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+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.
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+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.
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+

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+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. 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. These loopholes can be exploited by
+malicious attackers to jeopardize our daily use of WiFi devices. 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.
+REFERENCES
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+150 m
+50 m
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+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.
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+12
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+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)
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+22
+

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+1 
+ 
+ 
+Physical Realization of a Hyper Unclonable Function 
+ 
+Sara Nocentini*1,2, Ulrich Rührmair3,4, Mauro Barni5, Diederik S. 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 (L<Trans)
+PUF (L,Trans)
+Ril
+RiK
+RiN
+R
+Responses
+Responses 
+4 
+ 
+Figure 2. Polymer dispersed and polymer stabilized liquid crystals. a) Scheme of the polymeric film used as 
+disordered photonic medium to realize the HPUF. The LC droplets, whose molecular composition is reported on the 
+left, are randomly dispersed into the polymeric matrix (polydimethylsiloxane, PDMS). The polymer stabilized LC 
+formulation is made by a mesogen (5CB), a chromophore (Dispersed Red 1, DR1), and a bi-acrylate (cross-linker) 
+mesogen that enables a full recovery of the LC alignment. A scanning electron microscope image of the side view of 
+the film is reported on the right. b-c) Representative scheme of the molecule arrangement inside the PD&SLC droplets 
+for the switchable operation. b) Scheme of the LC molecular arrangement within the droplets in the nematic  and c) 
+isotropic phase.  d-f) Polarized Optical Microscope images of the PD&SLC before (d), during (e) and after (f) the blue 
+light illumination indicated by the dashed blue circle. The four dot cross pattern (d) is a signature of the LC radial 
+alignment in each droplet39 and it is lost under blue laser illumination. This is the indication that the stabilized LC 
+polymer does not prevent full LC disordering to the isotropic phase. Once the blue illumination is removed (e), the 
+system evolves in around 10 seconds to the previously aligned configuration with the same four dot feature that was 
+present before the transformation (as highlighted by the green circles in f). The scale bars are 10 µm in length. 
+The experimental characterization of a HPUF is illustrated in Fig. 3a. The system is illuminated with the 
+challenge CiProbe that is generated by modulating the Gaussian wavefront of an He-Ne laser using a digital 
+micro-mirror device (DMD) 40,41. Light is then scattered by the HPUF, generating in the far field the 
+response Rik (the speckle pattern) – a 2D image whose spatial features depend uniquely on the probe CiProbe 
+and transform challenge LkTrans  of the system. The response Rik is imaged on a CCD camera, then filtered 
+and binarized to generate the key. The raw speckle images (the optical responses Rik) are converted into 
+binary keys by using a Gabor filter to remove pixel-scale noise, averaging the undesired intensity variations 
+and extracting the independent bits3,27. The parameters of the Gabor filter have been tuned in order to 
+maximize the extractable entropy from the PUF response.  
+Switching between the levels of the HPUF is triggered by the bright blue profile LkTrans (spatially overlapped 
+on the bright red challenge CiProbe), generated by a standard projector. 
+
+a
+5CB
+PD&SLC
+CL
+DR1
+100μm
+Blue lightOFF:nematicphase
+C Blue light ON: isotropic phase
+Phase
+transition
+Full recovery
+5CB
+5CB
+CL
+PDMS
+CL
+PDMS
+DR1
+DR1
+8
+Blue light OFF
+Blue light ON
+Blue light OFF 
+5 
+ 
+ 
+Figure 3. Hyper PUF characterization: one and two-level operation. a) Schematics of the HPUF characterization 
+setup where a binary challenge CiProbe is incident on the physical hardware. The transmitted intensity profile Rik is 
+collected in the far field by a CCD camera and successively converted into a binary post-processed key. A second 
+light beam LkTrans, generated by a blue LED and spatially modulated by a DMD (integrated on a projector board), is 
+used to reversibly transform the HPUF. Among the RGB colors of the projector, the blue LED was chosen as it better 
+matches the dye absorption peak. The two optical images are overlapped on the token. b) Characterization of one-
+single level of the HPUF. The mean value of the like FHD shows that, on the average, the 12% of the bit between two 
+keys generated by the same challenges are different. The test has been made with 400 challenges. The unlike FHD 
+(orange histogram) is the result of 79800 pairwise comparisons that results from all possible comparisons of the 400 
+responses to random challenges. On average, each pair of generated keys differs in the 50% of its bits and the number 
+of the independent bits is N1Level =928 bits. c) Temporal evolution of the Pearson correlation coefficient between two 
+raw responses (two speckle images) acquired every two seconds by interrogating the sample with the same challenge 
+CiProbe while switching the configuration beam between L1Trans and L2 Trans. The three curves correspond to three 
+different values of the blue-light intensities (60, 85, 105 mW/cm2), indicating that higher intensities induce a more 
+efficient decorrelation as well as faster dynamics. We observe a full recovery of the LC alignment after every LC 
+phase transition. d) Like and unlike FHDs of the HPUF for a two-level configuration (intensity: 85mW/cm2). The 
+number of the independent bits of the generated key is N2Level =1323 bits. In the inset, we report the responses for the 
+two transformer challenges L1Trans and L2Trans to two different challenges, C1Probe and C2Probe.  
+The characterization of the HPUF has been performed by evaluating the entropic content of the keys 
+generated by an increasing number of levels: from one-single level PUF up to a ten-level HPUF. We firstly 
+characterize the one-single level system by interrogating the HPUF with challenges (CiProbe, LTrans=0) with 
+i={1,…,100}. Fig. 3b shows that the like and unlike FHDs distributions are well separated, and that the 
+authentication threshold can be safely set around 0.35. The number of the independent bits of the generated 
+keys is estimated to be N1Level=928 bits.  
+
+One-levelPUF
+b
+a
+0.14
+Challenge C,probe
+HPUF
+Response Rik
+1
+Intra-deviceunlikeFHD
+1
+0.12
+Intra-devicelikeFHD
+1
+He-Ne
+0
+0.1
+Laser &
+1
+DMD
+Hashing
+0
+Gabor
+0
+N=928bits
+1
+0.06
+0
+0.04
+Projector:
+0
+Blue LED&
+1
+0.02
+DMD
+0
+0
+0
+1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+FHD
+p
+c
+Two-level HPUF
+0.2
+60mW/cm
+UnlikeFHD
+85mW/cm²
+Trans
+LikeFHD
+105mW/cm
+0.8
+C,Probe
+0.15
+*0
+O口
+Frequency
+口
+0.1
+N=1323bits
+米
+米
+米
+0.05
+Lo
+0.2
+口
+口
+0
+0
+50
+100
+150
+200
+250
+300
+350
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Time (sec)
+FHD 
+6 
+ 
+The next step is to characterize the two-level HPUF. The first level is obtained by completely shading the 
+blue light, while the second one is configured by illuminating the PUF with a uniform blue wavefront. The 
+responses of the two levels interrogated with the same challenge are well decorrelated and also reproducible 
+(see Fig. 3c). The challenge-response characterization of a two-level HPUF is done by illuminating each 
+level with the same set of 100 random challenges (Fig. 3d). The unlike FHD distribution is obtained by 
+comparing all possible pairs of responses (roughly 2*104 pairwise comparisons), while the like FHD 
+distribution is obtained by comparing a defined set of 150 random challenges acquired multiple times for 
+each level. The two distributions do not overlap and the authentication threshold can be set around 0.4. We 
+also observe a net gain in the independent bits (entropy) of the keys generated by the two-level HPUF with 
+respect to a one-level HPUF, from N1Level= 928 bits to N2Levels= 1323 bits (Fig. 3b and Fig. 3d). This is the 
+indication that two keys generated by the two-level HPUFs have a greater probability of differing in 50% 
+of the bits (optimal situation), compared to two keys of one-level PUFs.  
+The natural question that arises is whether, and to which extent, a further increase in the number of levels 
+of the HPUF increases the entropy of the generated keys. To investigate this problem, we define a set of 10 
+transformer challenges (LkTrans, with k={1,..,10}) by choosing 10 elements of the Walsh-Hadamard binary 
+basis – i.e. a subset of a complete 16 orthogonal set of 4x4 macropixel images. Each level is interrogated 
+with the same set of randomly selected challenges CiProbe  with i={1,.., 100}. Fig. 4a shows the scheme of 
+the domain and codomain (Rik, with i={1,.., 100} and k={1,.., 10}) of the HPUF.  The whole codomain of 
+responses Rik can be compartmentalized by randomly joining the codomains of individual levels. For each 
+compartment, we evaluate the entropy per symbol of the keys, i.e. the entropy per bit. Fig. 4b left panel 
+shows that the number of independent bits is of around 950 (0.14 bit/bit). This value is almost independent 
+on the chosen level (in this case, each compartment is composed by the codomain of a single level). By 
+populating the compartments with the codomains of more levels (up to ten), the entropy of the generated 
+key increases up to around N10Levels= 1750 bits that correspond to 0.24 bit/bit (Fig. 4b, left panel, red circles).   
