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+arXiv:2301.13804v1  [cs.GT]  31 Jan 2023
+Fairness in the Assignment Problem with Uncertain Priorities∗
+Zeyu Shen†
+Zhiyi Wang∗
+Xingyu Zhu∗
+Brandon Fain∗
+Kamesh Munagala∗
+Abstract
+In the assignment problem, a set of items must be allocated to unit-demand agents who ex-
+press ordinal preferences (rankings) over the items. In the assignment problem with priorities,
+agents with higher priority are entitled to their preferred goods with respect to lower priority
+agents. A priority can be naturally represented as a ranking and an uncertain priority as a
+distribution over rankings. For example, this models the problem of assigning student appli-
+cants to university seats or job applicants to job openings when the admitting body is uncertain
+about the true priority over applicants. This uncertainty can express the possibility of bias in
+the generation of the priority ranking. We believe we are the first to explicitly formulate and
+study the assignment problem with uncertain priorities. We introduce two natural notions of
+fairness in this problem: stochastic envy-freeness (SEF) and likelihood envy-freeness (LEF). We
+show that SEF and LEF are incompatible and that LEF is incompatible with ordinal efficiency.
+We describe two algorithms, Cycle Elimination (CE) and Unit-Time Eating (UTE) that satisfy
+ordinal efficiency (a form of ex-ante Pareto optimality) and SEF; the well known random serial
+dictatorship algorithm satisfies LEF and the weaker efficiency guarantee of ex-post Pareto op-
+timality. We also show that CE satisfies a relaxation of LEF that we term 1-LEF which applies
+only to certain comparisons of priority, while UTE satisfies a version of proportional allocations
+with ranks. We conclude by demonstrating how a mediator can model a problem of school
+admission in the face of bias as an assignment problem with uncertain priority.
+1
+Introduction
+Consider a motivating example of the assignment problem where a number of university admission
+slots (the items) must be assigned to student applicants (the agents). The university slots could be
+at a single university or several. Applicants might have preferences over different universities, or
+might have preferences over different slots at the same university (for example, some slots might be
+associated with merit-based financial aid, or include admission to particular academic programs).
+Applicants are unit-demand, meaning they only need to be assigned a single slot (and derive no
+benefit from being assigned multiple).
+Most university systems employ some form of priority-based admissions; this can be expressed
+through a ranking over applicants. For example, a priority might rank applicants by standardized
+exam scores, or perhaps by some more complex holistic assessment. Given any deterministic priority
+(a ranking), one might naturally solve the assignment problem using the serial dictatorship rule,
+so that students choose their most preferred remaining university slot one at a time in order of
+∗This work is supported by NSF grant CCF-2113798.
+†Computer
+Science
+Department,
+Duke
+University,
+Durham,
+NC
+27708-0129.
+Email:
+{zeyu.shen,zhiyi.wang,xingyu.zhu}@duke.edu, {btfain,kamesh}@cs.duke.edu.
+1
+
+their standardized exam score. Indeed, systems roughly like this are employed in several countries
+around the world such as the Indian Institutes of Technology [13].
+Despite the appeal of such a simple and ostensibly fair system, there is reason to suspect that
+any scoring or ranking system is based on imperfect noisy signals of the true underlying priority
+(whatever that might be). For example, an applicant A scoring 1 point higher on a standardized
+exam or holistic assessment than another applicant B is not, in general, 100% more likely to be a
+better student than B. Even more worryingly, studies show that standardized exam performance is
+closely related to demographic factors such as race and income [8], leading to uncertainty based on
+social bias and inequality in addition to random noise like whether one had a good breakfast the day
+of an exam. More holistic assessments are further vulnerable to the well documented phenomenon
+of implicit bias against historically marginalized groups [5]. Ignoring these uncertainties may result
+in arbitrary decisions (deterministically preferring one applicant over another when the comparison
+is unclear and noisy) and systemic discrimination against historically marginalized groups.
+Previous work has attempted to solve the second problem of bias without explicitly modeling
+an uncertain priority by adapting the so-called “Rooney Rule” [15, 7]. There are variations, but
+roughly speaking these methods reserve a number of “minority” spots and prioritize this many
+“minority” applicants in some serial dictatorship assignment. This approach can lead to fairness
+gerrymandering [14] by which structured subgroups remain disadvantaged. In particular, Rooney
+Rule style approaches are predicated on a single binary distinction of the applicant population into
+“majority” (or privileged) and “minority” (or disadvantaged) applicants. But in reality, applicant
+identity is multidimensional (race, gender, income, disability, first language, etc.) and bias can
+compound along intersections. In fact, it is perfectly plausible that the vast majority of applicants
+are disadvantaged (that is, suffer from bias leading to underestimation of their priority) along
+one or more dimensions of identity, though not all to the same extent.
+In addition to group
+identity, there may sometimes be uncertainties related to the priority of individual applicants,
+unique circumstances that merit accounting.
+For these reasons, we consider the more general problem that takes as input an uncertain
+priority, expressed as a probability distribution over rankings of applicants. The generality of the
+input to our algorithms ensures that a decision maker can fully model the complexity of uncertainty
+and bias inherent in the creation of a priority. This modeling problem is outside the scope of this
+paper, though we do provide an example for our experiments in Section 6. Rather, our emphasis
+is on the question of characterizing fairness and efficiency given a random priority, and providing
+algorithms to compute random assignments that satisfy these desiderata.
+1.1
+Contributions
+We study an extension of the random assignment problem [6, 18, 2] in which a decision maker must
+allocate a number of items to unit-demand agents in a way that is consistent with an uncertain
+priority represented as a distribution over rankings of the agents. To the best of our knowledge,
+we are the first to characterize this more general problem.
+In general we want to compute a random assignment that is simultaneously efficient with respect
+to agent preferences over the items and fair with respect to the agent priorities. Ordinal efficiency
+(OE) [6] generalizes the concept of Pareto efficiency to the case of a random assignment. Our main
+contribution is to characterize two alternative notions of fairness for the random assignment problem
+with uncertain priorities in Section 3.
+The first notion, which we call stochastic envy-freeness
+(SEF), guarantees that any agent whose priority first-order stochastically dominates another agent’s
+2
+
+priority should prefer their own (random) assignment to that of the other agent. The second notion,
+which we call likelihood envy-freeness (LEF), guarantees that the likelihood (over the random
+assignment) that an agent prefers the assignment of another should be at most the likelihood (over
+the uncertain priority) that the latter agent has higher priority than the former.
+We introduce additional notions that helps more finely distinguish between algorithms that
+satisfy one of the above notions. The first is a relaxation of LEF called 1-LEF that holds only when
+an agent has higher priority than another with probability 1. The next is ranked proportionality
+(PROP), where the allocation of any agent should stochastically dominate the allocation where
+she gets her i-th preferred item with probability pi if she herself is ranked at position i with that
+probability.
+Formal definitions are provided in Section 3. We provide illustrative examples of these concepts
+as well as justification for why multiple definitions of fairness might be appropriate in Section 3.3.
+In Section 4 we show that it is impossible to guarantee OE and LEF simultaneously.
+We
+also show that it is impossible to guarantee SEF and LEF simultaneously. Given this, we focus
+on achieving OE and SEF. In Section 5 we describe two algorithms: Unit-time Eating (UTE) and
+Cycle Elimination (CE). We show that both of these algorithms satisfy OE and SEF. To more finely
+distinguish between these algorithms, we show that CE also satisfies the relaxed 1-LEF property,
+while UTE satisfies PROP. We also show that any algorithm achieving OE cannot achieve PROP
+and 1-LEF simultaneously, so that we cannot achieve a super-set of the properties achieved by
+these algorithms.
+It is straightforward to observe that the well known Random Serial Dictatorship (RSD) that
+samples a priority from Σ and then uses the serial dictatorship satisfies LEF, PROP, and is ex-post
+Pareto efficient, though it does not satisfy OE [6]. We obtain a nearly complete characterization of
+achievable subsets of our efficiency and fairness properties, as shown in Table 1.
+Algorithm
+OE
+SEF
+LEF
+1-LEF
+PROP
+RSD
+✓
+✓
+✓
+UTE (new)
+✓
+✓
+✓
+CE (new)
+✓
+✓
+✓
+Table 1: Summary of fairness properties achieved.
+In Section 6 we return to a consideration of our motivating application of biased school ad-
+missions. We provide a practical example modeling an uncertain priority in the presence of bias
+and compare our CE and UTE algorithms with previous approaches to address bias using “Rooney
+Rule” style approaches [15, 7].
+1.2
+Related Work
+Random Assignment.
+There is a large body of work studying the problem of random assign-
+ment with no priority (or, in our framework, when the priority is uniform). The work of [1] proposed
+a random serial dictatorship mechanism, which draws an ordering of agents uniformly at random
+and let them choose items in that order, and showed that this mechanism is ex-post efficient. The
+work of [23] observed that though random serial dictatorship is fair, it is not efficient when the
+agents are endowed with Von Neumann-Morgenstern preferences over lotteries. The work of [6]
+3
+
+introduced a notion of efficiency that is stronger than ex-post efficiency, namely ordinal efficiency,
+and showed that random serial dictatorship is not ordinally efficient. They proposed the probabilis-
+tic serial rule that is ordinally efficient. Moreover, probabilistic serial is (stochastically) envy-free
+while random serial dictatorship is not. The work of [2] studied the relationship between ex-post
+efficiency and ordinal efficiency, showing that a lottery induces an ordinally efficient random as-
+signment if and only if each subset of the full support of the lottery is undominated (in a specific
+sense).
+Subsequent works investigated natural extensions of the canonical setup. The work of [18] con-
+sidered the problem of random assignment in the case where agents can opt out, and characterised
+probabilistic serial by ordinal efficiency, envy-freeness, strategyproofness, and equal treatment of
+equals in this setting. The work of [10] studied the notion of rank efficiency, which maximises the
+number of agents matched to their first choices.
+Fair Ranking.
+The assignment problem with priority is closely related to the subset selection
+problem that has been studied extensively as a problem in fair ranking [15, 16, 19, 7, 9, 17, 12]
+where the goal is to optimize some latent measure of utility for the algorithm designer subject to
+group fairness constraints on the resulting ranking. Recent work considers explicitly modeling the
+uncertainty from bias when estimating a ranking based on observed utilities [22], similar to our
+approach in modeling an uncertain priority. Our work differs from the fair ranking literature in that
+we study a more general assignment problem in which agents may not all have the same preferences
+over items. Of course, one can always translate a given ranking into an assignment by employing
+the serial dictatorship rule, but this need not be ordinally efficient [6]. Instead, we formulate our
+desiderata more explicitly in the wider context of the assignment problem itself.
+Two-sided matching.
+School choice problems are often studied in the context of two-sided
+matching, where applicants have preferences over schools and schools have preferences over ap-
+plicants.
+For example, the deferred acceptance algorithm (and its extensions) calculates stable
+matchings and has been extensively studied and deployed in the real world [11, 20, 21, 4, 3]. Our
+problem is different in two ways. First, the “items” in our problem (eg., school seats) share a single
+common priority over applicants, so the notion of stability simply means no applicant of lower
+priority is assigned an item preferred by an agent of higher priority. However, our setting is more
+complex in the second sense: The shared priority is uncertain, and the assignment will be random,
+requiring an extension of existing fairness properties and algorithms.
+2
+Preliminaries
+We are given n unit demand agents A = {1, 2, . . . , n} and a set of m items I. We assume without
+loss of generality that m ≥ n (if not, one can create additional “dummy” items that are least
+preferred by all agents). We write a ≻i b to denote that agent i prefers item a to item b. Each
+agent has ordinal preferences represented as a total order over I, that is, for every agent i we have
+a permutation πi : I → {1, . . . , n} such that πi(a) < πi(b) if and only if a ≻i b.1
+1In general, results extend trivially to the case where agents may have objective indifferences between items,
+meaning that if any agent is indifferent between two items then all agents are indifferent between those items.
+However, our results do not necessarily extend straightforwardly if agents have subjective indifferences, see [6].
+4
+
+A simple priority over agents is a permutation σ : A → {1, . . . , n} where σ(i) < σ(j) means that
+i has higher priority than j. A random priority is a probability distribution over simple priorities
+which we denote as Σ = {(σk, ρk)} where each σk is a simple priority, ρk ≥ 0, and �
+k ρk = 1.
+A simple assignment is a matching f : A → I. A lottery is a probability distribution over simple
+assignments which we denote as L = {(fk, pk)} where each fk is a simple assignment, pk ≥ 0, and
+�
+k pk = 1.
+Following [6], we call a probability distribution over [m] itself a random allocation to an agent.
+It is important to note that agents have ordinal preferences over deterministic items which only
+induces a partial order over random allocations. That is, given πi, it may be unclear whether i
+would prefer one random allocation to another. We denote by P = {pij} a random assignment,
+the n by m matrix where Pi, the i-th row, is agent i’s random allocation, and where �
+i pij = 1
+for all columns j. In general, a random assignment P can be induced by one or more lotteries, the
+existence of which is guaranteed by the Birkhoff-von Neumann Theorem, but a particular lottery
+induces a unique random assignment P.
+In the assignment problem with uncertain priorities we are given a random priority Σ and agent
+preferences {πi} and we must compute a random assignment.
+3
+Desiderata
+In this section we introduce the normative properties that an algorithm for the random assignment
+with uncertain priorities problem should satisfy. Broadly speaking, these desiderata require that
+the algorithm be efficient with respect to agent preferences and fair with respect to agent priorities.
+3.1
+Efficiency
+A simple assignment f is Pareto efficient (or Pareto optimal) if it is not dominated by any other
+simple assignment, which simply means that there is no alternative such that no agent is worse off
+and at least one agent is better off.
+Definition 1 (Pareto Efficiency). A simple assignment f is Pareto efficient if for all simple as-
+signments g one of the following holds: (i) ∃i ∈ A such that f(i) ≻i g(i), or (ii) g(i) ⊁ f(i) for all
+i ∈ [n].
+A lottery L is ex-post Pareto efficient if every simple assignment in the support of L (i.e., every
+simple assignment fk with pk > 0) is Pareto efficient.
+A stronger efficiency property for a random assignment is ordinal efficiency (OE) [6]. To define
+ordinal efficiency we must first define the notion of stochastic dominance.
+Definition 2 (Stochastic Domination). A probability distribution X stochastically dominates an-
+other distribution Y under permutation π (denoted X ≻sd
+π Y ) if for all t ∈ {1, . . . , n} it holds that
+�t
+r=1 Xπ−1(r) ≥ �t
+r=1 Yπ−1(r), where π−1 is the inverse permutation. A random assignment P is
+stochastically dominated by a random assignment Q ̸= P if the random allocation induced by Q
+stochastically dominates the random allocation induced by P under preferences πi for every agent
+i ∈ [n].
+Note that this implies the following: If random assignment Q stochastically dominates random
+assignment P, then every agent prefers Q to P under any Von Neumann-Morgenstern utility func-
+tion consistent with their ordinal preferences. Now we can define ordinal efficiency, following [6].
+5
+
+Definition 3 (Ordinal Efficiency, OE). We say that a random assignment P is ordinally efficient
+if it is not stochastically dominated by any other random assignment.
+At a high level, a random assignment is ordinally efficient if there is no other random assignment
+that is better for all agents and all utility functions consistent with their ordinal preferences. The
+property is not trivial: Some natural algorithms such as random serial dictatorship are Pareto
+efficient but not ordinally efficient.
+3.2
+Fairness
+We define fairness in terms of envy. We say that one agent envies another if the former prefers
+the item assigned to the latter. Envy of a lower priority agent constitutes a justified complaint
+against an assignment; ideally we would like to compute an envy-free assignment with respect to
+the priority.
+Definition 4 (Envy-Freeness). We say that a simple assignment f is envy-free with respect to a
+simple priority σ if for all i, j ∈ [n], σ(i) < σ(j) =⇒ f(i) ≻i f(j).
+However, it is immediately evident that it is impossible to compute a single simple assignment
+that is envy-free in this sense for every simple priority in the support of a random priority (for
+example, if there are two agents with uncertain priority who both prefer the same item). Instead,
+we need to compute a random assignment so that each agent is fairly treated ex-ante (for example,
+so that each agent has a fair probability of receiving the preferred good).
+There are two natural ways to generalize the concept of envy to a random assignment with a
+random priority. One is to imagine that one agent envies another if the random allocation of the
+latter stochastically dominates that of the former under the former’s ordinal preferences. Envy of
+this type forms a justified complaint if the envying agent also stochastically dominates the envied
+agent in terms of the random priority. More formally,
+Definition 5 (Stochastic Envy-Freeness, SEF). Consider a random assignment P generated under
+a random priority Σ. Let Si be the probability distribution over [n] induced by Σ for agent i, that
+is, for r ∈ [n], Sir = �
+k:σk(i)=r ρk.
+Let σ∗ be the identity permutation, i.e., σ∗(i) = i.
+P is
+stochastically envy-free (SEF) with respect to Σ if for all i, j ∈ [n], Si ≻sd
+σ∗ Sj =⇒ Pi ≻sd
+πi Pj.
+Loosely speaking, the implication of stochastic envy-freeness can be read as “if agent i probably
+has higher priority than j then i should prefer their random allocation to j’s under all utility
+functions consistent with i’s ordinal preferences.”
+A second way to generalize envy is by considering the likelihood of envy (in the simple sense)
+with respect to a lottery inducing a given random assignment. Envy of this type is justified if the
+likelihood of agent i envying another agent j is greater than the likelihood over the random priority
+that i has lower priority than agent j. We call a random assignment likelihood envy-free if there is
+a lottery which induces it and has no envy of this kind.
+Definition 6 (Likelihood Envy-Freeness, LEF). A random assignment P satisfies likelihood envy-
+freeness (LEF) under Σ if P can be induced by a lottery L such that for all i, j ∈ [n], Prσ∼Σ[σ(i) <
+σ(j)] ≤ Prf∼L[f(i) ≻i f(j)].
+In other words, LEF means that an agent i who is ℓ-likely to have higher priority than another
+agent j should be at least ℓ-likely to prefer their assigned item to j’s.
+6
+
+We say an algorithm satisfies OE (resp. SEF, LEF) if it always produces random assignment
+that satisfies OE (resp. SEF, LEF). As we show in Section 4, it is not possible to guarantee SEF
+and LEF simultaneously.
+3.3
+Relationship between LEF and SEF
+The relationship between SEF and LEF is subtle; neither implies the other and it is not immediately
+evident which is the “better” or more “natural” fairness property. We present two examples to
+illustrate that an assignment satisfying only one of SEF and LEF might still be unfair, so that both
+properties are useful competing notions of fairness, and neither is strictly stronger than the other.
+We first present an example which shows that an assignment that satisfies SEF can be unfair.
+Consider n = 2 agents and m = 2 items which we label a, b for clarity. Both agents prefer a to b,
+and the random priority is simply Σ = {(σ, 1)} with σ(1) < σ(2), i.e. agent 1 has higher priority
+than agent 2 with probability 1. In this setup, allocating 1
+2 unit of a and b to both agent yields
+an assignment that satisfies SEF. However, this assignment is clearly unfair, because even though
+agent 1 has higher priority than agent 2, they are getting the same assignment. Notice that this
+assignment does not satisfy LEF. In this instance, LEF could be used to characterize how much
+one agent is prioritized over the other.
+The next example shows that an assignment that only satisfies LEF can also be unfair. Consider
+n = 2 agents and m = 100 items which we label i1, . . . , i100 for clarity. The preferences of both
+agents are i1 ≻ · · · ≻ i100. The random priority is given by Σ = {(σ1, 1
+2), (σ2, 1
+2)} with σ1(1) < σ1(2)
+and σ2(2) < σ2(1). In other words, both agents have the same priority. In this setup, allocating
+1
+2 unit of i1 and 1
+2 unit of i100 to agent 1 and 1
+2 unit of i99 and 1
+2 unit of i100 to agent 2 yields
+an assignment that satisfies LEF. Notice that this assignment can be induced by a lottery L =
+{(f1, 1
+2), (f2, 1
+2)} where f1(1) = i1, f1(2) = i100, f2(1) = i100, f2(2) = i99. However, this assignment
+is clearly unfair, because even though the two agents have the same priority, agent 1 gets a strictly
+better assignment than agent 2.
+This shows that LEF alone has limitations as well, and the
+appropriate concept here is SEF.
+The above examples show that SEF and LEF provide reasonable competing notions of fairness.
+When combined with the efficiency notion of OE, we will show in Section 4 that LEF and OE
+are incompatible. If OE is replaced by the weaker notion of Pareto-efficiency, then it is easy to
+check that random serial dictatorship (RSD), which simply samples a priority of agents from the
+distribution and allocates each agent their favorite remaining item in this priority order, satisfies
+LEF2 and pareto efficiency. Thus, in our work, we will focus on the more non-trivial part of finding
+algorithms that satisfy SEF and OE.
+3.4
+Additional Fairness Criteria
+As we show in Section 5, there can be multiple algorithms that satisfy the same subset of the
+fairness criteria. We therefore consider two additional notions to more finely distinguish between
+them.
+The first criterion is the following relaxation of LEF: If agent i with probability 1 has higher
+priority than another agent j then agent i should certainly (again, with probability 1) not envy j.
+2To see why RSD satisfies LEF, suppose the random priority is given by Σ = {(σk, ρk)}, then the random
+assignment produced by RSD can be induced by the lottery L = {(fk, ρk)}, where fk is the deterministic assignment
+produced by letting agents successively choose an item based on the order given by σk.
+7
+
+Definition 7 (1-LEF). A random assignment P under random priority Σ satisfies 1-LEF if there
+exists some lottery L which induces P such that for all agents i ̸= j ∈ [n], if Prσ∼Σ[σ(i) < σ(j)] = 1,
+then Prf∼L[f(i) ≻i f(j)] = 1.
+The next criterion is called Ranked Proportionality (PROP), which captures stochastic dom-
+inance over an allocation that matches the probability an agent gets her ith ranked item to the
+probability of she being ranked at position i. Note that if all rankings of agents were equally likely,
+this captures stochastic dominance to an allocation that assigns every item to every agent uniformly
+at random.
+Definition 8 (PROP). Given a random priority Σ = {(σk, ρk)}, we define the baseline allocation
+P i for agent i by P iπ−1
+i
+(r) = Sir = �
+k:σk(i)=r ρk for all r ∈ [n]. In other words, if an agent i ranks
+the r-th in the random priorities with probability p, then we add p fraction of the r-th preferred item
+of agent i to her baseline allocation. For an allocation to satisfy ranked proportionality (PROP),
+it should stochastically dominate this baseline for each agent.
+4
+Impossibility Results
+In this part, we present several impossibility results. We note that these are existential hardness
+results, not computational. We begin by observing that LEF is incompatible with OE.
+Theorem 1. LEF is incompatible with OE.
+Proof. We present an instance in which no random assignment can satisfy both LEF and OE. There
+are n = 4 agents and m = 4 items which we label a, b, c, d for clarity. Agent preferences are given
+by
+π1, π3 : a ≻ b ≻ c ≻ d,
+π2, π4 : b ≻ a ≻ c ≻ d
+Moreover, we consider the priority Σ = {(σ1, 1
+2), (σ2, 1
+2)} where
+σ1(4) < σ1(2) < σ1(3) < σ1(1),
+σ2(3) < σ2(1) < σ2(4) < σ2(2).
+In other words, with probability 1
+2 under σ1, agent 4 has the highest priority, then agent 2, then
+agent 3, finally agent 1. Similarly for σ2 with probability 1
+2. Assume for contradiction that there
+exists a random assignment P = [pij], together with a lottery L which induces P, satisfying LEF
+and OE. By definition of LEF, we note that
+Pr
+f∼L[f(3) ≻3 f(1)] ≥ Pr
+σ∼Σ[σ(3) < σ(1)] = 1,
+so it must be that Prf∼L[f(3) ≻3 f(1)] = 1. Thus, we must have p1a = 0, because otherwise there
+would exist a simple assignment in the lottery in which agent 1 is assigned with a and agent 3 is
+assigned with some less preferred item under π3. By the same reasoning, we note that p2b = 0.
+Also by definition of LEF, observe that
+Pr
+f∼L[f(2) ≻2 f(3)] ≥ Pr
+σ∼Σ[σ(2) < σ(3)] = 1
+2.
+8
+
+This implies p3a < 1, as otherwise we have f(3) = a for all f ∼ L; combined with the fact that
+p2b = 0, we would have f(3) ≻2 f(2) for all f ∼ L, which contradicts Prf∼L[f(2) ≻2 f(3)] ≥ 1
+2.
+Since p1a = 0, p3a < 1, and �
+i pia = 1, it follows that p2a + p4a > 0. Similarly, we have
+p1b +p3b > 0. Without loss of generality, we assume that p2a > 0 and p1b > 0 (if p4a > 0 or p3b > 0,
+the proof proceeds similarly). Let pmin = min(p2a, p1b); define random assignment Q = [qij] by
+qij =
+
+
+
+
+
+pij if i /∈ {1, 2} and j /∈ {a, b}
+pij + pmin if (i, j) = (1, a) or (2, b)
+pij − pmin if (i, j) = (1, b) or (2, a)
+We can see that Q stochastically dominates P. In particular, all that is different in Q is that
+agent 1 swaps agent 2 some of agent 2’s allocated probability mass on item a in exchange for an
+equivalent amount of agent 1’s probability mass on item b. Since a ≻1 b and b ≻2 a and nothing else
+changes, agents 1 and 2 prefer Q, and nothing has changed for agents 3 and 4. This contradicts with
+the fact that P satisfies OE. Thus, we can conclude that no random assignment in this instance
+satisfies LEF and OE.
+Theorem 1 can be interpreted as a fundamental tradeoff between efficiency and fairness con-
+ceived as LEF. Next, we show that LEF and SEF are two fundamentally different notions of fairness
+that are incompatible with one another. As we will see later in Section 5, each of LEF and SEF
+independently can be guaranteed. Thus, neither notion of fairness is subsumed by the other.
+Theorem 2. LEF is incompatible with SEF.
+Proof. We present an instance in which no random assignment can satisfy both LEF and SEF.
+There are n = 5 agents and m = 5 items which we label a, b, c, d, e for clarity. Preferences are given
+by
+π1, π3 : a ≻ b ≻ c ≻ d ≻ e,
+π2, π4 : b ≻ a ≻ c ≻ d ≻ e,
+π5 : a ≻ c ≻ b ≻ d ≻ e.
+We consider the priority Σ = {(σ1, 1
+2), (σ2, 1
+2)} defined by
+σ1(3) < σ1(5) < σ1(1) < σ1(4) < σ1(2),
+σ2(4) < σ2(5) < σ2(2) < σ2(3) < σ2(1).
+In other words, with probability 1
+2 under σ1, agent 3 has the highest priority, then agents 5, 1,
+4, and finally 2. Similarly for σ2.
+Assume for contradiction that there exists a random assignment P = [pij], together with a
+lottery L which induces P, that satisfies LEF and SEF. Since agent 3 always has higher priority
+than agent 1 and agent 3 prefers a over all other items, LEF implies that p1a = 0. Similarly, since
+agent 4 always has higher priority than agent 2 prefers b over all other itmes, LEF implies that
+p2b = 0.
+Recall that Si is the probability density over [n] induced by Σ for agent i and σ∗ is the identity
+permutation. Since S1 ≻sd
+σ∗ S2 by construction and P satisfies (SEF) by assumption, we have
+P1 ≻sd
+π1 P2. Combined with the fact that p1a = 0, we must have p2a = 0. Similarly, p1b = 0.
+9
+
+We next show p1c = p2c = 1
+2. First, observe that LEF guarantees
+Pr
+f∼L[f(4) ≻4 f(2)] ≥ Pr
+σ∼Σ[σ(4) < σ(2)] = 1,
+Thus, since e is the least preferred item by agent 4, we must have p4e = 0. Also by LEF, we have
+Pr
+f∼L[f(1) ≻1 f(4)] ≥ Pr
+σ∼Σ[σ(1) < σ(4)] = 1
+2,
+i.e. Prf∼L[f(1) ≻1 f(4)] ≥ 1
+2. On the other hand, since p4e = 0, the worst item that agent 4 can
+get under π4 is d, so
+Pr
+f∼L[f(1) ≻1 f(4)] ≤ p1a + p1b + p1c = p1c,
+since we earlier found that p1a = p1b = 0.
+Recall Prf∼L[f(1) ≻1 f(4)] ≥
+1
+2, we get p1c ≥
+1
+2.
+Similarly, we have p2c ≥ 1
+2. Since �
+i pic = 1, it must be the case that p1c = p2c = 1
+2. We deduce
+that for any f ∼ L, either f(1) = c or f(2) = c, because on one hand, for any fixed f, we should
+have f(1) ̸= f(2), while on the other hand, p1c + p2c = 1.
+Observe that p5a ≤ 1
+2. This follows directly from LEF, because
+Pr
+f∼L[f(3) ≻3 f(5)] ≥ Pr
+σ∼Σ[σ(3) < σ(5)] = 1
+2;
+if p5a > 1
+2, we would have Prf∼L[f(3) ≻3 f(5)] < 1 − p5a = 1
+2, leading to contradiction. What’s
+more, we have p5c = 0, since we already have p1c + p2c = 1.
+On one hand, we should have Prf∼L[f(5) ≻5 f(1) and f(5) ≻5 f(2)] = 1, since σi(5) < σi(1) and
+σi(5) < σi(2) for i ∈ {1, 2}; but on the other hand, we have Prf∼L[f(5) ≻5 f(1) and f(5) ≻5 f(2)] ≤
+p5a ≤ 1
+2, because for any f ∼ L, either f(1) = c or f(2) = c, so f(5) ≻5 f(1) and f(5) ≻5 f(2) if
+and only if f(5) = a. This leads to contradiction. Thus, LEF and SEF are incompatible.
+We finally show that OE, 1-LEF, and PROP are simultaneously incompatible. This will inform
+the design of algorithms in Section 5.
+Lemma 1. There is an instance where no allocation simultaneously satisfies OE, 1-LEF, and
+PROP.
+Proof. We use the instance in the proof of Theorem 1. Since agent 3 is ranked higher than agent
+1 with probability 1 and since their preferences over items is identical, 1-LEF implies agent 1 is
+allocated item a with probability 0. Now PROP implies agent 1 receives item b with probability at
+least 1/2. By a similar reasoning, agent 2 must get item a with probability at least 1/2. But any
+such allocation cannot be OE, completing the proof.
+5
+Algorithms
+As we have seen, LEF is a very strong notion of fairness which is incompatible with both OE and
+SEF. In the following, we present two algorithms – cycle elimination (CE) and unit time eating
+(UTE) – that satisfy both OE and SEF. In addition, we show that CE satisfies 1-LEF and UTE
+satisfies PROP. Given Lemma 1, we cannot design an algorithm that achieves OE and both these
+properties.
+10
+
+Therefore, both CE and UTE are reasonable fair allocation algorithms in that they satisfy
+efficiency (OE) and envy-freeness (SEF). The choice of which to implement depends on whether
+we care more about a form of proportionality in the resulting allocation (UTE satisfies PROP) or
+whether we care about additional envy-freeness in a deterministic sense (CE satisfies 1-LEF).
+5.1
+Cycle Elimination algorithm
+We first introduce a Cycle Elimination algorithm (CE), which works by constructing a directed
+graph based on the random priority, and allocate items based on this graph.
+To begin with, we introduce the Probabilistic Serial rule [6], a continuous algorithm which
+works as follows. Initially, each agent i goes to their favorite item j and starts “eating” it (that is,
+increasing pij) at unit speed. It is possible that several agents eat the same item at the same time.
+Whenever an item is fully eaten, each of the agents eating it goes to their favorite remaining item
+not fully allocated (that is, �
+i pij < 1) and starts eating it in the same way. This process continues
+until all items are consumed, or all the agents are full (that is, �
+j pij = 1). We use PS(A, I) to
+denote the assignment produced by running Probabilistic Serial rule on the set of agents A and
+items I.
+We construct a graph from Σ, which we call a Stochastic-Dominance graph (SD-graph), as
+follows: Start with a graph with n vertices, where the i-th vertex corresponds to the i-th agent.
+For any pair of distinct agents i and j, if Si ≻sd
+σ∗ Sj, then we draw a directed edge from i to j. The
+algorithm is now formally stated in Algorithm 1.
+Algorithm 1: Cycle Elimination, Eliminate(A, I, G)
+Input: Set of agents A, set of items I, SD-graph G;
+Let �G be the condensation3of G;
+Let �
+A be the set of agents that belong to a strongly connected component whose in-degree
+in �G is zero;
+if A = �
+A then
+Output PS(A, I);
+else
+A′ ← A \ �
+A; I′ ← I \ PS( �
+A; I); G′ ← G \ �
+A;
+Output PS( �
+A, I) + Eliminate(A′, I′, G′);
+end
+Analysis.
+Our main result is the following theorem.
+Theorem 3. The Cycle Elimination algorithm satisfies OE, SEF, and 1-LEF. It runs in O(n3 +
+nm + n|Σ|) time.
+Proof. Theorem 1 in [6] states that any simultaneous eating algorithm where each agent always
+eats from her favorite remaining item satisfies OE. Hence, CE satisfies OE.
+3Condensation of a graph is a directed acyclic graph formed by contracting each strongly connected component
+to a single vertex.
+11
+
+To show SEF, fix two agents i and j, and assume Si ≻sd
+σ∗ Sj. Let P be the random assignment
+produced by CE. We show that Pi ≻sd
+πi Pj. Since Si ≻sd
+σ∗ Sj, there exists an edge from i to j
+in the SD-graph. Thus, i and j either belong to the same strongly connected component, or the
+strongly connected component of i has higher topological order than that of j’s. Either way, we
+have Pi ≻sd
+πi Pj, from which we can conclude that CE satisfies SEF.
+To show 1-LEF, fix two agents i and j, and assume Prσ∼Σ[σ(i) < σ(j)] = 1.
+Let P be
+the random assignment produced by CE. We show that, for any lottery L inducing P, we have
+Prf∼L[f(i) ≻i f(j)] = 1. We use proof by contradiction. Assume that there exists a lottery L0
+which induces P such that Prf∼L0[f(i) ≻i f(j)] < 1. This implies that there exists two items a
+and b such that a ≻i b, pib > 0 and pja > 0. On the other hand, since Prσ∼Σ[σ(i) < σ(j)] = 1,
+so we have Si ≻sd
+σ∗ Sj; thus, the strongly connected component that agent i belongs to must have
+higher topological order than that of j’s. By CE, agent j could start eating only when agent i is
+completely full. Thus, pja > 0 implies that there is still item a remaining when agent i finishes
+eating; this leads to contradiction, because a ≻i b implies that i could eat a instead of b. Therefore,
+we must have Prf∼L[f(i) ≻i f(j)] = 1 for all lotteries L which induces P, from which we can
+conclude that CE satisfies 1-LEF.
+To show the running time, preprocessing Σ in order to compute the stochastic dominance
+relation between agents takes O(n|Σ| + n3) time. Constructing the SD-graph by the stochastic
+dominance relation between agents takes O(n2) time, as there are
+�n
+2
+�
+pair of agents. Given the
+SD-graph, running CE takes O(nm) time. This is because we only need to consider at most m
+time points: the time at which each item is eaten up. We divide this process into m time intervals.
+During each time interval, each agent keeps eating the same item, so it simply takes O(n) time to
+keep track of the state of each agent, and the running time over m intervals is O(nm). Hence, the
+total running time is O(n3 + nm + n|Σ|).
+5.2
+Unit-time Eating Algorithm
+We next introduce the Unit-time Eating Algorithm (UTE). Recall that Σ = {(σk, ρk)}. Essentially,
+the algorithm works by dividing the time into n units, each of duration one; in time unit t, the t-th
+ranked agent in σk eats their favorite item among those left over at rate ρk for all k. The procedure
+is formally stated in Algorithm 2.
+Algorithm 2: Unit-time Eating Algorithm
+for t = 1, . . . , n do
+The t-th ranked agent in each σk eats their favorite item among those left over at rate
+ρk for all (σk, ρk) ∈ Σ;
+end
+Analysis.
+We show the following theorem.
+Theorem 4. The Unit-time Eating Algorithm satisfies OE, SEF, and PROP. Further, it runs in
+O(n2|Σ| + nm) time.
+Proof. By Theorem 1 in [6], we have UTE satisfies OE.
+12
+
+To show SEF, fix two agents i and j; assume that Si ≻sd
+σ∗ Sj. Let P be the random assignment
+produced by UTE, we show that Pi ≻sd
+πi Pj. Let tk be the time when item π−1
+i
+(k) has been eaten
+up. Fix some k ∈ [m]; because Si ≻sd
+σ∗ Sj, we have
+⌊tk⌋
+�
+t=1
+Sit ≥
+⌊tk⌋
+�
+t=1
+Sjt
+and
+⌈tk⌉
+�
+t=1
+Sit ≥
+⌈tk⌉
+�
+t=1
+Sjt
+Combining these gives
+�
+tk − ⌊tk⌋
+�
+Si⌈tk⌉ +
+⌊tk⌋
+�
+t=1
+Sit ≥
+�
+tk − ⌊tk⌋
+�
+Sj⌈tk⌉ +
+⌊tk⌋
+�
+t=1
+Sjt.
