is used to examine cluster decay half-lives for trans-lead region: 211−231Rn, 213−226Fr, 214−235Ra, 215−233Ac, 216−237Th, 218−241Pa, 228−243U, 226−245Np, 226−245Pu, 227−248Am, and 231−252Cm isotopes leading to 208Pb daughter (doubly magic) and neighbouring nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' We have found the considerable probability of emission of various isotopes of Be, B, C, N, O, F, Ne, Na, Mg, and Si from above mentioned trans-lead nuclei, respectively, and many of them are found to be favorable for the measurement (T1/2 < 1030 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' This study reveals that doubly magic daughter nuclei play a crucial role in the cluster decay process and could serve as a stimulus to the experiments eyeing on cluster radioactivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Acknowledgement AJ and GS acknowledge 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Shi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Swiatecki, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 54 (1989) 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Royer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Moustabchir, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' A 683 (2001) 182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Poenaru, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Data Tables 95 (2009) 815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [19] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Saxena, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Jain, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Sharma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 96 (2021) 125304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [20] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Sharma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Jain, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Saxena, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' A 1016 (2021) 122318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [21] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Saxena, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Sharma, Prafulla Saxena, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Phys.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Ellithi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Adela, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Abbas, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' A 58 (2022) 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [28] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Cheng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Wu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Cao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Zhang, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' A 58 (2022) 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [29] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Price, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 39 (1989) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [30] Ning Wang, Min Liu, Xizhen Wu, Jie Meng, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Lett.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Myers, At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Data Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Data Tables 125 (2019) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [32] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Singh, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Saxena, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Seaborg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Inorg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 28 (1966) 741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [36] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Kondev, Meng Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Naimi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Audi Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' C 45 (2020) 030001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [37] Meng Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=', Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' C 45 (2021) 030003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [38] V.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=', At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Data Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Data Tables 18 (1976) 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' [40] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Saxena, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' Kumawat, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfbfcG/content/2301.00261v1.pdf'} diff --git a/zNFRT4oBgHgl3EQfjjf7/content/tmp_files/2301.13591v1.pdf.txt b/zNFRT4oBgHgl3EQfjjf7/content/tmp_files/2301.13591v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1a4fd2b89112bb94e5e9cdd62edd6ac370cda2c --- /dev/null +++ b/zNFRT4oBgHgl3EQfjjf7/content/tmp_files/2301.13591v1.pdf.txt @@ -0,0 +1,621 @@ +ZERO3D: SEMANTIC-DRIVEN MULTI-CATEGORY 3D SHAPES GENERATION +Bo Han, Yitong Fu, Yixuan Shen +Zhejiang University, Hangzhou, China +University of Sydney, Sydney, Australia +ABSTRACT +Semantic-driven 3D shape generation aims to generate 3D +objects conditioned on text. Previous works face problems +with single-category generation, low-frequency 3D details, +and requiring a large number of paired datasets for train- +ing. To tackle these challenges, we propose a multi-category +conditional diffusion model. Specifically, 1) to alleviate the +problem of lack of large-scale paired data, we bridge the +text, 2D image and 3D shape based on the pre-trained CLIP +model, and 2) to obtain the multi-category 3D shape feature, +we apply the conditional flow model to generate 3D shape +vector conditioned on CLIP embedding. 3) to generate multi- +category 3D shape, we employ the hidden-layer diffusion +model conditioned on the multi-category shape vector, which +greatly reduces the training time and memory consumption. +Index Terms— Conditional Diffusion, Text-to-Shape, +Multi-modal, Latent Vector +1. INTRODUCTION +As the core element in the Metaverse world [1], 3D objects +play a vital role in enhancing people’s interactive experience. +With the rapid development of AIGC technology [2, 3, 4], +people can easily create images, audio, video, etc. through +text prompts. But 3D objects are currently designed by man- +ually modeling software like Blender and Maya3D, which re- +quires a great deal of time and expertise. Therefore, how to +generate high-quality 3D objects through semantic informa- +tion becomes a practical task. +3D shape generation [17] is a challenging task, unlike 2D +images which can be viewed as arrays of pixel values. 3D ob- +jects have diverse and complex representations, such as vox- +els, point clouds, grids, and implicit representations. Each +representation has its own advantage and limitation. Differ- +ent representations require different processing methods. +Text-to-shape generation is also challenging [18, 19, 20] +since it is hard to jointly understand 3D shape and text at the +same time, resulting in it being difficult to represent them in a +common space. At the same time, unlike text-to-image gener- +ation, where paired data is abundant, text-to-shape generation +lacks large-scale paired text and shape data. +Recently, much work has been done on 3D shape gener- +ation [5, 6, 7, 14, 21]. DreamFusion [6] transforms the dif- +fusion and denoising process in the pixel space into the op- +erations in the NeRF parameter space. Since the supervision +signal in DreamFusion operates on very low-resolution im- +ages (64 × 64), therefore it cannot synthesize high-frequency +3D geometric and texture details. DPM [14] trains an en- +coder to generate a shape vector representing the point cloud +shape, which is then used to train a flow model. After that, +the pre-trained flow model can turn noise into the shape vec- +tor. Subsequently, the diffusion model part utilizes this shape +vector as a condition for 3D shape generation. DPM [14] is +trained on the specific category, therefore it can only generate +point cloud data of one type. +To tackle these challenges, we first pre-train a CLIP model +which establishes a superior correspondence between text and +2D image. +At the same time, we can get a large number +of high-resolution 2D images corresponding to 3D objects +through the blender. Therefore, the CLIP model bridges text, +2D images, and 3D objects, thus alleviating the problem of +lack of large-scale paired text-3D objects data. Thereafter, we +apply a condition flow model to generate the specific category +shape vector conditioned on CLIP embedding. Subsequently +we employ a condition diffusion model to generate 3D object +conditioned on the shape vector. Specifically, during train- +ing, the CLIP model is used to encode the 2D image as the +condition, so the corresponding relationship between the 2D +image and the 3D shape can be learned. During inference, the +CLIP model is used to encode the semantic information as +the condition, thus the 3D shape corresponding to the seman- +tic information can be generated. At the same time, in view of +the high time and memory consumption problems of the dif- +fusion model itself, we implement the diffusion and denoising +operations on the hidden layer. +To summarize, our main contributions are as follows: +• Considering the superior correspondence between images +and texts in the CLIP model, we use images as the interme- +diary to generate 3D objects with semantic information. +• We propose a conditional diffusion model based on hidden +layers and then use the model to generate multiple cate- +gories of point cloud data, which greatly reduces training +time and memory consumption. +arXiv:2301.13591v1 [cs.CV] 31 Jan 2023 + +2. BACKGROUND +3D object generation. 3D-GAN [12] uses a three-dimensional +convolutional neural network to gradually map a high- +dimensional hidden vector into a 3D object represented by +a voxel. However, due to the uncertainty of Generative Ad- +versarial Networks, the results are not ideal. PointFlow [13] +introduces a flow model to generate the shape distribution of +point clouds. It uses the hidden vector representing the shape +distribution as a conditional to guide the point cloud gener- +ation. Since point clouds are usually distributed on a two- +dimensional manifold, it is difficult to obtain better results +through a flow model assuming that the point cloud obeys a +three-dimensional prior distribution. In the 3D domain, DPM +[14] and PVD [15] use diffusion models to generate point +cloud data. Although they can generate satisfactory results, +they are all trained in a specific category. +Semantics-Driven 3D Object Generation. Text2Shape [21] +proposes an end-to-end association learning framework. It +encodes text and 3D shapes separately into the same latent +space. However, large-scale 3D-text datasets are still diffi- +cult to obtain, so ClipForge [22] bypasses this problem with +the aid of the CLIP model on text-image matching. CLIP- +Mesh [23] also uses the CLIP model to measure the match- +ing degree between the image rendered by the grid model +and the text, so as to optimize the entire model parameters. +Dreamfields [5], DreamFusion [6] and Magic3D [7] all use +NeRF [24] as an implicit representation of 3D objects, and +render images through differentiable renderers. They utilize +the matching degree between images and text to optimize the +entire network and finally adopt the optimized implicit neural +field representation to extract the 3D mesh model. +3. METHOD +The schematic overview of the proposed architecture is il- +lustrated in Fig.1. +The photo on the left is the training +architecture of our model. It mainly consists of four com- +ponents: shape encoder, CLIP model, condition flow model, +and condition diffusion model. We use the DPM model [14] +as our backbone model, which samples noise data from Gaus- +sian distribution and generates point cloud data through the +denoising process under the guidance of the shape vector. +Specifically, we separate our model into two tasks during +training. +First, we render the 3D objects to obtain high- +resolution 2D images, and then the 2D rendered images are +used as the pre-trained CLIP model input, thereafter the con- +ditional flow model is trained to establish the relationship +between the CLIP model output and the shape vector s. Next, +we adopt the shape vector as the condition to guide the 3D +shape generation. During inference, the text is used as the +CLIP model input. Based on the bi-directionality of the flow +model, we can obtain the shape vector s guided by the CLIP +model output. Subsequently the shape vector s guides the +diffusion model to generate multi-category point cloud. +Shape Encoder: It maps the point cloud data to a distri- +bution of shape vectors, namely the shape mean and shape +variance, and then samples a shape vector from the shape +mean and variance. The overall network includes the feature +extraction layer and distribution map layer. For the feature ex- +traction layer, the attribute values of point cloud data are only +3D coordinates. First, we use a series of 1D convolutional +layers to increase the dimension of the point cloud data, and +then select the maximum value of each dimension feature to +perform feature dimension reduction. Then for the distribu- +tion mapping layer, the data after feature dimension reduction +is mapped to the shape mean and variance respectively with +the fully connected layer to represent the distribution of the +point cloud shape vector. After obtaining the representation +of the shape mean and variance, randomly generate an offset +value ε to sample a shape vector s defined as equation 1. +z = µ + ϵ ∗ exp +� +0.5 ∗ log +� +σ2�� +(1) +CLIP Model: It encodes text and images into the same +latent space, i.e. matching images and text. Therefore, based +on the CLIP model, we learn the correspondence between 3D +point clouds and text using images as an intermediary. The +CLIP model is based on VisualTransformer [2]. We match +images to 16*16 text vectors using the ViT-B/32 model. Im- +ages and texts are passed through corresponding CLIP en- +coders to obtain a one-dimensional vector with a length of +256, which is normalized and input into the conditional flow +model as a condition. +Conditional Flow Model: Traditional VAE encodes data +into a standard normal distribution, while the flow model can +learn a more flexible and variable distribution. The shape vec- +tor is fed into the conditional flow model to learn the transfor- +mation from the Gaussian noise distribution to the distribution +of s, where the CLIP encoded vector is as the condition. Dur- +ing inference, the data is directly sampled from the Gaussian +distribution, and the corresponding shape vector is obtained +through the inverse transformation of the flow model, which +is then input into the diffusion model as a condition. We use +the affine transformation layer in the RealNVP network archi- +tecture [25] to build the flow model. The affine transformation +layer divides the input into two parts. The first part keeps the +same as before. For the second part, the scale scaling coeffi- +cient and the offset coefficient are used to transform the data. +Point Cloud Autoencoder : A point cloud autoencoder +consists of an encoder and a decoder. The encoder is mainly +based on the PointNet network architecture [9] and the graph- +based max pooling layer [10] to extract point cloud features. +The decoder is mainly based on the FoldingNet [26], which +transforms point cloud features into raw data. Figure 2 shows +the network architecture of the autoencoder. Similar to LDM +[8], a point cloud autoencoder is first trained, and the encoded +hidden vector is used as the input of the diffusion model for +training. During inference, the output of the diffusion model + +𝑫 +𝑿(𝟎) +Shape +encoder +Condition +flow +CLIP +𝑿(𝒕) +𝑿(𝑻) +ℇ +S +Point cloud +image +… +… +Gaussian prior +𝑿(𝟎) +𝑿(𝒕) +𝑿(𝑻) +… +… +Point cloud +A boeing 747 +CLIP +Condition +flow +Gaussian prior +Text +S +Fig. 1. An overview of our proposed model +after the inverse diffusion process is a vector in the hidden +space, which is decoded into point cloud data by the point +cloud decoder. +Conditional Diffusion Model: We transform noisy data +into point cloud data using a diffusion model whose condi- +tion is the shape vector s. The diffusion model is comprised +of the diffusion process and the denoised process. The dif- +fusion process of the point cloud gradually adds noise to the +point cloud hidden vector, thereby converting a point cloud +distribution of a specific shape into a random noise distribu- +tion. The diffusion process can be expressed as follows: +q +� +x(t) +i +| x(t−1) +i +� += N +� +x(t) | +� +1 − βtx(t−1), βtI +� +(2) +q +� +x1:T +i +| x(0) +i +� += +T +� +t=1 +q +� +x(t) +i +| x(t−1) +i +� +(3) +where β1...