+The increase of the entropy per symbol evaluated by the Daugman’s analysis is confirmed by modeling the 
+extracted keys with equivalent Markov chains, generated via transition matrices whose coefficients 
+represent the permanence and transition probabilities of the binary values of the keys. The entropy per 
+symbol of the Markov chains is then calculated analytically30. The fact that the experimental data analyzed 
+with two models show the same entropic trend suggests that the different levels behave like different 
+cryptographic primitives coexisting in the same hardware.  
+ 
+To validate this idea, we fabricated ten different cryptographic primitives. We applied the same 
+compartmentalization scheme making an analogy among each codomain of the ten different PUFs and each 
+codomain of the ten levels of the HPUF. We observe that the entropy of the generated keys as a function of 
+the number of PUFs, has absolute values and a dependence qualitatively similar to the HPUF (Fig. 4b, left 
+panel, black circles). This is the confirmation that the transformer challenges LkTrans induce different 
+microscopic configurations in the same region of the sample, mimicking different PUFs. It is important to 
+notice that the entropy increase does not depend on the number of responses of the PUFs but only on the 
+number of levels of the HPUF. Increasing the number of responses but considering a single configuration, 
+the entropy per symbol of the PUF remains constant.  
+The increase in entropy per symbol implies a greater unpredictability of the bit sequence. By analyzing the 
+properties of the equivalent Markov chains, we observe that the increase of the number of levels leads to 
+permanence and transition (α and 1-α respectively) probabilities of the Markov transition matrix, that  tend 
+towards a situation of equiprobability (α = 0.5). This implies the reduction of the correlation length in the 
+bit sequence (Fig. 4c-d). Indeed, the correlation length of the bit sequence gets shorter and shorter when 
+
+ 
+7 
+ 
+increasing the number of levels (Fig. 4c), until it reaches an asymptotic value and the entropy per symbol 
+saturates. The physical origin of the increase in the entropy per symbol is due to an increases of the 
+microscopic configurations of the system probed by the challenge CiProbe, that translates in an increase of 
+the variety of speckle patterns that form the codomain of the HPUF. The growing rate of the entropy is 
+reduced up to a saturation level when the compartment is populated by roughly 8-10 levels. For a given 
+size of the challenge CProbe and LTrans, the entropy per symbol saturates when light probes all the possible 
+accessible configurations of the system. 
+Figure 4. Hyper PUF characterization. a) Scheme of the domain (CiProbe, LkTrans) and the codomain Rik of the HPUF. 
+The whole codomain can be compartmentalized by joining the responses of different levels. b) Entropy per symbol 
+for different compartments of the HPUF codomain. The left panel (blue circles) shows the entropy per symbol for 
+each single level (the horizontal labels show the Hadamard basis configuration patterns). The right panel shows the 
+increase of the entropy per symbol by randomly joining the codomains of individual levels (black circles). Red circles 
+refer to the same analysis performed by populating the compartment by joining the responses of different PUFs. The 
+blue error bars refer to 10 different characterizations of each single level. The black and red error bars refer to the 
+standard deviation calculated over ten different random selections of the PUF or levels of HPUF, respectively. c) 
+Autocorrelation of the bit sequences generated by equivalent Markov chain. The correlation length decreases as the 
+number of levels increases, because the permanence probability of the equivalent key increases from α = 0.07 to α = 
+0.12. d) Representation of the binary keys generated by the equivalent Markov chains for one level (α = 0.07) and ten 
+levels (α = 0.12).  
+Conclusions  
+We developed new optical cryptographic primitives, named Hyper PUFs or HPUFs, that allow multi-level 
+operation thanks to fully reversible switching of their optical properties. The all-optical HPUF of this paper 
+is realized with polymer dispersed and stabilized liquid crystals, and the transformation of the levels is 
+enabled by a light pattern that can selectively and locally drive the reversible phase transition of the 
+embedded liquid crystals. The entropy of the HPUF’s keys has been studied using different methods, both 
+confirming its increase with the number of joint levels. These results show that the HPUF is equivalent to 
+combining several physical unclonable functions into a single hardware. The overall entropy per bit is 
+
+a
+b
+0.35
+o1LevelPUF
+HyperPUF
+Diff.PUFs
+0.3
+Trans
+R,10
+0.25
+HPUF
+0.2
+1
+Ri,k
+10 levels
++++
+klevels
+1 level
+C
+Trans
+R,1
+0.1
+domain
+codomain
+8
+2
+4
+6
+8
+10
+Numberof Levels/Numberof PUFs
+d
+α=0.07
+α=0.12
+C
+1Lev:Q=0.07
+3Lev:α=0.95
+0.8
+10 Lev: α=0.12
+20
+20
+40
+40
+Bits
+Bits
+0.4
+60
+α / # Levels
+60
+0.2
+80
+80
+..
+0
+:
+100
+0
+5
+10
+15
+20
+20
+40
+60
+80
+100
+20
+40
+60
+80
+100
+Lag (bits)
+Eg.MCkeys
+Eg.MC keys 
+8 
+ 
+affected by the unavoidable presence of spatial correlations between the microscopic configurations and 
+reaches a saturation levels when the challenge light probes all the microscopic configurations of the system. 
+We believe that the concept described in this paper allows for the development of a new generation of all-
+optical security devices. Amongst the advantages is the unique possibility to create multi-level PUFs 
+integrated into one and the same material, thus enabling a practical implementation of quantum secure 
+authentication of data10. This not only opens up to novel quantum protocols via strong optical PUF but 
+significantly increases their security level and also allows to have multi-user key generators and hence 
+multiple clients on one device. 22,33 
+ 
+Acknowledgements 
+The research leading to these results has received funding from Ente Cassa di Risparmio di Firenze 
+(2018/1047), AFOSR/RTA2 (A.2.e. Information Assurance and Cybersecurity) project “Highly Secure 
+Nonlinear Optical PUFs” (Award No. FA9550-21-1-0039) and Fondo premiale FOE to the project “Volume 
+photography: 
+measuring 
+three 
+dimensional 
+light 
+distributions 
+without 
+opening 
+the 
+box” 
+(E17G17000300001). 
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+

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+EFFICIENT ATTACK DETECTION IN IOT DEVICES 
+USING FEATURE ENGINEERING-LESS MACHINE 
+LEARNING  
+ARSHIYA KHAN1 AND CHASE COTTON2 
+1University of Delaware, Newark, USA 
+arshiyak@udel.edu 
+2University of Delaware, Newark, USA 
+ccotton@udel.edu 
+ABSTRACT 
+Through the generalization of deep learning, the research community has addressed critical challenges in 
+the network security domain, like malware identification and anomaly detection. However, they have yet to 
+discuss deploying them on Internet of Things (IoT) devices for day-to-day operations. IoT devices are often 
+limited in memory and processing power, rendering the compute-intensive deep learning environment 
+unusable. This research proposes a way to overcome this barrier by bypassing feature engineering in the 
+deep learning pipeline and using raw packet data as input. We introduce a feature engineering-less 
+machine learning (ML) process to perform malware detection on IoT devices. Our proposed model,” 
+Feature engineering-less ML (FEL-ML),” is a lighter-weight detection algorithm that expends no extra 
+computations on “engineered” features. It effectively accelerates the low-powered IoT edge. It is trained 
+on unprocessed byte-streams of packets. Aside from providing better results, it is quicker than traditional 
+feature-based methods. FEL-ML facilitates resource-sensitive network traffic security with the added 
+benefit of eliminating the significant investment by subject matter experts in feature engineering. 
+KEYWORDS 
+Feature engineering-less, AI-enabled security, 1D-CNN, Internet-of-Things, Botnet Attack 
+1. INTRODUCTION 
+Cyber Security experts have found pivotal features in network traffic, including packet captures 
+(pcap). Data scientists have used them to fashion impressive models capable of differentiating 
+malicious traffic from benign [1]. However, most network traffic is emitted over encrypted 
+channels in the current scheme. This security measure has limited experts’ ability to contrive 
+meaningful features for machine learning (ML), which can soon become obsolete. This challenge 
+has given birth to analyzing raw bytes to detect malicious behavior in internet flows. 
+In the Internet of Things (IoT) domain, devices are sensors that interact with the environment. 