+(1)
+Observe that
+k
+�
+r=1
+Piπ−1
+i
+(r) =
+�
+tk − ⌊tk⌋
+�
+Si⌈tk⌉ +
+⌊tk⌋
+�
+t=1
+Sit,
+k
+�
+r=1
+Pjπ−1
+i
+(r) ≤
+�
+tk − ⌊tk⌋
+�
+Sj⌈tk⌉ +
+⌊tk⌋
+�
+t=1
+Sjt,
+which gives �k
+r=1 Piπ−1
+i
+(r) ≥ �k
+r=1 Pjπ−1
+i
+(r). Because this holds for all k ∈ [m], we conclude that
+Pi ≻sd
+πi Pj. Hence, UTE satisfies SEF.
+To show PROP, fix some agent i. Suppose the allocation produced by UTE for this agent is Pi,
+and the baseline allocation for this agent is P i. We will show that Pi ≻sd
+πi P i. Let tk be the time
+when item π−1
+i
+(k) has been eaten. Fix some k ∈ [m]. Clearly, we have tk ≥ k, because in order to
+eat up π−1
+i
+(k), we have to eat up π−1(r) for all r < k. We observe that
+k
+�
+r=1
+Piπ−1
+i
+(r) =
+�
+tk − ⌊tk⌋
+�
+Si⌈tk⌉ +
+⌊tk⌋
+�
+t=1
+Sit.
+Combined with tk ≥ k, we have
+k
+�
+r=1
+Piπ−1
+i
+(r) ≥
+k
+�
+t=1
+Sit =
+k
+�
+r=1
+P iπ−1
+i
+(r).
+Because this holds for all k ∈ [m], we conclude that that Pi ≻sd
+πi P i, and hence UTE satisfies PROP.
+To show running time, preprocessing Σ to obtain the eating speed of each agent in each unit
+time interval takes O(n2|Σ|) time. Then, running UTE takes O(nm) time, as we similarly only
+need to consider at most m time points and keeping track of the state of each agent in each time
+interval takes O(n) time. Therefore, the total running time is O(n2|Σ| + nm).
+6
+Generating Random Priorities and Empirical Results
+In this section, we will demonstrate how one could obtain random priorities in practical settings
+using an example of school admission under implicit bias. In several environments based on such
+13
+
+generative model, we will compare our proposed algorithms, namely Cycle Elimination (CE) and
+Unit-time Eating (UTE), with other common bias mitigating allocation algorithms such as “the
+Rooney Rule” [7]. We empirically demonstrate that all existing algorithms induce stochastic envy.
+To show this, using the same notation as in Definition 5, we say a pair of agents i and j form a
+stochastic envy pair if Zi ≻sd
+σ∗ Zj but Pi ⊁sd
+πi Pj, and we will count the number of stochastic envy
+pairs produced by each algorithm.
+6.1
+Random Priority in School Admission
+Consider a group of N students, including n disadvantaged students with indices {1, . . . , n} and
+N − n advantaged students with indices {n + 1, . . . , N}.
+Suppose that they are competing for
+admission priorities of ℓ schools with capacities c1, . . . , cℓ ∈ N that �ℓ
+i=1 ci = N, in which process
+disadvantaged students are subjected to implicit bias on their capability. We will quantify the effect
+of implicit bias in the experiments. This is equivalent to allocating N items to N agents, where
+items correspond to seats, and the agents’ preferences are their school choices.
+Denote the j-th seat of school i as sij, then the set of seats is S ≜ �ℓ
+i=1{si1, si2, . . . , sici}. For
+any ordinal preference �π : [ℓ] → [ℓ] over the schools, it induces an ordinal preference π : S → [N]
+over the seats such that for any sij ∈ S, π(sij) = j + �
+k:�π(k)<�π(i) ck. In other words, if a student
+prefers school a to school b, then all seats of school a are preferred over the seats of school b. For the
+seats in the same school, smaller indices are preferred. In the following, we describe how random
+priorities over students are generated.
+For each student, we assign a “capability score” xi that is drawn from the same distribution
+D, and students with higher capability score should have higher priority. Moreover, assume every
+student from the disadvantaged group is subjected to a multiplicative implicit bias bi, which is
+independently sampled from some distribution B. A disadvantaged student with capability score
+xi is perceived to have a biased score �xi ≜ bixi. We will also consider additive bias �xi ≜ xi + bi in
+our experiments. The admission committee make decisions based on the perceived scores (which
+are biased for disadvantaged students and equal to the true scores for advantaged students).
+For each experiment, we fix a set of unbiased capability scores {xi}N
+i=1 for the students, where
+xi
+iid
+∼ D. Then, we take n bias parameters {bi}n
+i=1 independently from B. The perceived scores of
+the students are {�x1, . . . , �xn, xn+1, . . . , xN}, where �xi = bixi. Now imagine we are the admission
+committee. We know B, D, and the perceived scores of the students. The goal is to approximately
+recover the underlying true scores of the students. To do this, we compute a posterior distribution
+for the bias factor of each disadvantaged student given B, the biased score of this student, and
+D. Concretely, the density of the posterior distribution for the bias factor of the ith disadvantaged
+student, which we denote by bi, is fbi(b) =
+fB(b)fD(�xi/b)
+� ∞
+0
+fB(u)fD(�xi/u)du. Given {bi}n
+i=1, we independently
+draw q sets of bias parameters for disadvantaged students, where we denote the jth set of bias
+parameters as {b(j)
+i }n
+i=1, i.e. b(j)
+i
+iid
+∼ bi. Let the ordinal relationship induced by {b(j)
+i }n
+i=1 be σ(j).
+We consider the random priority {(σ(j), 1
+q)}q
+j=1. We denote the random priority induced by q sets
+of bias parameters as Σ(q).
+6.2
+Algorithms for Comparison
+To compare with CE and UTE, we consider four alternative solutions to the allocation problem
+under implicit bias. Fix a set of biased scores {�x1, . . . , �xn, xn+1, . . . , xN}, let �σ denote its induced
+14
+
+ℓ
+β
+N
+RN
+R
+RR
+CE
+UTE
+0.2
+3.4
+0
+3.4
+10
+0
+0
+1
+0.5
+1.2
+0
+1.2
+10
+0
+0
+0.8
+0.6
+0
+0.6
+10
+0
+0
+0.2
+14.3
+42.8
+2.6
+3.8
+0
+0
+2
+0.5
+14.5
+42.8
+1.0
+3.8
+0
+0
+0.8
+19.7
+42.8
+0.6
+3.8
+0
+0
+0.2
+88.8
+175.7
+1.6
+2.5
+0
+0
+3
+0.5
+98.9
+175.7
+0.7
+2.5
+0
+0
+0.8
+103.5
+175.7
+0.4
+2.5
+0
+0
+ℓ
+β
+N
+RN
+R
+RR
+CE
+UTE
+0.2
+7.0
+0
+15.4
+25.4
+0
+0
+1
+0.5
+7.3
+0
+6.2
+36.9
+0
+0
+0.8
+7.8
+0
+2.8
+42.2
+0
+0
+0.2
+38.2
+36.2
+11.2
+16.8
+0
+0
+2
+0.5
+37.0
+38.3
+4.5
+22.9
+0
+0
+0.8
+37.4
+39.6
+2.2
+25.6
+0
+0
+0.2
+156.2
+183.3
+7.3
+9.8
+0
+0
+3
+0.5
+141.4
+200.5
+3.4
+12.6
+0
+0
+0.8
+127.6
+205.5
+1.9
+15.5
+0
+0
+Figure 1: Number of stochastic envy pairs under multiplicative bias (left) and additive bias (right).
+ordinal relationship. For a deterministic priority σ over the students and ordinal preferences Π ≜
+{πi}i∈[N] of students over the seats, let GS(σ, Π) denote the deterministic assignment produced by
+the Gale-Shapley algorithm [11] which produces a stable matching between students and seats.
+The algorithms that we compare with are as follows:
+1. Naive Stable Matching (N) takes deterministic priority �σ and returns the assignment PN(�σ, Π) ≜
+GS(�σ, Π).
+2. Random Naive Stable Matching (RN) takes the random priority Σ(q) = {(σi, pi)}q
+i=1 and
+outputs a lottery based on (N), namely {(PN(σi, Π), pi)}q
+i=1.
+3. Rooney Stable Matching (R) takes in the deterministic priority �σ as input. Using the Rooney
+constraint in Theorem 3.3 of [7], it creates a new priority �σR.
+We present this formally
+in Algorithm 3.
+Using �σR, Rooney Stable Matching returns the assignment PR(�σR, Π) ≜
+GS(�σR, Π).
+4. Random Rooney Stable Matching (RR) takes the random priority Σ(q) = {(σi, pi)}q
+i=1 and
+outputs a lottery based on (R), namely {(PR(σi, Π), pi)}q
+i=1.
+6.3
+Prevalence of Stochastic Envy
+We now demonstrate that with random priority induced by the generative model described in
+Section 6.1, stochastic envy exists for the bias mitigating algorithms N, RN, R, RR.
+15
+
+Algorithm 3: Proportional Rooney-rule-like Constraint [7]
+Let A, B be the ordered sub-sequences of disadvantaged and advantaged candidates in �σ
+respectively, i.e. p < q ⇐⇒ �σ(A[p]) < �σ(A[q]);
+i, j ← 0;
+while i + j < N do
+if ⌊
+i
+i+j ⌋ < n
+N or �σ(i) < �σ(n + j) then
+�σR(A[i]) = i + j and i ← i + 1;
+else
+�σR(B[j]) = i + j and j ← j + 1;
+end
+end
+return �σR
+We consider an admission problem with ℓ schools each with ⌊ N
+ℓ+1⌋ seats. Every student i ∈ [N]
+has a uniformly random preference order over the ℓ schools. There is also a ”dummy school” with
+N − ℓ⌊ N
+ℓ+1⌋ seats representing no admission. Every student prefers seats in the dummy school the
+least. For seats in the same school, all students have the same preference order. This represents
+the situation in which schools may distribute educational resources to students based on their rank
+when admitted.
+We take N = 35 and n = 10, and experiment with ℓ = 1, 2, 3. For each choice of k, we experiment
+with β = 0.2, 0.5, 0.8. For multiplicative bias, we take D = Exponential(1) and B = Exponential(β);
+for additive bias, we take D = Uniform(0, 2) and B = Uniform(0, β). Figure 1 presents the number
+of stochastic envy pairs for each algorithm averaged over 100 experiments. For each experiment,
+the random priority is computed with 1000 sets of bias parameters.
+Except for RN in the one school setting, stochastic envy exists in all other scenarios for N,
+RN, R, RR. While empirically Rooney-rule-like constraints do significantly reduce the number of
+stochastic envy pairs compared to applying no mitigation mechanism at all, we still need CE or
+UTE to obtain guaranteed SEF.
+7
+Conclusion
+We conclude with some open questions. First, even though SEF and LEF are incompatible, we
+do not know whether they can be compatible under certain natural generative assumptions on the
+random priorities and agent preferences. Second, it is known [6] that OE (and hence CE and UTE)
+is incompatible with strategyproofness under natural assumptions. However, it is interesting to
+explore whether SEF alone is also incompatible with strategy-proofness. Finally, can our framework
+be extended to the scenario where the agent preferences are random as well, i.e. each agent reports
+a distribution over preferences instead of a deterministic preference?
+16
+
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+

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+Precise certification of a qubit space
+Tomasz Białecki1, Tomasz Rybotycki2,3, Josep Batle4, Jakub Tworzydło1, and Adam Bednorz1, ∗
+1Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL02-093 Warsaw, Poland
+2Systems Research Institute, Polish Academy of Sciences, 6 Newelska Street, PL01-447 Warsaw, Poland
+3Center for Theoretical Physics, Polish Academy of Sciences,
+Al.
+Lotników 32/46, PL02-668 Warsaw, Poland
+4CRISP - Centre de Recerca Independent de sa Pobla, 07420 sa Pobla, Balearic Islands, Spain
+We demonstrate an implementation of the precise test of dimension on the qubit, using the
+public IBM quantum computer, using the determinant dimension witness. The accuracy is below
+10−3 comparing to maximal possible value of the witness in higher dimension. The test involving
+minimal independent sets of preparation and measurement operations (gates) is applied both for
+specific configurations and parametric ones.
+The test is be robust against nonidealities such as
+incoherent leakage and erroneous gate execution. Two of the IBM devices failed the test by more
+than 5 standard deviations, which has no simple explanation.
+I.
+INTRODUCTION
+Physics is an exact science, which is confirmed by pre-
+cise measurements of fundamental constants and estab-
+lishing definition of SI units by precise quantum experi-
+ments [1–6]. Precision is also required from every com-
+puter, also quantum.
+Unfortunately, current quantum
+technologies suffer from inevitable sources of errors, both
+just from mechanical limitations and inseparable physical
+environment. Of course, there are methods to mitigate
+and correct the errors. Such approach relies, however,
+on assumptions about the controllable space of possible
+actions.
+The basic building block of a quantum computer is a
+qubit, a generic two-level system. Since the goal is to
+manipulate accurately many qubits, it is necessary to as-
+certain whether or not the qubit space is reliable, i.e.
+not combined with a larger space. The most promising
+implementations of qubits keep them detuned from en-
+vironment and other states, except for small incoherent
+disturbance.
+On the other hand, the potential contri-
+bution of external states can lead to systematic errors,
+hard to correct. Operations on qubits, gates, realized by
+microwave pulses, suffer from distortions due to nonlin-
+earities of waveform generators [7], so a simple deviation
+of the probability distribution from the theoretical pre-
+diction is not yet a proof of extra space [8]. Therefore,
+to increase the quality of classical and quantum com-
+putation and communication, these systems need precise
+certification, robust against imperfections of physical im-
+plementations.
+The dimension of the quantum space can be checked
+by a dimension witness [9–14]. The construction of the
+witness is based on the two-stage protocol, the initial
+preparation and subsequent final measurement, which are
+chosen from independent sets. The preparation must be
+completed before the start of the measurement. A precise
+∗Electronic address: Adam.Bednorz@fuw.edu.pl
+witness must be based on equality, i.e. a quantity, which
+is exactly zero up to a certain dimension, and nonzero
+otherwise. Such a good witness test is the linear inde-
+pendence of the specific dichotomic outcome probability
+p(M|N) for the preparation N and measurement M, see
+Fig. 1, tested by a suitable determinant [15–17]. It has
+been been already performed on optical states [18]. It be-
+longs to a family to equality-based tests, like the Sorkin
+equality [19] in the three-slit experiment [20–22] testing
+Born’s rule [23], benchmarking our trust in fundamental
+quantum models and their actual realizations.
+In this paper, we apply the test to several IBM quan-
+tum device. While some results agree with the 2−level
+model, taking a large statistics revealed signature of the
+failure by more than 5 standard deviations. Of course it
+does not immediately mean a larger space but the prob-
+lem needs urgent further investigation to determine the
+cause, which may be also another assumption of the test
+(e.g. lack of independence of the operations).
+II.
+THEORY
+We apply a test of the qubit space d = 2 with the
+witness constructed for p(M|N) = trMN, N = N † ≥ 0,
+trN = 1 and measurement 1 ≥ M = M † ≥ 0. Taking
+5 preparations Nj, j = 1..5 and 4 measurements Mk,
+k = 1..4.
+Then the determinant W = det p, for the
+5 × 5 matrix p with entries pkj = p(Mj|Nk) and p5j = 1,
+must be equal to zero if all Nj and Mk are represented
+in the same two-level space. In addition, it remains zero
+also if all preparations and measurements contain some
+constant incoherent leakage term, i.e.
+N ′
+j = Nj + Ne
+and M ′
+k = Mk + Me, with Ne and Me independent of
+j and k and commuting with Nj and Mk. In this way,
+the common leakage to higher states does not affect the
+test [14]. For d = 2 we have W = 0, but d = 3 gives
+maximally 27
+√
+2/64 ≃ 0.6 in the real space and ≃ 0.632
+in the complex space [16]. For d = 4 the maximum (real
+and complex) is 212/37 ≃ 1.87. Even higher dimensions
+are saturated by the classical maximum 3.
+arXiv:2301.03296v1  [quant-ph]  9 Jan 2023
+
+2
+The IBM Quantum Experience cloud computing of-
+fers several devices, collections of qubits, which can be
+manipulated by a user-defined set of gates (operations)
+– either single qubit or two-qubit ones, also paramteric.
+One can put barriers (controlling the order of operations)
+or additional resets (nonunitary transition to the ground
+state). The qubits are physical transmons [24], the arti-
+ficial quantum states existing due to interplay of super-
+conductivity (Josephson effect) and capacitance. Due to
+anharmonicity one can limit the working space to two
+states. The decoherence time (mostly environmental) is
+long enough to perform a sequence of quantum opera-
+tions and read out reliable results.
+The ground state |0⟩ can be additionally assured by
+a reset operation.
+Gates are implemented by time-
+scheduled microwave pulses prepared by waveform gen-
+erators and mixers (time 30 − 70ns with sampling at
+0.222ns), tuned to the drive frequency (energy difference
+between qubit levels) [25] (about 4−5Ghz). The rotation
+Z is not a real pulse, but an realized by an instantaneous
+virtual gate VZ(θ), which adds a rotation between in-
+and out-of-phase components of the next gates [26]. The
+readout is performed another long microwave pulse of
+frequency (different from the drive) to the resonator and
+measuring the populated photons [25, 27].
+In the following, we assume the two-level description
+of the qubits, expecting W = 0 up to statistical er-
+ror.
+Larger deviation would be an evidence that this
+description is inaccurate. The states and operators will
+be can be described either in a two-dimensional Hilbert
+space with basis |0⟩, |1⟩ or in the Bloch sphere with
+V = (v0 + v · σ)/2, with the 3-component Bloch vec-
+tor v and standard Pauli matrices
+σ1 =
+�
+0 1
+1 0
+�
+, σ2 =
+�
+0 −i
+i
+0
+�
+, σ3 =
+�
+1
+0
+0 −1
+�
+.
+(1)
+Then the initial state |0⟩⟨0| corresponds to the vector
+(0, 0, 1) while n0 = 1 and |n| ≤ 1 and 2 − |m| ≥ m0 ≥
+|m|. A microwave pulse tuned to the interlevel drive fre-
+quency corresponds to parametric gates, π/2 rotations,
+Sγ = Z†
+γSZγ, Zθ =
+�
+e−iγ/2
+0
+0
+eiγ/2
+�
+,
+S = RX(π/2) =
+√
+X =
+1
+√
+2
+�
+1
+−i
+−i
+1
+�
+,
+(2)
+in the basis |0⟩, |1⟩ while
+S =
+�
+�
+1 0
+0
+0 0 −1
+0 1
+0
+�
+� , Zγ =
+�
+�
+cos γ − sin γ 0
+sin γ
+cos γ
+0
+0
+0
+1
+�
+� ,
+(3)
+on the Bloch vector, i.e. SγV S†
+γ.
+Physically the experiment is sequence of preparation in
+the state |0⟩, two gates Sα, Sβ for the preparation, two
+gates Sφ, Sθ, and the the readout pulse for the measure-
+ment of the state |0⟩ again, see Fig. 2. There are 5 pairs
+of angles αj, βj to be chosen independently of the 4 pairs
+N
+M
+1
+0
+FIG. 1: Preparation and measurement scenario; the state is
+prepared as N and measured by M to give an outcome of
+either 1 or 0.
+|0⟩
+Sα
+Sβ
+Sφ
+Sθ
+FIG. 2: The quantum circuit for the dimension test.
+The
+initial state |0⟩ and four gates Sγ, split into preparation and
+measurement stages, are followed by the final dichotomic mea-
+surement
+θk, φk. Then N = SβSα|0⟩⟨0| and M = S†
+φS†
+θ|0⟩⟨0|SθSφ.
+The actual pulse waveform of a sample sequence of gates
+is depicted in Fig. 3.
+III.
+EXPERIMENT
+In a perfect theory, we can predict a probability for
+every choice of α, β, θ, φ. The experimental results can
+differ for a variety of reasons. Firstly, the test is random
+and we have to estimate the error due to finite statistics.
+For T times the experiment is repeated, the variance of
+W can be estimated as
+T⟨W 2⟩ ≃
+�
+kj
+pkj(1 − pkj)(Adj p)2
+jk,
+(4)
+where Adj is the adjoint matrix (matrix of minors of p,
+with crossed out a given row and column, and then trans-
+0
+30
+60
+90
+119
+149
+Time (ns)
+VZ(2.28)
+VZ(
+4.37)
+VZ(7.33)
+VZ(
+1.57)
+D0
+5.26 GHz
+FIG. 3:
+The actual waveform of the pulse on IBM quan-
+tum computer (nairobi), with four subsequent gates Sγ, with
+γ = α, β, φ, θ, consecutively. The discretization unit time is
+dt = 0.222ns. Driving (level gap) frequency is denoted by D0.
+The light/dark shading corresponds to in-phase/out-of-phase
+amplitude component, respectively. The element VZ(ξ) is a
+zero-duration virtual gate Zξ for subsequent gates SγSδ with
+ξ = γ − δ [26].
+
+3
+posed). Note that the identity p−1 det p = Adjp makes
+no sense here as W = det p = 0 in the limit T → ∞.
+Secondly, the implementation of gates may be not faith-
+ful.
+Our test is capable to take them into account as
+long as the leakage to external states (e.g. |2⟩) is inco-
+herent and does not depend on the parameters α, β, θ, φ.
+Lastly, we have to assume that the pulse does not depend
+on the previous ones. In other words, we can only test
+the combination of assumptions, dimension of the space
+and independence of operations. We have calculated W
+in two ways, (i) determining p for each job and then find-
+ing W (see the values for each job in Fig. 10) and finally
+averaging W, (ii) averaging first p from all jobs and then
+finding W.
+There is no a priori best selection of preparations and
+measurements but they should not lie on a single Bloch
+circle. We decided to make two kinds of tests: (I) two
+special configurations corresponding to either the same
+Bloch vectors for preparation and measurements or max-
+imal ⟨W 2⟩ for a given R; (II) a family of configurations
+with one preparation vector at one of the 5 directions on
+the Bloch circle. In both cases the corresponding Bloch
+vectors are derived explicitly in Appendix A. The sets
+of angles in the case (I) are given in the Table I, and
+the corresponding Bloch vectors are visualized in Fig.
+4.
+We have run the test on lima and lagos, qubit 0.
+The probability matrix, compared to the ideal expecta-
+tion is depicted in Fig. 5. The deviation from zero and
+the statistical error is given in Fig. 6. The number of
+T = #jobs·#shots·#repetitions. Technically, one sends
+a list of jobs to execute, each job contains up to 300 cir-
+cuits, to be distributed between experiments repeated the
+same number of times. Each circuit is run the number
+of shots. The readout counts for each circuit is the value
+returned after the job execution is accomplished.
+The sets of angles in the case (II) are prepared dif-
+ferently. Four preparations and measurements are fixed
+while the last preparation is parameter-dependent. The
+fixed angles are specified in Table II. The last prepara-
+tion angles are α5 = 2πi/5 = β5 − π/2 for i = 0..4. The
+corresponding Bloch vectors are depicted in Fig. 7. We
+have run the test on nairobi and perth, qubit 0.
+The
+probability matrix, compared to the ideal expectation is
+depicted in Fig. 8. The deviation from zero and the sta-
+tistical error is given in Fig. 9. The deviation from the
+expected 0 is more than 5 standard deviations. The data
+and scripts are available at the public repository [28].
+A.
+Nonidealities
+There are several factors that can affect the correct-
+ness of the experiment. (A) The daily calibration. The
+drive frequency and the gate waveforms are corrected so
+different jobs can rely on different realizations of gates.
+There first order effect of calibrations is cancelled out.
+Nevertheless, we made more detailed estimates on sec-
+ond order effects in Appendix B. Only large, unexpected
+j 1
+2’
+3’
+4’
+5’
+2”
+3”
+4”
+5”
+α 0 2π/3 2π/3 4π/3 4π/3
+0 η − π η + 5π/3 η + π/3
+β 0 π/6 −π/6 π/6 −π/6 π
+0
+2π/3
+−2π/3
+k
+1’
+2’
+3’
+4’
+1”
+2”
+3”
+4”
+θ 5π/3 5π/3 π/3
+π/3
+π π/2 7π/6 −π/6
+φ 7π/6 5π/6 7π/6 5π/6 0
+π
+5π/3
+π/3
+TABLE I: The angles for the special two special cases, ’ and
+”, with η = acos(1/3) and 1=1’=1”.
+FIG. 4: The Bloch vectors for the preparations (red) and
+measurements (blue) corresponding to the angles from Table
+I, top ’ and bottom ”. For the case ’, the four measurement
+direction are identical to four preparations.
+j 1
+2
+3
+4
+α 0 η − π η + 5π/3 η + π/3
+β 0
+0
+2π/3
+−2π/3
+k 1
+2
+3
+4
+θ π π/2 7π/6 −π/6
+φ 0
+π
+5π/3
+π/3
+TABLE II: The angles for the parametric case (II) for prepa-
+rations and measurements 1..4
+
+14
+1
+2
+3
+4
+ideal k
+1
+2
+3
+4
+lima k
+1
+2
+3
+4
+5
+j′
+1
+2
+3
+4
+lagos k
+1
+2
+3
+4
+5
+j′′
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+FIG. 5: Results of the test (I) with probabilities pkj for the an-
+gles from Table I, for lima and lagos, compared to the ideal ex-
+pectation. Lagos: 60 jobs, 32000 shots, 15 repetitions. Lima:
+521’/194” jobs, 20000 shots, 5 repetitions
+failures could be a problem. (B) Amplitude-dependent
+leakage and distortion of the waveform. The leakage to
+higher states, e.g.
+|2⟩ is small, of the order 10−4 and
+incoherent [8, 14], see details in Appendix C. It is pos-
+sible that distortion of amplitude to the waveform de-
+pends on the rotation angle (phase) but we expect this
+effect to be very small, 10−3, based on the deviations ob-
+served in our previous work, and so the net effect is 10−7.
+(C) Memory of the waveform between successive gates.
+Highly unlikely, a residual voltage amplitude can persist
+up to the next gate. In principle it can be mitigated by
+delay-separated gates if the effect fades out with time.
+(D) Other qubits. They are usually detuned but some
+crosstalk may remain. As in the case of leakage, we ex-
+pect the crosstalk to be incoherent and so irrelevant for
+the witness. As a sanity check we have run simulations,
+using the noise models from nairobi and perth, and no
+significant deviation have been found, see Appendix D.
+lima′
+lima′′
+lagos′
+lagos′′
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+×10
+4
+FIG. 6: Results of the test (I) the witness W = det p, for the
+angles from Table I, for lima and lagos, with the error given
+by (4). Red – W for p from each job and then averaged, blue
+– p averaged from all jobs to give W.
+FIG. 7: The Bloch vectors for the parametric case (II) with
+fixed preparations (red) and measurements (blue) correspond-
+ing to the angles from Table I, and a parametric preparation
+(green).
+IV.
+DISCUSSION
+A test of linear independence of quantum operations
+reveals subtle deviations, invisible in more crude tests.
+Further tests are necessary to identify the origin of the
+deviations, to exclude e.g. exotic many world/copies the-
+ories [29, 30]. We suggest: (i) an extreme statistics col-
+lected in a relatively short time to avoid corrections due
+to calibrations, (ii) a time separation between gates to ex-
+clude potential overlap of the effects, (iii) a scan through
+a large set of Bloch vectors to maximize the potential
+deviation, (iv) run the test on a single-qubit devices to
+avoid cross-talks. It is also possible to develop more so-
+phisticated tests, with different assumptions, or involv-
+ing different qubits. In any case, a precise diagnostics of
+qubits must become a standard in quantum technologies.
+
+5
+1
+2
+3
+4
+j
+1
+2
+3
+4
+k
+theory
+0
+1
+2
+3
+4
+i
+j = 5
+0.0
+0.5
+1.0
+1
+2
+3
+4
+j
+1
+2
+3
+4
+k
+nairobi
+0
+1
+2
+3
+4
+i
+j = 5
+0.0
+0.5
+1.0
+1
+2
+3
+4
+j
+1
+2
+3
+4
+k
+perth
+0
+1
+2
+3
+4
+i
+j = 5
+0.0
+0.5
+1.0
+FIG. 8: Results of the test (II) with probabilities pkj for the
+angles from Table II, for nairobi and perth, compared to the
+theory expectation. Nairobi/perth: 115/93 jobs, both 100000
+shots and 8 repetitions
+Appendix A: Bloch sphere representations
+Using vectors n to represent the state N = |n⟩⟨n| =
+(ˆ1 + n · σ)/2, we have SαNS†
+α = Nα and S†
+θMSθ = Mθ
+with
+nα =
+�
+�
+cos2 α
+− cos α sin α − sin α
+− cos α sin α
+sin2 α
+− cos α
+sin α
+cos α
+0
+�
+� n,
+mθ =
+�
+�
+cos2 θ
+− cos θ sin θ sin θ
+− cos θ sin θ
+sin2 θ
+cos θ
+− sin θ
+− cos θ
+0
+�
+� n,
+(A1)
+For n = m = (0, 0, 1) and m0 = 1, we have Nαβ =
+SβSαNS†
+αS†
+β with
+n′
+αβ = (sin(β − α) cos β, sin(α − β) sin β, − cos(β − α))
+(A2)
+while Mθφ = S†
+φS†
+θMSθSφ wirh
+M θφ = (sin(θ − φ) cos φ, sin(φ − θ) sin φ, − cos(θ − φ)).
+(A3)
+0
+1
+2
+3
+4
+−2.5
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+nairobi
+×10−4
+0
+1
+2
+3
+4
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+perth
+×10−4
+FIG. 9: Results of the test (II) the witness W = det p, for
+the angles from Table II, for nairobi and perth, with the error
+given by (4). Red – W for p from each job and then averaged,
+blue – p averaged from all jobs to give W.
+0
+1
+2
+3
+4
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+nairobi
+×10−4
+0
+1
+2
+3
+4
+−0.75
+−0.50
+−0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+perth
+×10−3
+FIG. 10: Results of the test (II) the witness W = det p, for the
+angles from Table II, for nairobi and perth, for individual jobs.
+Two values for nairobi are beyond the picture boundaries,
+(3, 0.0023) and (4, 0.003)
+
+6
+1
+2
+3
+4
+j
+1
+2
+3
+4
+k
+nairobi-sim
+0
+1
+2
+3
+4
+i
+j = 5
+0.0
+0.5
+1.0
+1
+2
+3
+4
+j
+1
+2
+3
+4
+k
+perth-sim
+0
+1
+2
+3
+4
+i
+j = 5
+0.0
+0.5
+1.0
+FIG. 11: Results of the simulations of the test (II) with proba-
+bilities pkj for the angles from Table II, for nairobi and perth.
+0
+1
+2
+3
+4
+−2
+0
+2
+4
+nairobi-sim
+×10−5
+0
+1
+2
+3
+4
+−8
+−6
+−4
+−2
+0
+2
+4
+6
+perth-sim
+×10−5
+FIG. 12: Results of the simulations of the test (II) the witness
+W = det p, for the angles from Table II, for nairobi and perth
+noise models. Note that the two ways of calculation of W
+almost coincide (the blue one covers the red one), which is
+consistent with our explanation of averaged out first order
+difference in Appendix B.
+0
+1
+2
+3
+4
+−1.0
+−0.5
+0.0
+0.5
+nairobi-sim
+×10−4
+0
+1
+2
+3
+4
+−7.5
+−5.0
+−2.5
+0.0
+2.5
+5.0
+7.5
+perth-sim
+×10−4
+FIG. 13: Results of the simulations of the test (II) the witness
+W = det p, for the angles from Table II, for nairobi and perth
+noise models, for individual jobs
+Then the probability matrix elements read
+pkj = TrMkNj = (1 + n · m)/2
+(A4)
+while p5j = 1.
+In this way we can represent the choices used
+in our experiment.
+In the first choice,
+prepara-
+tions n′
+1 = (0, 0, −1), n′
+2 = (−
+√
+3/2, 1/2, 0), n′
+3 =
+(−
+√
+3/4, −1/4,
+√
+3/2), n′
+4 = (
+√
+3/4, −1/4,
+√
+3/2), n′
+5 =
+(
+√
+3/2, 1/2, 0) and measurements m′
+k = n′
+k−1. In the sec-
+ond choice, n′′
+1 = −n′′
+2 = (0, 0, −1), n′′
+3 = (2
+√
+2, 0, 1/3),
+n′′
+4,5 = (−
+√
+2/3, ∓
+�
+2/3, 1/3) and measurements m′′
+1 =
+(0, 0, 1), m′′
+2 = (1, 0, 0), m′′
+3,4 = (−1/2, ∓
+√
+3/2, 0).
+For the parametric test we have n1 = (0, 0, 1), n2 =
+(2
+√
+2, 0, 1/3), n3,4 = (−
+√
+2/3, ∓
+�
+2/3, 1/3) while ni
+5 =
+(− sin(2πi/5), − cos(2πi/5), 0).
+Appendix B: Bounds on daily calibrations
+Suppose that the calibration from job to job can al-
+ter the matrix of probabilities. Assuming that each job
+n = 1..N satisfies W (n) = 0 for probabilities p(n), we
+ask if W for p = �
+n p(n)/N can be nonzero. Suppose
+δp(n) = p(n) − p(0) is small for some reference matrix p(0)
+and |δp(n)
+kj | ≤ ϵ for all kj and some small bound ϵ. Then,
+in the first order of δp we have still W ≃ 0 from expand-
+ing determinant in linear combinations of single columns
+
+7
+p(n) and the rest of columns kept equal p(0). The nonva-
+nishing contribution is of the second order, when replac-
+ing either of two columns by δp(n). Their length is ≤ 2ϵ.
+The last row contains 0 for the replaced columns and 1
+for the rest. Subtracting 1/2 of that row from the other
+rows. The moduli of remaining elements are ≤ 1/2 for
+the length of the remaining 3 columns is ≤
+√
+2. From
+Hadamard inequality | det A| ≤ �
+j |Aj| with |Aj being
+the length of the vector (column) Aj of the matrix A,
+we have the upper bound |W| ≤ 80
+√
+2ϵ2 as we have 10
+choices of 2 columns out of 5.
+Appendix C: Corrections from higher states
+The generic Hamiltonian, in the basis states |n⟩, n =
+0, 1, 2, ... (ℏ = 1) reads
+H =
+�
+n
+ωn|n⟩⟨n| + 2 cos(ωt − θ) ˆV (t)
+(C1)
+with energy ωn eigenstates levels and the external drive
+V at frequency ω and phase shift θ (the second term). In
+principle free parameters ω, θ and ˆV (t) can model a com-
+pletely arbitrary evolution. We can estimate deviations
+by perturbative analysis, setting ω0 = 0, ω1 = ω (reso-
+nance), ω2 = 2ω+ω′ (anharmonicity, i.e. ω′ ≪ ω, in IBM
+about 300Mhz compared to drive frequency ∼ 5GHz).
+The state |2⟩ should give the most significant potential
+contribution. We can incorporate rotation and phase into
+the definition of states, |n⟩ → |n′⟩ = e−in(θ+ωt)|n⟩ so that
+H′ =
+�
+�
+2 cos(ωt − θ)V00
+(1 + e−2i(θ+ωt))V01 (e−i(θ+ωt) + e−3i(θ+ωt))V02
+(1 + e2i(ωt+θ))V10
+2 cos(ωt + θ)V11
+(1 + e−2i(θ+ωt))V12
+(e−i(θ+ωt) + e3i(ωt+θ))V20
+(1 + e2i(ωt+θ))V21
+2 cos(ωt + θ)V22 + ω′
+�
+�
+(C2)
+Extracting the Rotating Wave Approximation (RWA) part from H′ = HRW A + ∆H,
+HRW A =
+�
+�
+0
+V01
+0
+V10
+0
+V12
+0
+V21
+ω′
+�
+� ,
+(C3)
+the correction reads
+∆H =
+�
+�
+2 cos(ωt + θ)V00
+e−2i(θ+ωt)V01
+(e−i(θ+ωt) + e−3i(θ+ωt))V02
+e2i(ωt+θ)V10
+2 cos(ωt + θ)V11
+e−2i(θ+ωt)V12
+(e−i(θ+ωt) + e3i(ωt+θ))V20
+e2i(ωt+θ)V21
+2 cos(ωt + θ)V22
+�
+�
+(C4)
+Evolution due to RWA has the form
+U(t) = T exp
+� t
+−∞
+HRW A(t′)dt′/i,
+(C5)
+where T means chronological product in Taylor expan-
+sion. Then the 1st order correction to U reads
+∆U = U(+∞)
+�
+dtU †(t)∆H(t)U(t)/i
+(C6)
+where the full rotation is U(+∞). All θ-dependent terms
+in ∆H, contain also eiωt, which exponentially damps
+slow-varying expressions. The 2nd order correction reads
+∆2U = −U(+∞)×
+�
+dtU †(t)∆H(t)U(t)
+� t
+dt′U †(t′)∆H(t′)U(t′).
+(C7)
+Most of components get damped exponentially, too, ex-
+cept when ∆H(t) contains eikωt and ∆H(t′) contains
+e−ikωt, k = 1, 2, 3, so kθ cancels.
+The nonnegligible
+part of ∆2U is therefore independent of θ giving slowly
+Bloch-Siegert shift [31]. Stroboscopic corrections to RWA
+[32] can be neglected due to a very short sampling time,
+dt = 0.222ns,
+Appendix D: Simulations
+As a cross-check of our test we have run the identical
+programs on IBM simulator of a quantum computer with
+the noise model taken from the real devices perth and
+nairobi. However, in contrast to real devices, the results
+are in agreement with the theory as shown in Figs. 11,
+12, 13.