βT are hyperparameters at each time step that +controls the noise addition process. +The denoised process is to recover the original point +cloud hidden vector from the noise. First, the point cloud hid- +den vector is sampled from the noise distribution, and then +through the reverse Markov chain, the noise is gradually sub- +tracted. Under the condition of shape vector s, the denoised +diffusion process can be expressed as follows: +pθ +� +x(t−1) | x(t), s +� += N +� +x(t−1) | µθ +� +x(t), t, s +� +, βtI +� +(4) +pθ +� +x(0:T ) | s +� += p +� +x(T )� T +� +t=1 +pθ +� +x(t−1) | x(t), s +� +(5) +Among them, µθ is a mean value estimated by the neural +network, s is the shape vector, and the initial data of inverse +diffusion obeys the standard normal distribution N(0, I). +The training objective is to maximize the likelihood func- +tion of the generated point cloud data E +� +log pθ +� +X(0)�� +. +Similar to the VAE model, the specific optimization goal is +still to maximize its variational lower bound (ELBO). +E[log pθ(X(0))] ≥ E +� +log +pθ(X(0:T ), s) +q(X(1:T ), s|X(0)) +� += E +� +log p(XT ) ++ +T +� +t=1 +log pθ(X(t−1)|X(t), s) +q(X(t)|X(t−1)) +− log qφ(s|X(0)) +p(s|c) +� +(6) +Where c is the condition of the flow model, i.e., the vector +encoded by the CLIP model. s is the condition of the dif- +fusion model, i.e., the shape vector. To simplify the above +variational bound, [14] propose training on pairs of (xt, x0) +to learn to parameterize this process with a simple squared +L2 loss. The following objective is simpler to train, resem- +bles denoising score matching and was found to yield higher- +quality samples: +L(θ) = +���ϵ − ϵθ +� +x(t) +i , t, s +���� +2 +, ϵ ∼ N(0, I) +(7) +where t is sampled uniformly between 1 and, and ϵθ is the +learned diffusion model. +MLP +Graph +Layer +Max +Pool +MLP +Fold +Fold +Chamfer loss +Decoder +Encoder +Fig. 2. Point Cloud Autoencoder Network Architecture + +4. EXPERIMENTS +4.1. Dataset +We use the ShapeNet (v2) dataset [27], which contains 13 cat- +egories of data, and a single sample contains point cloud data +and the corresponding rendered images of each 3D object. +4.2. Evaluation Metrics +1-NNA: It uses the nearest neighbor classifier to test the gen- +erated data separately, similar to the discriminator in GAN. +If it classifies the generated data close to random guessing, +i.e., the accuracy rate is close to 50%, and the quality of the +generated data is considered to be relatively high. +CLIP R-precision: It [5] can evaluate the generation ef- +fect with the composite text. CLIP R-precision ranks retrieval +results between generated model renderings and text to mea- +sure the visual-semantic similarity between textual descrip- +tions and generated images. The higher the ranking of the +real text, the higher the quality of the generated data. +We generate point cloud data instead of images, so we +need to convert point cloud data into images first. We use the +pre-trained SAP model to convert the point cloud data into a +grid model, and then render the grid model to obtain an image. +4.3. Results +The point cloud data generated using the words corresponding +to the category as text is shown in Fig 3. The point cloud data +generated using composite text is shown in Fig. 4. +(a) airplane +(b) car +(c) chair +(d) lamp +(e) table +(f) display +Fig. 3. Point cloud generated from the corresponding word +5. CONCLUSION +In this paper, we combine the CLIP model and the flow model +to propose a zero-shot learning method to establish the rela- +Boeing 747 +Triangular plane +Fighter plane +Square chair +Round chair +Sofa chair +Thin table +Rectangular table +Square table +Fig. 4. Point cloud generated from composite text +Airplane +chair +Car +Method +CD +EMD +CD +EMD +CD +EMD +r-GAN +97.25 +95.21 +86.24 +89.57 +91.68 +98.65 +PointFlow +78.24 +78.12 +67.15 +67.24 +58.14 +54.65 +PVD +76.81 +67.84 +60.54 +56.87 +58.95 +54.52 +baseline +80.12 +71.21 +64.82 +58.23 +65.32 +61.74 +ours +77.27 +65.68 +61.82 +54.35 +57.07 +53.93 +Table 1. 1-NNA metrics tested on three categories of Air- +plane, Chair, and Car +tionship between 3D shape and text through the intermediary +of 2D images, which can generate multi-category point cloud +and alleviate large-scale Insufficient sample data pairs. Due +to the diffusion and denoising on the hidden layer, the training +speed and memory usage are greatly optimized. +CLIP R-precision +Method +CLIP B/32 +CLIP B/16 +CLIP L/14 +DreamFusion +42.5 +44.6 +58.5 +ours +32.8 +34.1 +41.5 +Table 2. CLIP R-precision results + +6. 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Springer, 2016, pp. 628–644. + diff --git a/zdFLT4oBgHgl3EQfny-J/content/tmp_files/2301.12129v1.pdf.txt b/zdFLT4oBgHgl3EQfny-J/content/tmp_files/2301.12129v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6181280be3c6d97ce6f7caa058630456e9986d5f --- /dev/null +++ b/zdFLT4oBgHgl3EQfny-J/content/tmp_files/2301.12129v1.pdf.txt @@ -0,0 +1,1956 @@ +Decentralized Energy Market Integrating Carbon +Allowance Trade and Uncertainty Balance in Energy +Communities +Yuanxi Wua, Zhi Wua,∗, Wei Gua, Zheng Xua, Zheng Shub, Qirun Suna +aSchool of Electrical Engineering, Southeast University, Nanjing, 210096, China +bNARI Technology Co,. Ltd., Nanjing, 2111062, China +Abstract +With the sustained attention on carbon neutrality, the personal carbon trading (PCT) +scheme has been embraced as an auspicious paradigm for scaling down carbon emis- +sions. To facilitate the simultaneous clearance of energy and carbon allowance inside +the energy community while hedging against uncertainty, a joint trading framework +is proposed in this article. The energy trading is implemented in a peer-to-peer (P2P) +manner without the intervention of a central operator, and the uncertainty trading is ma- +terialized through procuring reserve of conventional generators and flexibility of users. +Under the PCT scheme, carbon allowance is transacted via a sharing mechanism. Pos- +sible excessive carbon emissions due to uncertainty balance are tackled by obliging +renewable agents to procure sufficient carbon allowances, following the consumption +responsibility principle. A two-stage iterative method consisting of tightening Mc- +Cormick envelope and alternating direction method of multipliers (ADMM) is devised +to transform the model into a mixed-integer second-order cone program (MISOCP) and +to allow for a fully decentralized market-clearing procedure. Numerical results have +validated the effectiveness of the proposed market model. +Keywords: personal carbon trade, uncertainty balance, peer-to-peer, coordinated +market design +1. Introduction +Traditionally, a large proportion of distribution network load is supplied by central- +ized fossil-fired power plants, resulting in considerable emissions of greenhouse gas +carbon dioxide [1]. The ever-worsening climate change has escalated the urgent need +of distributed energy resources (DERs) in the distribution network, including micro- +turbines (MT), rooftop photovoltaic (PV) panels and small wind turbines. However, +current policies such as feed-in tariff fail to promote the integration of DERs [2] and +are insufficient to fulfill the goal of carbon neutrality [3]. Recently, the technologi- +cal advance in energy system management enables a novel electricity market design +∗corresponding author: Zhi Wu, E-mail address: zwu@seu.edu.cn +arXiv:2301.12129v1 [eess.SY] 28 Jan 2023 + +named peer-to-peer (P2P) energy market [4, 5], which facilitates the consumption of +renewable energy. Decentralized platforms for P2P energy trading transactions with +the aid of blockchain technology are developed in [6, 7]. Besides, generalized Nash +game formulation is widely adopted to formulate the energy sharing mechanism [8, 9]. +As for the decentralized optimization algorithm to clear the P2P market, the P2P mar- +ket is designed as a social welfare maximization problem and the alternating direction +multiplier method (ADMM) is employed to achieve consensus among market play- +ers [10, 11, 12]. Other approaches include primal-dual gradient method [13], Relaxed +Consensus + Innovation [14], etc.. Within the aforementioned P2P trading frameworks, +individual participants are more inclined to trade directly with their counterparts in the +energy community [15] rather than with the upstream grid. Therefore, the energy com- +munity can reduce the energy loss due to the long-distance transmission and is expected +to scale down carbon emissions. +Regarding carbon neutrality, researchers have made endeavors to shed light on low- +carbon operations in the power industry. A straightforward approach is to consider +low-carbon factors by means of objective functions or price signals. In [16, 17], the +goal of minimizing energy cost is combined with the minimization of CO2 emissions +and the problem is further formulated as a multi-objective optimization program. In +contrast, a energy-carbon integrated price is coined in [18] based on carbon emission +flow and further incentives multiple energy systems to operate in a low-carbon mode +implicitly, but it ignores the energy sharing among different entities. As a supplement, +reference [19] considers multi-energy sharing among energy communities and incor- +porates carbon tax policy to curb carbon emissions. +Another alternative is to introduce a carbon transactive market which is similar to +the practice in the energy sector. Carbon market usually refers to a cap-and-trade mar- +ket [20] where all market participants can trade carbon emission allowances and should +surrender corresponding proportion of allowances for the CO2 emissions. Convention- +ally, the production responsibility principle [21] is adopted, which means energy pro- +ducers should be accountable for carbon emissions. Recent works have combined the +P2P energy market with the carbon market based on this accounting method. In [22], +all microgrids are motivated to form a grand coalition to transact energy and carbon al- +lowances. Nevertheless, market clearance is solved by the distribution system operator +(DSO) and individual privacy concerns may occur. A three-layer framework to trade +energy and carbon allowances is established in [23]. Notwithstanding the decentral- +ized settling procedure, the exchange of carbon allowances is launched in each time +slot, which is scarce in practice. +In recent years, personal carbon trading (PCT) has been viewed as a promising +scheme targeted at reducing carbon emissions at the individual and household level +[24]. The difference is that PCT applies the consumption responsibility principle, i.e., +consumers are responsible for carbon emissions precipitated by energy usage [21, 19]. +In a PCT scheme, each consumer is assigned with an initial allocation of carbon al- +lowances and can trade with other consumers. A coupled electricity and emission +trading market considering end-users’ carbon responsibility is introduced in [25], but +the electricity market is centralized and consumers are penalized for excessive car- +bon emissions instead of exchanging allowances. The carbon allowances trading is +proposed in [26], while the transactive energy trading is omitted and the identities of +2 + +allowance sellers/buyers are assigned beforehand. +All of the aforementioned references do not tackle the threat of uncertainty, which is +imposed by the presence of increasing penetration of renewable energy sources. Exist- +ing works have looked into different approaches to compensate for these uncertainties +[27, 28, 29, 30]. In [27], node-to-node balancing participation factors are leveraged +to procure reserve of controllable generators to keep the bulk power system balanced. +Flexibility of users is exploited to accommodate deviations of renewable energy out- +puts in the real-time market via a P2P energy sharing mechanism [28]. As for the +day-ahead P2P market, the uncertainty is traded with conventional generators or end- +users in [29, 30]. Nevertheless, the process of balancing uncertainty is possible to +induce more carbon emissions (i.e., emissions resulted from upward reserve supplied +by conventional generators), which should be addressed in the carbon market. +Summing up the above, the following issues still need to be further addressed: 1) +how to establish a day-ahead decentralized market that can trade energy, uncertainty +and carbon allowances jointly in the energy community. 2) how to take into account +the exceeding carbon emissions incurred by conventional generators’ upward reserve. +To this end, this paper proposes a novel community-level P2P market which can trade +day-ahead energy, uncertainty and carbon allowances simultaneously. The market par- +ticipants are constituted of three parts, i.e., renewable agents, conventional generators +and users. Renewable agents are supposed to compensate for their forecast errors by +procuring reserve from conventional generators and flexibility from users. The defi- +nition of flexibility in this paper is the same as that in [28], which is the adjustable +capacity the demand can provide in the demand response program. Besides, the carbon +market is established under the PCT scheme, and the need to predetermine the par- +ticipants’ identities (sellers or buyers) is obviated through a carbon allowance sharing +mechanism. The main contributions of this paper are summarized as follows. +1) A joint energy, uncertainty and carbon allowance trading market is developed +for the energy community. The proposed framework not only enables energy clearing +and carbon allowance sharing simultaneously, but also hedges against the uncertainty. +2) We leverage the consumption responsibility principle and propose that renewable +agents are responsible for acquiring sufficient allowances, which effectively covers po- +tential carbon emissions precipitated due to uncertainty balance and ensures the total +emissions are within the prescribed limit. +3) A fully decentralized optimization method is developed based on a combination +of a modified tightening McCormick method and ADMM, ensuring accuracy while +excluding privacy concerns. +The remainder of this paper is organized as follows. Section 2 presents the proposed +trading framework and market formulations. Section 3 provides the distributed solution +techniques. Case studies are conducted in Section 4. Finally, the conclusions of this +paper follow in Section 5. +2. Trading Framework and Market Formulation +2.1. Trading Framework +In this paper, a set Ω of participants are considered in the joint market, which can +be split into three categories, i.e., Ωu for users, Ωr for renewable agents (RES), such +3 + +Figure 1: Proposed market framework in the energy community +as photovoltaics, and Ωg for conventional generators (CG), such as MTs and combined +heat and power units (CHP). The joint market is proposed for the day-ahead market +and the time interval is 1h. The trading framework is depicted in Fig. 1. +In the energy market, users choose to buy clean energy from RESs or fossil energy +from CGs alternatively. The renewable source generation is featured with uncertainty +and thus, RESs need to procure regulating capacity from CGs or users to balance po- +tential forecast errors in the real-time stage, otherwise they will be punished for not +fulfilling the contract made in the day-ahead market. +Regarding carbon allowances transactions, users and RESs trade allowances to +cover the incurred carbon emissions. Under the PCT scheme, based on individual +consumption profiles, users who fail to cover emissions need to purchase allowances +in the market, while others with excessive allowances can choose to sell them in the +market. The flexibility of users and reserve of CGs contracted in the day-ahead mar- +ket should be dispatched by RESs who deviate from their predictions at the real-time +stage. Then the dispatched reserve becomes another source of carbon emissions. To +deal with emissions induced during uncertainty balance, we propose that RESs are ac- +countable and should purchase adequate carbon allowances, which is consistent with +the consumption responsibility principle. +Moreover, the users with a surfeit of allowances can sell the allowances to the com- +munity manager, which can incentivize users to lead a low-carbon life. The renewable +generation not consumed inside the community can be accommodated by the manager +as well. All market participants communicate with the community manager to clear +the market. +2.2. Market Formulation +2.2.1. Modelling Uncertainty +Firstly, we model the uncertainty in order to quantify forecast errors. Only the +energy deficiency case is considered in this paper since the surplus generation can be +curtailed or accommodated by the system operator in the real-time stage [30]. +Instead of assuming Gaussian distributed forecast errors, here we only adopt mean +and standard deviation of the error to capture uncertainty. Let ωt0 +i denote the random +4 + +Energy Community +Users +Information Flow +Energy Flow +Allowance Flow +Reserve Flow +Flexibility Flow +Community +Conventional Generators +Renewable Agents +Manager +由！！