+They conditionally react to changes in the environment and exchange information over the 
+internet about these changes. A typical example of an IoT device is a doorbell camera. We can 
+check any activity at our door using the camera from anywhere on earth. It can also alert us of 
+any break-ins. For an IoT device to operate, it must continuously communicate with its users, 
+other devices, and servers without fail. This communication is done over the internet, where 
+information travels in the form of packets. This flow of packets is commonly known as network 
+traffic. For an IoT grid to function, the network traffic must be secure from cyber criminals. 
+Intrusion Prevention Systems (IPS) (like Cisco Firewall and McAfee), and Intrusion Detection 
+Systems (IDS) (like SolarWinds and Snort), can scan the network traffic and determine if they 
+are secure or insecure. Historically, packets are validated using a pre-defined rule book. However, 
+
+since the introduction of ML, many attempts have been made to detect insecure packets using 
+ML. Both detection systems display substantial hardware constraints like processing power and 
+sizeable memory requirements to store moving traffic and identify anomalous packets. The 
+evolving world of smart sensors and IoT devices demands precision and speed simultaneously. 
+Researchers have used machine learning to detect traffic anomalies in several studies [2,3]. They 
+have followed the standard pipeline of typical ML modeling. By extracting features from raw 
+data, they have trained their models. In the network traffic specialization, they have manipulated 
+values in the packets to create features. This conventional approach is not only ineffective over 
+encrypted channels but is also task-sensitive. Attack scenarios on network traffic have evolved 
+rapidly. Now, attackers can hit multiple levels of network architecture [4], rendering the task-
+sensitive approach defenseless. In 2020, a group of researchers published a survey [5] on the use 
+of ML to enable security in IoT devices. They divided the survey into five sections dedicated to 
+understanding the enormity of security challenges. The survey discussed current ML solutions 
+like anomaly detection, attack evasion, and mitigation. However, it is only possible to use these 
+methods if we already know the type of attack being launched. The survey also discussed the 
+complexity of IoT networks and multi-level attacks [6] suffered by these devices. They can cause 
+the IoT infrastructure to fail anywhere from the application layer to the physical layer. Such a 
+complicated structure makes it impossible to design a defense model that can guard against all 
+possible attack scenarios. 
+Therefore, it is imperative to leverage the generalized scope of ML, which can learn even minor 
+irregularities from the dataset. Traditional attack-specific feature-based detection models can crop 
+or manipulate the dataset in an unhelpful way. 
+The remaining paper is organized as follows. Section 2 is dedicated to a detailed review of 
+contemporary work. We have discussed the use of deep learning models for network traffic in 
+section 2.1 and their feasibility on IoT devices in Section 2.2. Section 3 goes into detail about our 
+proposed methodology. It is divided into three parts. 3.1 discusses our proposed approach in 
+detail. 3.2 talks about the dataset and preparation of the experiment, while 3.3 talks about our 
+deep learning architecture. Section 4 explains the experiment results and performance 
+comparison. Section 5 covers concluding remarks and future work. 
+2. BACKGROUND AND RELATED WORK 
+2.1. The advent of 1D-CNN in Malware Detection 
+The last decade has seen exemplary use of machine learning to classify network traffic. Studies 
+based on these features usually use three approaches: 1) Port-based, 2) Deep packet inspection 
+(DPI)-based, and 3) Behavior-based [7].  
+A port-based classifier classifies reliable ports as benign and unreliable ports malicious. However, 
+this classification technique has been rendered unusable by employing port hiding techniques 
+such as port camouflaging. Port randomization is also used to hide ports.  
+The port-based approach uses only headers to classify traffic; however, the DPI-based approach 
+examines both header and the payload. The header is inspected to collect information about the 
+sender application, and the payload signature is examined to ensure they are not invalid or 
+blacklisted. Due to high computational costs, DPI-based is an inefficient method. Lastly, in the 
+behavior-based approach, flow or session trends are observed. Features to train an ML model are 
+fashioned from the statistical inferences of these trends. 
+Velan et al. [7] published a comprehensive survey of 26 papers in 2015. It investigated studies 
+focused on encrypted traffic classification published between the period 2005 to 2014. Flow 
+features were used in 12 of the 26 papers, packet features in 5, a combination of packet and flow 
+features in 7, and other features in two. 
+
+Wang et al. [8] published another survey in 2019, which addressed the rise in raw packet content 
+to train deep learning models. As more parts of the packets were encrypted, extracting features 
+became difficult. Both headers and payload were concealed, forcing researchers to use raw data 
+in the training set. From 2015 to 2018, 8 out of 12 publications used raw data from the packet for 
+encrypted traffic classification, and two used both derivative features and raw data. One of the 
+remaining two papers solely used packet-based features, while the other used flow-based features. 
+Wei Wang et al. [9] used raw bytes collected in groups of flows and sessions. This study employed 
+the Representation Learning technique and converted raw bytes into images to create the training 
+dataset. It is called USTC-TFC2016 [9], a private dataset pulled from the Stratosphere IPS Project 
+Malware Dataset [10]. It used a two-dimensional convolutional neural network (2D-CNN) to 
+classify the images. This experiment saved a small amount of computational power as it did not 
+require feature extraction. However, it spent a lot on byte-to-image conversion and image training. 
+The binary classifier (malware or benign) was 100% accurate, the 10-label classifier was 99.23% 
+accurate, and the 20-label classifier was 99.17% accurate. The classifiers achieved an average 
+accuracy of 99.41%. For the 10-label classifier, the lowest and the highest precision score were 
+90.7% and 100%, respectively. Similarly, the highest and the lowest recall score were 91.1% and 
+100%, respectively, for the 10-label classifier.  
+Wang et al. [11] conducted a similar study in 2017, where they used the ISCX VPN-non-VPN 
+[12] dataset to perform encrypted traffic classification. This dataset contains six traffic types: 
+chats, emails, file transfer, peer-to-peer (P2P), streaming, and Voice over Internet Protocol (VoIP) 
+captured over both Virtual Private Network (VPN) and non-VPN settings. This study performed 
+binary classification on VPN and non-VPN traffic and multi-label classification on the six traffic 
+types. It used raw bytes of flows and sessions of the ISCX dataset. To train on raw bytes, it used 
+a one-dimensional convolutional neural network (1D-CNN). The 2-label (binary) classification 
+achieved 100% precision, and the 6-label classification achieved 85.5% accuracy. Six traffic types 
+were branched into 12 classes using VPN as a factor (VPN and non-VPN traffic). The 12-label 
+classification achieved 85.8% accuracy. This study compared their results with [9], which used 
+2D-CNN. In a series of 4 experiments, 1D-CNN outperformed 2D-CNN by as much as 2.51%.  
+DeepPacket [13], introduced in 2017, is a deep learning framework for network traffic 
+characterization and application identification. It employed two deep learning techniques: a) 
+stacked autoencoder (SAE) neural network and b) 1D-CNN. Both were trained on ISCX VPN-
+non-VPN [12] dataset. The traffic characterization task classified VPN and non-VPN traffic. With 
+SAE, the binary classifier's average precision and recall were 92%. However, with 1D-CNN, the 
+same classification resulted in average precision of 94% and an average recall of 93%. The 
+application identification task classified packets into these six applications: chats, emails, file 
+transfer, peer-to-peer (P2P), streaming, and Voice over Internet Protocol (VoIP). With the SAE 
+classifier, the average precision of the 6-label classification was 96%, and the average recall was 
+95%. With the 1D-CNN classifier, the average precision and recall were 98%.  
+In 2020, Rezaei and Liu [14] employed multi-task learning to classify network traffic. It divided 
+the classification task into two tasks: a) bandwidth requirement and b) predicting the duration of 
+traffic flow. It used two datasets for training: ISCX VPN-non-VPN and QUIC [15]. The multi-
+task learning model was trained on a 1D-CNN using these three time-series features: i) packet 
+length, ii) inter-arrival time, and iii) direction of the traffic. On the ISCX dataset [12], their model 
+classified traffic with 80.67% accuracy. The same experiment classified bandwidth requirement 
+and prediction of flow duration with 88.67% and 90% accuracies, respectively. On the QUIC 
+dataset [15], the model classified traffic with 94.67% accuracy. In this experiment, bandwidth 
+requirement and flow duration prediction attained 90.67% and 91.33% accuracies, respectively. 