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+[2] T. A. Wagner, S. Schlamminger, J. H. Gundlach, and E.
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+

49E1T4oBgHgl3EQfmQS7/content/tmp_files/load_file.txt

@@ -0,0 +1,512 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf,len=511
+page_content='Precise certification of a qubit space  Tomasz Białecki1, Tomasz Rybotycki2,3, Josep Batle4, Jakub Tworzydło1, and Adam Bednorz1, ∗  1Faculty of Physics, University of Warsaw, ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Pasteura 5, PL02-093 Warsaw, Poland  2Systems Research Institute, Polish Academy of Sciences, 6 Newelska Street, PL01-447 Warsaw, Poland  3Center for Theoretical Physics, Polish Academy of Sciences,  Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Lotników 32/46, PL02-668 Warsaw, Poland  4CRISP - Centre de Recerca Independent de sa Pobla, 07420 sa Pobla, Balearic Islands, Spain  We demonstrate an implementation of the precise test of dimension on the qubit, using the  public IBM quantum computer, using the determinant dimension witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The accuracy is below  10−3 comparing to maximal possible value of the witness in higher dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The test involving  minimal independent sets of preparation and measurement operations (gates) is applied both for  specific configurations and parametric ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  The test is be robust against nonidealities such as  incoherent leakage and erroneous gate execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Two of the IBM devices failed the test by more  than 5 standard deviations, which has no simple explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  INTRODUCTION  Physics is an exact science, which is confirmed by pre-  cise measurements of fundamental constants and estab-  lishing definition of SI units by precise quantum experi-  ments [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Precision is also required from every com-  puter, also quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Unfortunately, current quantum  technologies suffer from inevitable sources of errors, both  just from mechanical limitations and inseparable physical  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Of course, there are methods to mitigate  and correct the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Such approach relies, however,  on assumptions about the controllable space of possible  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  The basic building block of a quantum computer is a  qubit, a generic two-level system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Since the goal is to  manipulate accurately many qubits, it is necessary to as-  certain whether or not the qubit space is reliable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  not combined with a larger space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The most promising  implementations of qubits keep them detuned from en-  vironment and other states, except for small incoherent  disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  On the other hand, the potential contri-  bution of external states can lead to systematic errors,  hard to correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Operations on qubits, gates, realized by  microwave pulses, suffer from distortions due to nonlin-  earities of waveform generators [7], so a simple deviation  of the probability distribution from the theoretical pre-  diction is not yet a proof of extra space [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Therefore,  to increase the quality of classical and quantum com-  putation and communication, these systems need precise  certification, robust against imperfections of physical im-  plementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  The dimension of the quantum space can be checked  by a dimension witness [9–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The construction of the  witness is based on the two-stage protocol, the initial  preparation and subsequent final measurement, which are  chosen from independent sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The preparation must be  completed before the start of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' A precise  ∗Electronic address: Adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='Bednorz@fuw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='pl  witness must be based on equality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' a quantity, which  is exactly zero up to a certain dimension, and nonzero  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Such a good witness test is the linear inde-  pendence of the specific dichotomic outcome probability  p(M|N) for the preparation N and measurement M, see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 1, tested by a suitable determinant [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' It has  been been already performed on optical states [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' It be-  longs to a family to equality-based tests, like the Sorkin  equality [19] in the three-slit experiment [20–22] testing  Born’s rule [23], benchmarking our trust in fundamental  quantum models and their actual realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  In this paper, we apply the test to several IBM quan-  tum device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' While some results agree with the 2−level  model, taking a large statistics revealed signature of the  failure by more than 5 standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Of course it  does not immediately mean a larger space but the prob-  lem needs urgent further investigation to determine the  cause, which may be also another assumption of the test  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' lack of independence of the operations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  THEORY  We apply a test of the qubit space d = 2 with the  witness constructed for p(M|N) = trMN, N = N † ≥ 0,  trN = 1 and measurement 1 ≥ M = M † ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Taking  5 preparations Nj, j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='.5 and 4 measurements Mk,  k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='.4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Then the determinant W = det p, for the  5 × 5 matrix p with entries pkj = p(Mj|Nk) and p5j = 1,  must be equal to zero if all Nj and Mk are represented  in the same two-level space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' In addition, it remains zero  also if all preparations and measurements contain some  constant incoherent leakage term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  N ′  j = Nj + Ne  and M ′  k = Mk + Me, with Ne and Me independent of  j and k and commuting with Nj and Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' In this way,  the common leakage to higher states does not affect the  test [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' For d = 2 we have W = 0, but d = 3 gives  maximally 27  √  2/64 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='6 in the real space and ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='632  in the complex space [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' For d = 4 the maximum (real  and complex) is 212/37 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Even higher dimensions  are saturated by the classical maximum 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='03296v1  [quant-ph]  9 Jan 2023  2  The IBM Quantum Experience cloud computing of-  fers several devices, collections of qubits, which can be  manipulated by a user-defined set of gates (operations)  – either single qubit or two-qubit ones, also paramteric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  One can put barriers (controlling the order of operations)  or additional resets (nonunitary transition to the ground  state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The qubits are physical transmons [24], the arti-  ficial quantum states existing due to interplay of super-  conductivity (Josephson effect) and capacitance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Due to  anharmonicity one can limit the working space to two  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The decoherence time (mostly environmental) is  long enough to perform a sequence of quantum opera-  tions and read out reliable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  The ground state |0⟩ can be additionally assured by  a reset operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Gates are implemented by time-  scheduled microwave pulses prepared by waveform gen-  erators and mixers (time 30 − 70ns with sampling at  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='222ns), tuned to the drive frequency (energy difference  between qubit levels) [25] (about 4−5Ghz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The rotation  Z is not a real pulse, but an realized by an instantaneous  virtual gate VZ(θ), which adds a rotation between in-  and out-of-phase components of the next gates [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The  readout is performed another long microwave pulse of  frequency (different from the drive) to the resonator and  measuring the populated photons [25, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  In the following, we assume the two-level description  of the qubits, expecting W = 0 up to statistical er-  ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Larger deviation would be an evidence that this  description is inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The states and operators will  be can be described either in a two-dimensional Hilbert  space with basis |0⟩, |1⟩ or in the Bloch sphere with  V = (v0 + v · σ)/2, with the 3-component Bloch vec-  tor v and standard Pauli matrices  σ1 =  �  0 1  1 0  �  , σ2 =  �  0 −i  i  0  �  , σ3 =  �  1  0  0 −1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  (1)  Then the initial state |0⟩⟨0| corresponds to the vector  (0, 0, 1) while n0 = 1 and |n| ≤ 1 and 2 − |m| ≥ m0 ≥  |m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' A microwave pulse tuned to the interlevel drive fre-  quency corresponds to parametric gates, π/2 rotations,  Sγ = Z†  γSZγ, Zθ =  �  e−iγ/2  0  0  eiγ/2  �  ,  S = RX(π/2) =  √  X =  1  √  2  �  1  −i  −i  1  �  ,  (2)  in the basis |0⟩, |1⟩ while  S =  �  �  1 0  0  0 0 −1  0 1  0  �  � , Zγ =  �  �  cos γ − sin γ 0  sin γ  cos γ  0  0  0  1  �  � ,  (3)  on the Bloch vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' SγV S†  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Physically the experiment is sequence of preparation in  the state |0⟩, two gates Sα, Sβ for the preparation, two  gates Sφ, Sθ, and the the readout pulse for the measure-  ment of the state |0⟩ again, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' There are 5 pairs  of angles αj, βj to be chosen independently of the 4 pairs  N  M  1  0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 1: Preparation and measurement scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' the state is  prepared as N and measured by M to give an outcome of  either 1 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  |0⟩  Sα  Sβ  Sφ  Sθ  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 2: The quantum circuit for the dimension test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  The  initial state |0⟩ and four gates Sγ, split into preparation and  measurement stages, are followed by the final dichotomic mea-  surement  θk, φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Then N = SβSα|0⟩⟨0| and M = S†  φS†  θ|0⟩⟨0|SθSφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  The actual pulse waveform of a sample sequence of gates  is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  EXPERIMENT  In a perfect theory, we can predict a probability for  every choice of α, β, θ, φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The experimental results can  differ for a variety of reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Firstly, the test is random  and we have to estimate the error due to finite statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  For T times the experiment is repeated, the variance of  W can be estimated as  T⟨W 2⟩ ≃  �  kj  pkj(1 − pkj)(Adj p)2  jk,  (4)  where Adj is the adjoint matrix (matrix of minors of p,  with crossed out a given row and column, and then trans-  0  30  60  90  119  149  Time (ns)  VZ(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='28)  VZ(  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='37)  VZ(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='33)  VZ(  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='57)  D0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='26 GHz  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 3:  The actual waveform of the pulse on IBM quan-  tum computer (nairobi), with four subsequent gates Sγ, with  γ = α, β, φ, θ, consecutively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The discretization unit time is  dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='222ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Driving (level gap) frequency is denoted by D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  The light/dark shading corresponds to in-phase/out-of-phase  amplitude component, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The element VZ(ξ) is a  zero-duration virtual gate Zξ for subsequent gates SγSδ with  ξ = γ − δ [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  3  posed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Note that the identity p−1 det p = Adjp makes  no sense here as W = det p = 0 in the limit T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Secondly, the implementation of gates may be not faith-  ful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Our test is capable to take them into account as  long as the leakage to external states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' |2⟩) is inco-  herent and does not depend on the parameters α, β, θ, φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Lastly, we have to assume that the pulse does not depend  on the previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' In other words, we can only test  the combination of assumptions, dimension of the space  and independence of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' We have calculated W  in two ways, (i) determining p for each job and then find-  ing W (see the values for each job in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 10) and finally  averaging W, (ii) averaging first p from all jobs and then  finding W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  There is no a priori best selection of preparations and  measurements but they should not lie on a single Bloch  circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' We decided to make two kinds of tests: (I) two  special configurations corresponding to either the same  Bloch vectors for preparation and measurements or max-  imal ⟨W 2⟩ for a given R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' (II) a family of configurations  with one preparation vector at one of the 5 directions on  the Bloch circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' In both cases the corresponding Bloch  vectors are derived explicitly in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The sets  of angles in the case (I) are given in the Table I, and  the corresponding Bloch vectors are visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  We have run the test on lima and lagos, qubit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  The probability matrix, compared to the ideal expecta-  tion is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The deviation from zero and  the statistical error is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The number of  T = #jobs·#shots·#repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Technically, one sends  a list of jobs to execute, each job contains up to 300 cir-  cuits, to be distributed between experiments repeated the  same number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Each circuit is run the number  of shots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The readout counts for each circuit is the value  returned after the job execution is accomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  The sets of angles in the case (II) are prepared dif-  ferently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Four preparations and measurements are fixed  while the last preparation is parameter-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The  fixed angles are specified in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The last prepara-  tion angles are α5 = 2πi/5 = β5 − π/2 for i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='.4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The  corresponding Bloch vectors are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' We  have run the test on nairobi and perth, qubit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  The  probability matrix, compared to the ideal expectation is  depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The deviation from zero and the sta-  tistical error is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The deviation from the  expected 0 is more than 5 standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The data  and scripts are available at the public repository [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Nonidealities  There are several factors that can affect the correct-  ness of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' (A) The daily calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The  drive frequency and the gate waveforms are corrected so  different jobs can rely on different realizations of gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  There first order effect of calibrations is cancelled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Nevertheless, we made more detailed estimates on sec-  ond order effects in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Only large, unexpected  j 1  2’  3’  4’  5’  2”  3”  4”  5”  α 0 2π/3 2π/3 4π/3 4π/3  0 η − π η + 5π/3 η + π/3  β 0 π/6 −π/6 π/6 −π/6 π  0  2π/3  −2π/3  k  1’  2’  3’  4’  1”  2”  3”  4”  θ 5π/3 5π/3 π/3  π/3  π π/2 7π/6 −π/6  φ 7π/6 5π/6 7π/6 5π/6 0  π  5π/3  π/3  TABLE I: The angles for the special two special cases, ’ and  ”, with η = acos(1/3) and 1=1’=1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 4: The Bloch vectors for the preparations (red) and  measurements (blue) corresponding to the angles from Table  I, top ’ and bottom ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' For the case ’, the four measurement  direction are identical to four preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  j 1  2  3  4  α 0 η − π η + 5π/3 η + π/3  β 0  0  2π/3  −2π/3  k 1  2  3  4  θ π π/2 7π/6 −π/6  φ 0  π  5π/3  π/3  TABLE II: The angles for the parametric case (II) for prepa-  rations and measurements 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='.4  14  1  2  3  4  ideal k  1  2  3  4  lima k  1  2  3  4  5  j′  1  2  3  4  lagos k  1  2  3  4  5  j′′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 5: Results of the test (I) with probabilities pkj for the an-  gles from Table I, for lima and lagos, compared to the ideal ex-  pectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Lagos: 60 jobs, 32000 shots, 15 repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Lima:  521’/194” jobs, 20000 shots, 5 repetitions  failures could be a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' (B) Amplitude-dependent  leakage and distortion of the waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The leakage to  higher states, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  |2⟩ is small, of the order 10−4 and  incoherent [8, 14], see details in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' It is pos-  sible that distortion of amplitude to the waveform de-  pends on the rotation angle (phase) but we expect this  effect to be very small, 10−3, based on the deviations ob-  served in our previous work, and so the net effect is 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  (C) Memory of the waveform between successive gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Highly unlikely, a residual voltage amplitude can persist  up to the next gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' In principle it can be mitigated by  delay-separated gates if the effect fades out with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  (D) Other qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' They are usually detuned but some  crosstalk may remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' As in the case of leakage, we ex-  pect the crosstalk to be incoherent and so irrelevant for  the witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' As a sanity check we have run simulations,  using the noise models from nairobi and perth, and no  significant deviation have been found, see Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  lima′  lima′′  lagos′  lagos′′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  ×10  4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 6: Results of the test (I) the witness W = det p, for the  angles from Table I, for lima and lagos, with the error given  by (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Red – W for p from each job and then averaged, blue  – p averaged from all jobs to give W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 7: The Bloch vectors for the parametric case (II) with  fixed preparations (red) and measurements (blue) correspond-  ing to the angles from Table I, and a parametric preparation  (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  DISCUSSION  A test of linear independence of quantum operations  reveals subtle deviations, invisible in more crude tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Further tests are necessary to identify the origin of the  deviations, to exclude e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' exotic many world/copies the-  ories [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' We suggest: (i) an extreme statistics col-  lected in a relatively short time to avoid corrections due  to calibrations, (ii) a time separation between gates to ex-  clude potential overlap of the effects, (iii) a scan through  a large set of Bloch vectors to maximize the potential  deviation, (iv) run the test on a single-qubit devices to  avoid cross-talks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' It is also possible to develop more so-  phisticated tests, with different assumptions, or involv-  ing different qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' In any case, a precise diagnostics of  qubits must become a standard in quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  5  1  2  3  4  j  1  2  3  4  k  theory  0  1  2  3  4  i  j = 5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  1  2  3  4  j  1  2  3  4  k  nairobi  0  1  2  3  4  i  j = 5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  1  2  3  4  j  1  2  3  4  k  perth  0  1  2  3  4  i  j = 5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 8: Results of the test (II) with probabilities pkj for the  angles from Table II, for nairobi and perth, compared to the  theory expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Nairobi/perth: 115/93 jobs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' both 100000  shots and 8 repetitions  Appendix A: Bloch sphere representations  Using vectors n to represent the state N = |n⟩⟨n| =  (ˆ1 + n · σ)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' we have SαNS†  α = Nα and S†  θMSθ = Mθ  with  nα =  �  �  cos2 α  − cos α sin α − sin α  − cos α sin α  sin2 α  − cos α  sin α  cos α  0  �  � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  mθ =  �  �  cos2 θ  − cos θ sin θ sin θ  − cos θ sin θ  sin2 θ  cos θ  − sin θ  − cos θ  0  �  � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  (A1)  For n = m = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 1) and m0 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' we have Nαβ =  SβSαNS†  αS†  β with  n′  αβ = (sin(β − α) cos β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' sin(α − β) sin β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' − cos(β − α))  (A2)  while Mθφ = S†  φS†  θMSθSφ wirh  M θφ = (sin(θ − φ) cos φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' sin(φ − θ) sin φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' − cos(θ − φ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  (A3)  0  1  2  3  4  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  nairobi  ×10−4  0  1  2  3  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  perth  ×10−4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 9: Results of the test (II) the witness W = det p, for  the angles from Table II, for nairobi and perth, with the error  given by (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Red – W for p from each job and then averaged,  blue – p averaged from all jobs to give W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  0  1  2  3  4  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  nairobi  ×10−4  0  1  2  3  4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='75  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='50  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='00  perth  ×10−3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 10: Results of the test (II) the witness W = det p, for the  angles from Table II, for nairobi and perth, for individual jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Two values for nairobi are beyond the picture boundaries,  (3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0023) and (4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='003)  6  1  2  3  4  j  1  2  3  4  k  nairobi-sim  0  1  2  3  4  i  j = 5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  1  2  3  4  j  1  2  3  4  k  perth-sim  0  1  2  3  4  i  j = 5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 11: Results of the simulations of the test (II) with proba-  bilities pkj for the angles from Table II, for nairobi and perth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  0  1  2  3  4  −2  0  2  4  nairobi-sim  ×10−5  0  1  2  3  4  −8  −6  −4  −2  0  2  4  6  perth-sim  ×10−5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 12: Results of the simulations of the test (II) the witness  W = det p, for the angles from Table II, for nairobi and perth  noise models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Note that the two ways of calculation of W  almost coincide (the blue one covers the red one), which is  consistent with our explanation of averaged out first order  difference in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  0  1  2  3  4  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  nairobi-sim  ×10−4  0  1  2  3  4  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='5  perth-sim  ×10−4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 13: Results of the simulations of the test (II) the witness  W = det p, for the angles from Table II, for nairobi and perth  noise models, for individual jobs  Then the probability matrix elements read  pkj = TrMkNj = (1 + n · m)/2  (A4)  while p5j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  In this way we can represent the choices used  in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  In the first choice,  prepara-  tions n′  1 = (0, 0, −1), n′  2 = (−  √  3/2, 1/2, 0), n′  3 =  (−  √  3/4, −1/4,  √  3/2), n′  4 = (  √  3/4, −1/4,  √  3/2), n′  5 =  (  √  3/2, 1/2, 0) and measurements m′  k = n′  k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' In the sec-  ond choice, n′′  1 = −n′′  2 = (0, 0, −1), n′′  3 = (2  √  2, 0, 1/3),  n′′  4,5 = (−  √  2/3, ∓  �  2/3, 1/3) and measurements m′′  1 =  (0, 0, 1), m′′  2 = (1, 0, 0), m′′  3,4 = (−1/2, ∓  √  3/2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  For the parametric test we have n1 = (0, 0, 1), n2 =  (2  √  2, 0, 1/3), n3,4 = (−  √  2/3, ∓  �  2/3, 1/3) while ni  5 =  (− sin(2πi/5), − cos(2πi/5), 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Appendix B: Bounds on daily calibrations  Suppose that the calibration from job to job can al-  ter the matrix of probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Assuming that each job  n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='.N satisfies W (n) = 0 for probabilities p(n), we  ask if W for p = �  n p(n)/N can be nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Suppose  δp(n) = p(n) − p(0) is small for some reference matrix p(0)  and |δp(n)  kj | ≤ ϵ for all kj and some small bound ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Then,  in the first order of δp we have still W ≃ 0 from expand-  ing determinant in linear combinations of single columns  7  p(n) and the rest of columns kept equal p(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The nonva-  nishing contribution is of the second order, when replac-  ing either of two columns by δp(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Their length is ≤ 2ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  The last row contains 0 for the replaced columns and 1  for the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Subtracting 1/2 of that row from the other  rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The moduli of remaining elements are ≤ 1/2 for  the length of the remaining 3 columns is ≤  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' From  Hadamard inequality | det A| ≤ �  j |Aj| with |Aj being  the length of the vector (column) Aj of the matrix A,  we have the upper bound |W| ≤ 80  √  2ϵ2 as we have 10  choices of 2 columns out of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Appendix C: Corrections from higher states  The generic Hamiltonian, in the basis states |n⟩, n =  0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' (ℏ = 1) reads  H =  �  n  ωn|n⟩⟨n| + 2 cos(ωt − θ) ˆV (t)  (C1)  with energy ωn eigenstates levels and the external drive  V at frequency ω and phase shift θ (the second term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' In  principle free parameters ω, θ and ˆV (t) can model a com-  pletely arbitrary evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' We can estimate deviations  by perturbative analysis, setting ω0 = 0, ω1 = ω (reso-  nance), ω2 = 2ω+ω′ (anharmonicity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' ω′ ≪ ω, in IBM  about 300Mhz compared to drive frequency ∼ 5GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  The state |2⟩ should give the most significant potential  contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' We can incorporate rotation and phase into  the definition of states,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' |n⟩ → |n′⟩ = e−in(θ+ωt)|n⟩ so that  H′ =  �  �  2 cos(ωt − θ)V00  (1 + e−2i(θ+ωt))V01 (e−i(θ+ωt) + e−3i(θ+ωt))V02  (1 + e2i(ωt+θ))V10  2 cos(ωt + θ)V11  (1 + e−2i(θ+ωt))V12  (e−i(θ+ωt) + e3i(ωt+θ))V20  (1 + e2i(ωt+θ))V21  2 cos(ωt + θ)V22 + ω′  �  �  (C2)  Extracting the Rotating Wave Approximation (RWA) part from H′ = HRW A + ∆H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  HRW A =  �  �  0  V01  0  V10  0  V12  0  V21  ω′  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  (C3)  the correction reads  ∆H =  �  �  2 cos(ωt + θ)V00  e−2i(θ+ωt)V01  (e−i(θ+ωt) + e−3i(θ+ωt))V02  e2i(ωt+θ)V10  2 cos(ωt + θ)V11  e−2i(θ+ωt)V12  (e−i(θ+ωt) + e3i(ωt+θ))V20  e2i(ωt+θ)V21  2 cos(ωt + θ)V22  �  �  (C4)  Evolution due to RWA has the form  U(t) = T exp  � t  −∞  HRW A(t′)dt′/i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  (C5)  where T means chronological product in Taylor expan-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Then the 1st order correction to U reads  ∆U = U(+∞)  �  dtU †(t)∆H(t)U(t)/i  (C6)  where the full rotation is U(+∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' All θ-dependent terms  in ∆H, contain also eiωt, which exponentially damps  slow-varying expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' The 2nd order correction reads  ∆2U = −U(+∞)×  �  dtU †(t)∆H(t)U(t)  � t  dt′U †(t′)∆H(t′)U(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  (C7)  Most of components get damped exponentially, too, ex-  cept when ∆H(t) contains eikωt and ∆H(t′) contains  e−ikωt, k = 1, 2, 3, so kθ cancels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  The nonnegligible  part of ∆2U is therefore independent of θ giving slowly  Bloch-Siegert shift [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Stroboscopic corrections to RWA  [32] can be neglected due to a very short sampling time,  dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='222ns,  Appendix D: Simulations  As a cross-check of our test we have run the identical  programs on IBM simulator of a quantum computer with  the noise model taken from the real devices perth and  nairobi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' However, in contrast to real devices, the results  are in agreement with the theory as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' 11,  12, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Eötvös, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Pekár, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Fekete, Beiträge zum Gesetz  der Proportionalität von Trägheit and Gravität, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' (Berlin) 373, 11, 1922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  8  [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Wagner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Schlamminger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Gundlach, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Adelberger, Torsion-balance tests of the weak equiva-  lence principle, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=' Quantum Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content=', 29:184002, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
+page_content='  [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'}
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+1 
+ 
+How Does Traffic Environment Quantitatively Affect 
+the Autonomous Driving Prediction? 
+ 
+Wenbo Shao, Yanchao Xu, Jun Li, Chen Lv, Senior Member, IEEE, Weida Wang and Hong Wang☒, Senior Member, 
+IEEE
+Abstract—An accurate trajectory prediction is crucial for safe 
+and efficient autonomous driving in complex traffic environments. 
+In recent years, artificial intelligence has shown strong capabilities 
+in improving prediction accuracy. However, its characteristics of 
+inexplicability and uncertainty make it challenging to determine 
+the traffic environmental effect on prediction explicitly, posing 
+significant challenges to safety-critical decision-making. To 
+address these challenges, this study proposes a trajectory 
+prediction framework with the epistemic uncertainty estimation 
+ability that outputs high uncertainty when confronting 
+unforeseeable or unknown scenarios. The proposed framework is 
+used to analyze the environmental effect on the prediction 
+algorithm performance. In the analysis, the traffic environment is 
+considered in terms of scenario features and shifts, respectively, 
+where features are divided into kinematic features of a target 
+agent, features of its surrounding traffic participants, and other 
+features. In addition, feature correlation and importance analyses 
+are performed to study the above features’ influence on the 
+prediction error and epistemic uncertainty. Further, a cross-
+dataset case study is conducted using multiple intersection 
+datasets to investigate the impact of unavoidable distributional 
+shifts in the real world on trajectory prediction. The results 
+indicate that the deep ensemble-based method has advantages in 
+improving prediction robustness and estimating epistemic 
+uncertainty. The consistent conclusions are obtained by the 
+feature correlation and importance analyses, including the 
+conclusion that kinematic features of the target agent have 
+relatively strong effects on the prediction error and epistemic 
+uncertainty. Furthermore, the prediction failure caused by 
+distributional shifts and the potential of the deep ensemble-based 
+method are analyzed. 
+ 
+Index Terms—Artificial intelligence, autonomous driving, 
+distributional shift, epistemic uncertainty, traffic environment, 
+trajectory prediction. 
+I. INTRODUCTION 
+RAJECTORY prediction is an indispensable part of the 
+autonomous driving pipeline [1]. To drive safely and 
+efficiently 
+in 
+complex 
+traffic 
+environments, 
+autonomous vehicles (AVs) are required to have the ability to 
+predict the future motion of surrounding traffic participants 
+ 
+This work has been submitted to the IEEE for possible publication. Copyri
+ght may be transferred without notice. This work was supported in part by the 
+National Science Foundation of China Project: 52072215and U1964203, and t
+he National Key R&D Program of China:2020YFB1600303. (Corresponding 
+authors: Hong Wang) 
+Wenbo Shao, Jun Li and Hong Wang are with Tsinghua Intelligent Vehicle 
+Design and Safety Research Institute, School of Vehicle and Mobility, Tsingh
+ua University, Beijing 100084, China. (e-mail: swb19@mails.tsinghua.edu.cn;
+ lijun1958@tsinghua.edu.cn; hong_wang@tsinghua.edu.cn). 
+(TPs), such as vehicles and pedestrians, accurately and reliably. 
+In recent years, with the accumulation of large-scale driving 
+data and rapid development of algorithms, artificial intelligence 
+(AI) has been widely applied to autonomous driving trajectory 
+prediction [2, 3], and promising results have been achieved. 
+However, trajectory prediction has still been challenging, 
+particularly in urban driving scenarios, where an agent's 
+movement is influenced by a combination of its historical state 
+and its complex interactions with the surrounding environment. 
+Many recent studies [4-6] have considered multiple factors 
+simultaneously to improve trajectory prediction algorithms, but 
+there has still been certain performance degradation of a 
+prediction model in complex traffic environments.  
+With the improvement in prediction accuracy, the 
+complexity of AI-based models has also increased gradually. 
+Highly elaborated models pose a great challenge to explaining 
+the operation and failure mechanisms of prediction algorithms, 
+which in turn reduces the credibility of a prediction model. In 
+addition, AI has its inherent uncertainty and faces many 
+problems, such as insufficient training data, imperfect model 
+architecture, and limited training process, which may lead to 
+functional insufficiencies of the model under specific 
+environmental conditions, potentially causing severe traffic 
+accidents [7]. The existing research mainly focuses on 
+improving the dataset-level accuracy of prediction algorithms 
+[8, 9], and little attention has been paid to the changes in 
+prediction 
+performance 
+under 
+different 
+environmental 
+conditions. However, this is not conducive to addressing the 
+practical challenges that a prediction algorithm confronts. 
+For a target agent (TA) moving in a specific scenario, a 
+trajectory prediction model predicts its future trajectory by 
+modeling time series, interaction, and other relationships based 
+on its historical state, surrounding TPs' features, and other 
+environmental features. Correspondingly, various traffic 
+environmental factors may have different effects on trajectory 
+prediction, but fewer studies have quantitatively investigated 
+these effects. Further, data-driven methods strongly depend on  
+Yanchao Xu and Weida Wang is with the School of Mechanical Engineerin
+g, Beijing Institute of Technology, Beijing 100081, China. (e-mail: 31202004
+10@bit.edu.cn, wangwd0430@163.com) 
+Chen Lv is with the School of Mechanical and Aerospace Engineering, 
+Nanyang 
+Technological 
+University, 
+Singapore 
+639798 
+(e-mail: 
+lyuchen@ntu.edu.sg). 
+T 
+
+1 
+Traffic environmental features and distributional shifts
+Trajectory prediction with epistemic uncertainty estimation
+Graph 
+Representation
+Deep Ensemble-based prediction method
+Graph 
+Convolution 
+Model
+RNN-based 
+Trajectory 
+Prediction 
+Model
+Graph Feature
+How dose traffic environment affect the 
+autonomous driving prediction?
+Distributional shifts
+Surrounding traffic 
+participants
+The target agent
+Different environmental features
+Scenario features analysis & research across intersection datasets
+Prediction error & epistemic 
+uncertainty estimation
+quantitative analysis
+Answer
+As independent 
+variables
+qualitative analysis
+ 
+Fig. 1. Illustration of the traffic environment effect on the trajectory prediction algorithm performance. The traffic environmental 
+data include various TAs’ states, their surrounding TPs’ states, and other contextual information, which may affect the prediction 
+differently. In addition, variations in time and place can lead to distributional shifts, which may further degrade the prediction 
+performance. This study focuses on extracting these factors and analyzing their influence on prediction performance. 
+ 
+training data, and a model trained on one dataset may not 
+perform well on other datasets. In real-world applications, the 
+operating environment of AVs may change significantly with 
+different factors, such as time, geography, country, and weather 
+conditions. This may cause certain distributional shifts, posing 
+additional challenges to trajectory prediction. Therefore, it is 
+increasingly important to study how distributional shifts [10] in 
+a real environment affect trajectory prediction. As shown in Fig. 
+1, this work focuses on the effects of both specific scenario 
+features and scenario shifts on the prediction algorithm. 
+As for the prediction algorithm performance, previous 
+studies have generally focused on prediction error. In recent 
+years, there has been an increasing interest in extracting the 
+uncertainty of AI-based models [11, 12], thus empowering the 
+models with a self-awareness ability. Epistemic uncertainty [13] 
+is a recurring suggestion that helps to represent the model's 
+confidence in its current predictions; namely, these models tend 
+to have greater epistemic uncertainty when they encounter 
+challenging environments. Therefore, the epistemic uncertainty 
+of a prediction model is extracted and considered a type of 
+performance metric in this work. As shown in Fig. 1, based on 
+both the prediction error and epistemic uncertainty, the effect 
+of the traffic environment on the prediction algorithms’ 
+performances can be analyzed. 
+The main contributions of this work can be summarized as 
+follows: 
+ 
+A trajectory prediction framework that integrates 
+epistemic uncertainty estimation is proposed. The 
+proposed framework performs the TA’s future state 
+prediction and estimates the epistemic uncertainty 
+simultaneously; 
+ 
+The potential of the proposed deep ensemble-based 
+trajectory prediction framework for improving the 
+prediction 
+algorithm 
+robustness 
+and 
+estimating 
+epistemic uncertainty is demonstrated; 
+ 
+For the trajectory prediction task, the key features of a 
+traffic environment are extracted, and methods for 
+feature correlation analysis and feature importance 
+analysis are proposed to obtain the relationship between 
+the traffic environment and the trajectory prediction 
+algorithm performance; 
+ 
+The distributional shifts between different intersection 
+datasets and the resulting trajectory prediction 
+degradation are investigated. The features of multiple 
+datasets and their prediction difficulty levels are 
+analyzed, and it is demonstrated that the deep ensemble 
+is helpful in improving the trajectory prediction 
+robustness against cross-dataset evaluation. 
+The remainder of this paper is organized as follows. Section 
+II presents the existing work related to this paper. Section III 
+introduces the proposed method. Section IV describes the 
+datasets and evaluation metrics used in this work, as well as 
+implementation details. Section V analyzes and discusses the 
+experimental results. Section VI concludes the paper. 
+II. RELATED WORK 
+A. Trajectory Prediction 
+There have been numerous studies on improving the 
+trajectory prediction algorithms, and according to the modeling 
+principles, they can be mainly divided into physics-based 
+
+2 
+methods, maneuver-based methods, and interaction-aware 
+methods [14]. Physics-based methods [15] consider only the 
+historical motion state of an object while ignoring the influence 
+of surrounding TPs. Therefore, they are mainly suitable for 
+short-term trajectory predictions. Maneuver-based methods [16] 
+learn prototype trajectories from the observed agent behaviors 
+to predict future motion, but they lack consideration of 
+interactions between TPs. Interaction-aware methods [17] have 
+shown better performance compared to the other two types of 
+methods through learning the interaction between a TA and 
+surrounding TPs. 
+In recent works, many methods have been used to model 
+interactions between agents, providing valuable information for 
+trajectory prediction improvement [3, 9]. For instance, social 
+pooling (S-pooling) [8] pools hidden states of a TA’s neighbors 
+within a certain spatial distance to model interactions with the 
+surrounding environment. Convolutional social pooling [18] 
+combines the convolutional and max-pooling layers to model 
+interactions 
+between 
+agents 
+in 
+the 
+occupancy 
+grid. 
+Subsequently, the grid representation is further modified to 
+consider only eight neighbors that have the most critical impact 
+on the TA [19]. In addition, recent research has focused on the 
+rasterized representation of scenes; the historical state of a TA 
+and scene context were co-encoded in a raster map [20-22], and 
+various information was distinguished by different channels 
+and colors. In addition, convolutional neural networks (CNNs) 
+were used to extract desired features from raster graphs. Graph 
+models have attracted great interest recently due to their good 
+performance in modeling inter-agent interactions. In graph 
+models, a node represents an agent, and an edge represents an 
+interaction between two agents. Diehl et al. [23] modeled 
+interactions between vehicles as a homogeneous directed graph 
+to achieve high computational efficiency and large model 
+capacity. They evaluated graph convolutional networks and 
+graph attention networks and introduced several adaptations for 
+specific scenarios. Mo et al. [2] employed a heterogeneous 
+edge-enhanced graph attention network to handle the 
+heterogeneity of TAs and TPs. The GRIP [24] represents the 
+input as a specific grid and uses an undirected graph to model 
+interactions between agents within a certain range, where fixed 
+graphs are considered in the graph convolution submodule. The 
+GRIP++ [5] improves the above-mentioned method by 
+adopting trainable graphs, which overcomes the shortcoming 
+that fixed graphs based on manually designed rules cannot 
+model interaction properly. In addition to the interaction 
+modeling, another important requirement of trajectory 
+prediction relates to time series processing. Recently, recurrent 
+neural networks (RNNs), including the long short-term memory 
+(LSTM) and gated recurrent unit (GRU) models, have been 
+widely used in modeling sequential problems, and significant 
+results have been achieved. Accordingly, these models have 
+been used as sub-modules in many trajectory prediction 
+algorithms [5, 6]. 
+Neural networks have been demonstrated to be highly 
+efficient in trajectory prediction for different classes of TPs. 
+Research on pedestrian intent modeling and motion prediction 
+has been conducted for decades. The Social-LSTM [8] is a 
+typical success case in early research in this field, which 
+combines S-pooling and LSTM to predict the future trajectory 
+of pedestrians in crowded scenes. The Social-GAN [25] uses 
+generative 
+adversarial 
+networks 
+(GANs), 
+sequence-to-
+sequence models, and pooling mechanisms and employs the 
+corresponding generators and recursive discriminators to 
+predict pedestrians’ socially feasible future. However, the GAN 
+model training is difficult and may not converge and can lead 
+to mode collapsing and dropping. Therefore, the Social-Ways 
+uses the Info-GAN, which does not apply the mean square error 
+loss (L2 loss) to force the generated samples to be close to real 
+data but adds another item to consider mutual information, thus 
+alleviating the above-mentioned problems. Since vehicles have 
+higher running speeds and need to obey more road constraints 
+than pedestrians, predicting their future movements is a 
+prerequisite for realizing safe and efficient autonomous driving. 
+A number of studies have designed specialized networks for 
+vehicle trajectory prediction [26]. For instance, vehicle 
+trajectory prediction in highway scenarios, which are relatively 
+simple and where the motion pattern of a vehicle is relatively 
+fixed, has received early attention [18, 24, 27]. With the 
+collection of large-scale datasets [28, 29] and the development 
+of autonomous driving in urban scenes, much research has been 
+focused on motion prediction in complex urban environments 
+[4, 30-32]. TrafficPredict [33] adopted a four-dimensional 
+graph to model the interaction in the instance and category 
+layers, thus realizing the heterogeneous traffic-agent trajectory 
+prediction. The GRIP++ achieved joint trajectory prediction of 
+all observed objects while considering multiple classes of TPs, 
+thus greatly improving real-time prediction performance. 