forecast error of RES i, which can be divided into two parts: negative component +denoted as ωt− +i +and positive component denoted as ωt+ +i , and it is assumed that P(ωt0 +i ≤ +0) = 0.5, P(ωt0 +i ≥ 0) = 0.5. Next, to model the case that only generation deficiency is +considered, a mixed random variable is defined as follows: +ωt +i = +������� +0, +if ωt0 +i ≥ 0, +ωt− +i , +if ωt0 +i ≤ 0. +(1) +Thus, it can be easily deduced that E(ωt +i) = 1 +2µt +i, Var(ωt +i) = 1 +2(δt +i)2 + 1 +4(µt +i)2. +2.2.2. Energy Trading +The proposed energy market is a bilateral trading market where each participant +decides its trading quantity with its neighborhoods. The market equilibrium is repre- +sented by the following balancing constraints: +Est +ij + Ebt +i j = 0, +∀i ∈ Ωu, j ∈ Ωg ∪ Ωr +(2) +Each user determines the row vector Ebt +i[·], while each RES/CG determines the column +vector Est +[·]i. Besides, the trade quantities of sellers are restricted to be non-negative: +Est ⪰ 0 +(3) +2.2.3. Uncertainty Trading +In this paper, the participation factor is adopted to model the bilateral uncertainty +trading: αt +ij denotes the participation factor based on which CG i produces energy to +compensate the uncertainty ωt +j, and βt +i j denotes the participation factor based on which +user i is willing to curtail its flexible load to compensate ωt +j. RESs and CGs, as well +as users, should achieve consensus on these uncertainty transactions when reaching the +equilibrium: +αr,t +ij + αt +i j = 0, +∀i ∈ Ωg, j ∈ Ωr +(4) +βr,t +ij + βt +i j = 0, +∀i ∈ Ωu, j ∈ Ωr +(5) +Each RES decides column vectors Ar,t +[·]i and Br,t +[·]i, while each CG/user decides the row +vector At +i[·]/Bt +i[·]. Similarly, the participation factors cannot be negative: +At ⪰ 0 +(6) +Bt ⪰ 0 +(7) +RES j needs to match the forecast error with the participation factors through uncer- +tainty trading, which means the sum of the participation factors must equal to minus +one (since αr,t +ij /βr,t +ij and αt +ij/βt +ij are opposite in sign): +� +i∈Ωg +αr,t +i j + +� +i∈Ωu +βr,t +i j = −1, +∀ j ∈ Ωr +(8) +5 + +2.2.4. Carbon Market +As is stated before, the players in the carbon market are users and RESs. The +initial daily carbon allowances Ψ0 +i are allocated to users, and then they purchase/sell +allowances, respectively, to satisfy individual constraints. Meanwhile, RESs who own +no initial allocation have to purchase allowances to counterbalance emissions result- +ing from upward reserve provided by CGs. Via the sharing mechanism, the carbon +allowance trading process can therefore be represented by the balancing constraint be- +low: +� +i∈Ωu∪Ωr +ci = 0 +(9) +The user who sells allowances in the market can sell them to the community man- +ager alternatively: +0 ≤ cs +i ≤ M ∗ idi +(10) +− M ∗ idi ≤ ci ≤ M ∗ (1 − idi) +(11) +where idi is a binary variable denoting the identity of the user, i.e., 1 for seller while 0 +for buyer. +After the clearance of the carbon market, each participant possesses a certain amount +of carbon allowances Ψi: +Ψi = +������� +ci, +if i ∈ Ωr, +Ψ0 +i + ci − cs +i , +if i ∈ Ωu +(12) +Remark: Note that the participants are not permitted to purchase more allowances from +the manager since the total initial allocation is set as a cap for the whole community. +2.2.5. Individual Constraints +At the equilibrium of the energy market, the power set-point of each participant is +equal to the summation of its trade quantity: +pt +u,i = −(1)⊺ · Ebt +i[·], +∀i ∈ Ωu +(13) +pt +r/g,i = 1 · Est +[·]i, +∀i ∈ Ωr ∪ Ωg +(14) +which is also bounded by the following constraints: +pt +g,i ≤ pt +g,i ≤ pt +g,i, +∀i ∈ Ωg +(15) +pt +u,i ≤ pt +u,i ≤ pt +u,i, +∀i ∈ Ωu +(16) +pt +r,i + ˆpt +r,i = Pt +r,i, +∀i ∈ Ωr +(17) +Following (17), we assume that the ”green energy” not consumed in the community +ˆpt +r,i can be accommodated by the community manager. +However, participating in uncertainty balancing induces deviations in the output of +CGs and energy consumption of users, which are given by: +�pt +g,i = pt +g,i − ωt · At +i[·], +∀i ∈ Ωg +(18) +6 + +�pt +u,i = pt +u,i + ωt · Bt +i[·], +∀i ∈ Ωu +(19) +where ωt = +� +ωt +1 +ωt +2 +· · · +ωt +|Ωr| +� +is a random row vector containing all RESs’ uncer- +tainties at time t, and �pt +g,i/�pt +u,i denotes the actual energy set-point of CGs/users, which +is a random variable. Therefore, to ensure the limits are respected, chance-constraints +are introduced: +P(�pt +g,i ≤ pt +g,i) ≥ 1 − εg,i, +∀i ∈ Ωg +(20) +P(�pt +u,i ≥ pt +u,i) ≥ 1 − εu,i, +∀i ∈ Ωu +(21) +Constraints (18) and (19) enforce that the power limits should be respected with a +predefined probability 1 − εg/r,i, which can be further converted into second-order cone +formulations with the aid of Chebyshev approximation[31]: +pt +g,i − Mt · At +i[·] + zg,iS t +g,i ≤ pt +g,i, +∀i ∈ Ωg +(22) +− pt +u,i − Mt · Bt +i[·] + zu,iS t +u,i ≤ −pt +u,i, +∀i ∈ Ωu +(23) +where Mt = E(ωt) is the mean value of ωt, and zg/u,i = �(1 − εg/u,i)/εg/u,i. The +covariance matrix of ωt is denoted as Σt and the formulations for S t +g,i/S t +u,i are: S t +g,i = +���(Σt)1/2(At +i[·])⊺���2, S t +u,i = +���(Σt)1/2(Bt +i[·])⊺���2. +Users’ carbon allowances should cover their corresponding emissions: +CEi = − +� +t +� +j∈Ωg +σjEbt +i j ≤ Ψi, +∀i ∈ Ωu +(24) +While for RESs, potential carbon emissions incurred by dispatching upward reserve +can be calculated as follows: +� +CEi = +� +t +� +j∈Ωg +σjαr,t +ji · ωt +i, +∀i ∈ Ωr +(25) +RES i must procure sufficient allowances to cover the above emissions: +P(� +CEi ≤ Ψi) ≥ 1 − εr,i, +∀i ∈ Ωr +(26) +Similarly, the above constraint can be transformed into a second-order cone constraint: +− E(ωi) · mi + +� +(1 − εr,i)/εr,i +���Ξ1/2 +i +(mi)⊺���2 ≤ Ψi +(27) +mt +i = − +� +j∈Ωg +σjαr,t +ji , +∀i ∈ Ωr +(28) +where ωi = +� +ω1 +i +ω2 +i +· · · +ωT +i +� +is a row vector containing RES i’s uncertainties +throughout the scheduling horizon and mi = +� +m1 +i +m2 +i +· · · +mT +i +� +. Ξi is the covari- +ance matrix of ωi. +7 + +2.2.6. Expected Social Welfare Maximization Problem +It is assumed that all market participants collaboratively minimize the overall cost +of the group. Therefore, the objective function can be formulated as follows: +obj = +� +t +[E( +� +i∈Ωg +Ci(�pt +g,i)) − E( +� +i∈Ωu +Ui(�pt +u,i)) − +� +i∈Ωr +rt +e ˆpt +r,i] − +� +i∈Ωu +rs +ccs +i +(29) +where Ci(p) = c2,ip2 + c1,ip + c0,i, Ui(p) = d2,ip2 + d1,ip. rt +e is the selling price +for renewable generation at time t, and rs +c is the selling price for carbon allowances. +Substituting (18) and (19) into (29), the objective can be further converted into the +following expression: +obj = +� +t +[ +� +i∈Ωg +(c2,i(pt +g,i)2 + c1,ipt +g,i + c0,i − (2c2,ipt +g,i + c1,i)(Mt · At +i[·]) + c2,i((Mt · At +i[·])2 ++ (S t +g,i)2)) − +� +i∈Ωu +(d2,i(pt +u,i)2 + d1,ipt +u,i + (2d2,ipt +u,i + d1,i)(Mt · Bt +i[·]) ++ d2,i((Mt · Bt +i[·])2 + (S t +u,i)2)) − +� +i∈Ωr +rt +e ˆpt +r,i] − +� +i∈Ωu +rs +ccs +i +(30) +Summing up the above, the problem can be formulated as: +min +obj +s.t. +(2) − (17), +(22) − (24), +(27) − (28) +(31) +3. Distributed Solution Techniques +In order to solve (31) in a privacy-preserving manner, two obstacles need to be +addressed: 1) The uncertainties bring bilinear terms into the objective function (30), +which makes it nonconvex function. 2) The constraints (2), (4)-(5) and (9) are coupled +among different participants. In this section, we will provide a two-stage iterative +method that includes a Relax-ADMM-Contraction loop as described below. +3.1. Convexification of the Objective Function— Relax +The bilinear terms are normally eliminated through McCormick envelopes [32]. +Firstly, the following auxiliary variables are introduced for simplicity: +πt +g,i = Mt · At +i[·], +∀i ∈ Ωg +(32) +πt +u,i = Mt · Bt +i[·], +∀i ∈ Ωu +(33) +χt +i = pt +g,iπt +g,i, +∀i ∈ Ωg +(34) +ϕt +i = pt +u,iπt +u,i, +∀i ∈ Ωu +(35) +8 + +The lower and upper bounds of πt +g,i and πt +u,i can be easily deduced as follows: +πt +g,i = πt +u,i = − +���Mt���∞ +(36) +πt +g,i = πt +u,i = 0 +(37) +Then the McCormick envelope is employed to reformulate the objective as a convex +function: +obj = +� +t +[ +� +i∈Ωg +(c2,i(pt +g,i)2 + c1.ipt +g,i + c0,i − 2c2,iχt +i − c1,iπt +g,i ++ c2,i((πt +g,i)2 + (S t +g,i)2)) − +� +i∈Ωu +(d2,i(pt +u,i)2 + d1,ipt +u,i + 2d2,iϕt +i ++ d1,iπt +u,i + d2,i((πt +u,i)2 + (S t +u,i)2)) − +� +i∈Ωr +rt +e ˆpt +r,i] − +� +i∈Ωu +rs +ccs +i +(38) +Additional constraints need to be incorporated: +χt +i ≥ pt +g,iπt +g,i + πt +g,ipt +g,i − pt +g,iπg,i +(39a) +χt +i ≥ pt +g,iπt +g,i + πt +g,ipt +g,i − pt +g,iπg,i +(39b) +χt +i ≤ pt +g,iπt +g,i + πt +g,ipt +g,i − pt +g,iπg,i +(39c) +χt +i ≤ pt +g,iπt +g,i + πt +g,ipt +g,i − pt +g,iπg,i +(39d) +ϕt +i ≥ pt +u,iπt +u,i + πt +u,ipt +u,i − pt +u,iπu,i +(39e) +ϕt +i ≥ pt +u,iπt +u,i + πt +u,ipt +u,i − pt +u,iπu,i +(39f) +ϕt +i ≤ pt +u,iπt +u,i + πt +u,ipt +u,i − pt +u,iπu,i +(39g) +ϕt +i ≤ pt +u,iπt +u,i + πt +u,ipt +u,i − pt +u,iπu,i +(39h) +Following the above procedure, the objective function is transformed into a convex +function. +3.2. Distributed Negotiation Mechanism— ADMM +A decentralized market mechanism is essential for keeping transparency and pri- +vacy of the joint market and is expected to motivate players in the community to partic- +ipate. In this paper, a distributed optimization method based on ADMM is adopted to +split the global optimization problem into smaller, individual optimization problems. +These local problems are solved by market players with limited information exchanges +with the community manager. Based on the exchange form of ADMM [33], the whole +procedure for solving (31) is presented as follows. +3.2.1. Local Optimization of Each Player +In the remainder, the cost/utility of each CG/user in (38) will be denoted as ˆCt +i/ ˆUt +i +for simplicity. +9 + +For each user i, its decision variable set is ξu +i = {pu,i, Ebi[·], Bi[·], πu,i, ϕi, ci, cs +i , idi}. +The local optimization problem of user i at a given iteration k is: +ξu(k+1) +i += arg min +� +t +� − ˆUt +i + +� +j∈Ωr +λt(k) +i j βt +i j + +� +j∈Ωr +ρ +2(βt +i j − βt(k) +i j + ˆβt(k) +i j )2 ++ +� +j∈Ωr∪Ωg +υt(k) +ij Ebt +i j + +� +j∈Ωr∪Ωg +γ +2(Ebt +i j − Ebt(k) +i j + ˆEt(k) +i j )2� ++ θ(k)ci + φ +2(ci − c(k) +i ++ c(k))2 − rs +ccs +i +s.t. +(7), (10) − (13), (16), (23) − (24), (33), (39) +(40) +For each RES i, its decision variable set is ξr +i = {pr,i, Es[·]i, Ar +[·]i, Br +[·]i, ci}. The local +optimization problem of RES i at a given iteration k is: +ξr(k+1) +i += arg min +� +t +� − rc,t ˆpt +r,i + +� +j∈Ωu +λt(k) +ji βr,t +ji + +� +j∈Ωu +ρ +2(βr,t +ji − βr,t(k) +ji ++ ˆβt(k) +ji )2 ++ +� +j∈Ωg +ηt(k) +ji αr,t +ji + +� +j∈Ωg +τ +2(αr,t +ji − αr,t(k) +ji ++ ˆαt(k) +ji )2 + +� +j∈Ωu +υt(k) +ji Est +ji ++ +� +j∈Ωu +γ +2(Est +ji − Est(k) +ji + ˆEt(k) +ji )2� + θ(k)ci + φ +2(ci − c(k) +i ++ c(k))2 +s.t. +(3), (8), (12), (14), (17), (27) − (28) +(41) +For each CG i, its decision variable set is ξg +i = {pg,i, Es[·]i, Ai[·], πg,i, χi}. The local +optimization problem of CG i at a given iteration k is: +ξg(k+1) +i += arg min +� +t +� ˆCt +i + +� +j∈Ωr +ηt(k) +i j αt +i j + +� +j∈Ωr +τ +2(αt +i j − αt(k) +i j + ˆαt(k) +i j )2 ++ +� +j∈Ωu +υt(k) +ji Est +ji + +� +j∈Ωu +γ +2(Est +ji − Est(k) +ji + ˆEt(k) +ji )2� +s.t. +(3), (6), (14) − (15), (22), (32), (39) +(42) +3.2.2. Global Variable Update +After gathering all the local information from market players, the community man- +ager is in charge of updating the global variables and then broadcasting the results to +all the players. To be specific, the update procedure at a given iteration k is as follows: +ˆαt(k+1) +ij += 1 +2(αt(k+1) +i j ++ αr,t(k+1) +i j +) +(43a) +ˆβt(k+1) +ij += 1 +2(βt(k+1) +i j ++ βr,t(k+1) +i j +) +(43b) +ˆEt(k+1) +ij += 1 +2(Ebt(k+1) +i j ++ Est(k+1) +i j +) +(43c) +c(k+1) = +1 +|Ωu ∪ Ωr| +� +i +c(k+1) +i +(43d) +10 + +3.2.3. Dual Price Update +At the end of each iteration, the dual prices need to be updated following the steps +below: +θ(k+1) = θ(k) + φc(k+1) +(44a) +λt(k+1) +i j += λt(k) +i j + ρˆβt(k+1) +i j +(44b) +ηt(k+1) +i j += ηt(k) +i j + τˆαt(k+1) +i j +(44c) +υt(k+1) +i j += υt(k) +i j + γ ˆEt(k+1) +i j +(44d) +3.2.4. Stopping Criteria +The above problem is a convex one except for the nonconvex constraints (10) and +(11). Nevertheless, since the non-convexity arises from Boolean constraints and only +exists in each user’s local problem, the ADMM procedure can still be carried out [34, +33]. The proposed distributed mechanism converges as long as the total local residuals +fall below the global stopping criteria: +se(k) = +� +t +∥Est(k) + Ebt(k)∥2 +F ≤ ϵ pri +e +(45a) +sr(k) = +� +t +∥At(k) + Ar,t(k)∥2 +F ≤ ϵ pri +r +(45b) +sd(k) = +� +t +∥Bt(k) + Br,t(k)∥2 +F ≤ ϵ pri +d +(45c) +sc(k) = ( +� +i +c(k) +i )2 ≤ ϵ pri +c +(45d) +te(k) = +� +t +∥ ˆE +t(k) − ˆE +t(k−1)∥2 +F ≤ ϵdual +e +(45e) +tr(k) = +� +t +∥ ˆA +t(k) − ˆA +t(k−1)∥2 +F ≤ ϵdual +r +(45f) +td(k) = +� +t +∥ ˆB +t(k) − ˆB +t(k−1)∥2 +F ≤ ϵdual +d +(45g) +tc(k) = (c(k) − c(k−1))2 ≤ ϵdual +c +(45h) +where ∥ · ∥F denotes the Frobenius norm, and ϵ pri +e +∼ ϵdual +c +are the corresponding thresh- +olds. +3.3. Tightening Bound— Contraction +Traditional McCormick envelope usually relax the bilinear term at the sacrifice of +accuracy and feasibility. The relaxed version of the market model above renders a +lower-bound solution without promising the feasibility of the original model. Hence, +a heuristic bound contraction algorithm modified from [35] is adopted in this paper to +improve the precision of the traditional McCormick envelopes, which can iteratively +strengthen the bounds of pt +g,i, pt +u,i, πt +g,i and πt +u,i. This is achieved by using a decreasing +scalar to tighten the bounds according to the solutions from the last iteration. Besides, +11 + +as stated in [35], the updated bounds should be the intersection of the result-oriented +bounds and the initial bounds to ensure the feasibility of the original model. Therefore, +at a given iteration n, the bounds should be updated based on the following rules: +pt +g,i = max{(1 − ϵn)pt∗ +g,i, pt,ini +g,i } +(46a) +pt +g,i = min{(1 + ϵn)pt∗ +g,i, pt,ini +g,i } +(46b) +pt +u,i = max{(1 − ϵn)pt∗ +u,i, pt,ini +u,i } +(46c) +pt +u,i = min{(1 + ϵn)pt∗ +u,i, pt,ini +u,i } +(46d) +πt +g,i = max{(1 + ϵn)πt∗ +g,i, πt,ini +g,i } +(46e) +πt +g,i = min{(1 − ϵn)πt∗ +g,i, πt,ini +g,i } +(46f) +πt +u,i = max{(1 + ϵn)πt∗ +u,i, πt,ini +u,i } +(46g) +πt +u,i = min{(1 − ϵn)πt∗ +u,i, πt,ini +u,i } +(46h) +where ϵn = ϵn−1−κ is a decreasing scalar, (·)∗ denotes the solution from the last iteration +and (·)ini denotes the bound used in the first iteration. The discrepancy between signs +in (48) and (49) is because πt +u,i and πt +g,i are always non-positive while pt +u,i and pt +g,i are +always non-negative. +Upon the updates of the bounds of the decision variables, the McCormick envelopes +(39) and (40) are updated accordingly. +The update procedure will terminate once the maximal relative error of the bilinear +constraints (34) and (35) fall below a reasonable level: +errg = max +t,i |(χt +i − pt +g,iπt +g,i)/χt +i| ≤ δg +(47a) +erru = max +t,i |(ϕt +i − pt +u,iπt +u,i)/ϕt +i| ≤ δu +(47b) +To sum up, the whole procedure of the Relax-ADMM-Contraction loop is presented +in Algorithm 1. +Remark: To accelerate the convergence, the solutions of dual prices and global vari- +ables from the outer iteration n − 1 will be adopted as initial values at the iteration n. +The effectiveness of this warm-start method will be illustrated in Section 4. +We now state that the dual prices exactly constitute the competitive market equilib- +rium prices. +Proposition 1. Using πe,t +ij = −υt +i j, πu,t +i j += −ηt +i j, πd,t +i j += −λt +i j, πc = θ as the bilateral +energy prices, upward reserve prices, flexibility prices and carbon allowance price, +respectively, constitutes a competitive market equilibrium. +Proof. We firstly define the individual profit-maximizing problem for RES i as follows: +min +� +t +� − re,t ˆpt +r,i − +� +j∈Ωu +πd,t +ji βr,t +ji − +� +j∈Ωg +πu,t +ji αr,t +ji +− +� +j∈Ωu +πe,t +ji Est +ji +� + πcci +s.t. +(3), (8), (12), (14), (17), (27) − (28) +(48) +12 + +Algorithm 1 Relax-ADMM-Contraction +Output: ˆE +t, ˆA +t, ˆB +t, ci, pt +g,i, pt +u,i, pt +r,i +1: repeat +▷ Tightening McCormick Envelope +2: +Derive the relaxed original problem (38)-(39). +3: +Initialization: k ← 0, dual prices, global variables +4: +repeat +▷ ADMM Procedure +5: +Local optimization of each player: +6: +for all i ∈ Ωu do +7: +Solve (40) and obtain ξu(k+1) +i +8: +for all i ∈ Ωr do +9: +Solve (41) and obtain ξr(k+1) +i +10: +for all i ∈ Ωg do +11: +Solve (42) and obtain ξg(k+1) +i +12: +Community manager coordination: +13: +Updates and broadcasts global variables: (43) +14: +Updates and broadcasts dual prices: (44) +15: +k ← k + 1 +16: +until convergence conditions (45) is satisfied +17: +Bound Contraction: +18: +for all i ∈ Ωu ∪ Ωg do +19: +Update local decision variables (46) +20: +Update local constraints (39) +21: +ϵn+1 = ϵn − κ +22: +n ← n + 1 +23: until convergence condition (47) is satisfied +It can be inferred from (41) that the market outcome will solve the following local +optimization problem for RES i after the convergence of the market: +min +� +t +� − re,t ˆpt +r,i + +� +j∈Ωu +λt +jiβr,t +ji + +� +j∈Ωg +ηt +jiαr,t +ji + +� +j∈Ωu +υt +jiEst +ji +� + θci +s.t. +(3), (8), (12), (14), (17), (27) − (28) +(49) +Applying πe,t +ij = −υt +ij, πu,t +ij = −ηt +i j, πd,t +i j = −λt +i j, πc = θ, it can be found that the out- +come of the market will solve the individual profit-maximizing problem (49) likewise. +The same procedure can be applied to users and CGs as well. Based on [36, Definition +1], the set of prices {πe,t +ij , πu,t +ij , πd,t +i j , πc} constitutes a competitive equilibrium. +4. Case Study +4.1. Case Parameters +In this study, the energy community consisting of eight market participants, i.e., +three MTs, three users and two PV generators is presented. The parameters of MTs +13 + +and users are listed in Table 1&2. The forecast output curves of PVs are depicted in +Fig. 2 and the standard deviations σt +i for the PV prediction errors are set as 10% of the +predicted outputs. Then, the standard deviation and mean value of the negative error +component are generated based on the following rules[27]: +δt +i = σt +i +� +π − 2 +π +, +µt +i = σt +i +� +2 +π +(50) +Table 1: Parameters of micro-turbines +Parameters +MT1 +MT2 +MT3 +c0 +�$� +2.01 +2.01 +2.03 +c1 +�$/kW� +0.045 +0.050 +0.052 +c2 +�$/kW2� +0.00021 +0.00021 +0.00019 +pg +�kW� +260 +270 +220 +pg +�kW� +0 +0 +0 +σi +�kg · kW−1� +0.870 +0.935 +0.910 +Table 2: Parameters of users +Parameters +U1 +U2 +U3 +d1 +�$/kW� +0.0870 +0.0765 +0.0600 +d2 +�$/kW2� +-0.00014 +-0.00014 +-0.000125 +00:00 +03:00 +06:00 +09:00 +12:00 +15:00 +18:00 +21:00 +0 +50 +100 +150 +200 +Load/PV Output (kW) +Time + U1 + U2 + U3 + PV1 + PV2 +Figure 2: Day-ahead PV forecast output and upper bound of loads +The upper bound of the load profile of each user is shown in Fig. 2 and the lower +bound is set to 40% of the upper bound. The selling price for carbon allowances rs +c is +14 + +set as $0.003/kg and the day-ahead selling price for electricity rt +e is set as $0.06/kWh. +The initial carbon allowance allocation is based on an equal per capita allocation, which +is 1800 kg per user in this study. All the confidence levels for chance constraints are +set as 0.95. In the ADMM procedure, the tolerance levels for primal residuals are set +to 10−6 for sr and sd, and 10−4 for se and sc. Tolerance levels for dual residuals are +chosen the same as the corresponding primal residuals. The stopping criteria for the +bound contraction are chosen to be 10−2. All the optimization problems are carried +out in MATLAB 2021b platform using Gurobi [37] solver along with Yalmip[38]. The +simulations run on a computer featuring AMD Ryzen 7 5800H @3.20GHz and 16 GB +of RAM. +4.2. Market Outcome +In this part, we will firstly discuss the outcomes of the proposed joint market. Fig. +3 and Fig. 4 present the outcomes of electricity trading and uncertainty balance, re- +spectively. Fig. 3 illustrates consumption profile for each user, including the load, +lower bound of the load (Lb) and components of the load. It can be seen that for each +user, the electricity purchase and demand reach the balance in each time slot. In ad- +dition, since MT3 is more expensive and has a relatively high carbon intensity, users +procure the least energy from MT3 as revealed in Fig. 3. Conversely, MT1, who has +the least cost parameter and carbon intensity, possesses the largest percentage among +all three MTs. This statement also explains the absence of MT3 in uncertainty balance +as depicted in Fig. 4, which illustrates the participation factors of users and MTs in +compensating for the uncertainty. It is obvious that for each PV, the sum of the partici- +pation factors of users and MTs is equal to 1, which implies that the forecast error can +be fully offset. Meanwhile, user3 does not participate in uncertainty balance during all +time periods because it has reached its lower bound in the energy market as shown in +Fig. 3, excluding itself from providing flexibility. +To achieve carbon allowance balance, PVs are required to purchase allowances +from the market and users can decide whether to procure allowances to consume more +energy or sell superfluous allowances to make profits based on their consumption pro- +files. The outcome of the carbon market is provided in Table 3. It is revealed that in +this case, all users can sell allowances to both PVs and the community manager by +adjusting their consumption behaviors. Besides, following KKT conditions, it is trivial +to state that the price in the allowance sharing is the same as the selling price to the +community manager. +Next, the influences of the carbon allowance price and electricity price on the mar- +ket outcomes are discussed. It is assumed that the carbon allowance price ranges from +$0.001/kg to $0.006/kg and the electricity selling price is increased from $0.04/kWh to +$0.08/kWh. The variations in social welfare, total allowances sold to the manager and +total PV generations sold to the manager are displayed in Fig. 5. +It can be observed from Fig. 5 (a) that the social welfare is improved along with +the increment in electricity and allowance price since the whole community can sell +allowances and PV generations at a more favorable price. Fig. 5 (b)&(c) reveal that +the total allowances sold to the community manager are reduced with the increase of +electricity price and the decrease of allowance price whereas the total PV generations +sold to the manager soar. The decline in allowance price undermines the economic +15 + + MT3 + Load + Lb +(a) User 1 + MT3 + Load + Lb +(b) User 2 +00:00 +04:00 +08:00 +12:00 +16:00 +20:00 +0 +20 +40 +60 +80 +100 +Energy (kWh) +Time + PV1 + PV2 + MT1 + MT2 + MT3 + Load + Lb +(c) User 3 +Figure 3: Optimal energy consumption of each user in the day-ahead market + U2 + U3 +(a) PV 1 +09:00 +13:00 +17:00 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Reserve Requirement Percentage +Time + MT1 + MT2 + MT3 + U1 + U2 + U3 +(b) PV 2 +Figure 4: Uncertainty balance of each PV in the day-ahead market +Table 3: The outcome of the carbon market +Participants +U1 +U2 +U3 +PV1 +PV2 +Allowance Sharing +Price �$/kg� +0.003 +Quantity �kg� +204.06 +204.06 +204.06 +-307.05 +-305.13 +Sold to the Community Manager +Price �$/kg� +0.003 +Quantity �kg� +584.20 +987.95 +716.55 +/ +/ +16 + +04:00L!WG +00:80 +IS:000 +JQ:00IV +bA50:00ITM +TM00:00 +0 +5O +一40 +0 +000 +J00 +JSO04:00 +08L!G +00:8 +JS:00bI Q:00IV +bA50:00ITM +STM00:005O +40一 +80L!G +13:00ITM +STM +M1J:00&TI +IU00:00 +0.0 +BG2GLIG +BednLeGI +0°4- +0.0Jf beLceUfsae +8.0 +0.I0.001 +0.002 +0.003 +0.004 +0.005 +0.006 +0.04 +0.05 +0.06 +0.07 +0.08 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +Electricity Price r +e +($/kWh) +Social Welfare ($) +Allowance Price r +c +($/kg) +r +e +=0.06 +r +c +=0.003 +(a) Social Welfare +0.001 +0.002 +0.003 +0.004 +0.005 +0.006 +0.04 +0.05 +0.06 +0.07 +0.08 +800 +1000 +1200 +1400 +1600 +1800 +2000 +2200 +2400 +2600 +Electricity Price r +e +($/kWh) +Allowance (kg) +Allowance Price r +c +($/kg) +r +e +=0.06 +r +c +=0.003 +(b) Total allowances sold to the manager +0.001 +0.002 +0.003 +0.004 +0.005 +0.006 +0.04 +0.05 +0.06 +0.07 +0.08 +0 +400 +800 +1200 +1600 +2000 +Electricity Price r +e +($/kWh) +PV Generation Sold (kWh) +Allowance Price r +c +($/kg) +r +e +=0.06 +r +c +=0.003 +(c) Total PV generations sold to the manager +Figure 5: Social welfare, total allowances and PV generations sold to the manager as a function of electricity +and carbon allowance price +incentives for users to cut down daily consumption and to prefer green energy. Mean- +while, the increment in electricity price additionally stimulates the readiness of PVs to +sell energy to the community manager instead of users. +4.3. The Effect of Carbon Allowance Sharing and Load Flexibility +In this part, the effects of the carbon allowance sharing mechanism and load flex- +ibility are demonstrated. To make a comparison, three different cases are considered +here. +Case 1: The joint market devised in this study is discussed, where carbon allowance +sharing and load flexibility are both included. +Case 2: The proposed joint market is considered, except for load flexibility. +Case 3: The proposed joint market is adopted, while the carbon market model +is modified according to [22]. In this case, the participants can only purchase/sell +allowances to the operator at fixed prices and the allowance balancing constraint (9) +is omitted. The selling price herein is still $0.003/kg and the purchase price is set as +$0.009/kg. +The discrepancies in social welfare and total allowances held inside the community +among the above three cases are presented in Table 4. +17 + +Table 4: Results in different cases +Cases +Social Welfare [$] +Total Allowances Held Inside +the Community [kg] +1 +13.9858 +3111.3 +2 +13.5043 +3295.7 +3 +10.4972 +3064.5 +As can be vividly discerned from Table 4, the whole community has more demands +for carbon allowances in case flexibility is not considered. This increase results from +the need of PVs to balance uncertainty. In Case 2, the only access for PVs to bal- +ance uncertainty is purchasing upward reserve of MTs, which, therefore, inevitably +precipitates more carbon emissions and demands for allowances. The decrease in the +allowances sold to the community manager also lowers the social welfare. Hence, em- +ploying the flexibility of users occupies a crucial role in promoting social welfare and +reducing carbon emissions. +In Case 3, the conspicuous shrinkage in social welfare is mainly due to the high cost +of purchasing carbon allowances, which suppresses the need of PVs for allowances, +and thereby saps the will of PVs to procure upward reserve of MTs. Compared with +Case 1, it can be inferred that the carbon allowance sharing mechanism helps facilitate +allowance trading among participants, generate a more affordable purchase price, and +thus, improve total social welfare. +4.4. Convergence Analysis +The evolution of the primal residuals and dual residuals through the proposed al- +gorithm are plotted in Fig. 6. To accelerate the convergence speed, an adaptive penalty +factor method as put forward in [33, Chapter 3] is employed. As shown in Fig. 6, +the proposed algorithm reaches stopping criteria within 3 bound contraction iterations +(denoted as ’round’) and a total of 488 ADMM iterations, verifying its convergence +performance. Besides, the fluctuations in the social welfare are presented in Fig. 7. +The optimal social welfare in the proposed algorithm is $13.9858 and $13.9869 in the +centralized algorithm, which results in a negligible optimality gap of less than 0.01%. +This indicates that the proposed solution techniques can effectively cope with the de- +centralized trading in the energy community. +Next, we will herein illustrate the efficiency of the warm-start method mentioned in +Section 3.3. Although the convergence of ADMM is guaranteed for any initialization +points, a good starting point can dramatically reduce the iterations and computation +complexity. In this study, it is reasonable to adopt the results obtained from the last +round as an approximation of accurate ones since these results provide a lower bound +to the original problem. Based on the fixed penalty factor version of ADMM, the total +iterations required to converge under two different situations are provided in Table 5, +one for results without a warm start and the other for results with a warm start. For +the first round, both of the two methods involves 1969 ADMM iterations since they +start from the same initialization points. However, for the second round, a transpar- +ent reduction in the number of iterations appears when the warm start is employed, +18 + +0 +100 +200 +300 +400 +500 +10 +13 +10 +9 +10 +5 +10 +1 +10 +3 +Primal Residuals +Iterations + sr + sd + se + sc +Round 1 +2 +3 +(a) Primal residuals +0 +100 +200 +300 +400 +500 +10 +9 +10 +5 +10 +1 +10 +3 + tr + td + te + tc +Dual Residuals +Iterations +Round 1 +2 +3 +(b) Dual residuals +Figure 6: Primal and dual residuals during different rounds +0 +100 +200 +300 +400 +500 +0 +4 +8 +12 +16 +20 +Social Welfare ($) +Iterations + Algorithm 1 + Centralized Algorithm +Figure 7: Social welfare of the proposed algorithm and the centralized algorithm +19 + +Table 5: Iteration numbers with/without warm start +Iterations +Round 1 +Round 2 +Total +No Warm start +1969 +1518 +3487 +With Warm start +1969 +33 +2002 +demonstrating that the warm-start method contributes to the superior convergence per- +formance. By comparison with Fig. 6, it is also found that a marked increase in total +iterations occurs when adopting fixed penalty factor method, which verifies the superi- +ority of the adaptive penalty factor method. +5. Conclusion +This paper proposes a joint day-ahead market paradigm where participants simulta- +neously trade energy, uncertainty, and carbon allowances. Moreover, this paper consid- +ers the possible excessive carbon emissions caused during uncertainty balance, which +are non-negligible for the management of the carbon emission. Simulations have re- +vealed several merits of the devised market: 1) The introduction of carbon allowance +trading guarantees that the total carbon emission of the energy community does not ex- +ceed the prescribed limit; 2) Proper electricity price and allowance price are conducive +to the local consumption of renewable energy and the reduction of carbon emissions; +3) The renewable agents can fully offset the uncertainty by procuring reserve from +conventional generators and flexibility from users; 4) The proposed Relax-ADMM- +Contraction loop is privacy-friendly, and can simultaneously yield trading quantities +and trading prices. However, it should be noted that the Chebyshev approximation is +generally far too conservative, and thereby may lead to the decline in the market ef- +ficiency. A further study should therefore concentrate on the methods to bypass the +conservatism. +Acknowledgements +This work was supported in part by the National Key R & D Program of China (No. +2020YFE0200400), and in part by the National Natural Science Foundation of China +(No. 52177077). +References +[1] H. jiemin, W. chen, The impact of private sector energy investment, innovation +and energy consumption on china’s carbon emissions, Renewable Energy 195 +(2022) 1291–1299. +[2] Y. Chen, C. Zhao, S. H. Low, A. Wierman, An energy sharing mechanism consid- +ering network constraints and market power limitation (2022). doi:10.48550/ +ARXIV.2203.04503. +URL https://arxiv.org/abs/2203.04503 +20 + +[3] Q. Lu, M. U. Farooq, X. Ma, R. Iram, Assessing the combining role of public- +private investment as a green finance and renewable energy in carbon neutrality +target, Renewable Energy 196 (2022) 1357–1365. +[4] G. Sun, S. Shen, S. Chen, Y. Zhou, Z. 