+
+Table 1. Related Works of Network Traffic ML models 
+Work 
+DL 
+Technique 
+Model 
+Input 
+Dataset 
+ML Task 
+Accuracy 
+Year 
+[9] 
+2D-CNN 
+images 
+[9] 
+Malware detection 
+99.23% 
+2017 
+[11] 
+1D-CNN 
+bytes 
+[12] 
+Traffic 
+characterization 
+100% 
+2017 
+[13] 
+1D-CNN + 
+SAE 
+bytes 
+[12] 
+Traffic 
+characterization 
+98% 
+2017 
+[14] 
+1D-CNN 
+time series 
+[12, 15] 
+Traffic 
+characterization 
+94.67% 
+2020 
+[17] 
+1D-CNN + 
+LSTM 
+raw bytes 
+[9] 
+Malware detection 
+98.6% 
+2020 
+ 
+Huang et al. [16] published a survey in 2019. It was based on deep learning use cases in the time-
+series domain. Cyber Security was one of the emergent real-world disciplines in the time-series 
+domain, along with health, finance, and transportation. To achieve this conclusion, the paper 
+analyzed topics such as traffic classification, anomaly detection, and malware identification.  
+Marin, Casas, and Capdehourat [17] published a study in 2020 aiming to remove engineered 
+features, thus eliminating the need for domain experts. They reintroduced two types of malware 
+detection approaches using raw packet content: raw packets and raw flows. They used raw byte-
+streams from pcaps to train their deep learning model to achieve this goal. Their ML model was 
+a combination of a Long short-term memory (LSTM) network and a 1D-CNN. The raw byte-
+stream of flow performed (98.6% accuracy) better than the byte-stream of a packet (77.6% 
+accuracy). A comparative experiment was performed between traditional feature-based and raw 
+byte-based models. For the traditional model, they used a random forest (RF) and trained it on 
+200 in-flow features. Both raw byte-based models performed better than RF.  
+Discussion of the previous works in this section has revealed two significant points:  
+1. They indicate the advent of 1D-CNN in the network traffic characterization and malware 
+detection domain. It is also evident in Table 1, which displays several publications of our study. 
+The table is arranged chronologically and shows a clear trend of ML-based network traffic 
+classifiers preferring 1D-CNN over other techniques. In addition to its superior performance, 1D-
+CNN has the advantage of preserving the time-series nature of network traffic. Inspired by its 
+effectiveness, we have used it as the modeling baseline of our study.  
+2. Studies also suggest that in recent years, malware detection models have moved from 
+engineered feature ML to non-engineered feature ML using raw data. However, these models may 
+only be applicable to some network traffic use cases. It is practically impossible to employ 
+complex models on memory-constrained devices like IoT. More extensive models like LSTMs 
+and 2D-CNNs need several preprocessing steps to extract raw data and train a model on them. In 
+the next section, we will discuss the use of ML practices in the IoT environment. 
+2.2. ML Techniques for Malware Detection in IoT environment 
+Traditional appliances, devices, and machines have moved to smart sensor technologies in the 
+last few decades. In addition to routers and IPSs, these technologies are also a component of the 
+IoT. Uninterrupted internet access makes them susceptible to malicious attacks, which may result 
+in malfunction or failure. Several studies have tried to find security solutions for these IoT 
+devices.  
+
+In 2019, Shouran, Ashari, and Priyambodo [18] introduced a straightforward way of detecting 
+threats in IoT devices. It classified every device interaction with the internet into Low, Medium, 
+and High impact. The classification criteria included compromise in Confidentiality, Integrity, 
+and Authenticity (CIA).  
+Another research in 2019 [19] introduced an elaborate IDS for IoT devices. However, the IDS 
+was feature-based with three layers of ML algorithms.  
+Later in 2020, a paper published by Vinayakumar et al. [20] developed a two-level deep learning 
+model which discriminates malicious traffic from benign. The first level detected the most 
+frequent DNS queries, and the second level used a domain generation algorithm (DGA) to detect 
+illegal domains.  
+Sriram et al. [21] presented their deep learning ML system, which used network flows to find 
+statistical features. Two datasets were used and compared over different ML modeling techniques 
+like logistic regression, random forest, and LSTM. [17] also presented a 1D-CNN model for 
+botnet detection on IoT devices. Details of their work are discussed in section 2.1. 
+Table 2. Related Work in IoT domain 
+Work 
+IoT Malware detection Technique 
+Task 
+Year 
+[18] 
+Non-ML (Rule-based) 
+Malware detection 
+2019 
+[19] 
+ML-based IDS 
+Traffic characterization 
+2019 
+[20] 
+ML based DNS categorization 
+Malware detection 
+2020 
+[21] 
+Ensemble of ML models 
+Botnet detection 
+2020 
+[17] 
+1D-CNN 
+Malware detection 
+2020 
+ 
+Table 2 shows a task-based analysis of research works published on detecting malware on IoT 
+devices. Most of the methods presented here require multiple steps to perform one task. This 
+approach is not suitable for the IoT environment.  
+Our contribution is as follows:  
+1. Outright elimination of ‘engineered features’ that require additional computation. We introduce 
+a network traffic classification system in favor of more lightweight features which come directly 
+from the input data. As a result, it does not require domain-based expertise to perform feature 
+engineering.  
+2. Increase the speed of classification by using 1D-CNN to train the deep learning model. The 
+model will consume less memory during both the training and testing phases allowing it to be 
+deployed on IoT devices. 
+3. METHODOLOGY 
+3.1. Feature engineering-less ML 
+IoT devices at the edge, like voice-based virtual assistants (e.g., Amazon Echo), smart appliances, 
+and routers, must react to change at a very high speed. Consequently, they have to validate 
+incoming traffic in real-time. They also have a limited battery life to support these unremitting 
+transactions. In an internet-dependent environment like this, security from cyber attacks is non-
+negotiable; nevertheless, it can become an overhead. It can be in various forms, like malware 
+recognition, anomaly detection, or behavior classification. Detecting cyber attacks using ML has 
+shown promising results in the past [22, 23]. However, their deployment on IoT devices is 
+unrealistic, as they involve computationally extensive feature engineering and require deep 
+
+classifiers for precision. Feature engineering is a step-by-step process that incorporates: i) 
+extracting desired elements from raw data, ii) cleaning and converting them into features, iii) 
+standardizing features, and iv) aggregating them to be used in the classifier. For a time sensitive 
+IoT environment, it is a complex and time-consuming operation. We cannot rely on traditional 
+methods to make ML adequate on IoT devices. In this study, we propose skipping feature 
+engineering and developing an IoT-friendly deep learning technique called Feature Engineering-
+less ML.  
+Feature engineering-less ML or FEL-ML is a product of the featureless modeling technique. 
+Feature engineering-less modeling is machine learning without feature engineering. It is a lighter-
+weight detection algorithm where no extra computations are expended to compute ‘engineered 
+features,’ resulting in an adequate acceleration to the low-powered IoT. We eliminate the feature 
+extraction and processing step from the ML pipeline in FEL-ML. We store the streams of packets 
+that arrive at an IoT device in their raw state. We create our training dataset by converting the raw 
+streams to raw byte streams. The rest of the deep learning process remains the same. The 
+advantages of FEL-ML are that it conserves the properties of a stream of bytes and saves time 
+during the process. Omitting the feature extraction and generation step makes both model training 
+and testing efficient.  
+Feature engineering-less modeling eliminates two significant IoT overheads: computation cost 
+and human cost. Human cost involves using technical expertise to collect and clean traffic data. 
+It also employs domain expertise to manipulate them into meaningful representatives of the 
+dataset. Computation cost incorporates the computational overhead a device endures toward 
+statistical operations at a device’s central processing unit (CPU). Further down the ML pipeline, 
+tensor-based computations like convolution, batch processing, and running multiple epochs 
+contribute to computational overhead.  
+We have represented network traffic in three views: 1) Raw session, 2) Raw flow, and 3) Raw 
+packet. Based on these three representations, we have developed three unique ML models. In the 
+end, we will compare their performance to determine the most suitable model.  
+Raw session: In an IoT network, a session is a bi-directional stream of communication be- tween 
+two devices. All packets in a session share these 5-tuple attributes: a) source IP address, b) source 
+port, c) destination IP address, d) destination port, and e) transport protocol present in each packet 
+header. In this representation, we split the pcap files into unique sessions based on the 5-tuple. 
+We used MIT’s pcapsplitter tool [24]. A pcap file records the live traffic stream on a device. Each 
+individual session stream forms a unique component in the ground truth.  
+Raw flow: A flow is similar to a session, except that they are unidirectional. A flow is a batch of 
+packets going from one device to another. We used SplitCap [25] to prepare our dataset in this 
+representation. SplitCap splits pcap files into flows. A sample in the ground truth is a byte stream 
+of a flow.  
+Raw packet: A pcap file consists of one or more packets. In this representation, we used the byte 
+stream of each packet. A single packet in byte format is an individual ground truth component.  
+In a similar study [17], raw bytes were used to train the classifier. However, their experiment only 
+included packets and sessions. Further, they have not discussed any memory or system metrics to 
+demonstrate its efficiency after eliminating feature engineering. Their model constituted 1D-CNN 
+and LSTM layers, i.e., a deep model. Deeper models are more complex and can result in 
+overfitting. As a result, we have used a smaller but effective neural network to overcome this 
+obstacle. 