+However, the above work focuses on the improvement in the 
+dataset-level accuracy while ignoring the research on the 
+sensitivity of the prediction algorithm to environmental factors, 
+which is the focus of this work. 
+B. Epistemic Uncertainty Estimation 
+Due to the rapid development of neural networks and their 
+application to trajectory prediction tasks, it has become 
+increasingly important to estimate the network confidence in its 
+prediction accuracy. However, the original neural network 
+cannot provide an estimation of its epistemic uncertainty. To 
+address this shortcoming, some studies have considered and 
+quantified the epistemic uncertainty of neural networks [11, 34, 
+35], which represents an indicator that can express how 
+confident the network is in its current prediction result. The 
+main epistemic uncertainty estimation methods include the 
+Bayesian neural network (BNN), single-pass uncertainty 
+estimation, and ensemble-based methods. The BNN quantifies 
+the epistemic uncertainty of a neural network by introducing 
+uncertainty into its parameters. The key challenge of these 
+methods is to solve the posterior distribution of network 
+parameters. In the early research, variational inference (VI) [36], 
+which uses a prespecified family of distributions [37, 38], was 
+widely adopted as a method with a strong theoretical basis. 
+However, with the rapid growth in the neural network structure 
+complexity, VI has faced many challenges in terms of solving 
+difficulty and computational complexity. To address these 
+limitations, the Monte Carlo (MC) dropout [39, 40] was 
+proposed to approximate the results obtained by sampling, 
+assuming that the network weights conformed to a Bernoulli 
+distribution. It has been theoretically demonstrated that the MC 
+dropout has the ability to approximate epistemic uncertainty. In 
+
+3 
+single-pass uncertainty estimation, uncertainty is obtained 
+through one forward propagation, which has obvious 
+advantages in terms of computational complexity. The deep 
+evidence theory is a representative method and has been widely 
+used in classification [41] and regression [42] tasks. However, 
+these methods require that the original network output has a 
+specific form, which limits their scalability. In addition, these 
+methods do not consider the uncertainty of network weights. In 
+view of that, some studies [43] positioned the uncertainty they 
+extracted as distributional uncertainty, different from epistemic 
+uncertainty. In deep ensemble-based methods, the training 
+process is adjusted to obtain multiple different models, and 
+epistemic uncertainty is estimated by synthesizing the 
+prediction results of the models. Deep ensemble [44] is a simple 
+and scalable uncertainty estimation method, which has attracted 
+extensive attention due to its excellent performance in 
+estimating epistemic uncertainty [45]. Currently, this method 
+has become a mainstream paradigm. Subsequently, to reduce 
+the storage and computational costs of the practical application 
+of deep ensemble, many improved methods have been proposed 
+[46, 47]. For instance, the Batch-Ensemble [46] reduces 
+training and testing costs by defining each weight matrix as the 
+Hadamard product of the shared weights of all ensemble 
+members and the rank-one matrix of each member, but the 
+uncertainty estimation performance is slightly decreased. 
+However, previous studies on epistemic uncertainty have 
+usually involved tasks such as semantic segmentation and 
+object detection but have lacked detailed research in the field of 
+trajectory prediction. This work proposes a trajectory prediction 
+method with epistemic uncertainty estimation, where deep 
+ensemble and MC dropout are used separately to estimate 
+epistemic uncertainty and compared on the real intersection 
+dataset. 
+C. Relationship between Prediction Performance and Traffic 
+Environment 
+Previous studies have mainly focused on enhancing the 
+dataset-level accuracy of trajectory prediction. However, the 
+actual trajectory prediction performance can be strongly 
+dependent on a traffic environment. Therefore, it is of great 
+significance to determine the relationship between the 
+environment 
+and 
+prediction 
+model 
+to 
+improve 
+the 
+interpretability of trajectory prediction algorithms and 
+determine their limitations. This is essential for safety-critical 
+autonomous driving applications. Several works focused on 
+modeling and complexity calculation of a traffic environment 
+using different methods, such as five- and six-layer scene 
+models [48, 49], where layer elements can have a strong 
+correlation with the prediction algorithm. Wang et al. [50] 
+proposed a method to quantify scenario complexity in traffic 
+but did not explore its relationship with the autonomous driving 
+algorithm performance. The Shapley value is a feature 
+attribution method that helps to measure the contribution of 
+input variables to model performance. Makansi et al. [51] 
+proposed a variant of Shapley value and analyzed the problems 
+that some of the existing trajectory prediction models consider 
+only the past trajectory of a TA and are difficult to reason about 
+interactions. In addition, recent studies have gradually paid 
+attention to the cross-dataset performance of AI algorithms in 
+object detection and prediction applications [52, 53]. Gesnouin 
+et al. [54] evaluated the impact of differences in pedestrian 
+poses and detection box heights in different datasets on 
+pedestrian crossing prediction. Gilles et al. [10] compared the 
+accuracy of vehicle trajectory prediction algorithms on several 
+datasets containing mixed scenarios. However, there has still 
+been a lack of comprehensive analysis of traffic environmental 
+factors and their changes and quantitative research on their 
+impact on prediction algorithms. 
+n this work, the research scenario is the intersection, which 
+is a typical and challenging urban scenario. Distributional shifts 
+between different intersection datasets and their effect on 
+trajectory prediction performance, considering both error and 
+epistemic uncertainty, are analyzed. 
+III. PROPOSED METHOD 
+A. 
+Trajectory 
+Prediction 
+with 
+Epistemic 
+Uncertainty 
+Estimation 
+1) Trajectory Prediction 
+Trajectory prediction is a task that estimates a TA’s future 
+position based on historical data on its state and context in a 
+scenario. Particularly, at time 
+0
+t 
+, the historical input state S  
+of a TA (over previous 
+ time steps) is represented as follows: 
+ 
+
+
+
+
+ 
+1
+2
+0
+,
+,
+,
+,
+h
+h
+t
+t
+s
+s
+s
+
+
+
+
+
+ 
+
+S
+ 
+(1) 
+where
+( )t
+s
+is the state of a TA at t , and it is defined as 
+( )
+( )
+( )
+,
+t
+t
+t
+s
+x
+y
+
+
+ 
+ .  
+In addition, interactions between a TA and its surrounding 
+environment are modeled based on scene context information 
+C , which includes information on the states of TPs’ around the 
+TA and environmental conditions. 
+A trajectory prediction model f is trained on dataset 
+. 
+Based on the input 
+
+
+X = S,C , the trained prediction model 
+outputs an estimate ˆY  of the real future trajectory Y  of the TA 
+as follows: 
+ 
+ 
+
+
+
+ˆ
+ˆ =
+,
+,
+,
+f
+f
+f
+
+
+
+Y
+X
+X
+X
+ 
+(2) 
+where 
+ 
+ 
+ 
+1
+2
+ˆ
+ˆ
+ˆ
+ˆ
+[
+,
+,
+,
+]
+ft
+s
+s
+s
+
+Y
+, 
+ft  is the predicted horizon, and ˆ  
+represents the trained model parameters. 
+In this work, the GRIP++, which is an enhanced graph-based 
+interaction-aware trajectory prediction method, is used as a base 
+model. It uses both fixed and dynamic graphs to describe the 
+relationship between different TPs, considering the effect of 
+inter-agent interactions on a TA’s motion. Furthermore, this 
+method employs a GRU-based encoder-decoder architecture as 
+a sub-module and allows joint trajectory predictions for 
+multiple agents, achieving good performances in terms of 
+prediction speed and accuracy. 
+2) Epistemic Uncertainty Estimation 
+In the previous section, a neural network-based trajectory 
+prediction model is presented, but the original GRIP++ can 
+output only deterministic prediction results. However, real-
+world traffic scenarios are complex and variable, and it is 
+difficult to construct a training set that will effectively cover all 
+scenarios. In addition, deep learning-based models are 
+inherently uncertain and difficult to interpret, so they may not  
+ht
+
+4 
+ht
+Graph Convolutional Model
+64
+ht
+n
+Trajectory Prediction Module
+Predicted Trajectories
+prediction 
+error
+Random initialization, random shuffling...
+epistemic 
+uncertainty
+ADE
+FDE
+APE
+FPE
+ 
+Fig. 2. The trajectory prediction framework with epistemic 
+uncertainty estimation (deep ensemble-based method). 
+ 
+ 
+be reliable enough when confronted with unknown scenarios 
+(e.g., scenarios unseen during training or scenarios with only 
+limited available information). These problems can result in 
+unacceptable degradation in autonomous driving performance. 
+In this regard, the BNN models and learns the posterior 
+distribution of network weights 
+
+
+ˆ
+|
+P
+
+
+, which can be 
+used to estimate the epistemic uncertainty as follows: 
+ 
+
+
+
+ 
+
+ˆ
+,
+|
+,
+P
+|
+,
+f
+f
+X 
+
+
+
+Y
+X
+Y
+ 
+(3) 
+where the main issue is how to learn the posterior distribution 
+of parameters effectively.  
+The Bayesian approximate inference is a typical solution, 
+which learns an approximate distribution  
+q   of 
+
+P
+|
+
+. The 
+MC dropout has been shown to be an effective sample-based 
+method for approximate inference, where the network weights 
+are assumed to follow the Bernoulli distribution. After adding 
+appropriate regularization during training and turning on 
+dropout during testing, epistemic uncertainty can be estimated 
+by sampling multiple times. 
+In recent years, deep ensemble, as a simple, parallelizable, 
+and scalable method, has shown excellent uncertainty 
+estimation ability. In this work, a deep ensemble-based 
+uncertainty estimation framework for the trajectory prediction 
+model is proposed. Specifically, random initialization of neural 
+network parameters and random shuffling of a dataset are 
+performed because they have been proven to have enough good 
+performance in practice. After training, 
+K  models of 
+isomorphism and different parameters are obtained. Further, by 
+integrating the results of 
+ models, the final trajectory 
+prediction output is obtained by, 
+ 
+
+
+1
+1
+ˆ
+ˆ =
+,
+,
+K
+k
+k
+K
+f
+
+
+
+Y
+Y | X
+ 
+(4) 
+where ˆ
+k  denotes the post-training parameters of the kth model 
+among the ensemble models; similarly, in the MC dropout-
+based method, ˆ
+k  indicates the model parameters for the kth 
+dropout during testing. 
+The predictive entropy is employed to quantify the epistemic 
+uncertainty of the proposed prediction model, where entropy 
+increases with uncertainty. The proposed model outputs K 
+continuous trajectories 
+
+
+(1)
+(2)
+]
+ˆ
+ˆ
+ˆ
+ˆ
+[
+,
+,
+,
+ft
+k
+k
+k
+k
+s
+s
+s
+
+Y
+ in a prediction task, 
+each of which contains the predicted position at multiple future 
+moments. To realize a prediction-task-wise uncertainty 
+estimation, the predictive entropy at multiple moments is 
+integrated to obtain the average predictive entropy (APE) as 
+follows: 
+ 
+
+
+
+
+( )
+( )
+( )
+1
+1
+1
+1
+ˆ
+ˆ
+ˆ
+ˆ
+A
+.
+PE
+ln
+d
+f
+f
+t
+t
+t
+t
+t
+t
+i
+i
+f
+f
+s
+p s
+p s
+s
+t
+t
+
+
+
+
+
+
+
+
+
+
+ 
+ 
+(5) 
+Assuming that 
+( )
+( )
+( )
+ˆ
+ˆ
+ˆ
+,
+t
+t
+t
+s
+x
+y
+
+
+ 
+  obeys the two-dimensional 
+Gaussian distribution, 
+( )
+ˆ t
+x
+ and 
+( )
+ˆ t
+y
+ are independent of each 
+other, and the APE can be expressed as follows: 
+ 
+ 
+
+
+
+
+=1
+2
+( )
+2
+( )
+1
+1
+1
+ˆ
+APE
+=
+(ln2
+1)
+ln
+2
+1
+1
+ˆ
+ˆ
+(ln2
+1)
+ln
+.
+2
+f
+f
+l
+t
+i
+f
+t
+t
+t
+i
+f
+t
+x
+x
+t
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+(6) 
+Similarly, the final predictive entropy (FPE) is defined as, 
+ 
+
+
+
+
+
+
+
+
+
+
+(
+)
+(
+)
+(
+)
+2
+2
+1
+ˆ
+ˆ
+FPE
+ln2
+1
+ln
+2
+1
+ˆ
+ˆ
+ln2
+1
+ln
+.
+2
+f
+f
+f
+f
+t
+t
+t
+t
+s
+x
+x
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+(7) 
+B. Scenario Features Extraction 
+In prediction scenarios, a TA’s motion is related to its 
+historical state, interactions with surrounding TPs, and other 
+factors. Although the existing prediction algorithms have either 
+explicitly or implicitly considered different factors, their 
+performances may still be affected by the above-mentioned 
+features due to algorithm limitations. Therefore, three types of 
+scenario features are considered in this work: 1) dynamic 
+features of a TA, which include data on its historical or future 
+motion states; 2) features of surrounding TPs, which refer to 
+their states and interactions with the TA; 3) other scenario 
+features, which include the type, behavior pattern, compliance 
+with traffic rules, and current location of the TA. 
+1) Kinematic features of TA 
+The historical data on the TA motion state denote an 
+important input to a prediction model and have a direct impact 
+on the model output. In addition, the future motion state of a 
+TA is a key reference for evaluating the model’s prediction 
+accuracy. Therefore, in this work, the kinematic features of TA 
+are extracted to analyze their impact on the prediction algorithm 
+performance. 
+Velocity is one of the primary kinematic features, which 
+directly reflects the aggressiveness of TA movement. It also 
+represents the discrete degree of a continuous trajectory. 
+Considering the trajectory prediction model characteristics, 
+three velocity sub-features are extracted: 1) average velocity of 
+the historical trajectory (AVHT), which indicates the 
+aggressiveness of the model's input trajectory; 2) current 
+velocity (CV), which directly represents a TA’s current state 
+and has a key impact on the trajectory prediction output; 3) 
+average velocity of the future trajectory (AVFT), which reflects 
+the spatial span of the future trajectory points. 
+In addition, the velocity variations indicate trajectory 
+stationarity, which may have a significant influence on 
+prediction results. For instance, a sudden start of a parked 
+vehicle may be difficult for the model to predict timely and 
+accurately. Therefore, the acceleration value at each moment is 
+calculated to obtain the following sub-features: 1) average 
+acceleration of the historical trajectory (AAHT), which 
+K
+
+5 
+represents the speed mutation degree of the input trajectory; 2) 
+average acceleration of the future trajectory (AAFT), which 
+reflects the overall situation of a TA's future speed mutation; 3) 
+maximum acceleration of the future trajectory (MAFT), 
+considering that a sudden speed change at any moment can lead 
+to severe deformation of the overall trajectory, it is necessary to 
+extract the fastest speed change in the future as a feature for 
+analysis. 
+Similarly, changes in the TA moving direction denote a 
+potentially influential factor of prediction performance. For 
+instance, a vehicle going straight may suddenly swerve or make 
+a U-turn, thus posing a serious challenge to the prediction 
+algorithm. Therefore, the heading change speed (HCS) is 
+extracted for analysis. In detail, the absolute value of the change 
+speed of the heading angle at each moment is calculated and 
+used as a basic feature, and then the analysis of the following 
+parameters is performed: 1) average HCS of the historical 
+trajectory (AHCSHT); 2) average HCS of the future trajectory 
+(AHCSFT), which reflects the overall curvature or volatility of 
+the future trajectory; 3) maximum HCS of the future trajectory 
+(MHCSFT), which increases when there is a sudden large 
+change in direction at any point in the future. 
+2) Features of Surrounding TPs  
+Convoluted interactions with other agents increase the 
+difficulty in trajectory prediction, and although many of the 
+existing prediction methods can explicitly or implicitly model 
+interactions, it has not been fully discussed whether the 
+performance of these black-box models is sensitive to actual 
+interactions. To examine this situation, a set of hierarchical 
+prediction scenario complexity metrics is proposed to analyze 
+the effect of a TA’s interactions with surrounding agents on the 
+prediction algorithm performance. 
+First, with a TA as a center, the prediction scenario 
+complexity has a positive correlation with the number of its 
+surrounding TPs, and a basic feature, the number of TPs within 
+x meters from the TA (NTPx), is defined. 
+In addition, the distance between the TA and its surrounding 
+TPs directly affects the prediction scenario complexity. 
+Assuming that a set of TPs within x meters of a TA i is denoted 
+by 
+ 
+x
+N
+i
+, then for any 
+ 
+x
+j
+N
+i
+
+, its distance from i is 
+
+
+,
+j
+j
+i
+dist s
+d
+s
+
+. The density of TPs within x meters around TA 
+(DTPx) is given by, 
+ 
+ 
+1
+DTPx
+j
+x
+N
+i
+d
+j
+e
+
+
+
+ 
+， 
+(8) 
+where   is a scaling factor. 
+Furthermore, the potential conflicts due to the movement of 
+surrounding TPs are analyzed. As shown in Fig. 3, for a TP 
+ 
+x
+j
+N
+i
+
+ within x meters around a TA i , its current state is 
+given by 
+ 
+ 
+0
+0
+[
+,
+]
+j
+j
+s
+v
+; then, its position after t seconds is expressed 
+as 
+ 
+ 
+t
+t
+j
+j
+s
+v t
+
+, and the degree of conflict from TPs within x 
+meters around TA (DCTPx) is defined as follows, 
+ 
+ 
+
+ 
+T
+( )
+( )
+agg
+1
+DCTPx
+X
+t
+t
+j
+N
+i
+d
+j
+e
+
+
+
+
+ 
+， 
+(9) 
+where 
+
+
+( )
+( )
+( )
+,
+t
+t
+t
+j
+i
+j
+d
+dist s
+s
+
+; T is the time horizon used for 
+evaluation; 
+( )t
+
+ is the scaling factor for the distance at time t, 
+and in this study, it is set to grow faster over time to reinforce  
+
+
+
+
+location 
+,
+velocity 
+,
+j
+j
+jx
+jy
+x
+y
+v
+v
+
+
+
+
+
+
+location 
+,
+velocity 
+,
+i
+i
+ix
+iy
+x y
+v
+v
+
+
+ 
+Fig. 3. Schematic diagram of the conflict degree calculation 
+from TPs within x meters from a TA. 
+(a) Agent type
+(d) Agent location
+Stage-1 Stage-2
+Stage-3
+Stage-4
+Stage-5
+Stage-6
+gap
+(c) Compliance with traffic rules
+Red-light 
+running
+Yellow-light 
+running
+Obeying the 
+rules
+For going straight 
+and turning left 
+vehicles
+(b) Behavior pattern
+U-turn
+Going 
+straight
+Turning 
+left
+Turning 
+right
+ 
+Fig. 4. Illustration of the other extracted scenario features. 
+ 
+ 
+the focus on the short-term risk; 
+ 
+T
+agg
+ represents the 
+aggregation operation, which is used to synthesize the conflicts 
+at T times in the future. The two basic modes used in this study 
+include the mean value (DCTPx_mean) and the maximum 
+value (DCTPx_max). 
+3) Other Scenario Features 
+In addition to the above two categories of features, the impact 
+of several other representative features on the prediction error 
+and epistemic uncertainty is studied, as shown in Fig. 4. 
+The considered features include: 
+ TA type: The GRIP++ can simultaneously predict future 
+trajectories of multiple types of TAs, such as vehicles, 
+pedestrians, and cyclists. Each type of agent has its 
+movement pattern, which may cause different prediction 
+performances. Referring to the research presented in 
+[33], TAs can be divided into four types: small vehicles, 
+vehicles, pedestrians, motorcyclists, and bicyclists; 
+ TA behavior pattern: This study mainly focused on three 
+basic behavior patterns of vehicles at intersections: 
+going straight, turning left, and turning right. In addition, 
+this study extracts certain corner cases, such as U-turns. 
+The prediction errors and epistemic uncertainty under 
+different behavioral patterns are compared and analyzed 
+in a unified manner; 
+
+6 
+ TA's compliance with traffic rules: Traffic rules partially 
+constrain the behaviors of participants, but in real-world 
+scenarios, some of TAs may violate the rules, thus 
+affecting the trajectory prediction performance. For 
+instance, in a signalized intersection, the behaviors of 
+TAs can be classified based on their compliance with the 
+traffic signal into obeying the rules, yellow-light running, 
+and red-light running; 
+ TA location: The whole process of a vehicle passing 
+through the intersection is divided according to the time 
+sequence into six stages: stage 1: ex-entering an 
+intersection; stage 2: in the gap; stage 3: in the first 
+crosswalk; stage 4: inside an intersection; stage 5: in the 
+last crosswalk; stage 6: exiting an intersection. 
+C. Scenario Features Analysis 
+To analyze the relationship between the above-mentioned 
+features systematically, the qualitative and quantitative analysis 
+methods are adopted. The main methods include feature 
+correlation analysis and feature importance analysis based on 
+random forest regression. 
+1) Feature Correlation Analysis 
+Correlation analysis is to calculate the degree of correlation 
+between two or more feature variables using correlation 
+coefficients as quantitative indicators. Typical correlation 
+coefficients include the Pearson correlation coefficient and 
+Spearman rank correlation coefficient. The Pearson correlation 
+coefficient requires evaluated variables to conform to the 
+normal distribution, but this is a strong assumption that 
+experimental results can hardly satisfy. In contrast, the 
+Spearman rank correlation coefficient does not have such strict 
+requirements on data as the Pearson correlation. Namely, it 
+requires only that observed values of the two variables are 
+paired rank data or rank data transformed from continuous 
+variable observation data. The Spearman rank coefficient can 
+be mainly used in the monotonic relationship evaluation. 
+Specifically, it is assumed that the original data (
+ix , 
+iy ) are 
+arranged in ascending order, and  
+i
+R x  and  
+i
+R y  are defined 
+as the ranking of 
+ix  and 
+iy  in their corresponding data, 
+respectively; 
+( )
+R x  and 
+( )
+R y  denote the mean of the ranking of 
+the two groups of data, and n  is the number of data pairs. Then, 
+the Spearman rank correlation coefficient 
+ can be defined as 
+follows: 
+ 
+ 
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+1
+2
+2
+1
+1
+2
+1
+2
+( )
+( )
+( )
+( )
+6
+1
+.
+1
+n
+i
+i
+i
+n
+n
+i
+i
+i
+i
+n
+i
+i
+i
+R x
+R x
+R y
+R y
+R x
+R x
+R y
+R y
+R x
+R y
+n n
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+
+ 
+(10) 
+2) Feature Importance Analysis Based on Random Forest 
+Regression 
+Feature correlation analysis can be used to assess linear and 
+ordinal consistent correlations but cannot identify other types 
+of correlations. To address this limitation, this work proposes a 
+feature importance analysis method to evaluate the impacts of 
+different scenario features. 
+The decision tree is a non-parametric supervised learning 
+algorithm that can be used in solving both classification and 
+regression problems. It is a hierarchical tree structure mainly 
+composed of three types of nodes, root, internal, and leaf nodes. 
+Ensemble learning uses multiple models to obtain accurate 
+prediction results, and bootstrapping is one of its typical 
+application techniques that refers to the process of randomly 
+sampling of a sub-dataset through a given number of iterations 
+and variables. Random forest regression combines ensemble 
+learning with the decision tree framework, creating multiple 
+decision trees from data; then, multiple outputs are averaged to 
+obtain the final result, often achieving excellent performance 
+for regression problems. 
+Random forest regression can be used to evaluate feature 
+importance. Particularly, in this study, the extracted scenario 
+features are regarded as independent variables, while the 
+prediction error and epistemic uncertainty of the prediction 
+model under the corresponding conditions are regarded as 
+dependent variables, and a random forest regression model is 
+constructed. Then, the contribution of each feature to the trees 
+in the random forest is analyzed, as well as its importance to the 
+performance of trajectory prediction. The variable importance 
+measure is denoted as VIM, and it is assumed that there are J 
+features and I decision trees. Then, VIM j  denotes the average 
+change in node split impurity of the jth feature in all decision 
+trees, and it is calculated by: 
+ 
+ 
+ 
+1
+'
+' 1
+1
+VIM
+VIM =
+,
+VIM
+I
+i
+j
+i
+j
+J
+I
+i
+j
+j
+i
+
+
+
+
+
+ 
+(11) 
+where 
+ 
+VIM
+i
+j  represents the importance of the jth feature in the 
+ith decision tree, and it can be obtained by calculating the 
+difference of Gini indices of nodes before and after branching. 
+D. Prediction across Different Intersection Datasets 
+The cross-dataset analysis aims to analyze differences in 
+scenarios between multiple datasets and the corresponding 
+prediction algorithm performance disparity. In this study, the 
+scenario type is limited to the intersection, and multiple 
+intersection datasets involving various countries are studied. 
+First, the scenario features are extracted to analyze 
+distributional shifts between different intersection datasets. 
+Next, 
+comprehensive 
+cross-validation 
+experiments 
+are 
+performed to investigate the prediction algorithm performance 
+in terms of distributional shifts fully. Particularly, N 
+intersection datasets are selected, and each of them is divided 
+into training and test subsets. Then, for each of the training 
+subsets, a trajectory prediction model is developed and trained 
+using the training subset first and then evaluated on the 
+corresponding test subset. Finally, 
+2
+N  sets of results are 
+obtained. 
+During the analysis, the following factors are mainly studied: 
+1) Trajectory prediction performance degradation due to 
+distributional shifts: Combined with the differences in 
+statistical features between different datasets obtained 
+earlier, we analyze the impact of changes in the traffic 
+environment on the prediction algorithm. 
+
+
+7 
+2) Effect of the deep ensemble on prediction robustness 
+across different datasets: The improvement in prediction 
+accuracy and sensitivity of the estimated epistemic 
+uncertainty to distributional shifts are analyzed; 
+3) Availability and complexity of different intersection 
+datasets for trajectory prediction: By synthesizing 
+multiple groups of results, the model performance is 
+evaluated using different intersection datasets as a 
+training set and the prediction challenge with different 
+intersection datasets as a test set. 
+IV. EXPERIMENTAL SETUP  
+A. Intersection Datasets 
+Focusing on the urban intersection scenario, multiple 
+trajectory datasets were used for evaluation and analysis, 
+involving different periods, weather, countries and regions, and 
+many TP types. 
+1) SinD [55]: The SinD dataset is a typical drone dataset 
+collected from a signalized intersection in Tianjin, China. These 
+data were recorded from a static bird’s eye view at a sampling 
+frequency of 10 Hz. This dataset contains about 420 minutes of 
+traffic recordings, including over 13,000 TPs with seven types, 
+including cars, trucks, buses, pedestrians, tricycles, bikes, and 
+motorcycles;  
+2) INTERACTION [29]: The INTERACTION dataset contains 
+12 subsets covering merging, roundabout, and intersection 
+scenarios, of which five intersection subsets are used in this 
+study: USA_Intersection_EP1 (EP1), USA_Intersection_EP2 
+(EP2), USA_Intersection_MA (MA), USA_Intersection_GL 
+(GL) and TC_Intersection_VA (VA). They contain about 493 
+minutes of recordings in total. The first four relate to 
+unsignalized intersections in the US, mainly involving 
+trajectories of vehicles, pedestrians, and bicycles recorded by 
+drones. The VA denotes a signalized intersection in Bulgaria, 
+which involves trajectories of cars, buses, trucks, motorcycles, 
+and bicycles recorded by traffic cameras. 
+B. Prediction Error Metrics 
+The prediction error is a preferred metric for quantifying the 
+performance of prediction models. Following [2, 8, 25], the 
+proposed model was evaluated using two error metrics: 
+1) Average Displacement Error (ADE): This is the mean square 
+error of all predicted points of a trajectory compared to the 
+ground truth; 
+2) Final Displacement Error (FDE): This is the distance 
+between the predicted final destination and the true final 
+destination at 
+. 
+C. Predictive Uncertainty Evaluation 
+As stated previously, the APE and FPE reflect the epistemic 
+uncertainty of a prediction model, which can indicate situations 
+where prediction models are performing poorly. Therefore, 
+referring to [56, 57], this study uses the error-retention curves 
+to evaluate the ability of the extracted epistemic uncertainty to 
+detect prediction errors. The curves depict the error over a 
+dataset as a model’s predictions are replaced by ground-truth 
+labels in order of decreasing uncertainty. The abscissa value of 
+a point represents the proportion of the retained true error (i.e., 
+retention fraction), while the ordinate value represents the 
+comprehensive error under this proportion. Similarly, the 
+optimal and random curves are obtained by replacing the 
+predictions in order of decreasing error and random order, 
+respectively. The area under the retention curves (AUC) is an 
+evaluation metric of both the robustness of prediction models 
+and the quality of uncertainty estimation, and an efficient 
+uncertainty estimation is considered to achieve a low AUC. 
+D. Implementation Details 
+For achieving fair evaluation and transferability, the datasets 
+were standardized to use up to 3 s of the previous data and 3 s 
+of the future data. The trajectory data were resampled to 2 Hz. 
+The single GRIP++ model in the ensemble models was 
+implemented in PyTorch, and its implementation details, 
+including input preprocessing, graph convolution, and 
+trajectory prediction model, mainly refer to the settings in [5]. 
+In the implementation of MC dropout and deep ensemble, the 
+value of k was set to five, which was the result of a trade-off 
+between uncertainty estimation quality and computational cost 
+[45]. In addition, during the training process of the MC dropout-
+model, a regularization term was added to the loss to improve 
+its ability of uncertainty estimation [39], where the 
+regularization coefficient was set to 0.0001, and the dropout 
+rate was set to 0.5 
+In addition, the original dataset was further processed to 
+obtain the labels for the subsequent analysis. The locations of 
+TAs were classified by combining their raw coordinate data 
+with the map in the Lanelet2 format [58]. 
+V. RESULTS AND DISCUSSION 
+A. Evaluation of Trajectory Prediction and Epistemic 
+Uncertainty Estimation 
+The training and test performances of the proposed model 
+obtained by following the train-test process in the same 
+intersection dataset are presented in TABLE I. Although the 
+MC dropout can be used to estimate epistemic uncertainty, it 
+increases the prediction error, which might be due to the 
+modifications in the original loss function. In contrast, the deep 
+ensemble-based 
+method 
+improves 
+trajectory 
+prediction 
+accuracy while estimating epistemic uncertainty. As shown in 
+TABLE I, the error obtained by the deep ensemble-based 
+method was lower than that of the single models. Thus, by 
+integrating the results of multiple models, deep ensemble could 
+effectively avoid the prediction performance degradation 
+caused by the deviation of a single model, which is conducive 
+to improving the prediction algorithm robustness. The 
+evaluation results of the estimated epistemic uncertainty are 
+shown in Fig. 5. Compared with the MC dropout, the deep 
+ensemble had obvious advantages in improving the model 
+prediction accuracy and uncertainty estimation. Therefore, in 
+the subsequent analysis, the epistemic uncertainty estimation 
+framework based on the deep ensemble was adopted. 
+As shown in Fig. 5, both uncertainty quantification metrics 
+(i.e., APE and FPE) could accurately reflect the prediction 
+model error (i.e., ADE or FDE), and they showed a high degree 
+of consistency. By comparing the left and right sides of Fig. 5, 
+it can be concluded that although the prediction error on the test 
+sets was slightly increased compared to that on the training set, 
+there was a small difference in the epistemic uncertainty 
+ft
+
+1 
+TABLE I  
+TRAJECTORY PREDICTION ERROR COMPARISON1 
+Dataset 
+ADE1/FDE1 
+ADE1/FDE1 
+ADE1/FDE1 
+ADE1/FDE1 
+ADE1/FDE1 
+ADEdropout/FDEdropout 
+ADE/FDE 
+SinD 
+0.405/0.865 
+0.404/0.865 
+0.402/0.86 
+0.403/0.861 
+0.408/0.872 
+0.477/1.021 
+0.389/0.832 
+VA 
+0.652/1.401 
+0.641/1.378 
+0.655/1.428 
+0.657/1.421 
+0.654/1.402 
+0.651/1.327 
+0.615/1.327 
+EP0 
+0.811/1.822 
+0.815/1.844 
+0.803/1.809 
+0.809/1.814 
+0.822/1.850 
+1.036/2.351 
+0.745/1.678 
+EP1 
+0.881/1.982 
+0.842/1.894 
+0.816/1.826 
+0.853/1.915 
+0.857/1.930 
+1.147/2.590 
+0.775/1.743 
+MA 
+0.878/2.026 
+0.857/1.979 
+0.866//1.999 
+0.867/2.000 
+0.870/2.008 
+1.083/2.546 
+0.811//1.879 
+GL 
+0.619/1.448 
+0.627/1.464 
+0.622/1.452 
+0.625/1.458 
+0.627/1.466 
+0.709/1.646 
+0.591/1.386 
+ 
+(a) ADE for training set
+(b) FDE for training set
+(c) APE for test set
+(d) FPE for test set
+ 
+Fig. 5. The ADE/FDE-retention curves (top) and retention scores curves (bottom) on the training and test sets of the SinD dataset. 
+The optimal curve (solid green line) was obtained by replacing the model's predictions with the ground-truth labels in order of 
+decreasing error. Similarly, the random curve (blue dotted line) was obtained by replacing the model’s predictions with the ground-
+truth labels in random order. The red and yellow solid lines correspond to the results captured in order of decreasing APE and 
+decreasing FPE, respectively. The retention scores corresponding to each retention fraction can be calculated by: (errorrandom – 
+erroruncertainty)/(errorrandom – erroroptimal). 
+ 
+ 
+estimation performance. The results indicate that the deep 
+ensemble-based method had good generalization ability 
+In the second row in Fig. 5, a consistent trend where the 
+retention scores first increase and then decrease with the 
+retention fraction can be observed. 
+B. Scenario Features Analysis Results 
+1) Feature Correlation Comparison 
+In general, it has been considered that scenario complexity 
+increases with the values of the extracted TA’s kinematic 
+features and the features of the TA’s surrounding TPs. 
+Therefore, the correlation between the model performance and 
+these two types of features was calculated to analyze the 
+influence of scenario complexity on trajectory prediction 
+performance. Based on the model trained on the SinD training 
+set, the feature correlation analysis experiment was performed 
+on the SinD test set, where the prediction performance was 
+ 
+1 ADEk/FDEk represents the prediction error of model k among the ensemble models, ADEdropout/FDEdropout is the prediction error of the mc dropout-based 
+method, and ADE/FDE denotes the prediction error of the deep ensemble-based method. 
+represented by prediction error and epistemic uncertainty. For 
+the features of the TA’s surrounding TPs, four groups of 
+distances of x = 10, 20, 30, 50 were studied. The analysis results 
+are presented in Fig. 6, and based on them, the following 
+conclusions can be drawn: 
+ 
+The distribution trends of environmental feature 
+correlations corresponding to the two types of prediction 
+errors (ADE and FDE) were highly consistent, as well as 
+the 
+distribution 
+trends 
+of 
+environmental 
+feature 
+correlations corresponding to the two types of epistemic 
+uncertainty (APE and FPE) estimates; 
+ 
+The comparison of a TA’s kinematic features with the 
+features of its surrounding TPs shows that the former had 
+a strong positive correlation with the error and epistemic 
+uncertainty of the prediction model, while the latter had 
+weak correlations with the error and epistemic uncertainty 
+(-0.2 <   < 0.2).  
+
+1 
+ 
+Fig. 6. Comparison of the correlation between prediction model performance and scenario features 
+ 
+ 
+ 
+The comparison of the correlation between different 
+kinematic features and the prediction error shows that: 
+ 
+The sorting order in terms of the correlation was: 
+acceleration-related features > velocity-related 
+features > heading change speed-related features. 
+This order indicates that the mutation of a TA’s 
+speed had a relatively large impact on the prediction 
+error, while the change in the TA’s movement 
+direction had a relatively low impact; 
+ 
+The sorting order in terms of the correlation was: 
+features related to future trajectories > features 
+related to historical trajectories. This shows that the 
+proposed trajectory prediction model had low 
+adaptability to the speed and position mutations 
+occurring at some certain points in the future. 
+ 
+The correlation between different kinematic sub-features 
+of a TA and the predictive uncertainty showed that the 
+epistemic uncertainty was highly sensitive to the velocity 
+and acceleration features; namely, when a TA was driving 
+at high speed or had a speed mutation, the model tended 
+to show lower confidence in the predictions. 
+2) Feature Importance Comparison 
+As mentioned above, the feature correlation analysis only 
+shows whether the relationship between two variables conforms 
+to the order consistency. Therefore, the feature importance 
+analysis experiment was performed based on the random forest 
+regression to explore whether there were other types of 
+correlation between the above scenario features and the 
+prediction model. The datasets and training settings used for the 
+prediction algorithm were the same as in the feature correlation 
+analysis. The grid-based search was employed to obtain optimal 
+random forest regression model, then the feature importance 
+analysis was performed. The results are presented in Fig. 7. 
+As shown in Fig. 7, the distribution trend of feature 
+importance was basically consistent with the feature correlation. 
+For instance, the features obtained from the surrounding TPs 
+had little effect on the error and uncertainty of the prediction 
+model uniformly. In contrast, the kinematic features of a TA 
+had a stronger influence, where velocity- and acceleration-
+related features had higher importance. In random forest 
+regression for the ADE, the AAFT had the highest importance, 
+ 
+Fig. 7. Comparison of feature importance based on the random 
+forest regression 
+ 
+while in the random forest regression for the APE, the CV had 
+the strongest influence. 