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No. +04CH37508), IEEE, 2004, pp. 284–289. +23 + +Appendix A. +Nomenclature +Sets and Indices +Ωg +Set of conventional generators +Ωu +Set of users +Ωr +Set of renewable agents +ξg +i +Decision variable set of conventional generator i +ξu +i +Decision variable set of user i +ξr +i +Decision variable set of renewable agent i +t ∈ T +Index of time steps +Parameters and Matrices +µt +i +Expectation of ωt− +i +δt +i +Variance of ωt− +i +Ψ0 +i +Initial carbon allowance allocated to user i +pt +u/g,i +Lower bound of user/conventional generator i +pt +u/g,i +Upper bound of user/conventional generator i +Pt +r,i +Forecast generation of renewable agent i +σi +Carbon intensity of conventional generator i +Variables +Est +ij +Energy quantity trade from seller j to buyer i at time t +αt +ij +Participation factor of conventional generator i +βt +ij +Participation factor of user i +ci +Carbon allowance trade quantity inside the community +cs +i +Carbon allowance quantity sold to the manager +pt +u,i +Energy usage of user i at time t +pt +g,i +Output of conventional generator i at time t +pt +r,i +Energy traded in the community of renewable agent i at time t +24 + diff --git a/zdFLT4oBgHgl3EQfny-J/content/tmp_files/load_file.txt b/zdFLT4oBgHgl3EQfny-J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa8d70fe3d043dc6f32d9add12bd2981c7ea3198 --- /dev/null +++ b/zdFLT4oBgHgl3EQfny-J/content/tmp_files/load_file.txt @@ -0,0 +1,1048 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf,len=1047 +page_content='Decentralized Energy Market Integrating Carbon Allowance Trade and Uncertainty Balance in Energy Communities Yuanxi Wua, Zhi Wua,∗, Wei Gua, Zheng Xua, Zheng Shub, Qirun Suna aSchool of Electrical Engineering, Southeast University, Nanjing, 210096, China bNARI Technology Co,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=', Nanjing, 2111062, China Abstract With the sustained attention on carbon neutrality, the personal carbon trading (PCT) scheme has been embraced as an auspicious paradigm for scaling down carbon emis- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' To facilitate the simultaneous clearance of energy and carbon allowance inside the energy community while hedging against uncertainty, a joint trading framework is proposed in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The energy trading is implemented in a peer-to-peer (P2P) manner without the intervention of a central operator, and the uncertainty trading is ma- terialized through procuring reserve of conventional generators and flexibility of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Under the PCT scheme, carbon allowance is transacted via a sharing mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Pos- sible excessive carbon emissions due to uncertainty balance are tackled by obliging renewable agents to procure sufficient carbon allowances, following the consumption responsibility principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' A two-stage iterative method consisting of tightening Mc- Cormick envelope and alternating direction method of multipliers (ADMM) is devised to transform the model into a mixed-integer second-order cone program (MISOCP) and to allow for a fully decentralized market-clearing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Numerical results have validated the effectiveness of the proposed market model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Keywords: personal carbon trade, uncertainty balance, peer-to-peer, coordinated market design 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Introduction Traditionally, a large proportion of distribution network load is supplied by central- ized fossil-fired power plants, resulting in considerable emissions of greenhouse gas carbon dioxide [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The ever-worsening climate change has escalated the urgent need of distributed energy resources (DERs) in the distribution network, including micro- turbines (MT), rooftop photovoltaic (PV) panels and small wind turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' However, current policies such as feed-in tariff fail to promote the integration of DERs [2] and are insufficient to fulfill the goal of carbon neutrality [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Recently, the technologi- cal advance in energy system management enables a novel electricity market design ∗corresponding author: Zhi Wu, E-mail address: zwu@seu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='cn arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='12129v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='SY] 28 Jan 2023 named peer-to-peer (P2P) energy market [4, 5], which facilitates the consumption of renewable energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Decentralized platforms for P2P energy trading transactions with the aid of blockchain technology are developed in [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Besides, generalized Nash game formulation is widely adopted to formulate the energy sharing mechanism [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' As for the decentralized optimization algorithm to clear the P2P market, the P2P mar- ket is designed as a social welfare maximization problem and the alternating direction multiplier method (ADMM) is employed to achieve consensus among market play- ers [10, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Other approaches include primal-dual gradient method [13], Relaxed Consensus + Innovation [14], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='. Within the aforementioned P2P trading frameworks, individual participants are more inclined to trade directly with their counterparts in the energy community [15] rather than with the upstream grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Therefore, the energy com- munity can reduce the energy loss due to the long-distance transmission and is expected to scale down carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Regarding carbon neutrality, researchers have made endeavors to shed light on low- carbon operations in the power industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' A straightforward approach is to consider low-carbon factors by means of objective functions or price signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' In [16, 17], the goal of minimizing energy cost is combined with the minimization of CO2 emissions and the problem is further formulated as a multi-objective optimization program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' In contrast, a energy-carbon integrated price is coined in [18] based on carbon emission flow and further incentives multiple energy systems to operate in a low-carbon mode implicitly, but it ignores the energy sharing among different entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' As a supplement, reference [19] considers multi-energy sharing among energy communities and incor- porates carbon tax policy to curb carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Another alternative is to introduce a carbon transactive market which is similar to the practice in the energy sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Carbon market usually refers to a cap-and-trade mar- ket [20] where all market participants can trade carbon emission allowances and should surrender corresponding proportion of allowances for the CO2 emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Convention- ally, the production responsibility principle [21] is adopted, which means energy pro- ducers should be accountable for carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Recent works have combined the P2P energy market with the carbon market based on this accounting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' In [22], all microgrids are motivated to form a grand coalition to transact energy and carbon al- lowances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Nevertheless, market clearance is solved by the distribution system operator (DSO) and individual privacy concerns may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' A three-layer framework to trade energy and carbon allowances is established in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Notwithstanding the decentral- ized settling procedure, the exchange of carbon allowances is launched in each time slot, which is scarce in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' In recent years, personal carbon trading (PCT) has been viewed as a promising scheme targeted at reducing carbon emissions at the individual and household level [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The difference is that PCT applies the consumption responsibility principle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=', consumers are responsible for carbon emissions precipitated by energy usage [21, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' In a PCT scheme, each consumer is assigned with an initial allocation of carbon al- lowances and can trade with other consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' A coupled electricity and emission trading market considering end-users’ carbon responsibility is introduced in [25], but the electricity market is centralized and consumers are penalized for excessive car- bon emissions instead of exchanging allowances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The carbon allowances trading is proposed in [26], while the transactive energy trading is omitted and the identities of 2 allowance sellers/buyers are assigned beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' All of the aforementioned references do not tackle the threat of uncertainty, which is imposed by the presence of increasing penetration of renewable energy sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Exist- ing works have looked into different approaches to compensate for these uncertainties [27, 28, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' In [27], node-to-node balancing participation factors are leveraged to procure reserve of controllable generators to keep the bulk power system balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Flexibility of users is exploited to accommodate deviations of renewable energy out- puts in the real-time market via a P2P energy sharing mechanism [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' As for the day-ahead P2P market, the uncertainty is traded with conventional generators or end- users in [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Nevertheless, the process of balancing uncertainty is possible to induce more carbon emissions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=', emissions resulted from upward reserve supplied by conventional generators), which should be addressed in the carbon market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Summing up the above, the following issues still need to be further addressed: 1) how to establish a day-ahead decentralized market that can trade energy, uncertainty and carbon allowances jointly in the energy community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 2) how to take into account the exceeding carbon emissions incurred by conventional generators’ upward reserve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' To this end, this paper proposes a novel community-level P2P market which can trade day-ahead energy, uncertainty and carbon allowances simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The market par- ticipants are constituted of three parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=', renewable agents, conventional generators and users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Renewable agents are supposed to compensate for their forecast errors by procuring reserve from conventional generators and flexibility from users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The defi- nition of flexibility in this paper is the same as that in [28], which is the adjustable capacity the demand can provide in the demand response program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Besides, the carbon market is established under the PCT scheme, and the need to predetermine the par- ticipants’ identities (sellers or buyers) is obviated through a carbon allowance sharing mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The main contributions of this paper are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 1) A joint energy, uncertainty and carbon allowance trading market is developed for the energy community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The proposed framework not only enables energy clearing and carbon allowance sharing simultaneously, but also hedges against the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 2) We leverage the consumption responsibility principle and propose that renewable agents are responsible for acquiring sufficient allowances, which effectively covers po- tential carbon emissions precipitated due to uncertainty balance and ensures the total emissions are within the prescribed limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 3) A fully decentralized optimization method is developed based on a combination of a modified tightening McCormick method and ADMM, ensuring accuracy while excluding privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Section 2 presents the proposed trading framework and market formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Section 3 provides the distributed solution techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Case studies are conducted in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Finally, the conclusions of this paper follow in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Trading Framework and Market Formulation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Trading Framework In this paper, a set Ω of participants are considered in the joint market, which can be split into three categories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=', Ωu for users, Ωr for renewable agents (RES), such 3 Figure 1: Proposed market framework in the energy community as photovoltaics, and Ωg for conventional generators (CG), such as MTs and combined heat and power units (CHP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The joint market is proposed for the day-ahead market and the time interval is 1h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The trading framework is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' In the energy market, users choose to buy clean energy from RESs or fossil energy from CGs alternatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The renewable source generation is featured with uncertainty and thus, RESs need to procure regulating capacity from CGs or users to balance po- tential forecast errors in the real-time stage, otherwise they will be punished for not fulfilling the contract made in the day-ahead market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Regarding carbon allowances transactions, users and RESs trade allowances to cover the incurred carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Under the PCT scheme, based on individual consumption profiles, users who fail to cover emissions need to purchase allowances in the market, while others with excessive allowances can choose to sell them in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The flexibility of users and reserve of CGs contracted in the day-ahead mar- ket should be dispatched by RESs who deviate from their predictions at the real-time stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Then the dispatched reserve becomes another source of carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' To deal with emissions induced during uncertainty balance, we propose that RESs are ac- countable and should purchase adequate carbon allowances, which is consistent with the consumption responsibility principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Moreover, the users with a surfeit of allowances can sell the allowances to the com- munity manager, which can incentivize users to lead a low-carbon life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The renewable generation not consumed inside the community can be accommodated by the manager as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' All market participants communicate with the community manager to clear the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Market Formulation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Modelling Uncertainty Firstly, we model the uncertainty in order to quantify forecast errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Only the energy deficiency case is considered in this paper since the surplus generation can be curtailed or accommodated by the system operator in the real-time stage [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Instead of assuming Gaussian distributed forecast errors, here we only adopt mean and standard deviation of the error to capture uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Let ωt0 i denote the random 4 Energy Community Users Information Flow Energy Flow Allowance Flow Reserve Flow Flexibility Flow Community Conventional Generators Renewable Agents Manager 由！