+3.2. Experimental Design 
+We have used the Aposemat IoT-23 dataset [26] to perform experiments for this study. It is a 
+labeled dataset captured from 2018 to 2019 on IoT devices and published in 2020 by the 
+
+Stratosphere Research Lab. Several studies have either discussed it in their work or used it in their 
+model. Bobrovnikova et al. [27] used this dataset to perform botnet detection in 2020. They 
+extracted features to curate numerical features for classification. They attained 98% accuracy with 
+support vector machines (SVM) [28]. Blaise et al. [29] have mentioned that this dataset is similar 
+to what they required but not pertinent to their experiments. This dataset also contained 
+conn.log.labeled file generated by employing Zeek on this dataset. Stoian [30] used numerical 
+and categorical features from this file. They trained a Random Forest classifier [31] with this 
+dataset and attained 100% accuracy.  
+IoT-23 has 23 sets of scenarios, out of which three are benign and 20 are malicious. The benign 
+dataset is collected from three IoT devices: i) Amazon Echo, ii) Somfy smart door lock, and iii) 
+Philips hue LED lamp. These devices are then infected with an assortment of botnet attacks 
+executed in a simulation environment forming the malicious dataset. 
+ 
+Figure 1. Byte Distribution 
+IoT-23 contains raw pcap files and Zeek files of each scenario. For our experiments, we used the 
+pcap files as our training dataset. We integrated the three benign scenarios into one extensive 
+dataset. The malicious scenarios are prepared in a simulation environment where the three devices 
+are infected by 20 unique botnet attacks. Pcap files of several of these botnets range between 
+hundreds of GBs. To demonstrate the fitness of our proposed model on limited-memory devices, 
+we used attack scenarios with smaller sizes. We have selected five out of 20 attacks: i) Hide and 
+Seek, ii) Muhstik, and iii) Linux.Hajime, iv) Okiru, and v) Mirai. Byte distribution of the five 
+infected and one benign traffic data is shown in Figure 1. Using binary classification, we 
+distinguished between benign and malicious traffic, while multi-label classification distinguished 
+between botnet traffic.  
+As mentioned in section 3.1, we represented this dataset in three unique views: session, flow, and 
+packet. In order to use raw bytes, we converted all traffic representations into a hexadecimal 
+format using tshark [32]. As we take a deep dive into this experiment, we named these three 
+experiments as follows:  
+ExpS: The experiment used the session representation of the dataset in the hexadecimal format. 
+In the training data, each session is represented as one row of a byte stream. 
+ExpF: The experiment used the flow representation of the dataset in the hexadecimal format. In 
+the training data, each flow is represented as a byte stream. 
+ExpP: The experiment used the packet representation of the dataset in the hexadecimal format. 
+Each pcap instance is a sample in the labeled training set. 
+
+450
+400
+350
+300
+ytes
+250
+B
+200
+150
+100
+50
+0
+Hide and
+Muhstik
+Linux.Hajime
+Okiru
+Mirai
+Benign
+Seek
+Packet
+Session
+Flow 
+Figure 2. Training data generation 
+Studies [17] conducted in the past on raw byte streams have an additional step to remove ethernet 
+and TCP/IP headers from their dataset. We experimented with the use of headers to achieve our 
+goal of diminishing computational overhead further. We reformulated the dataset into four unique 
+categories for each ExpS, ExpF, and ExpP. The first category included all headers in the packet 
+along with the payload. We called this category: All headers. In the next category, we kept the 
+ethernet headers and discarded the IP headers; hence we called this category Ethernet headers 
+only. Intuitively, in the third category, we removed ethernet headers from the packet and called 
+this category: Without ethernet. Ultimately, we dropped both ethernet and IPv4 headers from the 
+packet. It was called the No headers category. As a result, all three representations of traffic 
+(session, flow, and packet) were split into these four categories, and each category was then 
+trained separately. As shown in Figure 2, there were 12 different experiments in our study. All 
+categories in each representation had the same size. 
+3.3. DL Architecture 
+At a device, packets arrive as instances of data distributed over time which puts our dataset into 
+the time-series domain. Until recent years 1D-CNNs have been primarily used in the Natural 
+Language Processing (NLP) models [33]. However, they have also successfully classified time 
+series data [34]. This study aims to develop a small neural network that can be installed on 
+resource constraint devices and detect malicious streams of packets. Since tensor computations in 
+1D-CNN take less space than 2D-CNN and other ML methods, it has motivated us to use this 
+lightweight neural network as our machine learning model. A smaller neural network size will 
+increase its practicability on IoT devices. 
+As shown in Figure 3, our smaller neural network comprises two 1D-CNN layers, a Maxpooling 
+layer, a Dropout layer, and a Dense layer. We started with the first convolution layer that 
+performed convolution with a kernel size of 64 and a stride of 3 on the input vectorized byte 
+stream. Small values of these parameters reduced the complexity of the model resulting in less 
+overfitting. The maxpooling layer was sandwiched between the two convolution layers. It 
+performed a 5-to-1 pooling operation to reduce the output tensor size from the upper layer without 
+losing significant properties. The second convolution layer was placed after the maxpooling layer 
+and performed the same job as the first convolution layer. It also had the same environmental 
+controls. Next, we used the dropout layer with a drop rate of 0.5 to reduce validation loss. In the 
+end, we added a fully connected Dense layer that helped the model learn any non-linear 
+relationship between features. 
+ 
+
+PcapFiles
+PacketView
+Flow View
+SessionView
+Packet
+Flow
+Session
+Bytestream
+Bytestream
+Bytestream
+Legends
+1:All headers. 2:Ethernetheaders only 3:WithoutEthernet header 4:No headers 
+Figure 3. DL Architecture 
+The left section of Figure 3 depicts binary classification. It used the binary loss as the loss function 
+and the softmax function for activation. On the other hand, multi-label classification, as depicted 
+in the right section of the figure, used categorical cross-entropy as a loss function and a sigmoid 
+function for activation. The binary classifier performed classification between benign and 
+malicious traffic. We trained the multi-label model to classify the five botnet categories 
+mentioned in section 3.2. The neural network trained over 60,000 hyperparameters in batches of 
+32 using Keras [35] for 50 epochs. For efficiency, we also used a checkpoint function to store the 
+best model whenever encountered during training. It enables TensorFlow [36] to stop training 
+when it achieves the best possible value of the evaluation metric, which is “Accuracy,” in this 
+case. When the hyperparameters are suboptimal, the resultant model becomes complex, leading 
+to overfitting and high validation loss. This will result in more power usage and potentially 
+incorrect classification. 
+We trained on Nvidia GeForce GTX 1060 GPU with a 12GB Ubuntu 16.04 server on an x86 
+architecture. 
+4. EVALUATION 
+4.1. Evaluation Metrics 
+Experiments were evaluated on two metrics: accuracy and f-1 score, as shown in Figure 4. 
+Accuracy is the percentage of correct results from the total results produced by the model. It is 
+calculated on both training and validation data. The f-1 score is the weighted average of precision 
+and recall values on the validation data. f-1 score is also suitable for datasets with uneven 
+distribution which makes it suitable for our experiments.  
+ 
+Figure 4. Evaluation Metrics 
+
+ByteStreams
+ByteStreams
+1-DCNN
+1-DCNN
+Maxpooling
+Maxpooling
+1-DCNN
+1-DCNN
+Dropout
+Dropout
+Dese
+Dense
+Malicious
+Benign
+M1
+M2
+M3
+M4
+M54.2. Experimental evaluations 
+Our experiments indicate that binary classification achieved better accuracy the multi-label. First, 
+we will discuss binary classification outcomes between malicious and benign traffic. Tables 3 and 
+4 display evaluation metrics in each ExpP, ExpS, and ExpF. 
+Table 3. Binary Performance on IoT-23 dataset 
+Representation 
+Header 
+Accuracy 
+f1-score 
+ExpS 
+All headers 
+1.00 
+0.97 
+Only Eth 
+1.00 
+0.96 
+Without Eth 
+1.00 
+0.96 
+No headers 
+1.00 
+0.94 
+ExpF 
+All headers 
+1.00 
+0.97 
+Only Eth 
+1.00 
+0.93 
+Without Eth 
+0.97 
+0.96 
+No headers 
+0.99 
+1.00 
+ExpP 
+All headers 
+1.00 
+0.96 
+Only Eth 
+0.98 
+0.96 
+Without Eth 
+0.98 
+0.97 
+No headers 
+0.99 
+0.95 
+ 
+Table 4. Multi-label Performance on IoT-23 dataset 
+Representation 
+Header 
+Accuracy 
+f1-score 
+ExpS 
+All headers 
+0.99 
+0.96 
+Only Eth 
+0.94 
+0.93 
+Without Eth 
+0.84 
+0.92 
+No headers 
+0.96 
+0.92 
+ExpF 
+All headers 
+0.93 
+0.92 
+Only Eth 
+0.72 
+0.85 
+Without Eth 
+0.79 
+0.91 
+No headers 
+0.91 
+0.90 
+ExpP 
+All headers 
+0.97 
+0.93 
+Only Eth 
+0.98 
+0.93 
+Without Eth 
+0.74 
+0.80 
+No headers 
+0.98 
+0.93 
+ 
+
+In pcap experiment ExpP, the “no-header” category achieved the maximum accuracy of 99%. 