+3) Other Scenario Features Analysis 
+In addition to the above-mentioned features, the impacts of 
+several other environmental features on the prediction 
+algorithm performance were explored, including the type, 
+behavior patterns, compliance with traffic rules, and location of 
+a TA. Without loss of generality, the ADE was adopted as an 
+error metric, and the APE was used for epistemic uncertainty 
+quantification. The prediction model was trained on the SinD 
+training set and analyzed on the SinD test set 
+The results indicated obvious differences in the prediction 
+error distribution between different TAs, as shown in Fig. 8. 
+Although the movement of pedestrians had high degrees of 
+freedom and randomness, their speed and acceleration were 
+generally low, so the corresponding prediction error was small. 
+Meanwhile, the epistemic uncertainty distribution for different 
+types of TAs showed high similarity with that of the prediction 
+error. 
+The results of the trajectory prediction error and epistemic 
+uncertainty of a vehicle under different behavior patterns are 
+presented in Fig. 9, where it can be seen that the trajectory 
+prediction performance tended to exhibit larger errors when the 
+TA was turning left and right compared to the going-straight 
+behavior. The error in the right-turn scenario was relatively 
+(a) ρ for ADE
+(b) ρ for FDE
+(a) ρ for APE
+(b) ρ for FPE
+(a) feature importance for ADE (b) feature importance for APE
+
+2 
+ 
+Fig. 8. The results of the trajectory prediction error and 
+epistemic uncertainty of different TA types; sv: small vehicle, 
+bv: large vehicle; bi: motorcyclist or bicyclist; pe: pedestrian. 
+ 
+Fig. 9. The results of the trajectory prediction error and 
+epistemic uncertainty under different behavioral patterns. 
+ 
+ 
+Fig. 10. The results of the trajectory prediction error and 
+epistemic uncertainty under different traffic rule compliance. 
+ 
+Fig. 11. The results of the trajectory prediction error and 
+epistemic uncertainty at different locations; 1: ex-entering the 
+intersection; 2: in the gap; 3: in the first crosswalk; 4: inside the 
+intersection; 5: in the last crosswalk; and 6: exiting the 
+intersection. 
+ 
+large, and the reasons may be as follows. First, the right-turn 
+trajectory had a large curvature, and second, the vehicle was 
+less affected by traffic lights and other TPs when turning right, 
+compared to turning left and going straight, resulting in a higher 
+driving speed. In addition, although the U-Turn represents a 
+typical corner case, the results showed that the overall error in 
+this pattern was small, which may be related to the generally 
+low speed during the U-turning process. Furthermore, the 
+epistemic uncertainty distributions under different behavioral 
+patterns were relatively consistent with the error. 
+The relationship between the vehicles’ compliance with 
+traffic lights and the trajectory prediction performance is 
+presented in Fig. 10, where it can be seen that the prediction 
+error was larger when the vehicle ran red or yellow light than 
+when there was no violation of traffic lights, and 
+simultaneously the proposed model could output higher 
+epistemic uncertainty. 
+The impact of a TA’s location on the trajectory prediction 
+performance is presented in Fig. 11. When the vehicle was in 
+the gap area or first crosswalk before entering the intersection, 
+there were many possible strategic options, referring to whether 
+to enter the intersection and how to pass through the 
+intersection, which increased the prediction complexity, and 
+further increased the prediction error and epistemic uncertainty. 
+When the vehicle was inside the intersection, the prediction 
+model exhibited considerable error and uncertainty due to a 
+large number of interactions with other TPs and higher freedom 
+of movement. In contrast, the predominant behavior of the 
+vehicles before entering and after exiting the intersection was 
+to follow the lane, so these stages had lower prediction error 
+and uncertainty than the others. 
+C. Prediction evaluation across Different Intersection Datasets 
+Different intersection datasets were collected at different 
+times and locations, and the corresponding environmental 
+conditions might be relatively different, resulting in shifts in 
+data distribution. The distributions of velocity, acceleration, 
+heading, and HCS of objects in six intersection datasets are 
+presented in Fig. 12, where it can be seen that there were 
+obvious differences in the trajectory features between the 
+datasets. For instance, the velocity and acceleration of a portion 
+of trajectories in the SinD dataset were concentrated around 
+zero. One of the main reasons was the stopping of vehicles and 
+pedestrians while waiting for the green light. Furthermore, the 
+velocity in the SinD dataset exhibited a distinct multimodal 
+distribution, which could be related to the multiple movement 
+patterns caused by various TPs in the dataset. In contrast, the 
+velocity and acceleration in the GL, MA, and VA datasets 
+tended to be higher, reflecting more aggressive motions in these 
+datasets. In addition, distributional shifts could also be observed 
+by comparing the distribution of heading and its speed of 
+change in different datasets. 
+The aforementioned differences in data distribution between 
+datasets may bring application challenges of the trajectory 
+prediction algorithms in real-world environments. Therefore, 
+experiments were performed on multiple intersection datasets. 
+Specifically, based on the six intersection datasets mentioned 
+above, a set of deep ensemble-based prediction models were 
+trained on each dataset and evaluated on all six datasets. As 
+shown in Fig. 13, distributional shifts in the real-traffic 
+
+1 
+ 
+Fig. 12. Distribution comparison of different intersection datasets. 
+(a) ADE1
+(b) ADE
+(c) APE
+ 
+Fig. 13. The cross-dataset error and uncertainty matrix: (a) ADE1 is the error obtained by evaluating one of the ensemble models; 
+(b) ADE is the prediction error of the deep ensemble-based model; (c) APE also relates to the deep ensemble-based model. The 
+ith row and jth column of each matrix represent the results obtained by evaluating the model on the test subset at intersection j 
+after training it on the training subset at intersection i. 
+ 
+ 
+Fig. 14. Comprehensive performances of the prediction 
+algorithms on different datasets. (a) Comparison of prediction 
+errors of all trained models on different test subsets; (b) results 
+of evaluating models trained on different training subsets on all 
+test subsets. 
+ 
+environments had a strong impact on trajectory prediction 
+performance. Even for the same type of scenario, for the model 
+trained on one intersection dataset, it was difficult to generalize 
+well to other intersection datasets directly. For instance, on the 
+test subset of the SinD dataset, the model that achieved the best 
+accuracy (ADE1/ADE: 0.405/0.389) was trained on the training 
+subset of the SinD dataset; in contrast, the prediction error 
+(ADE1/ADE) of the model trained on the other intersection 
+training subsets increased by 94.9%/8.61% - 265.0%/221.0%. 
+Comparing the single prediction model error (Fig. 13(a)) 
+with the error of the deep ensemble-based model (Fig. 13(b)), 
+it could be concluded that the deep ensemble-based approach 
+performed systematically better than a single model, achieving 
+significant improvements in many cases. 
+As presented in Fig. 13(c), the extracted epistemic 
+uncertainty could indicate distributional shifts between 
+different datasets. Particularly, the proposed model could 
+output higher epistemic uncertainty when the error increased 
+due to changes in the test scenario. 
+In Fig. 14, the comprehensive performances of the trajectory 
+prediction algorithm on specific single training or test set are 
+presented. As shown in Fig. 14(a), the models trained on the 
+training subsets of the SinD and GL datasets had better 
+generalization ability than the other models. This could be 
+because of a larger amount of data and more diverse motion 
+patterns of these two datasets compared to the other datasets. 
+Conversely, the overall error of the model based on the training 
+subset of VA was the highest, and the main reasons were as 
+follows. First, this dataset contained less data than the other 
+datasets. Second, this dataset was constructed by data collected 
+by roadside equipment, which might introduce more noise than 
+drone-based data collection. The comprehensive performances 
+of the prediction models on different intersection test subsets 
+are presented in Fig. 14(b). The GL and MA with higher 
+velocity and acceleration features showed greater prediction 
+difficulty, and the overall trajectory prediction error on the 
+SinD test subset was the smallest among all datasets. 
+D. Qualitative Results 
+The prediction results of the proposed framework in several 
+scenarios on different intersection datasets are presented in Fig. 
+15, where each row corresponds to one intersection. In Fig. 15,  
+
+isolg
+0
+2
+or
+J2
+50
+52
+00.0
+20.0
+0'T0
+0'T2
+AM
+0'52
+EbJ
+Ebo
+0E.0
+AV 
+2E.0
+1 2UD
+926b2HC
+00.0
+0'52
+0'20
+0'>2
+J'00
+J'52
+J'32
+5'00
+0
+AM
+3
+Eb
+Ebo
+AV
+2!UD
+J926J6bnoit19l96
+0
+e
+8
+JO
+00.0
+0'52
+0'20
+AM
+J'S2
+EbJ
+Ebo
+J'20
+AV 
+ 2IUD
+J'>2
+926b-5
+0
+A
+0.0
+S.0
+0'4
+D6U
+aer
+AM
+EbJ
+8.0
+Ebo
+AV
+J'O
+2!UD
+J926J6b1 
+SinD
+VA
+EP0
+EP1
+MA
+GL
+ 
+Fig. 15. Qualitative results of trajectory prediction and uncertainty estimation on different datasets (corresponding to different 
+rows). The gray thick solid line represents the historical trajectory, and the black thin solid line represents the true future trajectory. 
+Different colors are used to denote predictions for different types of TAs: blue - small vehicle; magenta - large vehicle; yellow – 
+pedestrian; green - motorcycle or cyclist. In addition, the focused object in each subgraph is highlighted in red, and the current 
+prediction error and uncertainty estimation results of the object are presented above the subgraph. 
+
+nipnonipno1..1 
+the first column shows cases where the prediction error was 
+small and epistemic uncertainty was low. These cases generally 
+appeared when the TA exhibited obvious future intentions and 
+moved relatively smoothly. The second and third columns show 
+scenarios with large trajectory prediction errors, which were 
+generally accompanied by higher estimates of epistemic 
+uncertainty. This situation may occur when the TA was about 
+to enter the intersection where it was difficult to determine its 
+future behavior pattern, or its future motion showed a large 
+pattern or speed change compared to the historical trajectory. 
+VI. CONCLUSION 
+In this paper, a trajectory prediction framework that 
+integrates the epistemic uncertainty estimation function is 
+proposed, and the effects of the traffic environment and its 
+changes on the prediction algorithm performance are studied. A 
+few typical scenario features are considered, and their 
+influences on the prediction performance are examined by the 
+feature correlation and importance analyses. Further, the 
+distributional shifts between different intersection datasets and 
+the resulting performance degradation of the prediction model 
+are analyzed. Based on the obtained results, the following 
+conclusions are drawn: 
+(1) The extracted epistemic uncertainty is valuable for 
+representing the model's confidence in its current predictions 
+accurately, which has the potential to be used to identify 
+unknown scenarios where the model may be underpowered. 
+Compared with the MC dropout-based method, the deep 
+ensemble-based method performs significantly better in 
+estimating epistemic uncertainty and improving the trajectory 
+prediction robustness; 
+(2) Regarding the influence of different scenario features on 
+the trajectory prediction performance, the feature correlation 
+and importance analyses show similar results. Namely, there is 
+a positive correlation between the kinematics of a TA and the 
+prediction model performance. Higher velocity, acceleration, 
+and speed of the heading change generally pose a greater 
+challenge to the trajectory prediction process, while a prediction 
+model tends to exhibit higher epistemic uncertainty. However, 
+one interesting finding is that the features of surrounding TPs, 
+which reflect the complexity of interactions in the scenario, 
+show little impact on the proposed prediction model 
+performance. In addition, the error and uncertainty of the 
+prediction model vary with other abstract features, such as TA’s 
+type, behavior pattern, compliance with traffic rules, and 
+location. The conducted analyses are helpful in locating the 
+limitations in the prediction algorithms, thus providing 
+guidance for the improvement of autonomous driving functions; 
+(3) For the intersection scenario, different datasets show 
+distributional shifts due to differences in local driving habits, 
+road structures, and national cultures, thus posing great 
+challenges to the prediction algorithms. Fortunately, the 
+proposed framework based on the deep ensemble is beneficial 
+to improving the trajectory prediction robustness, and the 
+extracted epistemic uncertainty can respond to the reduced 
+confidence of the proposed model in a new environment. This 
+improves the self-awareness ability of autonomous driving. 
+Although the used basic prediction algorithm has strong 
+representativeness, it is still difficult to avoid the specificity of 
+analysis conclusions completely. However, the proposed 
+method is promising to analyze other prediction algorithms, 
+subsequent work could consider combining more types of 
+algorithms for systematic analysis. Moreover, this work focuses 
+on the extraction and analysis of epistemic uncertainty, but the 
+existing works in the field of multimodal trajectory forecasting 
+could be considered in the follow-up research to distinguish the 
+epistemic uncertainty from the aleatoric uncertainty better and 
+explore their practical significance. Furthermore, the role of 
+uncertainty estimation of a trajectory prediction model in an 
+autonomous driving decision-making mechanism needs to be 
+studied further to improve the robustness against the risks of 
+insufficient functions. 
+ 
+ACKNOWLEDGMENT 
+This work was supported in part by the National Science 
+Foundation of China Project (Grant No. 52072215 and 
+U1964203), and the National Key R&D Program of China 
+under Grant NO. 2020YFB1600303. 
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+3 
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+2022-06-05, 
+2022. 
+[Online]. 
+Available: 
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+paristech.archives-ouvertes.fr/hal-03682452. 
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+1679, doi: 10.1109/ITSC.2018.8569929.  
+ 
+ 
+ 
+Wenbo Shao received his B.E. degree in 
+vehicle 
+engineering 
+from 
+Tsinghua 
+University, Beijing, China, in 2019. He is 
+currently working toward the Ph.D. degree 
+in Mechanical Engineering at Tsinghua 
+University. He is a member of Tsinghua 
+Intelligent Vehicle Design And Safety 
+Research Institute (IVDAS) and supervised 
+by Professor Jun Li and Associate Research 
+Professor Hong Wang. His research interests include safety of 
+the intended functionality of autonomous driving, trajectory 
+prediction, 
+decision-making, 
+uncertainty 
+theory 
+and 
+applications. 
+ 
+Yanchao Xu received B.E degree in vehicle 
+engineering from Hainan University in 
+China in 2020. He is currently pursuing the 
+M.S. degree in Mechanical Engineering at 
+Beijing Institute of Technology. He is also 
+one of the visiting students at IVDAS since 
+2020. His research interest includes 
+prediction, trajectory data mining, scenario 
+parameterization for autonomous driving. 
+ 
+ 
+Jun Li received the Ph.D. degree in 
+vehicle engineering from Jilin University, 
+Changchun, Jilin, China, in 1989. He is 
+currently an academician of the Chinese 
+Academy of Engineering, a Professor at 
+school of Vehicle and Mobility with 
+Tsinghua University, president of the 
+Society of Automotive Engineers of China, 
+director of the expert committee of China 
+Industry Innovation Alliance for the Intelligent and Connected 
+Vehicles. His research interests include internal combustion 
+engine, electric drive systems, electric vehicles, intelligent 
+vehicles and connected vehicles. 
+ 
+Chen Lv (Senior Member, IEEE) 
+received the Ph.D. degree from the 
+Department of Automotive Engineering, 
+Tsinghua University, China, in 2016. 
+From 2014 to 2015, he was a Joint Ph.D. 
+Researcher at the EECS Department, 
+University of California at Berkeley, 
+Berkeley, CA, USA. From 2016 to 2018, 
+he worked as a Research Fellow at the 
+Advanced Vehicle Engineering Center, 
+Cranfield 
+University, 
+U.K. 
+He 
+is 
+currently an Assistant Professor with the School of Mechanical 
+and Aerospace Engineering, and the Cluster Director of future 
+mobility solutions at ERI@N, Nanyang Technological 
+University, Singapore. His research interests include advanced 
+vehicles and human–machine systems, where he has 
+contributed over 100 articles and obtained 12 granted patents. 
+ 
+ 
+Weida Wang received the Ph.D. degree 
+from Beihang University, Beijing, China, 
+in 2009. He is currently a Professor with 
+the School of Mechanical Engineering, 
+Beijing Institute of Technology, Beijing. 
+He is also the Director of the Research 
+Institute of Special Vehicle, Beijing 
+Institute of Technology. His current 
+research interests include electric vehicle, 
+automated vehicle motion planning and control, and 
+electromechanical transmission control. 
+ 
+Hong Wang is Research Associate 
+Professor at Tsinghua University. She 
+received the Ph.D. degree in Beijing 
+Institute of Technology, Beijing, China, 
+in 2015. From the year 2015 to 2019, she 
+was working as a Research Associate of 
+Mechanical 
+and 
+Mechatronics 
+Engineering with the University of 
+Waterloo. Her research focuses on the 
+safety of the on-board AI algorithm, the 
+safe decision-making for intelligent vehicles, and the test and 
+evaluation of SOTIF. She becomes the IEEE member since the 
+year 2017. She has published over 60 papers on top 
+international journals. Her domestic and foreign academic part-
+time includes the associate editor for IEEE Transactions on 
+Vehicular Technology and Intelligent Vehicles Symposium, 
+Guest Editor of Special Issues on Intelligent Safety of 
+Automotive Innovation, Young Communication Expert of 
+Engineering, lead Guest Editor of Special Issues on Intelligent 
+Safety of IEEE Intelligent Transportation Systems Magazine. 
+ 
+

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+Incorporating time-dependent demand patterns in the
+optimal location of capacitated charging stations
+Carlo Filippi(1)
+Gianfranco Guastaroba(1)
+Lorenzo Peirano(1)
+M. Grazia Speranza(1)
+(1) University of Brescia, Department of Economics and Management, Brescia, Italy
+{carlo.filippi, gianfranco.guastaroba, lorenzo.peirano, grazia.speranza}@unibs.it
+January 13, 2023
+Abstract
+A massive use of electric vehicles is nowadays considered to be a key element of a sustainable
+transportation policy and the availability of charging stations is a crucial issue for their extensive
+use. Charging stations in an urban area have to be deployed in such a way that they can satisfy a
+demand that may dramatically vary in space and time. In this paper we present an optimization
+model for the location of charging stations that takes into account the main specific features
+of the problem, in particular the different charging technologies, and their associated service
+time, and the fact that the demand depends on space and time. To measure the importance
+of incorporating the time dependence in an optimization model, we also present a simpler
+model that extends a classical location model and does not include the temporal dimension. A
+worst-case analysis and extensive computational experiments show that ignoring the temporal
+dimension of the problem may lead to a substantial amount of unsatisfied demand.
+Keywords: Facility location, Charging stations, Electric vehicles, Demand patterns, Time-dependent
+optimization.
+1
+Introduction
+Sustainable transportation is one of the major challenges that modern countries are facing. Several
+sources indicate that the transportation sector generates the largest share of GreenHouse Gas
+(GHG) emissions. According to the United States Environmental Protection Agency1, in 2020 the
+transportation sector produced 27% of the total GHG emissions in the US, mostly generated from
+burning fossil fuels by cars, trucks, ships, trains, and planes. Domestic statistics issued by the UK
+government2 confirm that the transportation sector generated 27% of the total GHG emission. The
+majority (91%) came from road transport vehicles, where the biggest contributors were cars and
+taxis. Furthermore, data provided by the European Environment Agency3 highlight that in the
+EU more than 22% of the GHG emissions came from the transportation sector.
+1https://www.epa.gov/ghgemissions/sources-greenhouse-gas-emissions
+2https://www.gov.uk/government/statistics/transport-and-environment-statistics-autumn-2021/tran
+sport-and-environment-statistics-autumn-2021
+3https://www.eea.europa.eu/data-and-maps/data/data-viewers/eea-greenhouse-gas-projections-data-
+viewer
+arXiv:2301.05077v1  [math.OC]  12 Jan 2023
+
+Despite technical advances have made available a range of options for sustainable mobility, there
+are still important obstacles that must be overcome for their mass adoption. Among such options,
+Electric Vehicles (EVs) are considered one of the major directions to reduce the environmental
+impact of people mobility and make urban areas more sustainable.
+In the 2021 edition of the
+Global EV Outlook 20214, the International Energy Agency pointed out that at the end of 2020
+the global EVs stock hit 10 millions units, with 3 millions newly registered EVs. Europe was the
+fastest growing market, with a sales share equal to 10% and some leading countries, such as Norway,
+which registered a record high sales share of 75%. This trend was accelerated by many countries
+of the European Union through substantial financial incentives. However, the decision of potential
+EV buyers is still strongly affected by two major issues. On one hand, the purchase cost of an EV
+is still higher than that of a traditional internal combustion engine vehicle. On the other hand,
+the limited travel range of an EV and the long charging time are well-known to generate anxiety
+in the potential buyers (e.g., Pevec et al., 2020). In fact, the willingness of drivers to purchase an
+EV strongly depends on the availability of charging stations nearby their points of interests (e.g.,
+home and work). As the number of charging stations is growing, thanks to public and private
+investments, the location problem of such stations has attracted much attention (see Section 2).
+There are a number of factors that make the location of charging stations substantially different
+from other, more classical, location problems, in particular the choice of the charger to install (e.g.,
+slow, quick, fast), and the characteristics of the charging demand.
+The type of charger is a key factor to be taken into account, as it impacts the charging time. As of
+the end of 2021, there exist three main types of charger (see Moloughney, 2021). Level 1 chargers,
+also referred to as slow chargers, use common 120-volt outlets, and can take up to 40 hours to
+raise the level of a standard battery EV (with a 60 kWh sized battery) from 10% to 80% of the
+capacity. These chargers are most suitable for private usage. Level 2 chargers, sometimes called
+quick chargers, can charge up to 10 times faster than a level 1 charger, and are the most commonly
+used types for daily EV charging (see Moloughney, 2021). Given the same battery characteristics
+mentioned above, the charging time is about 4.5 hours. The level 3 or fast chargers can reduce the
+charging time to 40 minutes or even less. For a comprehensive study regarding the state of the art
+on charging stations, the interested reader can refer to Pareek et al. (2020). The type of charger
+demanded by EVs is affected by the urban layout. For example, slow chargers will be demanded
+in residential areas so that EVs can be recharged over the night at low cost (an interesting study
+of the factors influencing the charging demand is provided in Wolbertus et al., 2018).
+In the classical location models a customer is characterized by the distance from any potential
+location and by a single quantity - a measure of the demand.
+The models do not consider a
+temporal dimension of the problem which basically corresponds to assuming that the demand is
+uniformly distributed over the time period of interest of the location decision. On the contrary, the
+charging demand of EVs fluctuates over time, with peaks of demand in periods of time where the
+traffic volume is high. Neglecting the demand dynamics may lead to solutions where the charging
+capacity deployed is not sufficient to satisfy the demand during the peak times.
+In this paper, we study the problem of determining an optimal deployment of charging stations for
+EVs within an urban environment. Different types of chargers have to be located in pre-defined
+potential locations, modeled as nodes of a network. The urban area is partitioned in sections. A
+customer is associated with each section of the urban area. Its demand in a certain time interval
+is the number of EVs in that section that need to be recharged. The customer is located in the
+4https://www.iea.org/reports/global-ev-outlook-2021
+2
+
+center of gravity of the section and is modeled as a node of the network. The urban area is also
+partitioned in zones (e.g., commercial, industrial, or residential) which have different needs in terms
+of minimum number of each type of charger deployed in the zone.
+We have to determine, for each type of charger and each potential location, the number of chargers
+to be deployed. Two criteria have a key role in this location problem: the cost of installing the
+chargers and the distance the customers have to travel to be recharged.
+We present, over a discretized time horizon, an optimization model that introduces a temporal
+dimension which, to the best of our knowledge, has never been introduced in the literature on
+location problems and captures the dynamics of the charging demand. Assuming that a charger can
+take more than one period to fully recharge an EV, the proposed multi-period formulation includes
+constraints to keep track of the usage of chargers across consecutive time periods and to ensure that
+no other vehicles are assigned to any occupied charger. This novel approach guarantees a correct
+sizing of the solution, in terms of number of stations opened and number of chargers installed, and
+ensures that the demand is completely satisfied in all time periods. In order to assess the value of
+introducing the temporal dimension in the location problem, which makes the optimization model
+more complex, we present a single-period optimization model that captures the same specificities of
+the problem but ignores the temporal aspect. In both models, the objective is the minimization of a
+convex combination of two terms: the total cost of deploying the charging stations and installing the
+chargers, and the average distance traveled by the customers to reach the assigned charging station.
+The two optimization models turn out to be Mixed Integer Linear Programming (MILP) problems.
+We compare the two models through a theoretical and a computational analysis. We show, through
+worst-case analysis, that a solution to the single-period model may fail to satisfy a large portion of
+the charging demand. Extensive computational experiments are run on different classes of randomly
+generated instances. The results confirm the importance of explicitly considering the dependence
+on time of the demand.
+In fact, the single-period model is based on the common assumption
+that the charging demand is uniformly distributed across the planning horizon. In an application
+context such as the one at hand, where the demand fluctuates significantly during the day and
+across different zones of the same urban area, the single-period model produces solutions that are
+not capable of serving a large portion of the charging demand, especially in those time periods
+where the demand is prominently concentrated. The computational experiments also include a
+parametric analysis of the relative weight assigned to the objective function components.
+Structure of the paper.
+The remainder of the paper is organized as follows.
+In Section 2,
+the literature most closely related to our research is reviewed and the contribution of this paper
+is highlighted.
+In Section 3, after the presentation of the single-period extension of a classical
+location model, we provide the multi-period mathematical formulation. In Section 4, we analyze
+the worst-case performance of the single-period model in terms of portion of unsatisfied charging
+demand. Section 5 reports extensive computational experiments conducted on instances gener-
+ated to resemble demand dynamics frequently observed in different zones of a city. Finally, some
+concluding remarks are outlined in Section 6.
+2
+Literature review
+The problem of determining an optimal location and size of charging stations for EVs has recently
+attracted an increasing academic attention. Recent overviews of the main modeling and algorithmic
+approaches employed in this research area are available in Deb et al. (2018), Zhang et al. (2019), and
+3
+
+Kchaou-Boujelben (2021). For a general introduction on location problems the interested reader
+can refer to Laporte et al. (2019). In the following, we focus on the papers that are most closely
+related to our research, and refer the interested reader to the above-mentioned surveys and the
+references cited therein.
+A first broad classification of the literature is based on the type of network considered (cf. Deb et al.,
+2018). When only the distribution network is considered, the optimal location of charging stations
+must consider the potential adverse effects on the power grid, as an inappropriate placement of
+charging stations can be a threat to the power system security and reliability. On the other hand,
+when only the transportation network is taken into account, the main issue is to determine an
+optimal location of charging stations over a road network. This paper lies in the latter category.
+Within this category, the related literature can be further classified into two main streams of
+models called flow-based and node-based demand models (e.g., see Kchaou-Boujelben, 2021). In
+the literature, the majority of the research efforts are devoted to the flow-based demand models,
+whereas the number of papers adopting a node-based approach is still relatively limited. To the
+best of our knowledge, Anjos et al. (2020) are the only authors that integrated, within the same
+optimization model, both a node-based and a flow-based approach. The flow-based demand models
+are best suited for modeling long-haul (e.g., inter-urban) journeys where accounting for the limited
+driving range of EVs is important (cf. Anjos et al., 2020). Contributions to this line of research
+can be found, for example, in Kuby and Lim (2005), MirHassani and Ebrazi (2013), Yıldız et al.
+(2016), and Hosseini et al. (2017). The present paper adopts a node-based demand model.
+In the class of node-based demand models, drivers demanding to charge their EVs are associated
+with one/few fixed locations, which represent, for instance, their residence, workplace or specific
+service facilities (such as commercial activities). This approach is best suited for urban settings.
+In fact, in such case EVs do not move much from the location where they need to be charged and
+their limited driving range can be neglected (cf. Anjos et al., 2020). The most common modeling
+approaches applied in the literature are based on the extension of classic discrete location models
+(e.g., location-allocation as in Zhu et al. (2016), set covering as in Huang et al. (2016), and maximum
+coverage problems as in Dong et al. (2019)) to incorporate technical constraints specific to EVs.
+Characteristics of the charging demand (such as the population size, the penetration rate of EVs,
+the type of zone, and the time of the day) are known to have a crucial impact on the optimal
+location of charging stations. To position the present paper within the literature, we classify the
+mathematical formulations into single-period and multi-period. In single-period optimization mod-
+els all the decision variables are time independent. Although the spatial-temporal distribution of
+the charging demands is described by different authors (e.g., see Yi et al., 2020, and the references
+cited therein), only few authors have proposed multi-period optimization models where the alloca-
+tion of the demand to the charging stations is time-dependent. The related stream of literature can
+be classified according to the length of the planning horizon considered. A long planning horizon is
+considered by some authors. The basic rationale of these models is that locating charging stations is
+a long-term strategic decision. As a consequence, during these long periods of time the technology
+available, as well as the charging demand, may change significantly. Along this line of research,
+we mention the paper by Anjos et al. (2020) where it is assumed that the locating decisions taken
+in a period have an impact on the charging demand in the subsequent periods. In fact, potential
+EV buyers are influenced by the availability of charging opportunities. Some papers have proposed
+multi-period optimization models that consider a short horizon, usually a day, divided in time
+periods, usually hours. Our research belongs to this category of papers.
+4
+
+To the best of our knowledge, Cavadas et al. (2015) are the first authors to recognize the importance
+of incorporating into an optimization model the dynamics of the charging demand across the day.
+The aim of the proposed multi-period model is the maximization of the total demand served,
+subject to a constraint on the budget available. The authors consider only one type of charger (i.e.,
+a slow type) and the sizing of the charging stations is not part of the optimization. In the model we
+present in this paper, we address these shortcomings by considering multiple types of chargers and
+optimizing the quantities installed in each opened station. Rajabi-Ghahnavieh and Sadeghi-Barzani
+(2017) estimate the charging demand of EVs in different zones of a city and at different hours. The
+authors consider the deployment of an unlimited number of fast chargers only and propose a non-
+linear optimization model that includes three cost components: the total opening cost, the total
+cost for the drivers to reach the assigned charging stations, and the cost of connecting the charging
+stations to the electric grid substations. The variability of the demand across the day is taken
+into consideration when determining the number of chargers to install. Nevertheless, the variables
+assigning EVs to stations are not time-dependent, and, hence, drivers demanding to charge their
+EVs at different hours are all assigned to the same station. In our paper, we allow the demand
+arising from the same location during the day to be assigned to different stations, depending on
+the evolution of the overall demand and the available cherging resources. Moreover, we consider
+different types of chargers. Both short-term and long-term decisions are considered in Quddus et al.
+(2019). The main long-term decisions are related to the year, the location, and the type of charging
+stations to open. The short-term decisions are mainly related to the amount of power (provided
+by different sources, such as electric grid and renewable sources) to satisfy the hourly charging
+demand at a given location. Compared to our research, the drivers are, indirectly, pre-assigned to
+a charging station and, hence, the assignment is not part of the optimization model. The authors
+cast the problem as a two-stage stochastic programming model. Li and Jenn (2022) present an
+optimization model based on the concept of charging opportunities, which is measured through
+the time an individual stays at a given location within a day. The authors separate the charging
+opportunities into home and non-home (i.e., public) categories, and allow the same individual to
+charge the EV multiple times at different locations. The proposed optimization model determines
+the number of home and non-home chargers to install, as well as the times and locations for each
+individual to charge the EV. The model aims at minimizing the sum of the annual electricity cost
+for charging the EVs and the total cost of locating the home and non-home chargers. The number
+of chargers that can be installed in each location (called region by the authors) is unlimited.
+Finally, we mention the growing body of literature that addresses the problem of determining an
+optimal location of charging stations for EVs in car-sharing systems (e.g., cf. Brandst¨atter et al.,
+2017, 2020; Bekli et al., 2021). Although such problem has some characteristics in common with
+ours, it includes some operational characteristics that make it considerably different, for example
+the decisions about the number of EVs to acquire, the relocation of the EVs among stations, and
+the assumption that charging occurs only between two consecutive trips.
+Contributions of the paper. The contributions of this paper to the literature can be summarized
+as follows.
+✓ We present a node-based multi-period optimization model for the location of charging stations
+that captures the dependence on time of the charging demand;
+✓ the multi-period model takes into account several characteristics of the real problem: multiple
+types of chargers (each with its own charging speed and installation cost), the capacitated
+nature of the charging stations (in terms of maximum number of chargers that can be in-
+5
+
+stalled), a minimum number of chargers to be installed in different zones (e.g., commercial,
+residential, industrial);
+✓ the multi-period model is compared to a single-period model through a worst-case analysis;
+✓ extensive computational experiments are presented that show, in particular, the importance
+of incorporating the dependence on time of the charging demand.
+3
+Problem definition and mathematical formulations
+In this section, we first provide a general description of the location problem along with the notation
+that is common to the two optimization models that will follow. Then, the single-period MILP
+model is presented, together with the notation that is specific for the model, followed by the multi-
+period formulation.
+We consider the problem of determining, in an urban area, an optimal location of charging stations
+for EVs, along with the type and number of chargers to deploy in each station. A maximum number
+of chargers, of each type and in total, can be deployed in each station. The location for any station
+can be selected from a pre-defined set of potential locations. We introduce a complete bipartite
+network G = (I ∪ J , A), where I = {1, 2, . . . , I} is the set of demand nodes and J = {1, 2, . . . , J}
+is the set of potential locations for the stations. Let cij be the travel distance from demand node i
+to station j.
+A fixed opening cost Fj is associated with each station j. The opening cost does not include the
+cost of the chargers. We denote as K = {1, 2, . . . , K} the set of types of chargers considered, and
+as fjk the cost of installing one charger of type k ∈ K in location j ∈ J . Let ujk be the maximum
+number of chargers of type k that can be installed in station j. Similarly, uj denotes the maximum
+number of chargers that can be installed in total in station j. The latter two parameters define,
+implicitly, the maximum charging capacity of station j.
+Each node i is the center of gravity of a section of the urban area where the demand of the section
+is measured as the number of EVs that need to be recharged. We will introduce later, for each of
+the two optimization models, the planning horizon and the notation for the demand of a customer.
+For the sake of brevity, hereafter we refer to each potential location j simply as station j. The
+demand must be entirely satisfied by the chargers that will be deployed.
+To take into account that different parts of the urban area have different needs in terms of type of
+charger desired, the urban area is partitioned in zones (e.g., commercial, residential, industrial). We
+denote by L = {1, 2, . . . , L} the set of zones. We assume that, based on some preliminary analysis,
+in each zone ℓ ∈ L a minimum percentage ρℓk of chargers of type k must be deployed. Each station
+j ∈ J belongs to a zone as well as each customer i ∈ I. Thus, the zones imply a partition of both
+the stations and the demand points. This partition does not restrict the allocation of demand to
+stations, i.e., a demand point located in a zone can be assigned to a station located in a different
+zone.
+Two criteria have a key role in this location problem: the cost of opening the stations and installing
+the chargers and the distance the customers have to travel to be recharged. The objective function
+we consider, to be minimized, is a convex combination of these two criteria. The optimization
+problem is aimed at determining, for each type of charger and each station, the number of chargers
+to be deployed in such a way that the objective function is minimized.
+6
+
+Both MILP models include the following decision variables. Let zj ∈ {0, 1}, with j ∈ J , be a
+binary variable that takes value 1 if station j is opened, and 0 otherwise. Let yjk ∈ Z+, with j ∈ J
+and k ∈ K, be an integer variable that represents the number of chargers of type k installed in
+station j.
+3.1
+A single-period location model
+This section presents a single-period model for the location of the charging stations. The MILP
+formulation, denoted as SP-CFL, is an extension of a classical CFL model. Hereafter, we introduce
+the notation needed for the formulation, in addition to the one introduced above.
+We consider a single planning period of length H and denote as di the total demand in i ∈ I, that
+is, the total number of EVs demanding to be recharged in i during H. Let pk denote the average
+number of EVs fully recharged by one charger of type k during time period H. For the sake of
+simplicity, we assume that pk does not depend on the type of EV.
+The SP-CFL model also makes use of the following decision variables. Let xijk ∈ [0, 1], with i ∈ I,
+j ∈ J , and k ∈ K, be the fraction of the demand of node i assigned to a charger of type k in station
+j. Then, the SP-CFL model can be stated as the following MILP:
+[SP-CFL]
+min
+λ ·
+�
+�
+1
+�
+i∈I
+di
+�
+i∈I
+di
+�
+j∈J
+cij
+�
+k∈K
+xijk
+�
+� + (1 − λ) ·
+�
+��
+j∈J
+Fjzj +
+�
+j∈J
+�
+k∈K
+fjkyjk
+�
+�
+(1)
+s.t.
+yjk ≤ ujkzj
+j ∈ J , k ∈ K
+(2)
+�
+k∈K
+yjk ≤ ujzj
+j ∈ J
+(3)
+�
+j∈J
+�
+k∈K
+xijk = 1
+i ∈ I
+(4)
+�
+i∈I
+dixijk ≤ pkyjk
+j ∈ J , k ∈ K
+(5)
+xijk ≤ yjk
+i ∈ I, j ∈ J , k ∈ K
+(6)
+�
+j∈Aℓ
+yjk ≥ ρℓk
+�
+j∈Aℓ
+�
+k∈K
+yjk
+k ∈ K, ℓ ∈ L
+(7)
+zj ∈ {0, 1}
+j ∈ J ;
+yjk ∈ Z+
+j ∈ J , k ∈ K;
+xijk ∈ [0, 1]
+i ∈ I, j ∈ J , k ∈ K.