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='forecast error of RES i, which can be divided into two parts: negative component denoted as ωt− i and positive component denoted as ωt+ i , and it is assumed that P(ωt0 i ≤ 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='5, P(ωt0 i ≥ 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Next, to model the case that only generation deficiency is considered, a mixed random variable is defined as follows: ωt i = ������� 0, if ωt0 i ≥ 0, ωt− i , if ωt0 i ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' (1) Thus, it can be easily deduced that E(ωt i) = 1 2µt i, Var(ωt i) = 1 2(δt i)2 + 1 4(µt i)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Energy Trading The proposed energy market is a bilateral trading market where each participant decides its trading quantity with its neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The market equilibrium is repre- sented by the following balancing constraints: Est ij + Ebt i j = 0, ∀i ∈ Ωu, j ∈ Ωg ∪ Ωr (2) Each user determines the row vector Ebt i[·], while each RES/CG determines the column vector Est [·]i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Besides, the trade quantities of sellers are restricted to be non-negative: Est ⪰ 0 (3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Uncertainty Trading In this paper, the participation factor is adopted to model the bilateral uncertainty trading: αt ij denotes the participation factor based on which CG i produces energy to compensate the uncertainty ωt j, and βt i j denotes the participation factor based on which user i is willing to curtail its flexible load to compensate ωt j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' RESs and CGs, as well as users, should achieve consensus on these uncertainty transactions when reaching the equilibrium: αr,t ij + αt i j = 0, ∀i ∈ Ωg, j ∈ Ωr (4) βr,t ij + βt i j = 0, ∀i ∈ Ωu, j ∈ Ωr (5) Each RES decides column vectors Ar,t [·]i and Br,t [·]i, while each CG/user decides the row vector At i[·]/Bt i[·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Similarly, the participation factors cannot be negative: At ⪰ 0 (6) Bt ⪰ 0 (7) RES j needs to match the forecast error with the participation factors through uncer- tainty trading, which means the sum of the participation factors must equal to minus one (since αr,t ij /βr,t ij and αt ij/βt ij are opposite in sign): � i∈Ωg αr,t i j + � i∈Ωu βr,t i j = −1, ∀ j ∈ Ωr (8) 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Carbon Market As is stated before, the players in the carbon market are users and RESs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The initial daily carbon allowances Ψ0 i are allocated to users, and then they purchase/sell allowances, respectively, to satisfy individual constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Meanwhile, RESs who own no initial allocation have to purchase allowances to counterbalance emissions result- ing from upward reserve provided by CGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Via the sharing mechanism, the carbon allowance trading process can therefore be represented by the balancing constraint be- low: � i∈Ωu∪Ωr ci = 0 (9) The user who sells allowances in the market can sell them to the community man- ager alternatively: 0 ≤ cs i ≤ M ∗ idi (10) − M ∗ idi ≤ ci ≤ M ∗ (1 − idi) (11) where idi is a binary variable denoting the identity of the user, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=', 1 for seller while 0 for buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' After the clearance of the carbon market, each participant possesses a certain amount of carbon allowances Ψi: Ψi = ������� ci, if i ∈ Ωr, Ψ0 i + ci − cs i , if i ∈ Ωu (12) Remark: Note that the participants are not permitted to purchase more allowances from the manager since the total initial allocation is set as a cap for the whole community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Individual Constraints At the equilibrium of the energy market,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' the power set-point of each participant is equal to the summation of its trade quantity: pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = −(1)⊺ · Ebt i[·],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωu (13) pt r/g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = 1 · Est [·]i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωr ∪ Ωg (14) which is also bounded by the following constraints: pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i ≤ pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i ≤ pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωg (15) pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i ≤ pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i ≤ pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωu (16) pt r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + ˆpt r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = Pt r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωr (17) Following (17),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' we assume that the ”green energy” not consumed in the community ˆpt r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i can be accommodated by the community manager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' However, participating in uncertainty balancing induces deviations in the output of CGs and energy consumption of users, which are given by: �pt g,i = pt g,i − ωt · At i[·], ∀i ∈ Ωg (18) 6 �pt u,i = pt u,i + ωt · Bt i[·], ∀i ∈ Ωu (19) where ωt = � ωt 1 ωt 2 · · ωt |Ωr| � is a random row vector containing all RESs’ uncer- tainties at time t, and �pt g,i/�pt u,i denotes the actual energy set-point of CGs/users, which is a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' to ensure the limits are respected,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' chance-constraints are introduced: P(�pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i ≤ pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i) ≥ 1 − εg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωg (20) P(�pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i ≥ pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i) ≥ 1 − εu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωu (21) Constraints (18) and (19) enforce that the power limits should be respected with a predefined probability 1 − εg/r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' which can be further converted into second-order cone formulations with the aid of Chebyshev approximation[31]: pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i − Mt · At i[·] + zg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iS t g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i ≤ pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωg (22) − pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i − Mt · Bt i[·] + zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iS t u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i ≤ −pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωu (23) where Mt = E(ωt) is the mean value of ωt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' and zg/u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = �(1 − εg/u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i)/εg/u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The covariance matrix of ωt is denoted as Σt and the formulations for S t g,i/S t u,i are: S t g,i = ���(Σt)1/2(At i[·])⊺���2, S t u,i = ���(Σt)1/2(Bt i[·])⊺���2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Users’ carbon allowances should cover their corresponding emissions: CEi = − � t � j∈Ωg σjEbt i j ≤ Ψi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωu (24) While for RESs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' potential carbon emissions incurred by dispatching upward reserve can be calculated as follows: � CEi = � t � j∈Ωg σjαr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='t ji · ωt i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωr (25) RES i must procure sufficient allowances to cover the above emissions: P(� CEi ≤ Ψi) ≥ 1 − εr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωr (26) Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' the above constraint can be transformed into a second-order cone constraint: − E(ωi) · mi + � (1 − εr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i)/εr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i ���Ξ1/2 i (mi)⊺���2 ≤ Ψi (27) mt i = − � j∈Ωg σjαr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='t ji ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωr (28) where ωi = � ω1 i ω2 i · · ωT i � is a row vector containing RES i’s uncertainties throughout the scheduling horizon and mi = � m1 i m2 i · · mT i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Ξi is the covari- ance matrix of ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Expected Social Welfare Maximization Problem It is assumed that all market participants collaboratively minimize the overall cost of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Therefore, the objective function can be formulated as follows: obj = � t [E( � i∈Ωg Ci(�pt g,i)) − E( � i∈Ωu Ui(�pt u,i)) − � i∈Ωr rt e ˆpt r,i] − � i∈Ωu rs ccs i (29) where Ci(p) = c2,ip2 + c1,ip + c0,i, Ui(p) = d2,ip2 + d1,ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' rt e is the selling price for renewable generation at time t, and rs c is the selling price for carbon allowances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Substituting (18) and (19) into (29),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' the objective can be further converted into the following expression: obj = � t [ � i∈Ωg (c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i(pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i)2 + c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ipt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i − (2c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ipt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i)(Mt · At i[·]) + c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i((Mt · At i[·])2 + (S t g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i)2)) − � i∈Ωu (d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i(pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i)2 + d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ipt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + (2d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ipt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i)(Mt · Bt i[·]) + d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i((Mt · Bt i[·])2 + (S t u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i)2)) − � i∈Ωr rt e ˆpt r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i] − � i∈Ωu rs ccs i (30) Summing up the above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' the problem can be formulated as: min obj s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' (2) − (17), (22) − (24), (27) − (28) (31) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Distributed Solution Techniques In order to solve (31) in a privacy-preserving manner, two obstacles need to be addressed: 1) The uncertainties bring bilinear terms into the objective function (30), which makes it nonconvex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 2) The constraints (2), (4)-(5) and (9) are coupled among different participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' In this section, we will provide a two-stage iterative method that includes a Relax-ADMM-Contraction loop as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Convexification of the Objective Function— Relax The bilinear terms are normally eliminated through McCormick envelopes [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Firstly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' the following auxiliary variables are introduced for simplicity: πt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = Mt · At i[·],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωg (32) πt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = Mt · Bt i[·],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωu (33) χt i = pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωg (34) ϕt i = pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ∀i ∈ Ωu (35) 8 The lower and upper bounds of πt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i and πt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i can be easily deduced as follows: πt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = πt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = − ���Mt���∞ (36) πt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = πt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = 0 (37) Then the McCormick envelope is employed to reformulate the objective as a convex function: obj = � t [ � i∈Ωg (c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i(pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i)2 + c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ipt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i − 2c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iχt i − c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i((πt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i)2 + (S t g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i)2)) − � i∈Ωu (d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i(pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i)2 + d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ipt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + 2d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iϕt i + d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i((πt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i)2 + (S t u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i)2)) − � i∈Ωr rt e ˆpt r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i] − � i∈Ωu rs ccs i (38) Additional constraints need to be incorporated: χt i ≥ pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + πt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ipt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i − pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i (39a) χt i ≥ pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + πt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ipt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i − pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i (39b) χt i ≤ pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + πt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ipt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i − pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i (39c) χt i ≤ pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + πt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ipt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i − pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i (39d) ϕt i ≥ pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + πt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ipt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i − pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i (39e) ϕt i ≥ pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + πt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ipt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i − pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i (39f) ϕt i ≤ pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + πt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ipt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i − pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i (39g) ϕt i ≤ pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + πt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ipt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i − pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='iπu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i (39h) Following the above procedure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' the objective function is transformed into a convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Distributed Negotiation Mechanism— ADMM A decentralized market mechanism is essential for keeping transparency and pri- vacy of the joint market and is expected to motivate players in the community to partic- ipate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' In this paper, a distributed optimization method based on ADMM is adopted to split the global optimization problem into smaller, individual optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' These local problems are solved by market players with limited information exchanges with the community manager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Based on the exchange form of ADMM [33], the whole procedure for solving (31) is presented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Local Optimization of Each Player In the remainder, the cost/utility of each CG/user in (38) will be denoted as ˆCt i/ ˆUt i for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 9 For each user i, its decision variable set is ξu i = {pu,i, Ebi[·], Bi[·], πu,i, ϕi, ci, cs i , idi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The local optimization problem of user i at a given iteration k is: ξu(k+1) i = arg min � t � − ˆUt i + � j∈Ωr λt(k) i j βt i j + � j∈Ωr ρ 2(βt i j − βt(k) i j + ˆβt(k) i j )2 + � j∈Ωr∪Ωg υt(k) ij Ebt i j + � j∈Ωr∪Ωg γ 2(Ebt i j − Ebt(k) i j + ˆEt(k) i j )2� + θ(k)ci + φ 2(ci − c(k) i + c(k))2 − rs ccs i s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' (7), (10) − (13), (16), (23) − (24), (33), (39) (40) For each RES i, its decision variable set is ξr i = {pr,i, Es[·]i, Ar [·]i, Br [·]i, ci}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The local optimization problem of RES i at a given iteration k is: ξr(k+1) i = arg min � t � − rc,t ˆpt r,i + � j∈Ωu λt(k) ji βr,t ji + � j∈Ωu ρ 2(βr,t ji − βr,t(k) ji + ˆβt(k) ji )2 + � j∈Ωg ηt(k) ji αr,t ji + � j∈Ωg τ 2(αr,t ji − αr,t(k) ji + ˆαt(k) ji )2 + � j∈Ωu υt(k) ji Est ji + � j∈Ωu γ 2(Est ji − Est(k) ji + ˆEt(k) ji )2� + θ(k)ci + φ 2(ci − c(k) i + c(k))2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' (3), (8), (12), (14), (17), (27) − (28) (41) For each CG i, its decision variable set is ξg i = {pg,i, Es[·]i, Ai[·], πg,i, χi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The local