+However, in the “all-headers” category, ExpP achieved 97% accuracy, only 0.02% below the 
+highest but gave the highest 99% f-1 score.  
+In ExpS, binary classification achieved 100% accuracy in every header category. It also achieved 
+the highest f-1 score of 97% in the “all-headers” category.  
+In ExpF, the “all-headers” category again achieved 100% accuracy alongside the “only- 
+ethernet” category. In contrast, ExpF achieved a 100% f-1 score “no-header” category, while the 
+“all-headers” category achieved only a 97% f-1 score. 
+We now switch our attention to multi-label classification between five botnet attack scenarios. 
+Table 4 shows the performance of the evaluation metrics in all categories: ExpP, ExpS, and ExpF. 
+Overall, the 5-label classification achieved a maximum accuracy of 99% and an f-1 score of 96% 
+in the ExpS session scenario. 
+In ExpP, accuracy was 98%, and the f-1 score was 93% f-1 score in the “no header” category. 
+However, the “all-headers” category was only 0.01% behind with a 97% accuracy and the same 
+93% f-1 score.  
+In ExpS, the “all-headers” category achieved the highest accuracy of 99% with a 96% f-1 score. 
+In ExpF, the “all-headers” category again achieved the highest accuracy of 93%, along with the 
+highest f-1 score of 92%. 
+Table 5. Binary comparison between IoT-23 and ETF-IoT Performance 
+Representation 
+Header 
+IoT-23 f-1 
+ETF-IoT f-1 
+ExpS 
+All headers 
+0.97 
+0.89 
+Only Eth 
+0.96 
+0.87 
+Without Eth 
+0.96 
+0.87 
+No headers 
+0.94 
+0.90 
+ExpF 
+All headers 
+0.97 
+0.87 
+Only Eth 
+0.93 
+0.86 
+Without Eth 
+0.96 
+0.87 
+No headers 
+1.00 
+0.98 
+ExpP 
+All headers 
+0.96 
+0.88 
+Only Eth 
+0.96 
+0.89 
+Without Eth 
+0.97 
+0.89 
+No headers 
+0.95 
+0.89 
+ 
+Evidently, session representation or ExpS performed better in binary and multi-label 
+classifications. Overall, accuracy was highest when the dataset included all headers. Accuracy 
+monotonically decreased when either header was removed. However, it recovered when there 
+were no headers. The f-1 score was always the highest when all headers were included, with one 
+exception in multi-label ExpF.  
+This study shows that headers significantly influence the precision of anomaly detection models. 
+Unlike the custom of cropping them out of the training set [17] we achieve better results by 
+incorporating them into the training set. 
+
+4.3. Comparison with another Dataset 
+In this section, we compare the results of our model trained on another dataset named ETF IoT 
+Botnet [37]. ETF is the newest publicly available botnet dataset. It has 42 malicious botnet attack 
+scenarios collected on RaspberryPi devices and two benign scenarios. 
+Tables 5 and 6 show lateral comparisons between both datasets. We have shown f-1 score 
+comparisons. The results differ case by case depending on the placement of the header. 
+Table 6. Multi-label comparison between IoT-23 and ETF-IoT Performance 
+Representation 
+Header 
+IoT-23 f-1 
+ETF-IoT f-1 
+ExpS 
+All headers 
+0.96 
+0.95 
+Only Eth 
+0.93 
+0.93 
+Without Eth 
+0.92 
+0.94 
+No headers 
+0.92 
+0.93 
+ExpF 
+All headers 
+0.93 
+0.96 
+Only Eth 
+0.85 
+0.95 
+Without Eth 
+0.91 
+0.96 
+No headers 
+0.90 
+0.94 
+ExpP 
+All headers 
+0.97 
+0.96 
+Only Eth 
+0.93 
+0.95 
+Without Eth 
+0.80 
+0.96 
+No headers 
+0.93 
+0.93 
+ 
+Figures 5 and 6 show a comparative analysis of f-1 scores between the two datasets. Both datasets 
+show similar results in all categories with a few exceptions. Noticeably, the “all headers” 
+category did not show any change in trend. f-1 scores of this model are consistent with IoT-23, 
+which strengthens the claim of FEL-ML’s usability for malware detection. As the obvious next 
+step, we have tested our model’s feasibility to be deployed on IoT devices. 
+We selected five botnet classes using the same technique we used in IoT-23. They are: 1) 666, 2) 
+SNOOPY, 3) arm7.idopoc2, 4) z3hir arm7, and 5) arm7l 1. We also used the same tools and 
+scripts to extract session, flow, and packet representation from pcap files. We trained on the same 
+GPU setting. All parameters of the 1D-CNN were also the same. 
+ 
+ 
+
+ 
+Figure 5 f-1 score comparison of Binary Classification (a) All headers (b) No headers (c) With 
+Ethernet headers (d) Without Ethernet headers 
+ 
+Figure 6 f-1 score comparison of Multi Classification (a) All headers (b) No headers (c) With 
+Ethernet headers (d) Without Ethernet headers 
+ 
+
+1.00
+0.95
+0.90
+0.85
+0.80
+0.70
+0.65
+0.60
+0.55
+0.50
+ExpS
+ExpF
+ExpP
+loT-23
+■ETF-loT1.00
+0.95
+0.90
+0.85
+0.80
+0.75
+f-1
+0.70
+0.65
+0.60
+0.55
+0.50
+ExpS
+ExpF
+ExpP
+1oT-23
+ETF-loT1.00
+0.95
+0.90
+0.85
+0.80
+0.70
+0.65
+0.60
+0.55
+0.50
+ExpS
+ExpF
+ExpP
+loT-23
+ ETF-loT1.00
+0.95
+0.90
+0.85
+score
+0.80
+0.75
+0.70
+0.65
+0.60
+0.55
+0.50
+ExpS
+ExpF
+ExpP
+1oT-23
+ ETF-IoT1.00
+0.95
+0.90
+0.85
+0.80
+0.70
+0.65
+0.60
+0.55
+0.50
+ExpS
+ExpF
+ExpP
+loT-23
+ ETF-loT1.00
+0.95
+0.90
+0.85
+0.80
+0.70
+0.65
+0.60
+0.55
+0.50
+ExpS
+ExpF
+ExpP
+■loT-23
+ ETF-loT1.00
+0.95
+0.90
+二
+0.85
+0.80
+0.70
+0.65
+0.60
+0.55
+0.50
+ExpS
+ExpF
+ExpP
+loT-23
+ ETF-loT1.00
+0.95
+0.90
+0.85
+score
+0.80
+0.75
+f-1
+0.70
+0.65
+0.60
+0.55
+0.50
+ExpS
+ExpF
+ExpP
+loT-23
+ETF-IoT4.4. Applicability in IoT scenario 
+These devices are cost-sensitive, resulting in lower-performing CPUs and less memory, 
+demanding much lower-cost detection schemes. We couldn’t find a study that addresses this issue. 
+It prompted us to develop a faster ML model than the existing standards. In this section, we 
+compare our FEL-ML’s performance with an existing feature-based ML model. Both models 
+were trained and tested on the same GPU-enabled architecture.  
+Since our focus is on botnet attack detection, for this experiment, we selected an ML model called 
+n-BaIoT [2], which detects botnet attacks on IoT devices. We used the [38] GitHub repository to 
+reproduce their model and train on the dataset used in the original study. We extracted 115 
+numerical features from the dataset, as mentioned in the paper. We performed binary 
+classification between benign and malicious traffic to compare model performances. We used 
+three performance metrics in this experiment:  
+1. Time consumed on testing the dataset, measured in seconds, 
+2. System time, 
+3. CPU utilization. 
+We measured 1 using the “time” function in python. We recorded 2 using the Linux time function. 
+We measured 3 using the “perf” tool on Ubuntu [39]. perf is a profiling tool that provides kernel-
+level information about a program when it executes. 