+(8)
+The objective function in (1) comprises two terms. The first one represents the average distance
+traveled by the EVs to reach the assigned station. The second term is the total cost of opening the
+7
+
+stations and installing the chargers. The two terms represent criteria of a substantially different
+nature: the first measures the quality of the service provided by the deployed stations and chargers
+to the drivers, whereas the second the cost of the service. The two criteria are weighted by the
+trade-off parameter λ ∈ [0, 1], which is used to balance their importance.
+Constraints (2) and (3) limit the number of chargers that can be installed in station j. The former
+set bounds the number of chargers of type k to be lower than or equal to ujk, whereas the second
+set of constraints bounds the total number of chargers to be lower than or equal to uj. Both sets of
+constraints (2) and (3) impose that no charger can be installed if station j is not open (i.e., zj = 0).
+Constraints (4) ensure that the demand of each node i ∈ I is entirely satisfied. Constraints (5)
+guarantee that the number of EVs assigned to the chargers of type k deployed in station j is not
+greater than the charging capacity available (i.e., pkyjk). They also impose that no EV can be
+assigned to a type k of chargers in station j if no charger of that type is available (i.e., yjk = 0).
+Inequalities (6), which are redundant in this formulation, are well-known to yield a tighter Linear
+Programming (LP) relaxation than the equivalent formulation without them (e.g., see Filippi et al.,
+2021). Constraints (7) guarantee that the number of chargers of type k installed in zone ℓ is at least
+equal to the minimum percentage ρℓk. Finally, constraints (8) define the domain of the decision
+variables.
+3.2
+A multi-period location model
+This section presents the MILP formulation for the multi-period model, henceforth denoted as the
+MP-CFL model, for the problem defined at the beginning of this section.
+The planning period H of the single-period model is here partitioned into a number T of time
+periods. For example, if H is a day, we may partition the day in hours. Let T = {1, 2, . . . , T}
+denote the set of time periods. We denote as Rk the number of consecutive time periods needed to
+completely recharge a car using a charger of type k. Note that, similar to pk for the SP-CFL model,
+Rk does not depend on the type of EV but only on the type of charger. Furthermore, parameters
+pk and Rk are strictly related, as the latter is determined by dividing the length of the time horizon
+by pk, i.e. Rk = T
+pk .
+The demand of each node i ∈ I is no longer identified by a single value (di in the SP-CFL model)
+but by a time-dependent profile. Let dt
+i denote the demand of node i ∈ I at the beginning of time
+period t ∈ T . A more detailed discussion about the demand profiles can be found in Section 5.1.1.
+We assume that the demand of a time period t must be served in that time period, i.e., it cannot
+be postponed to a later time. We say that a node is served by a charger of type k at time t if a
+charger is available at time t to start the charging which will occupy the charger for a total of Rk
+time periods. The capacity installed in each station must be sufficient to serve the charging demand
+assigned to that station in a time period and the demand assigned to the station in a previous time
+period that has not yet completed the charging. Finally, let xt
+ijk ∈ [0, 1], with i ∈ I, j ∈ J , k ∈ K,
+and t ∈ T , be the fraction of the charging demand of node i to be served at time t that is assigned
+to a charger of type k in station j.
+The MP-CFL model is formulated as follows:
+[MP-CFL]
+8
+
+min
+λ ·
+�
+�
+1
+�
+t∈T
+�
+i∈I
+dt
+i
+�
+t∈T
+�
+i∈I
+dt
+i
+�
+j∈J
+cij
+�
+k∈K
+xt
+ijk
+�
+� + (1 − λ) ·
+�
+��
+j∈J
+Fjzj +
+�
+j∈J
+�
+k∈K
+fjkyjk
+�
+�
+(9)
+s.t. (2), (3), and (7)
+�
+j∈J
+�
+k∈K
+xt
+ijk = 1
+i ∈ I, t ∈ T
+(10)
+xt
+ijk ≤ yjk
+i ∈ I, j ∈ J , k ∈ K, t ∈ T
+(11)
+�
+i∈I
+t−1
+�
+τ=0
+dt−τ
+i
+xt−τ
+ijk ≤ yjk
+j ∈ J , k ∈ K, t ∈ T : t < Rk
+(12)
+�
+i∈I
+Rk−1
+�
+τ=0
+dt−τ
+i
+xt−τ
+ijk ≤ yjk
+j ∈ J , k ∈ K, t ∈ T : t ≥ Rk
+(13)
+zj ∈ {0, 1}
+j ∈ J ;
+yjk ∈ Z+
+j ∈ J , k ∈ K;
+xt
+ijk ∈ [0, 1]
+i ∈ I, j ∈ J , k ∈ K, t ∈ T .
+(14)
+The objective function in (9) is the multi-period extension of function (1). For each node i ∈ I,
+constraints (10) ensure that the charging demand arising in each time period t is fully satisfied.
+Akin to the objective function, also inequalities (11) are the multi-period extension of constraints
+(6).
+Constraints (12) and (13) guarantee that the number of EVs that are charging in time period t at
+a charger of type k in station j is smaller than or equal to the number of available chargers of that
+type (i.e., yjk). Note that the second sum in (12) and (13) is used to keep track of the EVs that
+started to recharge in a previous time period but have not completed the charging in t. Constraints
+(12) are defined for the first time periods in the planning horizon (such that t < Rk), whereas
+(13) are defined for the remaining time periods. Finally, constraints (14) define the domain of the
+decision variables.
+4
+Worst-case analysis
+In this section, we analyze the worst-case performance of the SP-CFL model in terms of the demand
+that cannot be satisfied if the optimal solution produced is implemented in a context where the
+demand fluctuates over time. In fact, in this case if an optimal solution to the SP-CFL model is
+implemented, there is no guarantee that all the charging demand is satisfied. As the SP-CFL model
+implicitly assumes that the charging demand is uniformly distributed across the planning horizon,
+when the demand fluctuates over time, there may be peak time periods where the chargers installed
+are not sufficient.
+9
+
+Theorem 1 When an optimal solution of the SP-CFL model is implemented, the fraction of the
+demand that does not find an available charger to be served may be up to 1 − 1
+T , where T is the
+number of time periods of the planning horizon. This bound is tight.
+Proof To prove the theorem, we build the following instance.
+Recalling constraint (4) and summing up all constraints (5), the following chain of inequalities
+holds:
+�
+i∈I
+di
+(4)
+=
+�
+i∈I
+di
+�
+j∈J
+�
+k∈K
+xijk =
+�
+j∈J
+�
+k∈K
+�
+i∈I
+dixijk
+(5)
+≤
+�
+j∈J
+�
+k∈K
+pkyjk
+(15)
+for any feasible solution to the SP-CFL model.
+Consider an instance where the travel distances are all negligible compared to the fixed opening and
+installing cost. In this situation, the SP-CFL model would open the minimum number of charging
+stations and install the minimum number of chargers that are strictly necessary to satisfy the total
+demand. As a consequence, the value of the right-hand side of the rightmost inequality in (15)
+would be as small as possible.
+Additionally, suppose there is a single type of charger (i.e., K = 1) and that the total demand
+�
+i∈I di is a multiple of p1. Recall that the latter parameter represents the number of EVs fully
+recharged by one charger during the planning horizon. Note that it can be determined by dividing
+the number of time periods T by the number of consecutive time periods needed to completely
+recharge an EV (i.e., R1). Hence, at optimality, the inequality in (15) can be reformulated as
+follows:
+�
+i∈I
+di =
+�
+j∈J
+p1yj1 = T
+R1
+�
+j∈J
+yj1.
+(16)
+Thus, the total number of chargers deployed is �
+j∈J yj1 =
+R1
+�
+i∈I di
+T
+, which, assuming that R1 = 1,
+becomes �
+j∈J yj1 =
+�
+i∈I di
+T
+.
+Consider an extreme situation where the whole demand �
+i∈I di arises in one time period, say ˆt,
+whereas it is zero in the remaining periods. The demand that can be satisfied in such time period
+is equal to the number of chargers installed (i.e., �
+j∈J yj1). Given the assumptions above, this
+value is also equal to
+�
+i∈I di
+T
+. Thus, the amount of demand that does not find an available charger
+is equal to:
+�
+i∈I
+di −
+�
+i∈I
+di
+T
+.
+The statement follows.
+Figure 1 illustrates the construction for the special case where �
+i∈I di = T, which implies that
+�
+j∈J yj1 = 1. The whole demand, equal to T, arises in time period ˆt (green bar), whereas it is zero
+in the remaining periods. The SP-CFL model assumes that such demand is uniformly distributed
+across the planning horizon (pink bars), and hence it opens one station equipped with one charger.
+As a consequence, the charging demand that is not satisfied is T − 1 or, in percentage, T−1
+T .
+10
+
+Figure 1: An instance where the demand arises in time period ˆt (green bar). The pink bars show
+a uniform distribution of the demand across the planning horizon, as implicitly assumed by the
+SP-CFL model.
+5
+Experimental Analysis
+This section is devoted to the presentation and discussion of the computational experiments. They
+were conducted on a Workstation HP Intel(R)-Xeon(R) at 3.5GHz with 64 GB RAM (Win 10 Pro,
+64 bits). The processor is equipped with 6 physical cores, and all threads were used while solving
+each instance. The MILP models were implemented in Java, compiled within Apache NetBeans
+12.3, and solved by means of CPLEX 20.1. Each instance was solved with a CPU time limit of
+3,600 seconds. All other CPLEX parameters were set at their default values.
+The section is organized as follows.
+First, we present the testing environment we used in our
+experiments, then we compare the optimal solutions for two illustrative examples generated ac-
+cording to two different urban structure models, and finally we provide detailed computational
+results comparing the solutions produced by the single-period and the multi-period models.
+5.1
+Testing environment
+The generation of the charging demand and potential station locations follows the procedure de-
+scribed in Section 5.1.1. All the remaining parameters defining the testing environment are detailed
+in Section 5.1.2.
+5.1.1
+Spatial and temporal charging demand generation
+As far as the urban structure is concerned, we considered two classic models, the concentric zone
+model and the sector model. The concentric zone model was proposed in 1925 by sociologist Ernest
+Burgess on the base of his human ecology theory, and was initially applied to the city of Chicago (cf.
+Burgess, 2008). It is, perhaps, the first theoretical model used to explain urban social structures.
+The model depicts urban land usage as concentric rings: the business district is located in the
+center, whereas the remainder of the city is expanded in rings, each corresponding to a different
+land usage (such as industrial or residential). The sector model was proposed in 1939 by land
+economist Homer Hoyt (see Hoyt, 1939). It is a modification of the Burgess’ model where the
+city zones devoted to a specific land usage (e.g., business, residential, and productive) develop
+in sectors expanding from the original city center. Though the actual structure of modern cities
+can hardly be captured by models as simple as Burgess’ and Hoyt’s, they are the basis of more
+11
+
+T
+:
+1
+0
+t+1
+t-1
+<t
+1
+2
+T-1
+T
+..
+..(a) COR instance
+(b) SEC instance
+Figure 2: Demand nodes generation for a COR (left) and a SEC (right) instance. Blue points
+are located in the commercial zone, orange points in the residential zone, and gray points in the
+industrial zone.
+complex structures (Hall and Barrett, 2012) and, on the other hand, can simplify the interpretation
+of the results. For these reasons, we considered two classes of instances, each associated with one
+urban model. Hereafter, the two classes are referred to as the concentric ring (COR) instances
+and the sector (SEC) instances. In both cases, we assume the urban structure comprises three
+possible zones: commercial, residential, industrial. With a little abuse of notation, we denote the
+set of zones as L = {C, R, I}, where C, R, and I refer to the commercial, residential, and industrial
+zones, respectively. Each zone is characterized by a different pattern of the charging demand during
+the planning horizon, as it will be detailed later. We consider as planning horizon a day, discretized
+into T = 24 hours (i.e., the time periods).
+In the COR instances, we assume the commercial zone is the central circle with ray 1000, the
+residential zone is a ring in the middle around the commercial zone with outer ray 2000, and the
+industrial zone is a most outer ring around the residential zone with outer ray 3000. In the SEC
+instances, we assume that commercial, residential, and industrial zones correspond to three slices
+of identical size that partition a circle of ray 3000.
+Then, for each zone, we uniformly generate the same number of demand nodes. More exactly, given
+the total number of demand nodes to be generated, such value is divided by three to obtain, after
+an integer rounding whenever necessary, the number of demand nodes to generate in each zone.
+Figure 2 gives an example.
+Both in COR and SEC instances, a given number J of potential stations is uniformly generated
+over the total area (i.e., the circle with ray 3000).
+For each pair of demand node i and potential station j, parameter cij is computed as the Euclidean
+distance between the two nodes.
+Concerning the demand pattern, this is specific for each zone. In the commercial zone we assume
+there is a high density of offices, shops, pubs, restaurants, and hotels. Hence, we expect a high
+12
+
+3000
+.
+2000
+.
+1000
+..
+.
+.
+-3opo
+-20Do
+10bo
+1d00
+2000
+3000
+.
+-1000
+:
+.
+-200
+30003000
+.2000
+*
+.
+.
+.
+-3dpo
+-2000
+-1000
+2.
+0
+1000
+.
+3000
+to
+.
+.
+.
+.
+1000
+*
+.
+:
+.
+.
+.
+.
+.
+.
+2000
+.
+.
+.
+.
+.
+30000Level
+0
+1
+2
+3
+Demand
+null
+low
+medium
+high
+Table 1: Standard levels of hourly demand.
+demand with two peaks, the first one at the beginning of the working day and the second one
+in the afternoon, higher than the first peak and slowly decreasing in the evening hours. In the
+industrial zone, we expect a high demand in the morning with one peak around lunch time and an
+almost null demand during the night. Finally, in the residential zone, we assume the presence of a
+high density of private houses, with a comparatively low demand in the morning and a peak in the
+evening and early night hours. These assumptions are consistent with several studies regarding the
+spatial-temporal distribution of the charging demands observed in urban areas (see, e.g., Yi et al.,
+2020; Straub et al., 2021).
+To generate the charging demand according to these patterns, we proceed as follows. We first
+generate three basic profiles of demand that mimic the patterns described above for the commercial,
+industrial, and residential zones, respectively. Such basic profiles are built according to the four
+standard demand levels shown in Table 1. The resulting basic demand profiles are depicted in
+Figure 3. For each i ∈ I and t = 1, . . . , 24, we randomly generate an initial demand value ˜dt
+i from
+a Poisson distribution with mean (and variance) equal to the standard level assigned to i and t in
+the corresponding basic profile. For example, if i is in the commercial zone and t = 8, then ˜dt
+i is a
+realization of a Poisson with mean 2, cf. Figure 3(a). We then set:
+dt
+i =
+�
+˜dt
+i
+�
+t∈T ˜dt
+i
+· 10
+�
+,
+where [·] denotes the nearest integer rounding operator. In this way, we obtain that: (1) the total
+daily demand from each demand node is around 10; (2) the total demand in each zone is consistent
+with the corresponding basic demand profiles shown in Figure 3.
+5.1.2
+Remaining parameters
+We generated a set of instances by varying the number of demand nodes I, of potential stations J,
+and of the maximum number of chargers to install in each station uj. All the remaining parameters
+take the same value across all instances. The name of each instance is I J uj, where:
+✓ I: The number of demand nodes ranges according to the following values: I = 50, 100, 150,
+200, 250, and 500.
+✓ J: The number of potential stations ranges according to the following values: J = 10, 20, 30,
+40, and 50.
+✓ uj = u: The maximum number of chargers to install is equal across all the stations, and
+ranges according to the following values: uj = 10, 20, and 30.
+For example, instance 50 20 30 comprises 50 demand nodes, 20 potential stations, and parameter
+uj is equal to 30. Note that the latter parameter is equal for each potential station j. We made
+13
+
+(a) Commercial
+(b) Residential
+(c) Industrial
+Figure 3: Basic hourly demand profile in each zone.
+14
+
+3
+2
+1
+0
+1
+2
+3
+4
+5
+6
+7
+8
+6
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+243
+2
+1
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+243
+2
+1
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24this choice to simplify the interpretation of the results. For the same reason, we decided to set
+ujk = uj = u, for each j ∈ J and k ∈ K.
+Each instance in our testbed has the following common characteristics:
+✓ The planning horizon considered is 1 day, discretized in time periods of one hour length.
+Consequently, T = {1, 2, . . . , 24}.
+✓ The cost of opening one charging station is Fj =100,000 ∀j ∈ J.
+✓ Two types of chargers are considered. Therefore, K = {1, 2}, where 1 denotes quick chargers,
+and 2 stands for fast chargers.
+✓ The cost of installing each type of chargers is fj1 = 3, 000 and fj2 =25,000 ∀j ∈ J for quick
+and fast chargers, respectively.
+✓ Quick chargers need R1 = 4 hours to fully recharge an EV, whereas fast chargers require
+R2 = 1 hour. Parameter pk for the SP-CFL model is determined as: pk = 24
+Rk .
+✓ The minimum percentage of chargers of each type to deploy in each zone is the following:
+– Commercial zone: at least 20% of quick chargers (ρC1 = 0.20), and at least 40% of fast
+chargers (ρC2 = 0.40).
+– Residential zone: at least 50% of quick chargers (ρR1 = 0.50), and at least 20% of fast
+chargers (ρR2 = 0.20).
+– Industrial zone: at least 25% of quick chargers (ρI1 = 0.25), and at least 25% of fast
+chargers (ρI2 = 0.25).
+Note that, in both MILP models, the two components in the respective objective function can take
+very different values, differing even by orders of magnitude. In our experiments we scaled the two
+components to make them comparable in value.
+We initially considered all possible combinations of the values mentioned above for I, J, and
+uj.
+Subsequently, we ruled out each instance that turned out to be infeasible for both MILP
+models.
+This situation happened especially for the largest numbers of the demand nodes, the
+smallest numbers of potential stations, and the smallest values of parameter uj. In such cases, the
+maximum charging capacity, obtained opening all potential stations and deploying uj chargers in
+each station j, turned out not to be sufficient to serve the total charging demand. All together, we
+analyzed 67 instances.
+5.2
+A comparison between COR and SEC instances
+To illustrate the solutions obtained by the two MILP models on the COR and SEC instances, we
+discuss the results obtained on two small instances. In both instances, the number I of demand
+nodes is equal to 21, equally divided among the three zones. The number J of potential stations is
+5. We assumed that the planning horizon comprises 8 time periods, and that the demand profile
+in each zone is the one depicted in Figure 4. These profiles are the same for both the COR and
+SEC instance.
+15
+
+(a) Commercial
+(b) Residential
+(c) Industrial
+Figure 4: Illustrative example: Basic hourly demand profile in each zone.
+An optimal solution produced by the MP-CFL model for the COR instance is depicted in Figure 5,
+where demand nodes are colored circles (blue for the commercial zone, orange for the residential
+zone, grey for the industrial zone) and potential locations are black triangles. Moreover, the color
+of the edge connecting a demand node to a black triangle represents the fraction of the charging
+demand assigned to the station thereby opened (black = 100% of the demand, yellow 75%, red
+66%, blue 50%, green 33% and gray 25%). Note that Figure 5 shows the assignments concerning
+only the significant time periods. In other words, the assignments in time periods 3 and 8 are not
+reported, the former since there is no demand in that time period, the latter because it is identical
+to time period 7. Figure 6 displays an optimal solution to the SP-CFL model for the same instance.
+The optimal solutions found by the two MILP models for the SEC instance are shown in Figures 7
+and 8.
+Comparing the optimal solutions for the COR instance obtained by the MP-CFL and the SP-CFL
+models (see Figures 5 and 6, respectively), one can notice that in the former all the potential
+stations are open and the charging demand is assigned, in the majority of the cases, to the nearest
+station. The solution found by the SP-CFL model opens only three stations. This small example
+highlights the limits of the latter model: it neglects that the charging demand is concentrated in
+few peak time periods, and, consequently, underestimates the charging need in those time periods.
+In fact, most of the demand is assigned to the charging station located in the central position
+(coordinates (0,0)), but the chargers deployed there are not sufficient to serve all the EVs during
+the peak hours. Due to the lower number of stations opened (3 against 5) and the number of
+chargers approximately 40% lower (30 against 52), we observe that 17.04% of customers cannot be
+served by the solution to SP-CFL.
+Similar conclusions can be drawn observing the optimal solutions for the SEC instance produced
+by the MP-CFL and the SP-CFL models (see Figures 7 and 8, respectively). As expected, there is no
+remarkable difference between the computational time on the COR and SEC instances. To keep
+a reasonable length of the paper, we conducted the extensive experiments on the COR instances
+only.
+16
+
+3
+2
+1
+0
+1
+2
+3
+4
+5
+6
+7
+83
+2
+1
+0
+1
+2
+3
+4
+5
+6
+7
+83
+2
+1
+0
+1
+2
+3
+4
+5
+6
+7
+8(a) Time Period 1
+(b) Time Period 2
+(c) Time Period 4
+(d) Time Period 5
+(e) Time Period 6
+(f) Time Period 7
+Figure 5: COR instance: An optimal solution to the MP-CFL model.
+17
+
+T1
+15
+20
+10
+15T2
+15
+10
+15
+20
+-10T4
+10
+15
+10
+20T5
+15
+OT
+15
+20T6
+10
+20
+10T7
+10
+20
+OTFigure 6: COR instance: An optimal solution to the SP-CFL model.
+5.3
+Computational results
+This section is devoted to the illustration and comment of the computational results. Before entering
+into the details of the results, we illustrate the solutions produced by the two MILP models for
+instance 200 30 30 and λ = 0.50. Figure 9 depicts, for each time period, the charging capacity
+installed and the demand satisfied by an optimal solution to model MP-CFL for instance 200 30 30.
+For each time period (vertical axis), the tornado diagram shows as bordered bars the number of
+chargers of each type deployed: the red bordered bars (left) are the fast chargers, whereas the
+black bordered bars (right) are the quick chargers. The solid bars represent the assignment of the
+charging demand. For each time period and type of charger, the bar indicates the total number
+of chargers assigned to EVs. Recall that quick chargers need multiple time periods to fully charge
+an EV. Hence, an EV assigned to a quick charger will use it for multiple consecutive time periods.
+From Figure 9, one can notice that the demand assigned to each type of chargers in each time
+period does not violate the charging capacity deployed. Note also that in several time periods the
+demand approaches the capacity installed, and these two quantities are sometimes even equal (see
+the fast chargers in time periods 12 through 16).
+18
+
+15
+10
+15
+-10
+5
+20
+10
+15(a) Time Period 1
+(b) Time Period 2
+(c) Time Period 4
+(d) Time Period 5
+(e) Time Period 6
+(f) Time Period 7
+Figure 7: SEC instance: An optimal solution to the MP-CFL model.
+19
+
+T1
+15
+20T2
+15
+10
+10
+20
+15T4
+15
+5
+20
+10T5T6
+15
+10
+20T7
+15
+OT
+15
+10Figure 8: SEC instance: An optimal solution to the SP-CFL model.
+Figure 9: An optimal solution to the MP-CFL model for instance 200 30 30: Charging capacity
+installed (bordered bars) and demand assigned (solid bars).
+The limits of the solution produced by the SP-CFL model are evident from Figure 10. The solution
+found by the SP-CFL model assigns, in several time periods, more EVs than the chargers actually
+available. For the fast chargers, this happens in time periods 8, 9, and from 12 to 22. On the other
+hand, for the quick chargers, this occurs in time periods 11 through 15. We observed this outcome
+for the majority of the instances tested.
+20
+
+15
+10
+10
+20
+10100
+50
+0
+50
+100
+150
+200
+250
+T1
+T2
+T3
+T4
+T5
+T6
+T7
+T8
+T9
+T10
+T11
+T12
+T13
+T14
+T15
+T16
+T17
+T18
+T19
+T20
+T21
+T22
+T23
+T24Figure 10: An optimal solution to the SP-CFL model for instance 200 30 30: Charging capacity
+installed (bordered bars) and demand assigned (solid bars).
+We now analyze more thoroughly the solutions produced by the two MILP models. To gain some
+insights about the two components of the objective functions, each instance is solved by each
+model for several values of the trade-o��� parameter λ. We tested the following values: λ = 0.0001
+(maximum weight on the minimization of the total opening and installing costs), 0.25, 0.5, 0.75,
+and 0.9999 (maximum weight on the minimization of the average distance traveled by the EVs).
+Table 2 provides, in the first three groups of columns, a summary of the charging capacity deployed
+by the solutions found by the two MILP models. For each group of instances and each model,
+Table 2 shows the average number of stations open (columns with header “Stations”), as well as
+the average number of quick and fast chargers installed. For each value of λ, we reported in bold
+the average value of each of the former statistics for the MP-CFL model, along with the average
+deviation from the latter value for the SP-CFL model. The last group of three columns provides
+some statistics about the solution of the SP-CFL model. In fact, the statistics refer to a modified
+solution obtained as follows. The deployed capacity remains unchanged. However, as the solution
+assigns the demand to open stations that may be overloaded in some peak periods of time, we
+modified the assignment of the demand to the stations with the goal of increasing the percentage
+of demand satisfied by the charging capacity deployed by the solution to the SP-CFL model.
+The procedure to modify the solution to the SP-CFL model iteratively considers one time period at
+a time, from 1 to T, and, for a given time period t, examines each demand node i, from 1 to I. The
+procedure checks whether the demand dt
+i of node i could be served in time period t according to
+the assignment indicated by the values of variables xijk. In this context, being served means that
+there is a number of vacant chargers k in station j greater than or equal to dt
+i · xijk.
+If such demand cannot be completely served, the unserved demand is reallocated among the vacant
+chargers of any type different from k available at the same station j, if any. Then, the procedure
+attempts to reallocate the remaining unserved demand among the other stations. If there are vacant
+chargers among multiple stations, priority is given to the one nearest to j. If at the station there
+21
+
+100
+75
+50
+25
+0
+25
+50
+75
+100
+125
+150
+175
+200
+T1
+T2
+T3
+T4
+T5
+T6
+T7
+T8
+T9
+T10
+T11
+T12
+T13
+T14
+T15
+T16
+T17
+T18
+T19
+T20
+T21
+T22
+T23
+T24Stations
+Quick
+Fast
+SP-CFL
+λ
+I
+MP-CFL
+SP-CFL
+MP-CFL
+SP-CFL
+MP-CFL
+SP-CFL
+Reall%
+Lost%
+Max Lost%
+0.0001
+50
+4.33
+2.67
+53.33
+33.00
+23.33
+15.00
+6.37%
+21.00%
+60.23%
+100
+8.20
+5.00
+101.07
+60.53
+45.87
+30.40
+9.82%
+21.07%
+58.33%
+150
+10.87
+7.53
+141.40
+91.93
+60.73
+45.93
+10.95%
+20.88%
+58.03%
+200
+14.64
+9.71
+186.00
+126.50
+94.14
+61.00
+10.41%
+21.10%
+58.60%
+250
+17.75
+12.00
+239.42
+153.93
+115.08
+77.36
+11.24%
+21.28%
+58.19%
+500
+31.25
+21.64
+418.56
+300.18
+198.78
+154.82
+13.24%
+21.82%
+55.89%
+Average
+12.90
+-28.93%
+171.01
+-30.32%
+80.46
+-25.87%
+10.19%
+21.16%
+58.32%
+0.25
+50
+4.80
+3.80
+55.47
+35.60
+22.80
+14.13
+12.23%
+21.15%
+55.90%
+100
+8.13
+5.33
+95.00
+61.73
+47.07
+29.93
+12.12%
+20.90%
+58.43%
+150
+11.07
+7.73
+137.00
+90.13
+68.57
+46.07
+11.66%
+20.98%
+57.90%
+200
+14.14
+9.79
+170.14
+120.64
+97.93
+62.14
+12.17%
+21.11%
+57.73%
+250
+17.50
+11.79
+227.58
+143.21
+117.75
+79.79
+12.54%
+21.35%
+54.93%
+500
+30.38
+21.27
+393.44
+287.27
+229.75
+159.00
+14.56%
+21.79%
+54.95%
+Average
+12.82
+-26.74%
+162.39
+-29.14%
+85.00
+-28.74%
+12.46%
+21.19%
+56.73%
+0.50
+50
+6.93
+6.60
+62.67
+36.60
+23.20
+14.40
+18.38%
+19.64%
+54.65%
+100
+8.67
+7.60
+97.73
+73.67
+46.20
+27.00
+14.02%
+20.56%
+62.05%
+150
+11.07
+8.80
+131.86
+97.87
+69.93
+44.53
+13.55%
+20.58%
+59.55%
+200
+14.21
+10.71
+166.71
+126.29
+98.86
+60.86
+13.93%
+20.82%
+58.03%
+250
+17.75
+12.50
+219.67
+150.29
+119.58
+78.00
+14.41%
+21.06%
+57.80%
+500
+30.88
+21.55
+441.50
+285.73
+230.50
+158.82
+16.00%
+21.69%
+54.80%
+Average
+13.44
+-19.64%
+163.51
+-26.20%
+85.68
+-30.81%
+14.86%
+20.68%
+57.95%
+0.75
+50
+12.13
+12.20
+78.60
+37.47
+31.07
+15.20
+26.82%
+17.09%
+51.54%
+100
+12.20
+11.40
+120.60
+70.27
+50.20
+28.53
+19.83%
+19.71%
+60.30%
+150
+13.64
+12.80
+152.93
+103.73
+73.14
+43.00
+18.00%
+20.06%
+60.58%
+200
+15.21
+13.50
+173.36
+137.07
+97.14
+58.14
+16.34%
+20.59%
+59.90%
+250
+18.00
+14.79
+223.25
+156.71
+120.58
+76.43
+16.13%
+20.88%
+58.69%
+500
+30.63
+21.91
+428.25
+285.36
+233.75
+157.91
+17.78%
+21.66%
+54.67%
+Average
+15.77
+-10.69%
+175.14
+-29.15%
+88.72
+-33.95%
+19.02%
+19.90%
+57.71%
+0.9999
+50
+21.20
+21.20
+96.87
+40.53
+41.27
+17.00
+31.11%
+12.33%
+47.73%
+100
+27.20
+27.20
+160.20
+74.67
+77.60
+31.40
+26.57%
+16.04%
+52.02%
+150
+29.71
+28.40
+212.93
+104.33
+113.86
+46.87
+24.51%
+16.62%
+57.10%
+200
+30.93
+30.07
+245.50
+141.43
+146.00
+61.86
+23.28%
+17.41%
+57.54%
+250
+33.58
+30.79
+276.00
+165.29
+158.29
+79.43
+22.05%
+17.92%
+57.05%
+500
+38.75
+34.55
+429.38
+285.45
+375.25
+165.27
+19.59%
+19.90%
+52.74%
+Average
+29.33
+-3.25%
+218.95
+-41.67%
+132.99
+-53.23%
+24.80%
+16.53%
+54.02%
+Table 2: MP-CFL vs. SP-CFL models: An analysis of the solutions found.
+22
+
+are vacant chargers of multiple types, priority is given to the type that has the largest number of
+vacant units. Let 0 ≤ ¯dt
+i ≤ dt
+i be the demand arising in node i at time period t that the procedure
+has reallocated. Eventually, the procedure computes the fraction of the total demand that has been
+reallocated, in percentage, producing statistic “Reall%”. For each instance, the fraction of the total
+demand reallocated is computed as 100 ·
+�
+t∈T
+�
+i∈I ¯dt
+i
+�
+t∈T
+�
+i∈I dt
+i .
+If, after the reallocation procedure, a portion of the demand is not served, the fraction of the total
+demand that remains unserved is computed, in percentage, producing statistic “Lost%”. For each
+instance, the latter is computed as 100 ·
+�
+t∈T
+�
+i∈I ˇdt
+i
+�
+t∈T
+�
+i∈I dt
+i , where 0 ≤ ˇdt
+i ≤ dt
+i is the demand arising in
+node i at time period t that is not served. Finally, the procedure computes statistic “Max Lost%”
+as the maximum fraction of demand not served across all time periods. The latter is computed for
+each instance as 100 · max
+t∈T
+� �
+i∈I ˇdt
+i
+�
+i∈I dt
+i
+�
+.
+From Table 2 we can gain the following insights:
+✓ the charging capacity deployed in the solutions to the SP-CFL model is significantly smaller
+than the capacity installed according to the MP-CFL model, both in terms of stations open
+and chargers installed (see statistics “Stations”, “Quick”, and “Fast”);
+✓ reducing the weight of the opening and installing costs (i.e., increasing the value of λ), the
+average deviation between the solutions to the two models in terms of stations open decreases
+steadily, from -28.93% for λ = 0.0001 to -3.25% for λ = 0.9999. Nevertheless, the average
+number of chargers installed (of each type) in the solutions found by the SP-CFL model is
+always remarkably smaller compared to those produced by the MP-CFL model;
+✓ given a value of λ, for both models the greater the number of demand nodes, the greater
+the number of stations open and chargers installed (see statistics “Stations”, “Quick”, and
+“Fast”);
+✓ the larger the value of λ, the larger the values of “Stations”, “Quick”, and “Fast”. This is an
+expected outcome, as more importance is given to the average distance term in the objective
+functions. In other words, the reduction of the average distance traveled can only be achieved
+by increasing the number of stations opened and chargers deployed;
+✓ due to the cheaper installation cost, both models install a larger number of quick compared
+to fast chargers.
+Before entering into the details of the statistics computed to measure the limits of the SP-CFL model,
+it is worth pointing out that in every solution found by the latter model, a part of the demand was
+reallocated, and a part was lost. The main insights we can gain from the three rightmost columns
+of Table 2 are the following:
+✓ “Reall%” takes, on average, large values. It ranges from 6.37% (see λ = 0.0001 and I = 50)
+to 31.11% (see λ = 0.9999 and I = 50);
+✓ “Lost%” takes, on average, large values as well. It ranges from 12.33% (see λ = 0.9999 and
+I = 50) to 21.82% (see λ = 0.0001 and I = 500);
+✓ “Max Lost%” takes, on average, extremely large values, always larger than 47%.
+23
+
+(a) Demand reallocated (“Reall%”)
+(b) Unserved demand (“Lost%”)
+(c) Maximum fraction of unserved demand across all time periods (“Max Lost%”)
+Figure 11: SP-CFL model: Box-and-wisker plots showing the distribution of the demand reallocated
+and lost (I = 200).
+24
+
+0.0001
+0.25
+入
+0.5
+0.75
+0.9999
+10
+15
+20
+25
+30
+%0.0001
+0.25
+Y
+0.5
+0.75
+0.9999
+16
+17
+18
+19
+20
+21
+22
+23
+%0.0001
+0.25
+入
+0.5
+0.75
+0.9999
+45
+50
+55
+60
+65
+%The statistics confirm that the SP-CFL model is not capable of capturing the characteristics of the
+problem and tends to underestimate the charging capacity to deploy.
+Further insights that can be obtained from Table 2 on the limits of the SP-CFL model are as follows:
+✓ for values of λ smaller than or equal to 0.25, the average value of “Reall%” tends to increase
+with the number of demand nodes;
+✓ for values of λ greater than or equal to 0.75, the average value of “Reall%” tends to decrease
+with the number of demand nodes;
+✓ the larger the value of λ, the larger the average value of “Reall%”, and the smaller tends to
+be the value of “Lost%”. This behavior can be explained by observing that increasing the
+value of λ, the number of stations opened and chargers installed increases as well, making it
+easier to find vacant chargers, and thereby reducing the unserved demand. of “Max Lost%”
+slightly decreases as the value of λ” increases.
+The box-and-whisker plots depicted in Figure 11 show the distribution of the values of the three
+statistics computed to determine the reallocated demand, and the unserved demand, for all the
+instances with I = 200 solved for different values of λ. The box-and-whisker plots confirm the
+insights previously drawn. When the main term in the objective function of the SP-CFL model is
+the opening and installing cost - i.e, for small values of λ - the charging capacity installed is small
+and little can be done to reallocate the unserved demand. As a consequence, large percentages of
+the charging demand are unserved. By increasing the weight given to the average distance traveled
+- i.e., for large values of λ - the charging capacity installed increases. Consequently, the percentage
+of the demand that can be reallocated increases, and, thereby, the percentage of the demand that
+is unserved becomes smaller. Nevertheless, the latter percentages always remain quite large (the
+values are always greater than 16%, and often greater than 20%). The performance is even worse
+if we analyze the distribution of “Max Lost%”. Its average value ranges from approximately 56%
+to roughly 60% (see the dotted lines inside the boxes). Similar conclusions can be drawn observing
+the results obtained when solving the instances with a different number of demand nodes. For the
+sake of readability, such results are not reported here.
+The results discussed above clearly show that the solutions found by the SP-CFL model, if imple-
+mented, would lead to a very poor quality of service provided to the EV drivers. Moreover, the
+results on demand reallocation imply that the objective function of SP-CFL would underestimate
+the total travel distance covered by the EV drivers to reach a free charging station.
+The following analysis is focused only on the solutions of the MP-CFL model. Figure 12 illustrates
+the distributions of the values of the average distance traveled (first term in the objective function)
+and the opening and installing cost (second term), for all the instances with I = 200 solved for
+different values of λ.
+From Figure 12, we can draw the following main insights:
+✓ as expected, the larger the value of λ, the smaller the average distance traveled by an EV to
+reach the assigned charger;
+✓ besides its largest value, increasing the value of λ produces only slight increases in the values
+of the total cost.
+25
+
+(a) Distribution of average distance traveled.
+(b) Distribution of total opening and installing costs.