optimization problem of CG i at a given iteration k is: ξg(k+1) i = arg min � t � ˆCt i + � j∈Ωr ηt(k) i j αt i j + � j∈Ωr τ 2(αt i j − αt(k) i j + ˆαt(k) i j )2 + � j∈Ωu υt(k) ji Est ji + � j∈Ωu γ 2(Est ji − Est(k) ji + ˆEt(k) ji )2� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' (3), (6), (14) − (15), (22), (32), (39) (42) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Global Variable Update After gathering all the local information from market players, the community man- ager is in charge of updating the global variables and then broadcasting the results to all the players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' To be specific, the update procedure at a given iteration k is as follows: ˆαt(k+1) ij = 1 2(αt(k+1) i j + αr,t(k+1) i j ) (43a) ˆβt(k+1) ij = 1 2(βt(k+1) i j + βr,t(k+1) i j ) (43b) ˆEt(k+1) ij = 1 2(Ebt(k+1) i j + Est(k+1) i j ) (43c) c(k+1) = 1 |Ωu ∪ Ωr| � i c(k+1) i (43d) 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Dual Price Update At the end of each iteration, the dual prices need to be updated following the steps below: θ(k+1) = θ(k) + φc(k+1) (44a) λt(k+1) i j = λt(k) i j + ρˆβt(k+1) i j (44b) ηt(k+1) i j = ηt(k) i j + τˆαt(k+1) i j (44c) υt(k+1) i j = υt(k) i j + γ ˆEt(k+1) i j (44d) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Stopping Criteria The above problem is a convex one except for the nonconvex constraints (10) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Nevertheless, since the non-convexity arises from Boolean constraints and only exists in each user’s local problem, the ADMM procedure can still be carried out [34, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The proposed distributed mechanism converges as long as the total local residuals fall below the global stopping criteria: se(k) = � t ∥Est(k) + Ebt(k)∥2 F ≤ ϵ pri e (45a) sr(k) = � t ∥At(k) + Ar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='t(k)∥2 F ≤ ϵ pri r (45b) sd(k) = � t ∥Bt(k) + Br,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='t(k)∥2 F ≤ ϵ pri d (45c) sc(k) = ( � i c(k) i )2 ≤ ϵ pri c (45d) te(k) = � t ∥ ˆE t(k) − ˆE t(k−1)∥2 F ≤ ϵdual e (45e) tr(k) = � t ∥ ˆA t(k) − ˆA t(k−1)∥2 F ≤ ϵdual r (45f) td(k) = � t ∥ ˆB t(k) − ˆB t(k−1)∥2 F ≤ ϵdual d (45g) tc(k) = (c(k) − c(k−1))2 ≤ ϵdual c (45h) where ∥ · ∥F denotes the Frobenius norm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' and ϵ pri e ∼ ϵdual c are the corresponding thresh- olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Tightening Bound— Contraction Traditional McCormick envelope usually relax the bilinear term at the sacrifice of accuracy and feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The relaxed version of the market model above renders a lower-bound solution without promising the feasibility of the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Hence, a heuristic bound contraction algorithm modified from [35] is adopted in this paper to improve the precision of the traditional McCormick envelopes, which can iteratively strengthen the bounds of pt g,i, pt u,i, πt g,i and πt u,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' This is achieved by using a decreasing scalar to tighten the bounds according to the solutions from the last iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Besides, 11 as stated in [35], the updated bounds should be the intersection of the result-oriented bounds and the initial bounds to ensure the feasibility of the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' at a given iteration n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' the bounds should be updated based on the following rules: pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = max{(1 − ϵn)pt∗ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ini g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i } (46a) pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = min{(1 + ϵn)pt∗ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ini g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i } (46b) pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = max{(1 − ϵn)pt∗ u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ini u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i } (46c) pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = min{(1 + ϵn)pt∗ u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ini u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i } (46d) πt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = max{(1 + ϵn)πt∗ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' πt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ini g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i } (46e) πt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = min{(1 − ϵn)πt∗ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' πt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ini g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i } (46f) πt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = max{(1 + ϵn)πt∗ u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' πt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ini u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i } (46g) πt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i = min{(1 − ϵn)πt∗ u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' πt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ini u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i } (46h) where ϵn = ϵn−1−κ is a decreasing scalar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' (·)∗ denotes the solution from the last iteration and (·)ini denotes the bound used in the first iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The discrepancy between signs in (48) and (49) is because πt u,i and πt g,i are always non-positive while pt u,i and pt g,i are always non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Upon the updates of the bounds of the decision variables, the McCormick envelopes (39) and (40) are updated accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The update procedure will terminate once the maximal relative error of the bilinear constraints (34) and (35) fall below a reasonable level: errg = max t,i |(χt i − pt g,iπt g,i)/χt i| ≤ δg (47a) erru = max t,i |(ϕt i − pt u,iπt u,i)/ϕt i| ≤ δu (47b) To sum up, the whole procedure of the Relax-ADMM-Contraction loop is presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Remark: To accelerate the convergence, the solutions of dual prices and global vari- ables from the outer iteration n − 1 will be adopted as initial values at the iteration n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The effectiveness of this warm-start method will be illustrated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' We now state that the dual prices exactly constitute the competitive market equilib- rium prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Using πe,t ij = −υt i j, πu,t i j = −ηt i j, πd,t i j = −λt i j, πc = θ as the bilateral energy prices, upward reserve prices, flexibility prices and carbon allowance price, respectively, constitutes a competitive market equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' We firstly define the individual profit-maximizing problem for RES i as follows: min � t � − re,t ˆpt r,i − � j∈Ωu πd,t ji βr,t ji − � j∈Ωg πu,t ji αr,t ji − � j∈Ωu πe,t ji Est ji � + πcci s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' (3), (8), (12), (14), (17), (27) − (28) (48) 12 Algorithm 1 Relax-ADMM-Contraction Output: ˆE t, ˆA t, ˆB t, ci, pt g,i, pt u,i, pt r,i 1: repeat ▷ Tightening McCormick Envelope 2: Derive the relaxed original problem (38)-(39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 3: Initialization: k ← 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' dual prices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' global variables ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='4: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='repeat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='▷ ADMM Procedure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='5: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Local optimization of each player: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='6: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='for all i ∈ Ωu do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='7: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Solve (40) and obtain ξu(k+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='8: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='for all i ∈ Ωr do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='9: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Solve (41) and obtain ξr(k+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='10: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='for all i ∈ Ωg do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='11: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Solve (42) and obtain ξg(k+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='12: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Community manager coordination: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='13: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Updates and broadcasts global variables: (43) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='14: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Updates and broadcasts dual prices: (44) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='15: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='k ← k + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='16: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='until convergence conditions (45) is satisfied ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='17: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Bound Contraction: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='18: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='for all i ∈ Ωu ∪ Ωg do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='19: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Update local decision variables (46) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='20: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Update local constraints (39) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='21: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='ϵn+1 = ϵn − κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='22: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='n ← n + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='23: until convergence condition (47) is satisfied ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='It can be inferred from (41) that the market outcome will solve the following local ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='optimization problem for RES i after the convergence of the market: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='� − re,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='t ˆpt r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i + � j∈Ωu λt jiβr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='t ji + � j∈Ωg ηt jiαr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='t ji + � j∈Ωu υt jiEst ji � + θci s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' (3), (8), (12), (14), (17), (27) − (28) (49) Applying πe,t ij = −υt ij, πu,t ij = −ηt i j, πd,t i j = −λt i j, πc = θ, it can be found that the out- come of the market will solve the individual profit-maximizing problem (49) likewise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The same procedure can be applied to users and CGs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Based on [36, Definition 1], the set of prices {πe,t ij , πu,t ij , πd,t i j , πc} constitutes a competitive equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Case Study 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Case Parameters In this study, the energy community consisting of eight market participants, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=', three MTs, three users and two PV generators is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The parameters of MTs 13 and users are listed in Table 1&2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The forecast output curves of PVs are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 2 and the standard deviations σt i for the PV prediction errors are set as 10% of the predicted outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Then, the standard deviation and mean value of the negative error component are generated based on the following rules[27]: δt i = σt i � π − 2 π , µt i = σt i � 2 π (50) Table 1: Parameters of micro-turbines Parameters MT1 MT2 MT3 c0 �$� 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='03 c1 �$/kW� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='052 c2 �$/kW2� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='00021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='00021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='00019 pg �kW� 260 270 220 pg �kW� 0 0 0 σi �kg · kW−1� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='870 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='910 Table 2: Parameters of users Parameters U1 U2 U3 d1 �$/kW� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='0870 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='0765 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='0600 d2 �$/kW2� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='00014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='00014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='000125 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 0 50 100 150 200 Load/PV Output (kW) Time U1 U2 U3 PV1 PV2 Figure 2: Day-ahead PV forecast output and upper bound of loads The upper bound of the load profile of each user is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 2 and the lower bound is set to 40% of the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The selling price for carbon allowances rs c is 14 set as $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='003/kg and the day-ahead selling price for electricity rt e is set as $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='06/kWh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The initial carbon allowance allocation is based on an equal per capita allocation, which is 1800 kg per user in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' All the confidence levels for chance constraints are set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' In the ADMM procedure, the tolerance levels for primal residuals are set to 10−6 for sr and sd, and 10−4 for se and sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Tolerance levels for dual residuals are chosen the same as the corresponding primal residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The stopping criteria for the bound contraction are chosen to be 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' All the optimization problems are carried out in MATLAB 2021b platform using Gurobi [37] solver along with Yalmip[38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The simulations run on a computer featuring AMD Ryzen 7 5800H @3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='20GHz and 16 GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Market Outcome In this part, we will firstly discuss the outcomes of the proposed joint market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 4 present the outcomes of electricity trading and uncertainty balance, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 3 illustrates consumption profile for each user, including the load, lower bound of the load (Lb) and components of the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' It can be seen that for each user, the electricity purchase and demand reach the balance in each time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' In ad- dition, since MT3 is more expensive and has a relatively high carbon intensity, users procure the least energy from MT3 as revealed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Conversely, MT1, who has the least cost parameter and carbon intensity, possesses the largest percentage among all three MTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' This statement also explains the absence of MT3 in uncertainty balance as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 4, which illustrates the participation factors of users and MTs in compensating for the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' It is obvious that for each PV, the sum of the partici- pation factors of users and MTs is equal to 1, which implies that the forecast error can be fully offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Meanwhile, user3 does not participate in uncertainty balance during all time periods because it has reached its lower bound in the energy market as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 3, excluding itself from