+Table 6. Performance applicability for IoT devices 
+Model 
+Accuracy 
+Time elapsed 
+(sec) 
+System time 
+(sec) 
+CPU utilization 
+(max:2) 
+Binary ExpS 
+1.00 
+2.813 
+0.71 
+1.171 
+Binary ExpF 
+1.00 
+7.269 
+0.82 
+0.513 
+Binary ExpP 
+1.00 
+29.626 
+2.42 
+1.350 
+Binary n-BaIoT 
+0.99 
+22.877 
+1.30 
+1.238 
+ 
+All the binary classifications of our model used the “all-headers” category of the dataset. As 
+displayed in Table 7, the n-BaIoT model trained on 115 features took 22.877 seconds to perform 
+testing. However, its accuracy remained at 99.96%. Our featureless model performed better in 
+session and flow (ExpS and ExpF) representations, where it used less Testing Time compared to 
+the feature-based model. Testing time of only ExpP pcap representation took 6.746 seconds more 
+than the n-BaIoT model. Similar trends were seen in system time. Similarly, session and flow 
+utilize less CPU compared to feature engineered models. 
+5. CONCLUSION 
+Contrary to traditional ML methodologies, FEL-ML does not require the complex processing 
+power deemed necessary. With the ease of implementation, more industrial domains can now 
+include it in their day-to-day operations. Security of IoT devices is one such domain. With IoT 
+devices running on battery power, more accurate results can be achieved with less computational 
+overhead by training raw bytes on 1D-CNN. As well as identifying anomalies in traffic with 100% 
+accuracy, this methodology is able to identify their types with 99% accuracy. Previous works on 
+this topic have scraped headers from their training set. However, our experiment compares models 
+trained with and without headers. This extensive experiment reinforces our argument that feature 
+engineering and removing headers from packets is a step in the traffic classification process that 
+is unnecessary.  
+
+As evident from Table 7, one challenge faced by our algorithm is that pcap representation could 
+perform better than flow and session. Flow and session require additional overhead in 
+consolidation before feature engineering. The next step of this research will attempt to discover 
+simpler ML systems that are efficient for direct packet captures. Speed of detection and accuracy 
+are of utmost importance to performing more granular detection of malware on IoT devices. 
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+UNB, Canadian Institute for Cybersecurity. 2016. “ICSX VPN-nonVPN Dataset.” 
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+Lotfollahi, Mohammad, Ramin Shirali hossein zade, Mahdi Jafari Siavoshani, and 
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+and K. Simran. 2020. “A Visualized Botnet Detection System Based Deep Learning for the 
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+on the IoT-23 data set.” July. http://essay.utwente.nl/81979/. 
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+Conference on Document Analysis and Recognition (Volume 1) - Volume 1, USA, 278. IEEE 
+Computer Society. 
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+[33] 
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+of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), Doha, 
+Qatar, 
+Oct., 
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+Association 
+for 
+Computational 
+Linguistics. 
+https://www.aclweb.org/anthology/D14-1181. 
+[34] 
+Anguita, Davide, Alessandro Ghio, Luca Oneto, Xavier Parra, and Jorge L. Reyes-Ortiz. 2012. 
+“Human Activity Recognition on Smartphones Using a Multiclass Hardware-Friendly Support 
+Vector Machine.” In Ambient Assisted Living and Home Care, edited by José Bravo, Ramón 
+Hervás, and Marcela Rodriguez, Berlin, Heidelberg, 216–223. Springer Berlin Heidelberg. 
+[35] 
+Chollet, Francois, et al. 2015. “Keras.” https://keras.io. 
+[36] 
+Abadi, Martín, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg 
+S. Corrado, et al. 2015. “TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems.” 
+Software available from tensorflow.org, https://www.tensorflow.org/.  
+
+[37] 
+Jovanovic, Pavle, Dorde; Vuletic. 2021. “ETF IoT Botnet Dataset.” https://data.mendeley.com/ 
+datasets/nbs66kvx6n/1, doi=10.17632/nbs66kvx6n.1. 
+[38] 
+Tsimbalist, Sergei. 2019. “Botnet Traffic Analysis.” https://github.com/sergts/ botnet-traffic-
+analysis/tree/master/classification. 
+[39] 
+www.kernel.org. 
+2009. 
+“Linux 
+profiling 
+with 
+performance 
+counters.” 
+https://perf.wiki.kernel.org/index.php/Main_Page. 
+Authors 
+Arshiya Khan is currently pursuing her Ph.D. in 
+Electrical and Computer Engineering 
+(Cybersecurity) at the University of 
+Delaware, Newark, DE, USA. Her areas of 
+interest include network security, artificial 
+general intelligence, and fair machine 
+learning. She wrote her M.S. thesis on 
+feature taxonomy of network traffic for 
+machine learning algorithms. 
+Over the past 35 years, Chase Cotton (Ph.D. EE, 
+UD, 1984; BS ME, UT Austin, 1975, CISSP) has 
+held a variety of research, development, and 
+engineering roles, mostly in telecommunications. In 
+both the corporate and academic worlds, he has been 
+involved in computer, communications, and security 
+research in roles including communication carrier 
+executive, product manager, consultant, and 
+educator for the technologies used in Internet and 
+data services. 
+ 
+Beginning in the mid-1980 Dr. Cotton's 
+communications research in Bellcore's Applied 
+Research Area involved creating new algorithms 
+and methods in bridging, multicast, many forms of 
+packet-based applications including voice & video, 
+traffic monitoring, transport protocols, custom VLSI 
+for communications (protocol engines and Content 
+Addressable Memories), and Gigabit networking. 
+ 
+In the mid-1990s, as the commercial Internet began 
+to blossom, he transitioned to assist carriers 
+worldwide as they started their Internet businesses, 
+including Internet Service Providers (ISPs), hosting 
+and web services, and the first large scale 
+commercial deployment of Digital Subscriber Line 
+(DSL) for consumer broadband services. In 2000, 
+Dr. Cotton assumed research, planning, and 
+engineering for Sprint's global Tier 1 Internet 
+provider, SprintLink, expanding and evolving the 
+network significantly during his 8-year tenure. At 
+Sprint, his activities include leading a team that 
+enabled infrastructure for the first large-scale 
+collection and analysis of Tier 1 backbone traffic 
+and twice set the Internet 2 Land Speed World 
+Record on a commercial production network. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Since 2008, Dr. Cotton has been at the University of 
+Delaware in the Department of Electrical and 
+Computer Engineering, initially as a visiting 
+scholar, and later as a Senior Scientist, Professor of 
+Practice, and Director of Delaware's Center for 
+Intelligent CyberSecurity (CICS). His research 
+interests include cybersecurity and high-availability 
+software systems with funding drawn from the NSF, 
+ARL, U.S. Army C5ISR, JPMorgan Chase, and 
+other industrial sponsors. As Director, Cybersecurity 
+Minor & MS Programs, he currently is involved in 
+the ongoing development of a multi-faceted 
+educational initiative at UD, where he is developing 
+new security courses and degree programs, 
+including a minor, campus and online graduate 
+Master's degrees, and Graduate Certificates in 
+Cybersecurity. 
+ 
+Dr. Cotton currently consults on communications 
+and Internet architectures, software, and security 
+issues for many carriers and equipment vendors 
+worldwide. 
+

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@@ -0,0 +1,686 @@
+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.maeda@oist.jp
+Abstract: Optical nanofiber cavity research has mainly focused on the fundamental mode. Here,
+a Fabry-Pérot fiber cavity with an optical nanofiber supporting the higher-order modes, TE01,
+TM01, HE𝑜
+21, and HE𝑒
+21, is demonstrated. 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. In situ tuning of the intracavity birefringence enables the desired profile and polarization
+of the cavity mode to be obtained. 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.
+1.
+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]. 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]. Most ONF research focuses
+on single-mode ONFs (SM-ONFs) that only support the fundamental mode, HE11. 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]. 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. 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].
+In principle, there are many interesting phenomena which can be explored with a HOM-ONF.
+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]. Additionally, it was proposed
+that a HOM-ONF could be used to trap and manipulate cold atoms [4, 15, 16]. 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].
+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].
+Introducing a cavity system to the ONF could further increase light-matter interactions due
+to cavity quantum electrodynamics (cQED) effects [22–24]. 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]. Strong light-atom coupling using
+SM-ONF-based Fabry-Pérot and ring resonators has already been achieved [34,35]. Superstrong
+coupling of cold atoms and multiple longitudinal modes of a long fiber-ring resonator consisting
+of a SM-ONF section was demonstrated [36].
+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. The advantages are
+arXiv:2301.13432v1  [physics.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.
+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. The generation of TE01 and TM01 modes in a
+laser cavity consisting of a microfiber directional coupler-based mode converter was demonstrated
+previously [39]. 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; 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.