+Figure 12: MP-CFL model: Box-and-wisker plots showing the distribution of the average distance
+and the cost (I = 200).
+In fact, we expected a sharper increase of the values of the total cost when less importance is given
+to the second term of the objective function. On the contrary, and neglecting the extreme case
+with λ = 0.9999, when the value of λ is increased the solutions obtained by the MP-CFL model
+significantly improved the average traveling distance, at the cost of only a small deterioration of
+the total opening and installing cost.
+We conclude our analysis by considering the computational burden required to solve each MILP
+model. Recall that each instance is solved with a time limit of 3,600 seconds. Table 3 summarizes
+the computational performance of the two MILP models. For each value of λ, the instances are
+clustered in groups according to the number of potential locations J. For each group of instances
+and each model, Table 3 provides the average CPU time (in seconds) spent to find the optimal (or
+best) solution (columns with header “CPU Time (secs.)”), the average optimality gap (“Gap%”)
+and the worst optimality gap (“Max Gap%”).
+The main insights that we can gain from Table 3 are as follows:
+✓ the solution to the MP-CFL model is, in general, more computationally expensive compared
+to the SP-CFL model (see the average values of “CPU Time (secs.)”, and the average values
+of “Gap%”);
+✓ the optimality gaps, for both models, are on average very small. In the majority of the cases,
+the solver found an optimal solution, or a solution very close to the optimum;
+✓ the worst gaps, for both models, are also very small. In only few instances, statistic “Max
+Gap%” took a value greater than 1%;
+✓ as expected, for a given value of λ, computing times for both models increase with the number
+of potential locations;
+26
+
+0.0001
+0.25
+入
+0.5
+0.75
+0.9999
+80
+100
+120
+140
+160
+180
+200
+2200.0001
+0.25
+入
+0.5
+0.75
+6666'0
+400
+500
+600
+700
+800
+900
+1000
+10kUSDCPU Time (secs.)
+Gap%
+Max Gap%
+λ
+J
+MP-CFL
+SP-CFL
+MP-CFL
+SP-CFL
+MP-CFL
+SP-CFL
+0.0001
+10
+1,456.48
+820.60
+0.13%
+0.09%
+0.45%
+0.76%
+20
+2,775.72
+2,404.48
+0.61%
+0.46%
+1.85%
+1.87%
+30
+3,308.19
+2,759.82
+1.03%
+1.08%
+2.90%
+3.69%
+40
+3,515.42
+3,142.17
+0.94%
+1.23%
+2.90%
+3.69%
+50
+3,513.11
+3,330.17
+1.08%
+1.43%
+2.90%
+3.69%
+Average
+3,037.11
+2,568.85
+0.81%
+0.90%
+0.25
+10
+660.83
+259.72
+0.03%
+0.01%
+0.17%
+0.11%
+20
+2,133.02
+922.11
+0.12%
+0.04%
+0.60%
+0.37%
+30
+2,923.03
+1,320.37
+0.31%
+0.13%
+0.92%
+0.82%
+40
+3,156.36
+1,790.20
+0.45%
+0.19%
+1.19%
+1.07%
+50
+3,406.24
+2,120.10
+0.55%
+0.18%
+1.77%
+0.84%
+Average
+2,614.44
+1,335.04
+0.32%
+0.11%
+0.50
+10
+263.90
+2.43
+0.00%
+0.00%
+0.00%
+0.00%
+20
+1,135.25
+338.01
+0.01%
+0.01%
+0.06%
+0.17%
+30
+2,544.90
+1,334.12
+0.19%
+0.07%
+0.67%
+0.66%
+40
+2,739.82
+2,051.00
+0.34%
+0.09%
+1.06%
+0.58%
+50
+2,978.22
+2,145.28
+0.97%
+0.10%
+6.85%
+0.49%
+Average
+2,094.62
+1,238.01
+0.34%
+0.06%
+0.75
+10
+6.45
+2.51
+0.00%
+0.00%
+0.00%
+0.00%
+20
+376.49
+1,183.04
+0.00%
+0.01%
+0.02%
+0.13%
+30
+1,678.43
+1,984.19
+0.04%
+0.05%
+0.41%
+0.22%
+40
+2,449.72
+2,298.07
+0.06%
+0.07%
+0.37%
+0.24%
+50
+2,926.84
+2,312.06
+0.40%
+0.06%
+4.28%
+0.22%
+Average
+1,648.46
+1,629.29
+0.11%
+0.04%
+0.9999
+10
+2.80
+1.97
+0.00%
+0.00%
+0.00%
+0.00%
+20
+966.18
+1,488.73
+0.00%
+0.00%
+0.00%
+0.00%
+30
+1,575.65
+2,581.28
+0.00%
+0.00%
+0.00%
+0.00%
+40
+1,864.13
+3,243.86
+0.00%
+0.00%
+0.00%
+0.00%
+50
+2,569.15
+2,957.01
+0.00%
+0.00%
+0.00%
+0.00%
+Average
+1,528.78
+2,152.78
+0.00%
+0.00%
+Table 3: MP-CFL vs. SP-CFL models: A summary of CPU times and optimality gaps.
+✓ the computational burden required to solve each MILP model decreases as the value of λ
+increases.
+In summary, while the MP-CFL model requires on average more computational time than the SP-CFL
+model, the additional time needed by the MP-CFL model is marginally small.
+6
+Conclusions
+In this paper, we studied the role of temporal and spatial distributions of charging demand in
+determining an optimal location of charging stations for electrical vehicles in an urban setting.
+This is an application context where the daily demand is well-known to be very dynamic and
+concentrated at some peak hours, and where the demand pattern is known to depend also upon
+the city zone.
+To highlight the need of considering explicitly the daily demand patterns, we presented a multi-
+27
+
+period optimization model, that captures the variability over time of the demand, and compared it
+with a single-period optimization model. By means of a worst-case analysis, we theoretically proved
+that the single-period model may produce solutions where a large portion of the demand cannot
+be served.
+Extensive computational experiments confirm the limits of the single-period model.
+The goal of the optimization models is to balance two objectives: the total cost of deploying the
+infrastructure and the average distance traveled by the customers to reach a charging station.
+The limits pointed out for the single-period model go beyond the specific application considered
+in this paper, and suggest the importance of incorporating time-dependency in location decisions
+when the demand fluctuations are remarkable during the planning horizon.
+Future developments of the research may concern the objective function. Since the two objectives
+considered are not homogeneous, a thorough analysis of the trade-off between infrastructure cost
+and drivers traveling distance may be of interest to a decision-maker. Moreover, we expect that
+replacing the average traveling distance with some equity measures may produce solutions that
+are more satisfactory for the customers, at the cost of a little increase in the infrastructure cost.
+Finally, to solve larger instances, in particular when an equity measure is considered, a heuristic
+approach would deserve to be studied.
+Acknowledgements
+This study has been made in the framework of the MoSoRe@UniBS (Infrastrutture e servizi per
+la Mobilit`a Sostenibile e Resiliente) Project 2020-2022 of Lombardy Region, Italy (Call-Hub ID
+1180965; bit.ly/2Xh2Nfr, https://ricerca2.unibs.it/?page id=8548).
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+

9NFQT4oBgHgl3EQf5jYf/content/tmp_files/2301.13435v1.pdf.txt

@@ -0,0 +1,2217 @@
+1 
+ 
+Exciton diffusion in amorphous organic semiconductors:  
+reducing simulation overheads with machine learning 
+ 
+Chayanit Wechwithayakhlung1,2, Geoffrey R. Weal3,4,2, Yu Kaneko5, Paul A. Hume3,4, 
+Justin M. Hodgkiss3,4, Daniel M. Packwood1,2* 
+ 
+1 Institute for Integrated Cell-Material Sciences (iCeMS), Kyoto University, Kyoto, Japan 
+ 
+2 Center for Integrated Data-Material Sciences (iDM), MacDiarmid Institute for Advanced 
+Materials and Nanotechnology, Wellington, New Zealand 
+ 
+3 MacDiarmid Institute for Advanced Materials and Nanotechnology, Wellington, New 
+Zealand 
+ 
+4 School of Chemical and Physical Sciences, Victoria University of Wellington, Wellington, 
+New Zealand 
+ 
+5 Daicel Corporate Research Center, Innovation Park (iPark), Daicel Corporation, Himeiji, 
+Japan 
+ 
+* Corresponding author. Email: dpackwood@icems.kyoto-u.ac.jp 
+ 
+Abstract 
+ 
+Simulations of exciton and charge hopping in amorphous organic materials involve 
+numerous physical parameters. Each of these parameters must be computed from costly 
+ab initio calculations before the simulation can commence, resulting in a significant 
+computational overhead for studying exciton diffusion, especially in large and complex 
+material datasets. While the idea of using machine learning to quickly predict these 
+parameters has been explored previously, typical machine learning models require long 
+training times which ultimately contribute to simulation overheads. In this paper, we 
+present a new machine learning architecture for building predictive models for 
+intermolecular exciton coupling parameters. Our architecture is designed in such a way 
+that the total training time is reduced compared to ordinary Gaussian process regression 
+or kernel ridge regression models. Based on this architecture, we build a predictive 
+model and use it to estimate the coupling parameters which enter into an exciton hopping 
+simulation in amorphous pentacene. We show that this hopping simulation is able to 
+achieve excellent predictions for exciton diffusion tensor elements and other properties 
+as compared to a simulation using coupling parameters computed entirely from density 
+functional theory. This result, along with the short training times afforded by our 
+architecture, therefore shows how machine learning can be used to reduce the high 
+computational overheads associated with exciton and charge diffusion simulations in 
+amorphous organic materials. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+2 
+ 
+1. Introduction 
+ 
+Efficient methods for simulating charge and energy transport in solid-state organic 
+materials will accelerate the development of novel organic electronics and energy 
+conversion devices. In organic materials, large reorganization energies tend to localize 
+electronic states, resulting in a transport mechanism in which charge or energy carriers 
+hop between molecules [1, 2]. Several methods for simulating carrier hopping exist, each 
+of which involve physical parameters such as electronic couplings that need to be 
+estimated beforehand via computationally intensive ab initio calculations. For the case 
+of amorphous organic materials, a considerable number of ab initio calculations are 
+typically required, as physical parameters need to be computed for each of the many 
+unique intermolecular interactions and local environments which exist in the disordered 
+system. This large number of calculations represents a huge computational overhead 
+which must be surmounted before charge or energy dynamics can be simulated in an 
+amorphous organic materials. 
+ 
+Excitons - electron-hole charge pairs that are bound together by Coulombic attraction - 
+are important energy carriers in solid-state organic materials [3, 4, 5]. The diffusion of 
+excitons between molecules is central to the function of organic semiconductor 
+technologies, including photovoltaics (solar cells) [3, 6, 7], light-emitting diodes (LEDs) 
+[8, 9, 10], lasers [11, 12], and photocatalysts [13 - 17]. For these technologies, the exciton 
+diffusion rate is an important figure-of-merit for assessing candidate materials. Indeed, 
+for the case of organic photovoltaics, high exciton diffusion rates are needed to ensure 
+that the exciton can reach the heterojunction before the exciton decays via electron-hole 
+recombination and other processes [3]. This effect in turn constrains the sizes of the 
+donor and acceptor domains. Similarly, hyperfluorescent LEDs rely on efficient exciton 
+diffusion in order to selectively funnel excitons to terminal emitter molecules [18, 19]. 
+While exciton diffusion rates obtained from hopping-type simulations have achieved 
+good agreement with spectroscopic measurements [20 - 22], the computational 
+overheads of these simulations must be addressed before they can facilitate the large-
+scale screening of materials for technological applications.  
+ 
+Over the last several years, machine learning has emerged as a powerful means of 
+reducing the demands of simulations in computational materials science [23, 24]. With 
+machine learning, one can bypass the resource-intensive parts of a simulation with 
+regression models trained on structure-property databases. For the case of exciton 
+diffusion in organic materials, one might imagine substituting machine-learned 
+regression models for the heavy ab initio calculations used to estimate exciton couplings 
+or other physical parameters which enter into a hopping model. These physical 
+parameters, which usually would be calculated one-by-one from first-principles, would 
+instead be obtained with negligible computational cost by feeding them through a simple 
+regression model.  
+ 
+Several groups have recently presented regression models for estimating intermolecular 
+coupling energies in a variety of organic materials as well as in related biological systems. 
+Lederer et al presented a kernel ridge regression (KRR) model trained to predict coupling 
+parameters for charge hopping in pentacene, and used the model predictions in a 
+subsequent kinetic Monte Carlo charge hopping simulation [25]. Similar works were 
+subsequently reported by Wang et al [26], Kramer et al [27], Farahvash et al [28], and 
+Gagliardi et al [29], which further demonstrated the ability of KRR-based models (and 
+
+3 
+ 
+their Bayesian generalizations, Gaussian process regression (GPR) models) for quickly 
+predicting coupling energy parameters in pentacene and other simple organic solids. In 
+addition to KRR or GPR-based models, neural network models for predicting such 
+coupling parameters have been presented by Farahvash et al [28], Wang [30], and Li et 
+al [31]. Related works for the case of biological systems have been reported by Maiti and 
+co-workers, in which neural network models were trained to predict coupling parameters 
+between DNA bases [32, 33], and by Cignoni et al, who presented a KRR model for 
+predicting exciton coupling energies between pigment molecules in a protein complex 
+[34]. The high prediction accuracies achieved by these models, as well as their 
+minuscule prediction times compared to ab initio calculations, point towards an optimistic 
+future in which exciton transport simulations could be performed with relatively small 
+computational overhead.  
+ 
+However, despite the success of the above studies, there is one important contribution 
+to the computational overhead of an exciton hopping simulation which has not been 
+addressed so far: the large training times required for typical machine learning models. 
+Indeed, for models based on GPR or KRR – two of the most popular methods in 
+computational materials science – model training times scale as N3, where N is the size 
+of the training data set. Most of the studies cited above used enormous amounts of 
+training data, ranging from a few thousand to a few hundred thousand instances, which 
+necessarily implies enormous training times. Large training times add to the 
+computational overhead of subsequent exciton diffusion simulations, reducing the overall 
+advantage of using machine-learned models over ab initio calculations. 
+ 
+In this paper, we present a new machine-learning architecture for predicting exciton 
+coupling parameters in amorphous organic semiconductors. For the training data sizes 
+considered here (N ~ 3000), our architecture allows one to build regression models within 
+around 25 % of the time required to build a standard GPR model. On the basis of a new 
+type of sensitivity analysis, we confirm that our constructed model can be interpreted in 
+terms of straightforward and consistent structural information, thereby supporting its 
+generality beyond the training data. Moreover, we show that this model is sufficiently 
+accurate for use in subsequent exciton diffusion simulations, in spite of the reduced 
+training times. We apply our architecture for the case of exciton transport in amorphous 
+pentacene, however it is not restricted to this particular material. This architecture can 
+also be applied to charge transport in organic materials. This work therefore provides a 
+means to reduce the computational overheads associated with hopping simulations of 
+exciton and charge dynamics.  
+ 
+This paper is organized as follows. In section 2 we describe our methods and our 
+machine-learning architecture for building predictive models of intermolecular exciton 
+coupling. Section 3 describes our results, including the outcome of our model sensitivity 
+analysis and the validation of our predicted coupling parameters in an exciton diffusion 
+simulation. Discussion and conclusions are left to section 4. 
+ 
+2. Method 
+ 
+2.1. Generation of amorphous pentacene system 
+ 
+We study exciton diffusion in the amorphous pentacene system shown in Figure 1A. The 
+amorphous pentacene configuration was generated by consecutive molecular dynamics 
+
+4 
+ 
+simulations as follows. First, 800 pentacene molecules were placed at random into a 100 
+x 100 x 100 Å box using the Packmol package [35]. An amorphous phase was then 
+generated in multiple steps. In the first step, a geometry optimization was performed. In 
+the second step, velocities were assigned to each atom by sampling from a Boltzmann 
+distribution at 600 K, and a sequence of 100 ps-long molecular dynamics simulations 
+were performed at 1000 atm and temperatures of 500, 400, 600, 700, 600, 500, 600, 
+600, 250, and 300 K, respectively. Another simulation at 1000 atm and 300 K was then 
+performed for 500 ps in order to induce molecule agglomeration. In the third step, a 
+sequence of 2 ns-long simulations were performed at 100 atm at temperatures of 300 K, 
+250 K, 300 K, and 200 K in order to relax the molecule geometries and local 
+environments around each molecule. In the fourth step, two 2 ns-long simulations were 
+performed at 1 atm and temperatures of 200 K and 250 K, respectively, in order to induce 
+the formation of amorphous pentacene. In the final step, the velocity vectors for each 
+molecule were examined to confirm the absence of high-energy molecules. Each 
+molecular dynamics simulation was performed using 2 fs time steps, and the simulation 
+box was allowed to relax in each case. Simulations were performed using the GAFF2 
+force field [36] and the GROMACS package [37]. Periodic boundary conditions were 
+applied throughout the entire procedure. Temperature and pressure regulation were 
+applied using the Nose-Hoover thermostat [38, 39] and Berendsen algorithm [40], 
+respectively. The amorphous pentacene configuration can be downloaded in CIF format 
+in Supporting Information 1.  
+ 
+Amorphous pentacene was selected as a model material because it is a representative 
+small molecule semiconductor with a flat aromatic structure typical of many high-mobility 
+cases. Pentacene also lacks the structural degrees of freedom arising from electronically 
+inert side chains, which facilitates the generation of realistic amorphous packing 
+structures. However, while our exciton coupling parameters and diffusion simulations 
+consider singlet excitons, in real thin films of crystalline pentacene triplet excitons also 
+play an important role when singlet excitons undergo singlet fission at particular sites 
+ 
+Figure 1. (A) Amorphous pentacene system considered in this study. Grey and white spheres 
+correspond to carbon and hydrogen atoms, respectively. The chemical structure of pentacene is shown 
+in the insert. (B) Summary of the machine-learning architecture for predicting exciton coupling energies 
+for pentacene dimers. SVM indicates the support vector machine component. GPR1 and GPR2 indicate 
+the two Gaussian process regression components. 
+
+B
+Dimer
+Feature Extraction
+SVM
+GPR1
+GPR2
+Prediction
+Prediction5 
+ 
+[41]. For the case of amorphous pentacene, singlet exciton fission rates should be 
+reduced due to the increased difficulty of reaching the single fission sites compared to 
+the crystalline case. Nonetheless, we emphasize that the amorphous pentacene model 
+used for these calculations is an ideal model system to test our machine-learning 
+architecture, because it embodies a situation in which thousands of electronic 
+configurations emerge from amorphous packing of a relatively simple molecular building 
+block. 
+ 
+2.2. Dimer extraction and density functional theory calculations 
+ 
+Pentacene dimers were extracted from the amorphous pentacene system using an in-
+house dimer extraction code. A pair of molecules was considered a dimer if any two 
+carbon atoms in the respective molecules were within 5.0 Å of each other. Dimers 
+involving molecules spanning across the cell boundary (due to periodic boundary 
+conditions) were included. In total, 4927 pentacene dimers were obtained. Structure files 
+for these dimers are provided in Supporting Information 2. Our extraction code was 
+written in Python 3 and used the Atomic Simulation Environment, NetworkX, and 
+Pymatgen packages [42 - 44]. 
+ 
+For a dimer composed of pentacene molecules a and b, the exciton coupling value is 
+defined as  
+ 
+ab
+b
+a
+v
+H
+
+
+
+, 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(1) 
+ 
+where a and b denote states in which a singlet excitation is localized on molecules a 
+and b, respectively, and H is the Hamiltonian operator for the dimer. For each dimer, the 
+exciton coupling energy was calculated using the electronic energy transfer (EET) 
+method as implemented in the Gaussian 16 software package [45]. In this method, the 
+exciton coupling energy is computed in a perturbative way by contracting the converged 
+transition densities of the isolated molecules via the linear response time-dependent 
+density functional theory (TDDFT) Hamiltonian of the dimer [46, 47]. Single-point excited 
+state energies were calculated using the CAM-B3LYP functional [48], along with the 6-
+311+G(2d,p) basis set. Integrals were evaluated with an ultrafine integration grid and the 
+accuracy of two-electron integrals set to 10-12. Linear response TDDFT calculations for 
+the lowest 10 excited states were performed within the Tamm–Dankoff approximation. 
+ 
+2.3. Machine learning architecture 
+ 
+A family of models for predicting exciton coupling values for molecular dimers was 
+constructed within a common machine learning architecture. This architecture consists 
+of the four components shown in Figure 1B, and each model differs in the settings (choice 
+of kernel functions, values of numerical parameters, etc.) which specify each component. 
+The four components are a feature extraction procedure, a support vector machine 
+(SVM), and two Gaussian procession regression estimators (GPR1 and GPR2).  
+ 
+In short, this architecture allows for reduced training times as follows. Rather than 
+constructing a single GPR component valid for all possible exciton coupling values, this 
+architecture uses two GPR components which are respectively trained to make 
+predictions in a ‘weak’ and a ‘strong’ coupling regime only. These two GPR components 
+
+6 
+ 
+can therefore be built using a smaller amount of training data than a single GPR 
+component which is valid for all cases. Moreover, because the time cost of building a 
+GPR estimator scales cubically (as N3, where N is the size of the training data), training 
+two GPR components with a small amount of training data can be less time consuming 
+than training a single component with a large amount of data. The SVM component 
+needs to be trained in addition to the two GPR components, however this can be 
+performed relatively quickly as explained below. 
+ 
+In order to select a model for predicting exciton coupling values, a two-stage procedure 
+was used. In the first stage, multiple models were built using a small set of training data 
+and a simplified training procedure. The results of this stage were then used in the 
+second stage, in which a final model was built using a larger amount of training data and 
+a rigorous training procedure.  
+ 
+In the following we describe the four components in detail. 
+ 
+Feature extraction. Feature extraction refers to the generation of a real-valued vector (a 
+feature vector) to describe a pentacene dimer. The feature extraction process in our 
+architecture involves three steps. Let x denote a pentacene dimer. In the first step, a 
+Coulomb matrix M(x) = [Mij(x)]n x n, where n = 72 is the number of atoms in the dimer, is 
+generated [49]. For a pair of atoms i and j contained in the same molecule, Mij(x) is set 
+to 0. If i and j are contained in different molecules, then Mij(x) is set to 1/Rij, where Rij is 
+the distance between i and j. In the second step, a vector W(x) is computed from the 
+matrix M(x). W(x) has length n, and is obtained by selecting the largest element of each 
+row of M(x) and sorting them in descending order. We write this vector as  
+ 
+ 
+ 
+ 
+
+
+1
+2
+( )
+,
+,...,
+n
+x
+W x W
+x
+W
+x
+
+W
+.  
+ 
+ 
+ 
+ 
+ 
+(2) 
+ 
+In the third step, principal component analysis is used to reduce the dimensions of W(x). 
+The resulting vector  
+ 
+ 
+ 
+ 
+ 
+
+
+1
+2
+,
+,
+,
+d
+n
+x
+U
+x U
+x
+U
+x
+
+U
+,  
+ 
+ 
+ 
+ 
+ 
+(3) 
+ 
+where Uk(x) is the kth principal component and nd < n, is the feature vector for dimer x.  
+ 
+In order to set the feature extraction component, only the parameter nd needs to be 
+specified. The models in this work used either nd = 4, 5, or 6. Larger values of nd are 
+unjustified, as the principal component analysis showed that over 99 % of the variation 
+in the data was accounted for by the first six principal components.  
+ 
+SVM. Support vector machines (SVM) are a type of classifier. In this work, they are used 
+to classify dimers as exhibiting either weak or strong exciton coupling values. Using the 
+predictions of the SVM, the set of extracted dimers can be divided into two sets exhibiting 
+weak and strong coupling, respectively. GPR1 and GPR2 can then be trained using data 
+sampled from the weak coupling and strong coupling sets, respectively. 
+ 
+Intuitively, a SVM performs classification by transforming the feature vector U(x) into a 
+high-dimensional space equipped with a linear hyperplane. This hyperplane is oriented 
+
+7 
+ 
+in such a way that dimers with weak and strong exciton coupling energies appear on 
+respective sides of the plane. Given the hyperplane, the classification for a dimer x can 
+be predicted according to the formula 
+ 
+ 
+
+
+
+
+
+
+1
+sign
+,
+,
+N
+i
+k
+k
+k
+k
+i
+k
+h x
+y
+y
+K x
+x
+K x
+x
+
+
+
+
+
+
+
+
+
+
+
+
+,  
+ 
+ 
+ 
+(4) 
+ 
+where yi is the classification of the ith dimer in the training data (equal to -1 or +1 for weak 
+and strong coupling cases, respectively), k is a coefficient which is set during the 
+construction of the hyperplane, and N is the size of the training data [50]. Note that 
+equation (4) holds for any choice of i. The function K is the so-called kernel function; in 
+essence, the transformation of the feature vectors takes place through K. 
+ 
+The SVM component requires specification of four settings. The first is the value of the 
+parameter vc, which is the value which defines which exciton couplings are classified as 
+weak and which are classified as strong. For a dimer x, the coupling energy vx is defined 
+as weak if |vx| < vc and as strong otherwise. For the models used in this work, vc was set 
+to 0.005 eV, 0.0075 eV, or 0.0082 eV. The second setting is the choice of kernel function. 
+For the models used in this work, linear, radial, third-order polynomial, and sigmoid 
+kernel functions were used (see reference [50] for their definitions). The third setting is 
+the value of the parameter used in the kernel function. Each of the kernel functions 
+involved here used a single parameter. The fourth setting is the value of the penalty 
+parameter for training points which locate on the wrong side the hyperplane.  
+ 
+Compared to the GPR components discussed next, the SVM component can be trained 
+relatively quickly even with a large amount of training data. For a specific choice of 
+settings, the time required for building the hyperplane scales as N, the size of the training 
+data [51]. In practice, however, the values of kernel parameter and cost parameter often 
+need to be optimized in order to obtain a hyperplane with sufficient accuracy.  
+ 
+GPR1 and GPR2. GPR1 and GPR2 are regression estimators which are trained to make 
+predictions for weak and strong exciton couplings, respectively. GPR1 and GPR2 are 
+trained using dimers which have been classified as weak- and strong-coupling cases, 
+respectively, by the SVM.  
+ 
+GPR1 and GPR2 are both constructed using the Gaussian process regression (GPR) 
+method. GPR involves constructing a probability density on the space of candidate 
+values for vx, where vx represents the exciton coupling for a dimer x [52]. This probability 
+density is given by 
+ 
+
+
+
+
+2
+2
+2
+1
+exp
+2
+2
+x
+x
+x
+x
+x
+v
+g v
+s
+s
+
+
+
+
+
+
+
+
+
+
+
+
+
+,  
+ 
+ 
+ 
+ 
+ 
+(5) 
+ 
+where g(vx) can be interpreted as the probability of the exciton coupling is equal to vx 
+given the training data. This probability maximizes at x, which can be deduced using a 
+so-called Bayesian procedure. Applying this procedure yields (see [53] for proof) 
+ 
+
+8 
+ 
+1
+x
+x
+
+
+ Σ Σ V .  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(6) 
+ 
+In equation (6),  = λIN’ + [σij]N’ x N’ , where λ is a positive parameter, IN’ is an N’ x N’ 
+identity matrix, N’ is the training data size and  
+ 
+
+
+,
+ij
+i
+j
+C x x
+ 
+  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(7) 
+ 
+is the covariance between xi and xj. Intuitively, C(xi, xj) measures the structural similarity 
+between training dimers xi and xj. The term λIN’ is included to ensure numerical stability 
+when computing the inverse of  and does not have a physical meaning. x is a 1 x N’ 
+row matrix of covariances between dimer x and the dimers in the training data, i.e., x = 
+(C(x,x1), C(x,x1), …, C(x,xN’)). V = (v1, v2, … ,vN’)T is a column matrix of exciton coupling 
+values from the training data. x in equation (6) can be interpreted as a weighted sum of 
+the coupling values in the training data, where the weights reflect the degree of structural 
+similarity between x and the dimers in the training data. The variance sx2 in equation (5) 
+is given by 
+ 
+
+
+2
+1
+,
+T
+x
+x
+x
+s
+C x x
+
+
+ Σ Σ Σ . 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(8) 
+ 
+For both GPR1 and GPR2, x is used as the predictor for the exciton coupling vx. For 
+both GPR1 and GPR2 in all models, we set C(xi, xj) equal to the squared exponential 
+function: 
+ 
+
+
+2
+0
+,
+ij
+d
+i
+j
+C x x
+b e
+
+
+, 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(9) 
+ 
+where 
+ 
+ 
+ 
+
+
+2
+2
+1
+d
+n
+ij
+k
+k
+i
+k
+j
+k
+d
+b U
+x
+U
+x
+
+
+
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(10) 
+ 
+and the constants b0, b1, …, bnd are parameters (known as hyperparameters in this 
+context) and nd is the feature dimension specified in the feature extraction component.  
+ 
+The GPR components are specified by setting the values of the parameters λ and 
+hyperparameters b0, b1, …, and bnd. In order to obtain a GPR component with sufficient 
+accuracy, the hyperparameters b0, b1, …, and bnd almost always need to be optimized in 
+some way. The time required for a single step of the optimization process scales as N’3, 
+where N’ is the size of the training data [54]. This scaling arises from need to invert the 
+covariance matrix when computing equation (6). Moreover, this optimization takes place 
+in a space of dimension 5 to 7 (one dimension for each of the nd + 1 hyperparameters) 
+and can therefore require very many iterations before convergence is reached. In order 
+to achieve short training times, it is important that GPR1 and GPR2 are trained using 
+small sets of training data. 
+ 
+Model selection. The models developed during the first stage of selection were trained 
+
+9 
+ 
+using a set of 100 pentacene dimers along with their DFT-calculated exciton coupling 
+energies. The same training data set was used to train each component. In order to 
+reduce training times and therefore compare a larger number of models, the kernel and 
+penalty parameters for the SVM component were set to 1/nd and 1, respectively, and 
+were not optimized. For training the subsequent GPR1 and GPR2 components, these 
+100 dimers were split into two subsets according to the predictions of the SVM 
+component. GPR1 (GPR2) was then built by using 80 % of the weak coupling (strong 
+coupling) subset as training data and the remaining 20 % as testing data. 
+Hyperparameters were optimized with respect to the mean-square error (MSE) of the 
+predictions with respect to the testing data. A full list of models tested during the first 
+stage of selection are listed in Supporting Information 3. 
+ 
+For the second stage of selection, we trained a single model based on the results 
+obtained above. The SVM, GPR1, and GPR2 components were each built using an 
+independently selected sample of 1000 dimers. For each component, 800 dimers were 
+used for training and 200 used as testing data. SVM kernel and penalty parameters were 
+optimized with respect to the SVM classification fail rate compared to a set of test data. 
+This optimization was performed using a grid search. The hyperparameters of the GPR 
+components were optimized as described above.  
+ 
+All calculations related to model selection were performed in the R environment using 
+custom code [55]. The e1071 package was used to fit the SVM components [56]. The L-
+BFGS-B gradient optimizer as implemented in R was used to optimize GPR 
+hyperparameters [57].  
+ 
+2.4. Kinetic Monte Carlo (kMC) 
+ 
+Kinetic Monte Carlo (kMC) is a method for simulating diffusion dynamics. This method 
+treats the exciton diffusion process as a random hopping process over a network of 
+molecules. In our kMC simulations, each molecule corresponds to one of the pentacene 
+molecules from the amorphous pentacene system in Figure 1A. Hopping takes place 
+within the dimers extracted above, and for each dimer the hopping rate constant is a pre-
+defined constant. The rate constant for hopping from molecule a to b was defined 
+according to Marcus theory [58, 59]: 
+ 
+
+
+
+
+2
+2
+1 2
+2
+exp
+4
+4
+ab
+ab
+r
+ab
+r
+B
+r
+B
+v
+E
+k
+k T
+k T
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+,  
+ 
+ 
+ 
+ 
+(11) 
+ 
+where vab is the exciton coupling between molecules a and b, ħ is the reduced Planck 
+constant, kB is the Boltzmann constant, T is temperature, ΔEab is the energy difference 
+between molecule a and molecule b, and λr is the reorganization energy associated with 
+excitation energy transfer. In most simulations of exciton diffusion ΔEab is treated as a 
+random variable sampled from a Gaussian distribution with mean zero [60]. In this work, 
+we set ΔEab to zero and neglect energetic disorder, thereby allowing us to focus on 
+comparing simulations using DFT-calculated excitonic couplings with those predicted 
+from our model. Likewise, λr was kept constant for all molecules, i.e. the effect of different 
+local environments on the reorganization energy is neglected. Note that kab is only non-
+zero for dimers extracted from our dimer extraction code (section 2.2). Dimers not 
+
+10 
+ 
+extracted from our code were considered far apart such that their coupling was negligible. 
+ 
+Our kMC simulations were initialized at time t = 0 by randomly placing the exciton on one 
+of the 800 pentacene molecules from the amorphous pentacene system. Denoting this 
+molecule as a, a residence time was calculated according to 
+ 
+~
+ln
+a
+ad
+d
+a
+z
+k
+
+
+  
+, 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(12) 
+ 
+where z is a random number sampled between 0 and 1 and d ~ a denotes all molecules 
+neighboring molecule a. The simulation time was advanced by Δτa and the exciton was 
+shifted to a neighboring molecule b with probability 
+ 
+~
+ab
+ab
+ad
+d a
+k
+p
+k
+ 
+.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(13) 
+ 
+The above process was iterated for molecule b, and so on, until the end of the simulation. 
+ 
+Periodic boundary conditions were applied so that the amorphous pentacene system 
+was repeated indefinitely in all directions. The reorganization energy of a pentacene 
+molecule was obtained as [61] 
+ 
+
+ 
+
+*
+*
+r
+g
+e
+g
+e
+E
+E
+E
+E
+ 
+
+
+
+,  
+ 
+ 
+ 
+ 
+ 
+ 
+(14) 
+ 
+where E* and E denote the energies of the excited state and ground state, respectively, 
+and subscripts g and e denote optimized geometries in the ground and existed states, 
+respectively. These energies were calculated using the CAM-B3LYP functional and 6-
+311+G(2d,p) basis set, resulting in a value of λr = 340.6 meV.  
+ 
+All simulations were run for a simulation time of 1 ns. kMC simulations were repeated 
+10,000 times independently in order to compute probability distributions and statistics 
+related to exciton diffusion. 
+ 
+3. Results 
+ 
+3.1. Comparison of candidate models (selection stage 1) 
+ 
+The results of the first stage of our model selection procedure are summarized in Figure 
+2A - E. Each point corresponds to one distinct choice of settings for the Feature 
+Extraction, SVM, and initial hyperparameters (before optimization) for the two GPR 
+components (a full list of model parameter values is provided in Supporting Information 
+3). Each point is presented as a pair of adjacent squares, where the color of the left- 
+(right-) hand square corresponds to the mean-square error (MSE) of the GPR1 (GPR2) 
+component when tested against test data. The lower the value of the mean-square error 
+(MSE) (i.e., the more blue the point in Figures 2A – E), the more accurately the GPR 
+components can predict the value of excitonic coupling. The points are positioned  
+
+11 
+ 
+t  
+Figure 2. (A – E). Results of the first stage of model selection. Each point corresponds to one choice of 
+settings for the Feature Extraction, SVM, and the two GPR components. The points are presented as 
+adjacent squares, where the left (right)-hand square is colored according to the mean-square error 
+(MSE) of GPR1 (GPR2) compared to test data. In each plot, we highlight the effect of certain settings 
+by increasing the size of the corresponding points. The models which achieved the lowest MSE for 
+GPR1 (GPR2) are indicated by the dotted (solid) circle (F) Performance of a selected model. This model 
+used nd = 5, a polynomial kernel for the SVM component, vc = 0.0075 eV, and GPR1 (GPR2) covariance 
+hyperparameters initialized at 0.1 (0.01). See text for details.  
+
+007
+2
+t-SNE dimension
+log(MSE)
+Polynomial
+nd = 5
+kernel
+200
+100
+100
+200
+300
+200
+100
+100
+200
+300
+Initial hyperparameters
+V.=0.0075 eV
+:0.01
+-200
+100
+100
+200
+006
+200
+-100
+100
+200
+300
+0.03
+Model prediction (eV)
+0
+0.01
+Initial hyperparameters
+.00
+= 0.1
+200
+-100
+100
+200
+300
+0.00
+0.01
+0.02
+0.03
+t-SNE dimension 1
+Testdata(ey)12 
+ 
+according to the dissimilarity of the model settings using the t-distributed stochastic 
+neighbor embedding (t-SNE) technique [62]. Here, the dissimilarity between models i 
+and j is defined as 
+ 
+,
+,
+,
+,
+i
+j
+d
+d
+c
+c
+i
+j
+i
+j
+i
+j
+ij
+K K
+n
+n
+v
+v
+h h
+D
+
+
+
+
+
+
+
+
+, 
+ 
+ 
+ 
+ 
+ 
+ 
+(15) 
+ 
+where δrs is the Kronecker delta, and Ki, ndi, vci, and hi refer to the SVM kernel type, 
+feature dimension, coupling strength cut-off (vc), and initial GPR hyperparameter values 
+used for the gradient optimizer, respectively, for model i.  
+ 
+In Figures 2A – C, we can identify the common features of the Feature Extraction and 
+SVM components of the low MSE models: nd = 5, vc = 0.0075 eV, and third-order 
+polynomial kernels for the SVM component. Models using these settings can be 
+identified by the enlarged points. For the final model, the Feature Extraction and SVM 
+compounds should therefore be built using these settings.  