providing flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' To achieve carbon allowance balance, PVs are required to purchase allowances from the market and users can decide whether to procure allowances to consume more energy or sell superfluous allowances to make profits based on their consumption pro- files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The outcome of the carbon market is provided in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' It is revealed that in this case, all users can sell allowances to both PVs and the community manager by adjusting their consumption behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Besides, following KKT conditions, it is trivial to state that the price in the allowance sharing is the same as the selling price to the community manager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Next, the influences of the carbon allowance price and electricity price on the mar- ket outcomes are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' It is assumed that the carbon allowance price ranges from $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='001/kg to $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='006/kg and the electricity selling price is increased from $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='04/kWh to $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='08/kWh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The variations in social welfare, total allowances sold to the manager and total PV generations sold to the manager are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' It can be observed from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 5 (a) that the social welfare is improved along with the increment in electricity and allowance price since the whole community can sell allowances and PV generations at a more favorable price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 5 (b)&(c) reveal that the total allowances sold to the community manager are reduced with the increase of electricity price and the decrease of allowance price whereas the total PV generations sold to the manager soar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The decline in allowance price undermines the economic 15 MT3 Load Lb (a) User 1 MT3 Load Lb (b) User 2 00:00 04:00 08:00 12:00 16:00 20:00 0 20 40 60 80 100 Energy (kWh) Time PV1 PV2 MT1 MT2 MT3 Load Lb (c) User 3 Figure 3: Optimal energy consumption of each user in the day-ahead market U2 U3 (a) PV 1 09:00 13:00 17:00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='0 Reserve Requirement Percentage Time MT1 MT2 MT3 U1 U2 U3 (b) PV 2 Figure 4: Uncertainty balance of each PV in the day-ahead market Table 3: The outcome of the carbon market Participants U1 U2 U3 PV1 PV2 Allowance Sharing Price �$/kg� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='003 Quantity �kg� 204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='06 204.' metadata={'source': 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+page_content='95 716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='55 / / 16 04:00L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='WG 00:80 IS:000 JQ:00IV bA50:00ITM TM00:00 0 5O 一40 0 000 J00 JSO04:00 08L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='G 00:8 JS:00bI Q:00IV bA50:00ITM STM00:005O 40一 80L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='G 13:00ITM STM M1J:00&TI IU00:00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='0 BG2GLIG BednLeGI 0°4- 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='08 0 5 10 15 20 25 30 35 40 45 Electricity Price r e ($/kWh) Social Welfare ($) Allowance Price r c ($/kg) r e =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='06 r c =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='003 (a) Social Welfare 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='08 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 Electricity Price r e ($/kWh) Allowance (kg) Allowance Price r c ($/kg) r e =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='06 r c =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='003 (b) Total allowances sold to the manager 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='08 0 400 800 1200 1600 2000 Electricity Price r e ($/kWh) PV Generation Sold (kWh) Allowance Price r c ($/kg) r e =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='06 r c =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='003 (c) Total PV generations sold to the manager Figure 5: Social welfare, total allowances and PV generations sold to the manager as a function of electricity and carbon allowance price incentives for users to cut down daily consumption and to prefer green energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Mean- while, the increment in electricity price additionally stimulates the readiness of PVs to sell energy to the community manager instead of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The Effect of Carbon Allowance Sharing and Load Flexibility In this part, the effects of the carbon allowance sharing mechanism and load flex- ibility are demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' To make a comparison, three different cases are considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Case 1: The joint market devised in this study is discussed, where carbon allowance sharing and load flexibility are both included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Case 2: The proposed joint market is considered, except for load flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Case 3: The proposed joint market is adopted, while the carbon market model is modified according to [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' In this case, the participants can only purchase/sell allowances to the operator at fixed prices and the allowance balancing constraint (9) is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The selling price herein is still $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='003/kg and the purchase price is set as $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='009/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The discrepancies in social welfare and total allowances held inside the community among the above three cases are presented in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 17 Table 4: Results in different cases Cases Social Welfare [$] Total Allowances Held Inside the Community [kg] 1 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='9858 3111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='3 2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='5043 3295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='7 3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='4972 3064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='5 As can be vividly discerned from Table 4, the whole community has more demands for carbon allowances in case flexibility is not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' This increase results from the need of PVs to balance uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' In Case 2, the only access for PVs to bal- ance uncertainty is purchasing upward reserve of MTs, which, therefore, inevitably precipitates more carbon emissions and demands for allowances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The decrease in the allowances sold to the community manager also lowers the social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Hence, em- ploying the flexibility of users occupies a crucial role in promoting social welfare and reducing carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' In Case 3, the conspicuous shrinkage in social welfare is mainly due to the high cost of purchasing carbon allowances, which suppresses the need of PVs for allowances, and thereby saps the will of PVs to procure upward reserve of MTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Compared with Case 1, it can be inferred that the carbon allowance sharing mechanism helps facilitate allowance trading among participants, generate a more affordable purchase price, and thus, improve total social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Convergence Analysis The evolution of the primal residuals and dual residuals through the proposed al- gorithm are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' To accelerate the convergence speed, an adaptive penalty factor method as put forward in [33, Chapter 3] is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 6, the proposed algorithm reaches stopping criteria within 3 bound contraction iterations (denoted as ’round’) and a total of 488 ADMM iterations, verifying its convergence performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Besides, the fluctuations in the social welfare are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' The optimal social welfare in the proposed algorithm is $13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='9858 and $13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='9869 in the centralized algorithm, which results in a negligible optimality gap of less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='01%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' This indicates that the proposed solution techniques can effectively cope with the de- centralized trading in the energy community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Next, we will herein illustrate the efficiency of the warm-start method mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Although the convergence of ADMM is guaranteed for any initialization points, a good starting point can dramatically reduce the iterations and computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' In this study, it is reasonable to adopt the results obtained from the last round as an approximation of accurate ones since these results provide a lower bound to the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Based on the fixed penalty factor version of ADMM, the total iterations required to converge under two different situations are provided in Table 5, one for results without a warm start and the other for results with a warm start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' For the first round, both of the two methods involves 1969 ADMM iterations since they start from the same initialization points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' for the second round,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' a transpar- ent reduction in the number of iterations appears when the warm start is employed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Primal Residuals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='sr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='sd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='se ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='sc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Round 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='(a) Primal residuals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='td ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='te ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='tc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Dual Residuals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Round 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='(b) Dual residuals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Figure 6: Primal and dual residuals during different rounds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Social Welfare ($) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Algorithm 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Centralized Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Figure 7: Social welfare of the proposed algorithm and the centralized algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Table 5: Iteration numbers with/without warm start ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Round 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Round 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='Total ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='No Warm start ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='1969 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='1518 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='3487 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='With Warm start ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='1969 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='2002 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='demonstrating that the warm-start method contributes to the superior convergence per- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' By comparison with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 6, it is also found that a marked increase in total iterations occurs when adopting fixed penalty factor method, which verifies the superi- ority of the adaptive penalty factor method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Conclusion This paper proposes a joint day-ahead market paradigm where participants simulta- neously trade energy, uncertainty, and carbon allowances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Moreover, this paper consid- ers the possible excessive carbon emissions caused during uncertainty balance, which are non-negligible for the management of the carbon emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Simulations have re- vealed several merits of the devised market: 1) The introduction of carbon allowance trading guarantees that the total carbon emission of the energy community does not ex- ceed the prescribed limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 2) Proper electricity price and allowance price are conducive to the local consumption of renewable energy and the reduction of carbon emissions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 3) The renewable agents can fully offset the uncertainty by procuring reserve from conventional generators and flexibility from users;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 4) The proposed Relax-ADMM- Contraction loop is privacy-friendly, and can simultaneously yield trading quantities and trading prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' However, it should be noted that the Chebyshev approximation is generally far too conservative, and thereby may lead to the decline in the market ef- ficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' A further study should therefore concentrate on the methods to bypass the conservatism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Acknowledgements This work was supported in part by the National Key R & D Program of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 2020YFE0200400), and in part by the National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 52177077).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' jiemin, W.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='gurobi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='com [38] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Lofberg, Yalmip: A toolbox for modeling and optimization in matlab, in: 2004 IEEE international conference on robotics and automation (IEEE Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 04CH37508), IEEE, 2004, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 284–289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' 23 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content=' Nomenclature Sets and Indices Ωg Set of conventional generators Ωu Set of users Ωr Set of renewable agents ξg i Decision variable set of conventional generator i ξu i Decision variable set of user i ξr i Decision variable set of renewable agent i t ∈ T Index of time steps Parameters and Matrices µt i Expectation of ωt− i δt i Variance of ωt− i Ψ0 i Initial carbon allowance allocated to user i pt u/g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i Lower bound of user/conventional generator i pt u/g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i Upper bound of user/conventional generator i Pt r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i Forecast generation of renewable agent i σi Carbon intensity of conventional generator i Variables Est ij Energy quantity trade from seller j to buyer i at time t αt ij Participation factor of conventional generator i βt ij Participation factor of user i ci Carbon allowance trade quantity inside the community cs i Carbon allowance quantity sold to the manager pt u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i Energy usage of user i at time t pt g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i Output of conventional generator i at time t pt r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'} +page_content='i Energy traded in the community of renewable agent i at time t 24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFLT4oBgHgl3EQfny-J/content/2301.12129v1.pdf'}