+The inhomogeneous polarization structure of HOMs needs to be taken into account when
+studying a fiber cavity system with a HOM-ONF. 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]. Polarization singularities
+are a subset of Stokes singularities, i.e., phase singularity points in Stokes phases [42,43]. 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.e., how the polarization is distributed in the
+cross-section of a given mode, is undefined [41]. 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]. Moreover, points of linear polarization can form continuous
+lines, which are classified as L-lines [41].
+The generation of all Stokes singularities within a single beam has been demonstrated using a
+free-space interferometer [43,46]. Modal interference in a birefringent crystal can facilitate the
+creation of polarization singularities [47,48]. As a result, the SOP can significantly vary along
+the propagation length, with C-points and L-lines propagating as C-lines, i.e., continuous lines of
+circular polarization, and L-surfaces, i.e., surfaces of linear polarization, respectively [47–49].
+Moreover, polarization singularities can appear, move or disappear from a given cross-sectional
+region with a smooth and continuous change of birefringence [50]. Birefringent media were used
+to create laser cavity modes containing a polarization singularity [51,52]. 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].
+A few-mode optical fiber cavity has the potential to generate complex laser modes by its highly
+variable degree of birefringence.
+Interference and birefringence are generally inseparable properties in fibers. 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]. This
+effect was used in a few-mode optical fiber to generate ellipse fields containing a C-point [54,55].
+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. 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].
+In this paper, we experimentally demonstrate a HOM-ONF-based Fabry-Pérot fiber cavity.
+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. We also
+demonstrate in situ intracavity manipulation of the modal birefringence to change the amplitude,
+frequency position, and the SOP of the modes. 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. 1. (a) Sketch of tapered optical fiber with trilinear shape, d𝑤𝑎𝑖𝑠𝑡: waist diameter.
+(b) Schematic of experimental setup. 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.
+for which such nanodevices could be used.
+2.
+Methods
+2.1.
+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. Tapered fibers were drawn
+from SM1250 (9/80) fiber (Fibercore) using an oxy-hydrogen flame pulling rig. The untapered
+fiber supports the LP01, LP11, LP21, and LP02 modes at a wavelength, 𝜆 = 776 nm. The modes
+supported by the tapered fiber depend on the tapering profile and the waist diameter. 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. The shape of the tapered fibers was chosen to be trilinear, see Fig. 1(a), with angles
+of Ω1 = 2 mrad, Ω2 = 0.5 mrad and Ω3 = 1 mrad in order to be adiabatic for the LP11 and LP01
+modes. Fiber transmission following the tapering process was >95% 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.
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+

ANE0T4oBgHgl3EQfPgBb/content/tmp_files/2301.02179v1.pdf.txt

@@ -0,0 +1,1237 @@
+Draft version January 6, 2023
+Typeset using LATEX twocolumn style in AASTeX63
+The GLASS-JWST Early Release Science Program. 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.
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+

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+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
+

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+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. The 
+proposed model can also be easily extended to the guidance of 
+MIMO antenna design in complex spaces by numerical discrete 
+and optimization methods. 
+ 
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+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. 
+
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+Propagation, vol. 61, no. 2, pp. 917-926. 2013. 
+  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Ruifeng Li received the B.S. degree in engineering from 
+University of Electronic Science and Technology of 
+China, Chengdu, China, in 2020. 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. 
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+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. 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. Additionally,
+folding along regions of paper with preexisting creases will
+also be a crucial step to achieving elegant folding tasks such
+as robotic origami. Moving forward, we anticipate exploring
+solutions to such problems that take advantage of generalized
+problem formulations with data-driven control schemes such
+as reinforcement learning.
+We acknowledge financial support from the National Science
+Foundation under Grant numbers IIS-1925360, CAREER-
+2047663, and OAC-2209782.
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+

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+YUILLE-POGGIO’S FLOW AND GLOBAL MINIMIZER OF
+POLYNOMIALS THROUGH CONVEXIFICATION BY
+HEAT EVOLUTION
+QIAO WANG
+Abstract. 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<j
+xixj,
+C(t) = −
+�
+i<j<k
+xixjxk,
+D(t) =
+�
+i<j<k<l
+xixjxkxl,
+E(t) = −x1x2x3x4x5.
+(5.24)
+Without loss of generality, we may set ξ = x1, η = x2. Then factoring the
+polynomial p(ξ, t) − p(η, t), we will obtain
+p(ξ, t) − p(η, t) = −(ξ − η)3
+10
+· K(ξ, η, x3, x4, x5, t),
+But this K(ξ, η, x3, x4, x5, t) can be represented as function of compositions
+of I1 = x3 + x4 + x5, I2 = x3x4 + x4x5 + x5x3 and I3 = x3x4x5, as well as
+ξ,η and t. Note that
+I1 = x3 + x4 + x5 = −A(t) − ξ − η,
+I2 = x3x4 + x4x5 + x5x3 = B(t) − (ξ + η) · I1,
+I3 = x3x4x5 = −C(t) − (ξ + η) · I2 − ξη · I1,
+(5.25)
+then we obtain the representation of p(x, t) on account of A(t), B(t), C(t) and
+ξ, η. Finally, representing K(ξ, η, x3, x4, x5, t) as functions of ξ, η, a, b, c, d, e, f, t
+will yield the required results.
+□
+Remark 11. Notice that ξ and η are symmetric in function K(ξ, η, t),
+thus we may observe the information in semi-plane ξ < η, then p(ξ, t) <
+p(η, t) ⇐⇒ K(ξ, η, t) < 0, and p(ξ, t) > 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<j
+xixj =
+�
+i
+x2
+i + b,
+(5.30)
+which results in b ≤ 0.
+□
+Proposition 2. Under the reduced form a = 0, the straight line ξ + η = 0
+is contained in Zt(K) (Z+
+t (K), or Z−
+t (K), resp. ) for all t if and only if
+c = 0 (c > 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
+

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+arXiv:2301.01039v1  [math.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. 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. 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.
+1. 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;
+0 ≤ k ≤ n
+0;
+k < 0 or k > 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.
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+[2] F. Altomare, Korovkin-type Theorems and Approximation by Positive Linear Oper-
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+[3] F. Altomare, M. Cappelletti Montano, V. Leonessa, I. Ra¸sa, A generalization of
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+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).
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+[14] G. G. Lorentz, Bernstein Polynomials, University of Toronto Press, Toronto, 1953
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+[26] V. I. Volkov, On the convergence of sequences of linear positive operators in the space
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+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
+

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+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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' 1 (2010), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content='  3, 359-385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content='  [2] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Altomare, Korovkin-type Theorems and Approximation by Positive Linear Oper-  ators, Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Theory 5 (2010), 92-164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content='  [3] 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, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Ra¸sa, A generalization of  Kantorovich operators for convex compact subsets, Banach J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' 11 (2017),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' 3, 591–614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content='  [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Bennett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Sharpley, Interpolation of Operators, Academic Press Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=', 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content='  [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Berens, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' DeVore, Quantitative Korovkin theorems for Lp-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' In: Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content='  Theory II, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=', Austin 1976, 289–298 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content='  [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Berens, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' DeVore, Quantitative Korovkin theorems for positive linear operators  on Lp-spaces, Transactions AMS 245 (1978), 349–361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
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+page_content=' ded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' to the memory of Vl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
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+page_content=' Volkov, On the convergence of sequences of linear positive operators in the space  of continuous functions of two variables (Russian), Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
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+page_content=' ),  115 (1957), 17–19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
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+page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Xiong, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Cao, Multivariate Stancu operators defined on a simplex, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content=', 138 (2003), 189–198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfG_ve/content/2301.01039v1.pdf'}
+page_content='  Ankara University, Faculty of Science, Department of Mathematics, Str.' metadata={'source': '/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'}

FtE1T4oBgHgl3EQfXARM/content/tmp_files/2301.03121v1.pdf.txt

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+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). 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].
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+

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+Predicting Drivers’ Route Trajectories in Last-Mile Delivery Using A Pair-wise
+Attention-based Pointer Neural Network
+Baichuan Moa, Qing Yi Wanga,∗, Xiaotong Guoa, Matthias Winkenbachb, Jinhua Zhaoc
+aDepartment of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
+bCenter for Transportation and Logistics, Massachusetts Institute of Technology, Cambridge, MA 20139
+cDepartment of Urban Studies and Planning, 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.
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+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.mustaniemi@oulu.fi
+Abstract. 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. We acquire building-scale 3D dataset
+(BS3D) and demonstrate its value for monocular depth estimation. BS3D is
+unique also because it includes active infrared images, which are often miss-
+ing in other datasets. We employ infrared images for visual-inertial odometry,
+discovering a promising new research direction.
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