+ 
+For the purpose of building a final model, it is more useful to compare the initial values 
+used for hyperparameter optimization rather than the final optimized hyperparameter 
+values directly. This is because the hyperparameters will need to be optimized again 
+when dealing with a new set of training data. According to Figures 2D – E, the 
+hyperparameters for GPR1 and GPR2 should be initialized with different values in order 
+to achieve good performance. In particular, the hyperparameters of GPR1 should be 
+initialized at 0.1 and those of GPR2 should be initialized at 0.01 in order for the gradient 
+optimizer to reach a good set of final values. In Figures 2A – E, two models that initialized 
+GPR1 and GPR2 in this way are indicated by the dotted and solid circles. For these two 
+models, we also find that λ = 0.0001 and 0.001, respectively. This indicates that the GPR 
+components should also be built using different values of λ. 
+ 
+The results above therefore suggest that a good model could be built using the following 
+settings: nd = 5 for the Feature Extraction component; vc = 0.0075 eV and third-order 
+polynomial kernels for the SVM component; λ = 0.0001 and initial hyperparameter values 
+of 0.1 for the GPR1 component; and λ = 0.001 and initial hyperparameter values of 0.01 
+for the GPR2 component. Figure 4F shows the predictive performance of such a model. 
+The agreement between predicted and DFT-calculated exciton couplings is impressive, 
+however due to the small size of the training and test sets, such agreement is unlikely to 
+hold for all dimers in the amorphous pentacene system. 
+ 
+3.2. Model interpretation using sensitivity analysis 
+ 
+We now investigate whether the high-performing model in Figure 2C can be interpreted 
+in a physically meaningful way. To this end we proceed in two steps. 
+ 
+In the first step, we plot the vectorized Coulomb matrices W(x) (section 2.3 equation (2)) 
+for each of the 100 dimers in the training data. This plot is shown in Figure 3A as a 100 
+x 72 matrix. The cell in row i and column r corresponds to Wr(xi), the rth element of the 
+vector for dimer xi. The cell is red if the Coulomb interaction in Vr(xi) is between two H 
+atoms, gray if between an H and a C atom, and blue if between two C atoms. The left 
+and right edges of the matrix in Figure 3A are clearly dominated by interactions involving 
+hydrogen atoms. Indeed, interactions involving carbon atoms do not appear on the left-
+
+13 
+ 
+hand side of the plot at all until column r = 10. For the columns r = 1, 2, and 3, interactions 
+between hydrogen atoms alone are observed for 64 %, 64 %, and 53 % of the dimers, 
+respectively. Similarly, for column r = 72 on the right-hand edge of the plot, 11 % of the 
+interactions are between carbon and hydrogen, and 89 % between pairs of hydrogen 
+atoms. 
+ 
+In the second step, we determine which Coulomb interactions GPR1 and GPR2 are most 
+sensitive to. To do this, we expand the features Uk(xi) as 
+ 
+ 
+ 
+1
+n
+k
+i
+kr
+r
+i
+r
+U
+x
+c W
+x
+
+
+, 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(16) 
+ 
+where ckr are expansion coefficients. Equation (16) follows from the definition of principal 
+components. Inserting equation (16) into equation (10) gives 
+ 
+ 
+
+
+
+
+2
+2
+1
+1
+1
+d
+n
+n
+ij
+kr
+r
+i
+r
+j
+k
+r
+k
+d
+a
+W
+x
+W
+x
+b
+
+
+
+
+
+
+
+
+
+
+
+
+, 
+ 
+ 
+ 
+ 
+ 
+(17) 
+ 
+where the constants akr = bkckr are referred to as sensitivities. Large akr means that dij is 
+sensitive to the difference Wr(xi) – Wr(xj). This sensitivity is transmitted to the predicted 
+exciton coupling through the covariances (see equations (6) and (9)). In Figure 3B, we 
+ 
+Figure 3. Sensitivity analysis of the selected high-performing model. (A) Plot of the sorted vectorized 
+Coulomb matrices for the dimers in the training set. Rows correspond to one of the dimers, and columns 
+correspond to the element of the vectorized Coulomb matrix. Colors correspond to the type of Coulomb 
+interaction. (B) Plot of the sensitivities of each of the model input features. Rows correspond to input 
+features and columns to the element of vectorized Coulomb matrix. The top plot is for the GPR1 
+component and the bottom one for the GPR2 component. Sensitivities are normalized by the maximum 
+and minimum values in each table. See text for details. (C) The five dimers from our training set which 
+exhibit the strongest exciton coupling. The pairs of atoms involved in the largest Coulomb interactions 
+are indicated by the colored circles. 
+
+C-C
+C-H
+H-H
+B
+GPR1
+Feature index (k)
+2
+34
+Tot
+Sensitivity
+Dimer
+Coulomb interaction index (r)
+72
+GPR2
+Feature index (k)
+2
+n
+Tot
+100
+Coulombinteractionindex(r)
+72
+Coulomb interaction index ()
+7214 
+ 
+plot the sensitivities akr as matrices for the components GPR1 (top) and GPR2 (bottom) 
+as obtained from the high-performing model constructed at the end of the previous 
+section. In both plots, the largest sensitivities tend to be found for small and large values 
+of the index r. As shown in the previous paragraph, these values of r are mainly 
+associated with hydrogen atoms. 
+   
+The predictions of GPR1 and GPR2 are therefore sensitive to changes in the distances 
+between hydrogen atoms of the two molecules. This shows that the high-performing 
+model makes its predictions on the basis of a low-dimensional structural representation 
+consisting of hydrogen atoms. These hydrogen atoms therefore act as a proxy for 
+describing the alignment of the two pentacene molecules in the dimer. Figure 3C shows 
+the five dimers from our training set that exhibit the strongest exciton coupling. The atoms 
+involved in the three largest Coulomb interactions are indicated. Each of these 
+interactions involves a hydrogen atom, which shows that hydrogen atom positions are 
+important for predicting coupling in the strong-coupling regime.  
+  
+The fact that this model has a straightforward physical interpretation suggests that its 
+performance is based on a general structure-property relationship extracted from the 
+training data. In other words, it suggests that the model accuracy is not based on 
+‘overfitting’ of the hyperparameters to the training data, and that the model can be 
+meaningfully applied to dimers outside of the training data. 
+ 
+3.3. Final model training times and performance (selection stage 2)  
+ 
+Our final model for predicting exciton coupling was built using the settings shown in 
+Supporting Information 3. These settings are identical to those of the high-performing 
+models found from the first stage of selection, however the optimized values of the 
+hyperparameters for GPR1 and GPR2 differ slightly due to the different training set used 
+here. The parameters for the SVM component also differ slightly from the models used 
+in the first stage of selection due to the use of the optimization procedure.  
+ 
+Training times were evaluated by running our script within the R console in a Kubuntu 
+environment running on an Intel Xeon 3.5 GHz processor. This entire script includes the 
+whole gamut of the calculation, from processing the structure files of the dimers, 
+generating the feature vectors, optimizing the SVM parameters, and optimizing the 
+hyperparameters of GPR1 and GPR2. The script runs these parts sequentially on a 
+single processor. This script required 19.0 hours to complete its run. In fact, this training 
+time could be reduced to around 9.5 hours if the script were parallelized so that GPR1 
+and GPR2 were trained on different processors, as the other parts of the script require 
+negligible time to complete. In order to compare this training time, we considered the 
+case of a single GPR model initialized with the same hyperparameter settings as GPR1 
+and using 2400 dimers for training and 600 for testing. This case therefore compares 
+training times afforded by our machine learning architecture to those of an architecture 
+consisting of a single GPR component, while holding the total amount of training data 
+constant (recall that in the final model, the three components SVM, GPR1, and GPR2 
+were each trained using independently sampled sets of 800 training data points and 200 
+test data points). This script required 76.4 hours to complete its run. The training time of 
+19 hours for our final mode therefore corresponds to 25 % of the training time for the 
+case of a single GPR model using the same amount of training data (or around 12.4 % 
+if GPR1 and GPR2 were trained in parallel). This reduction can be traced to the fact that 
+
+15 
+ 
+GPR training times scale as N3, where N is the size of the training data set. 
+ 
+When discussing training times, it is important to emphasize that GPR1 and GPR2 were 
+implemented with a more general Gaussian process framework than the GPR models 
+reported in previous papers. As is shown by equation (10), our GPR components contain 
+a different hyperparameter per feature dimension. In contrast, previous works have used 
+the more common GPR implementation in which equation (10) only uses one 
+hyperparameter (that is, b1 = ··· = bd = b) [25 - 29]. While our implementation of GPR is 
+advantageous when trying to model complex functions such as exciton coupling, model 
+training (hyperparameter optimization) necessarily takes places in a higher dimensional 
+space. Our total training times are therefore not directly comparable to the training times 
+of other GPR models reported so far. However, the N3 scaling of the time required per 
+iteration is independent of feature dimension, meaning that our observations of the faster 
+training times afforded by our architecture will hold for other types of GPR 
+implementations as well. 
+ 
+In Figure 4A, predictions of the final model are compared to the TDDFT-calculated 
+couplings for all extracted pentacene dimers. While not perfect, the correlation between 
+the predictions and DFT-calculation couplings is satisfactory: linear regression on the 
+data in Figure 4A yields the result y = (0.0024  0.0001) + (0.80  0.006)x, where errors 
+refer to one standard error, and R2 = 0.77. In Supporting Information 4 we also report the 
+results of a sensitivity analysis for the final model, which are similar to the ones reported 
+for the model in the previous section. The final model is therefore also physically 
+interpretable, suggesting that its predictions are based on a genuine structure-property 
+relationship extracted from the training data.  
+ 
+In Supporting Information 5, we compare the predictions of the final model above with 
+those of a single GPR model constructed using 800 dimers for training and 200 for testing. 
+We are therefore comparing the accuracy achieved by our machine learning architecture 
+with that of an architecture consisting of a single GPR component, while holding the total 
+training time roughly constant. While both models achieve similar accuracy in the weak 
+coupling regime, the single GPR model significantly underestimates coupling values in 
+the strong coupling regime. This would be a serious concern if one wished to use such 
+predictions in a subsequent exciton diffusion simulation, as the strongly coupled dimers 
+are expected to have a decisive influence on diffusion dynamics. The inability of the 
+single GPR model to make accurate predictions in the strongly coupled regime is 
+unsurprising, because in a random sample of dimers weak coupling cases will be much 
+more frequent than strong coupling cases, and the former will exert an oversized 
+influence during model training. This problem is avoided by our architecture, which builds 
+two GPR components trained specifically for the respective regimes.  
+ 
+3.4. Kinetic Monte Carlo simulation 
+ 
+The exciton coupling parameters predicted from the final model above were used as 
+inputs in the kinetic Monte Carlo (kMC) simulation. For comparison, a ‘benchmark’ 
+simulation using coupling parameters computed entirely from TDDFT was also 
+performed. Figure 4B plots the mean-square displacement (MSD) of the exciton, as 
+computed according to the formula 
+ 
+
+16 
+ 
+
+
+2
+0
+( )
+t
+MSD t 
+
+r
+r
+  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(18)  
+ 
+where rt denotes the position of the exciton at time t and the angular brackets indicate 
+averaging over the simulation runs. The mean-square displacement for the simulation 
+performed using the predicted exciton couplings compares well to the benchmark 
+simulation in both magnitude and time evolution. The exciton diffusion coefficient, which 
+was computed according to 
+ 
+ 
+1 lim
+6 t
+MSD t
+D
+t
+
+
+,  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(19) 
+ 
+was (1.547  0.005) x 10-3 cm2 s-1, which compares well to the value of (1.630  0.02) x 
+10-3 cm2 s-1 obtained from the benchmark calculation (see Table 1).  
+ 
+Figure 4C plots the probability distribution of the exciton position at different points in 
+time, as projected onto the xy plane. Good agreement with the benchmark results is 
+obtained: not only does the spread in the probability distributions look comparable, but 
+the shape of the distribution is preserved. In order to compare the probability distributions 
+in a quantitatively rigorous way, we calculate the eigenvalues of the diffusion tensor. 
+These three eigenvalues correspond to the three dominant directions along which the 
+probability distribution spreads, and their magnitudes correspond to the rate of spread in 
+these directions. For the case of the simulations performed using predicted exciton 
+couplings, these eigenvalues were computed to be (1.690  0.020), (1.492  0.007), and 
+(1.460  0.020) x 10-3 cm2 s-1, respectively (see Table 1). These values compare to 
+(1.820  0.02), (1.550  0.02), and (1.530  0.02) x 10-3 cm2 s-1, respectively, as obtained 
+from the benchmark simulation. The eigenvalues for the case of predicted couplings are 
+ 
+Figure 4. Final model performance and exciton diffusion simulations. (A) Predicted exciton coupling 
+energies of the final model (vML) compared to ab initio calculations (vDFT) for all extracted dimers. (B) 
+Mean-square displacement of an exciton from its initial position as obtained from kinetic Monte Carlo 
+simulations using model-predicted exciton coupling energies (orange lines) and ab initio-calculated 
+couplings (blue lines). (C) Snapshots of the probability distribution of exciton positions at various times 
+for the case of model-predicted and ab initio-predicted exciton couplings. For clarity, the probability 
+distribution is plotted in the xy plane only. The red box indicates the dimensions of the simulation cell. 
+ 
+ 
+
+A
+B
+300
+90
+DFT couplings
+ dashed)
+120
+ML couplings
+(pllos
+200
+60
+4
+100
+A
+(×103 )
+100
+30
+%
+80
+0
+0
+(meV)
+0
+200
+400
+600
+800
+1000
+time (ps)
+60
+c
+DFT couplings
+400 A
+40-
+200 ps
+400 ps
+600 ps
+800 ps
+1000 ps
+20
+ML couplings
+400A
+0
+0
+20
+40
+60
+80
+100
+120
+IvDFTI (meV)
+200 ps
+400 ps
+600 ps
+800 ps
+1000 ps17 
+ 
+of a slightly smaller magnitude than those for the benchmark simulation, suggesting that 
+the exciton coupling is underestimated by our model for the most strongly coupled dimers 
+in the system. However, the ratios of the eigenvalues relative to the smallest one were 
+quite similar (1.15:1.02:1.00 for the case of predicted couplings and 1.19:1.02:1.00 for 
+the case of the benchmark), confirming that the simulation using predicted couplings 
+correctly preserved the anisotropic nature of the exciton diffusion. The individual 
+elements of the diffusion tensor are compared in Supporting Information 6, where a 
+similar agreement with the benchmark case is observed: comparable but slightly smaller 
+magnitudes, but very similar relative values as expressed by ratios. 
+ 
+In Supporting Information 6, we consider the case of a model built using a much larger 
+training set of 2000 dimers for GPR1 and 2200 dimers for GPR2. KMC simulations using 
+the predictions of this model achieved very similar results to the ones obtained from the 
+model above. Interestingly, the magnitudes of the diffusion tensor eigenvalues and 
+elements remain slightly underestimated for this case as well. We will return to this point 
+in the next section. The agreement between the predictions of kMC using this model and 
+the model described above suggests that the latter has converged, in some sense, with 
+respect to training data size. The fact that this convergence can be reached even with 
+small sets of training data may relate to the observations in section 3.2, which suggested 
+that our models have a meaningful physical interpretation hence should generalize well 
+beyond the small training and test data sets.  
+ 
+4. Discussion and conclusions 
+ 
+In order for machine learning to accelerate simulations for organic photovoltaics and 
+other types of materials, it is important that the predictive models for simulation 
+parameters can be built with short training times. Long training times may offset the 
+reductions in computational time achieved through the final machine-learned model. In 
+this paper, we presented a new machine learning architecture for predicting 
+intermolecular exciton coupling values in amorphous organic solids. For the training set 
+sizes used here (around 3000), our architecture allowed for predictive models to be 
+trained within around 25 % of the time required to train a typical Gaussian process 
+regression model (or, equivalently, a typical kernel ridge regression model). Importantly, 
+ 
+Ab initio 
+ couplings 
+Model-predicted 
+couplings 
+Diffusion Coefficient  
+(× 10-3 cm2s-1) 
+1.630 ± 0.011 
+1.547 ± 0.005 
+ 
+Diffusion tensor eigenvalues 
+(× 10-3 cm2s-1) 
+ 
+Major 
+1.815 ± 0.014 
+1.686 ± 0.017 
+Middle 
+1.551 ± 0.016 
+1.492 ± 0.007 
+Minor 
+1.525 ± 0.012 
+1.462 ± 0.014 
+ 
+  
+ 
+Major: Middle: Minor 
+1.19: 1.02: 1.00 
+1.15: 1.02: 1.00 
+ 
+Table 1. Exciton diffusion parameters estimated from kinetic Monte Carlo (kMC) simulations using ab 
+initio-calculated and model-predicted exciton coupling energies. Error bounds correspond to one 
+standard deviation. 
+
+18 
+ 
+when these predicted coupling energies were used as inputs in a subsequent exciton 
+diffusion simulation, we obtained results which were in excellent agreement with a 
+benchmark simulation, in spite of the reduced training time of the underlying model. Thus, 
+for the purpose of predicting model parameters for subsequent exciton diffusion 
+simulations, highly accurate machine-learned models built with massive sets of training 
+data and long training times are not necessarily required. We partly attribute the accuracy 
+of our model to the results of our sensitivity analysis, which suggest that the model can 
+be interpreted in terms of hydrogen atom positions. This straightforward physical 
+interpretation suggests that the model’s predictions are based on a genuine structure-
+property relationship acquired from the data, rather than the result of over-training on the 
+small training data set.  
+ 
+The accuracy of the model in spite of the reduced computational times might also be 
+attributed to the special machine-learning architecture used to generate it. Instead of 
+attempting the difficult task of fitting a single regression model valid for all possible 
+molecule dimers, this architecture separates the possibilities into a ‘weak coupling’ and 
+a ‘strong coupling’ regimes and builds regression estimators for these regimes 
+separately. A similar strategy was used in the context of dimer interaction energies in 
+reference [63]. While this model was built for the special but illustrative case of 
+amorphous pentacene, the underlying machine-learning architecture is not restricted to 
+this case and could be applied to other types of amorphous organic solids, including 
+ones composed of large flexible molecules, or ones involving multiple molecular species. 
+ 
+While kinetic Monte Carlo (kMC) simulations using model-predicted exciton coupling 
+parameters achieved excellent agreement with a benchmark simulation, the magnitudes 
+of the diffusion tensor elements and eigenvalues were slightly underestimated. This 
+underestimation was observed both for the model described above and for another 
+model built using a much larger set of training data. For practical purposes such as 
+comparing candidate materials for organic photovoltaics, such a shortcoming is not 
+expected to be an issue. However, it suggests that multiple examples of dimers with very 
+high exciton couplings might need to be included in the training data in order to obtain 
+very accurate predictions of the diffusion tensor elements. On the one hand, these high-
+coupling cases may be more important in determining the magnitude of the diffusion 
+tensor elements than the weak coupling cases. On the other hand, such cases are very 
+rare, and might not have been included with sufficient frequency in the training sets used 
+to build our models. These points should be clarified further in the next stage of this 
+research. Concretely, additional theoretical work should be performed to determine what 
+types of dimers are needed to obtain accurate predictions for the diffusion tensor 
+elements, and novel sampling schemes for generating training sets of dimers developed 
+accordingly. Theoretical work should also be performed to derive uncertainty bounds for 
+diffusion tensor elements and other quantities in terms of training set sizes. The machine 
+learning literature contains numerous uncertainty bounds for various types of regression 
+and classification models [50], however these are expressed in terms of highly general 
+concepts such as hypothesis space complexity and are difficult to apply to the case of 
+exciton diffusion directly.  
+ 
+For applications in chemistry and physics, it is desirable that machine-learned models 
+can be interpreted in terms of straightforward structural information. Such physical 
+interpretations help clarify the structural motifs the model uses to make its predictions (in 
+this case, relative positions of the hydrogen atoms in the two molecules of the dimer). In 
+
+19 
+ 
+this paper, we employed a sensitivity analysis to probe for a physical interpretation of the 
+GPR components in our models. This amounts to performing a post-hoc check on model 
+after it is constructed. An alternative strategy to guarantee model interpretability is to 
+design the regression model by deliberately including some aspects of the excitonic 
+transfer mechanism and dimer atomic structure in the model’s structure. This is difficult 
+to do with the SVM or GPR models used in this paper, as much of the model’s internal 
+structure is hidden from view by the kernel and covariance functions. However, outside 
+of the field of organic semiconductors, several groups have proposed neural network 
+models which incorporate known physical relationships in their architectures (e.g., [64]). 
+Compared to the sensitivity analysis approach, such models have some disadvantages. 
+For example, the decision as to what physics should be incorporated into the model is 
+subjective. Moreover, these models may acquire additional physical information during 
+the training procedure. Ironically, such information could only be identified by performing 
+a sensitivity analysis on the model following training. Finally, these models are more 
+difficult to train due to the presence of additional parameters describing the incorporated 
+physics. Indeed, if one insists that these additional parameters only take values within a 
+physically sensible range, then it may become difficult to minimize model prediction 
+errors sufficiently during training. Until these problems are overcome, post-hoc sensitivity 
+analyses are probably the most practical means to ensuring that a machine-learned 
+model is physically interpretable.  
+ 
+It is also important to consider the extent to which machine learning can be applied to 
+simulate exciton diffusion in amorphous organic solids. In this work, a machine-learned 
+model was substituted for the resource-heavy time-dependent density functional theory 
+(TDDFT) method for computing exciton coupling parameters. However, these 
+parameters are not the only ones present in a hopping simulation. For the case of the 
+simulations performed here, which used a Marcus-type expression for the hopping rate, 
+an energetic disorder parameter is also present. Moreover, some simulations include the 
+so-called Huang-Rhys factor, which characterizes electron-phonon coupling strengths. 
+Such electron-phonon coupling parameters need to be properly incorporated into 
+simulations which describe exciton and charge dynamics as a function of temperature. 
+Hence, there exists scope to reduce the computational demands of exciton diffusion 
+simulations further by developing predictive models for these parameters on the basis of 
+machine learning as well. 
+ 
+This work has introduced the strategy of splitting the training data into two groups in 
+order to mitigate the unfavorable N3 scaling of GPR or KRR-based models of exciton 
+coupling. However, these kinds of models are widely used in materials informatics 
+beyond simulations of charge or energy transport. Could this strategy therefore be used 
+for materials discovery purposes, in which machine-learned models need to be fit to 
+massive databases of candidate materials? We will explore this direction in future 
+research.  
+ 
+Acknowledgements 
+ 
+This work has been supported by the Kyoto University On-Site Laboratory Initiative, the 
+Institute for Integrated Cell-Material Sciences (iCeMS), the MacDiarmid Institute for 
+Advanced Materials and Nanotechnology, and the Victoria University of Wellington high-
+performance computing cluster Rāpoi. GRW, PAH, and JMH acknowledge support from 
+the Marsden Fund of New Zealand. DMP acknowledges JSPS Kakenhi Grant 21K05003.  
+
+20 
+ 
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+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+26 
+ 
+Supporting Information for: 
+Exciton diffusion in amorphous organic semiconductors: reducing 
+simulation overheads with machine learning 
+ 
+Chayanit Wechwithayakhlung1,2, Geoffrey R. Weal3,4,2, Yu Kaneko5, Paul A. Hume3,4, Justin 
+M. Hodgkiss3,4, Daniel M. Packwood1,2* 
+ 
+1 Institute for Integrated Cell-Material Sciences (iCeMS), Kyoto University, Kyoto, Japan 
+ 
+2 Center for Integrated Data-Material Sciences (iDM), MacDiarmid Institute for Advanced 
+Materials and Nanotechnology, Wellington, New Zealand 
+ 
+3 MacDiarmid Institute for Advanced Materials and Nanotechnology, Wellington, New Zealand 
+ 
+4 School of Chemical and Physical Sciences, Victoria University of Wellington, Wellington, 
+New Zealand 
+ 
+5 Daicel Corporate Research Center, Innovation Park (iPark), Daicel Corporation, Himeiji, 
+Japan 
+ 
+* Corresponding author. Email: dpackwood@icems.kyoto-u.ac.jp 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+27 
+ 
+SI 1. Amorphous pentacene system 
+ 
+CIF file of the amorphous pentacene system (Figure 1A) 
+ 
+ 
+SI 2. Pentacene dimers 
+ 
+Zip file of extracted pentacene dimers in xyz format. 
+ 
+ 
+SI 3. Parameters of the tested models and parameters of the final model 
+ 
+Excel file of model parameters. 
+ 
+ 
+SI 4. Sensitivity analysis of final model 
+ 
+ 
+ 
+As for Figure 3B (main text), but with sensitivity analysis performed on the final model. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+GPR1
+Feature index (k)
+0
+Sensitivity
+Coulomb interaction index (r)
+72
+GPR2
+Feature index (k)
+Coulomb interaction index (r)
+7228 
+ 
+SI 5. Predictions of final model compared to a single GPR model 
+ 
+ 
+(A) Predicted excitonic coupling energies of the final model compared to ab initio values for 
+all extracted pentacene dimers (identical to Figure 4A). (B) Predicted energies of a model 
+consisting of a single Gaussian process regression component. See section 3.3 for details. 
+ 
+For clarity, each point is coloured according to the density of data in its vicinity. Concretely, 
+for each point, the number of other points within a 0.01 eV radius were counted. The ‘density’ 
+was defined as this number divided by 4927 (the total number of data points).  
+ 
+Linear regression on the data in Figure A yields the result y = (0.0024  0.0001) + (0.80  
+0.06)x, where errors refer to one standard error, and R2 = 0.77. For Figure B, we obtain y = 
+(0.0032  0.0001) + (0.69  0.006)x and R2 = 0.74. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+A
+B
+2
+0.10
+0.10
+coupling (eV)
+Predicted coupling (ev)
+80'0
+800
+90'0
+90°0
+Predicted
+oo
+0.02
+00:0
+T
+00'0
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+00'0
+0.06
+80'0
+0.10
+0.12
+ab initio coupling (eV)
+ab initio coupling (eV)29 
+ 
+SI 6. Predicted diffusion tensor elements 
+ 
+ 
+Ab initio-calculated 
+couplings  
+Final model-calculated 
+couplings 
+Large training set-
+calculated couplings* 
+Diffusion Coefficient  
+(× 10-3 cm2s-1) 
+1.630 ± 0.011 
+1.547 ± 0.005 
+1.544 ± 0.006 
+Eigenvalues of the Diffusion Tensor (× 10-3 cm2s-1) 
+Major 
+1.815 ± 0.014 
+1.686 ± 0.017 
+1.679 ± 0.012 
+Middle 
+1.551 ± 0.016 
+1.492 ± 0.007 
+1.512 ± 0.006 
+Minor 
+1.525 ± 0.012 
+1.462 ± 0.014 
+1.440 ± 0.006 
+ 
+ 
+ 
+ 
+Ratio (Major: Middle: Minor) 
+1.19: 1.02: 1.00 
+1.15: 1.02: 1.00 
+1.17: 1.05: 1.00 
+ 
+ 
+Diffusion Tensor (× 10-3 cm2s-1) 
+xx 
+1.593 ± 0.020 
+1.492 ± 0.012 
+1.500 ± 0.010 
+yy 
+1.682 ± 0.015 
+1.612 ± 0.008 
+1.613 ± 0.012 
+zz 
+1.616 ± 0.007 
+1.536 ± 0.010 
+1.518 ± 0.006 
+ 
+ 
+ 
+ 
+ 
+xy 
+-0.092 ± 0.009 
+-0.061 ± 0.012 
+-0.098 ± 0.007 
+yz 
+-0.108 ± 0.011 
+-0.079 ± 0.011 
+-0.039 ± 0.010 
+xz 
+0.056 ± 0.012 
+0.033 ± 0.006 
+0.008 ± 0.003 
+ 
+ 
+ 
+ 
+ 
+Ratio  
+(xx: yy: zz: xy: yz: xz)  
+28.5: 30.1: 28.9:       
+1.65: 1.93: 1.00 
+28.5: 30.8: 29.3:       1.16: 
+1.51: 0.64 
+28.5: 30.6: 28.8:       
+1.86: 0.75: 0.15 
+ 
+* This model used all dimers to train the SVM component, 2000 dimers to train GPR1, and 
+2200 dimers to train GPR2 
+ 
+

AtAyT4oBgHgl3EQf3_qJ/content/tmp_files/2301.00779v1.pdf.txt

@@ -0,0 +1,1333 @@
+arXiv:2301.00779v1  [math.AG]  2 Jan 2023
+GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN
+VARIETIES, WITH APPLICATION TO BRILL-NOETHER THEORY
+GIUSEPPE PARESCHI
+Abstract. We introduce a variant of global generation for coherent sheaves on abelian varieties
+which, under certain circumstances, implies ampleness. This extends a criterion of Debarre asserting
+that a continuously globally generated coherent sheaf on an abelian variety is ample. We apply this
+to show the ampleness of certain sheaves, which we call naive Fourier-Mukai-Poincar´e transforms,
+and to study the structure of GV sheaves.
+In turn, one of these applications allows to extend
+the classical existence and connectedness results of Brill-Noether theory to a wider context, e.g.
+singular curves equipped with a suitable morphism to an abelian variety. Another application is
+a general inequality of Brill-Noether type involving the Euler characteristic and the homological
+dimension.
+1. Introduction
+1.1. Motivation: continuous global generation, M-regularity, and ampleness. We work
+with projective varieties over an algebraically closed field k. On abelian (or, more generally, irreg-
+ular) varieties, the basic notion of global generation of a coherent sheaf (at a given point) admits
+a variant, introduced by M. Popa and the author, called continuous global generation (CGG for
+short), see [PP1, Definition 2.10]. This is as follows: given a coherent sheaf F on an abelian variety
+A, a closed point x ∈ A, and a Zariski open set U ⊂ Pic0A, one considers the sum of twisted
+evaluation maps
+evU(x) :
+�
+α∈U
+H0(A, F ⊗ Pα) ⊗ P −1
+α
+→ F|x ,
+(1.1)
+where: F|x := F ⊗ k(x) denotes the fiber of F at the point x, and Pα = P|A×{α} denotes the line
+bundle on A parametrized by a point α ∈ Pic0A via the (normalized) Poincar`e line bundle P on
+A × Pic0A.
+The sheaf F is said to be CGG at x if this map is surjective for all (non empty) Zariski open
+subsets of Pic0A. Of course, if this holds for every x ∈ A the sheaf F is said to be CGG.1
+The condition of being CGG neither implies nor is implied by the usual global generaton
+(GG). For example, the line bundle associated to a theta divisor on a p.p.a.v. is CGG but not GG
+and the structure sheaf of an abelian variety is GG but not CGG. However the following three facts
+make the CGG property significant:
+Supported by the MIUR Excellence Department Project awarded to the Department of Mathematics, University
+of Rome Tor Vergata, CUP E83C18000100006”.
+1There is a more general version of these notions and their consequences holding for sheaves on any variety
+equipped with a morphism to an abelian variety, see Definition 2.1.1. In this paper we mostly consider sheaves on
+abelian varieties
+1
+
+2
+G.PARESCHI
+(i) Ampleness. Let µN : A → A be the multiplication by N. If F is a CGG coherent sheaf on A
+then there is a N >> 0 such that µ∗
+N(F ⊗ Pα) is GG for all α ∈ Pic0A. Consequently, a CGG
+sheaf on an abelian variety is ample ([D, Proposition 3.1 and Corollary 3.2]).2
+(ii) Criterion for global generation. If F and L are respectively a CGG coherent sheaf on A, and
+a CGG line bundle on a subvariety of A, then F ⊗ L is GG (this follows by considering sections of
+the form sα · t−α with sα ∈ H0(F ⊗ Pα) and t−α ∈ H0(L ⊗ P −1
+α ), [PP1, Proposition 2.12]).
+(iii) Cohomological criterion. There is a natural vanishing condition on the higher cohomology,
+named M-regularity, implying CGG ([PP1]).
+This package has found various applications. To name a few: effective projective normality
+and syzygies of abelian varieties (e.g.
+[PP3],[I]); effective birationality of pluricanonical maps
+of irregular varieties (e.g.
+[PP3], [JLT], [BLNP], [JS], [CCCJ]); positivity of direct images of
+pluricanonical sheaves of varieties mapping to abelian varieties (e.g. [D], [CJ], [LPS], [M1],[M2],
+[MP]); birational geometry and volume of irregular varieties via eventual maps (e.g. [J]).
+The property of being CGG, as well as the M-regularity condition implying it, are quite
+strong and not often verified. In this paper we introduce the following weaker condition with the
+purpose of enlarging the range of applicability of this circle of ideas, especially Debarre’s criterion
+for ampleness mentioned in (i).3
+1.2. Generation by a set of subvarieties of the dual abelian variety.
+Definition. Given a finite set of irreducible subvarieties of Pic0A, say Z = {Zi} (such that no
+subvariety is contained in another one), a coherent sheaf F on A is said to be generated by Z (at
+a given point x ∈ A) if the map (1.1) is surjective for all open sets U of Pic0A such that U meets
+all subvarieties Zi ∈ Z.
+In this language, to be CGG means to be generated by the trivial set Z = {Pic0A}. On
+the other hand, a GG sheaf is generated by the set formed by the origin alone: Z = {ˆ0} (but the
+converse does not hold in general). We say that a coherent sheaf is generated if it is generated by
+some set of subvarieties (this notion of generation coincides with the notion of algebraic generation
+of [LY, Definition 3.2]).
+There is a natural notion of irredundant generating set (see Remark 2.1.2), and it can be
+shown that, for any finite set of subvarieties Z = {Zi} as above, there are generated sheaves F
+having Z as irredundant generating set (Example 6.2.4 below).
+Next, an irreducible subvariety Z of an abelian variety B is said to span B if
+N
+�
+��
+�
+Z + · · · + Z = B
+for some N. A finite set of irreducible subvarieties Z = {Zi} is said to strongly span B if each
+subvariety Zi spans B.
+Items (i) and (ii) of the previous subsection generalize as follows:
+2(a) The notion of ampleness have been extended to coherent sheaves by Kubota [K], and many properties holding
+for vector bundles extend to this setting, see [D].
+(b) The quoted result is stated in loc cit over the complex numbers, but the proof works over any algebraically closed
+field
+3In the same spirit, a related notion, namely weak CGG, was introduced by M. Popa and the author in [PP2,
+Definition 2.1], but the notion given here is better behaved
+
+GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES
+3
+Theorem A (Nefness/ampleness). If F is generated then it is nef. If F is generated by a set of
+subvarieties Z strongly spanning Pic0A then F is ample (Theorem 2.2.2).
+Theorem B (Non-generation locus). If F is generated by a set Z = {Zi} and L is a CGG line
+bundle on a subvariety X of A, then the non-generation locus of F ⊗L (namely the locus B(F ⊗L)
+where F ⊗L is not globally generated) is explicitly controlled in function of the base loci of the line
+bundles L ⊗ Pα, with α ∈ Zi (Proposition 2.3.1).
+(As expected, it turns out that the bigger are
+the Zi, the smaller is B(F ⊗ L)).
+1.3. Relationship with the Fourier-Mukai-Poincar´e transform. We are left with the prob-
+lem of finding criteria or applicable methods for proving that a given coherent sheaf is generated.
+In view of Theorem A, it is especially relevant to understand whether a given sheaf F is generated
+by some set of subvarieties strongly spanning Pic0A. The author hopes that this method will be
+helpful in addressing the ampleness of vector bundles on subvarieties of abelian varieties.
+The next result is a step in this direction. Namely we provide a sufficient condition (close
+to be a characterization) ensuring that a given coherent sheaf F is generated, and we identify
+the irredundant set of subvarieties doing the job. Roughly speaking, this is a subset of the set of
+irreducible components of the supports of the sheaves appearing in the torsion filtration ([HuLe,
+Definition 1.1.4]) of a certain coherent sheaf on Pic0A associated to F, referred to as the naive
+Fourier-Mukai-Poincar´e (FMP) transform of F. However, except for some cases, this is just a first
+step toward the solution of the above problem, since in general the torsion filtration of the naive
+FMP transform of F is not easy to describe.
+Passing to a more detailed description, we will consider the FMP functor in the following
+form. Let P be the normalized Poincar`e line bundle on A × Pic0A. We consider the Fourier-Mukai
+equivalence
+ΦA
+P−1 : D(A) → D(Pic0A),
+ΦA
+P−1( · ) = Rq∗(p∗( · ) ⊗ P−1)
+(1.2)
+where p and q are the projections on A and Pic0A. We will also make use of the dualization functor
+in the following (unshifted) form:
+F∨ := RHom(F, OA).
+(1.3)
+The above mentioned naive FMP transform of a given coherent sheaf F on A is defined as
+follows
+T (F) := RgΦA
+P−1(F∨).
+(1.4)
+where g = dim A. By Serre-Grothedieck duality, Hi(A, F∨ ⊗ P −1
+α ) = 0 for all i > g and for all
+α ∈ Pic0A. Therefore, by base change and duality, we have the canonical isomorphisms
+RgΦA
+P−1(F∨)|α ∼= Hg(A, F∨ ⊗ P −1
+α ) ∼= H0(A, F ⊗ Pα)∨
+for all α ∈ Pic0A (note that the above Hg is usually a hypercohomology group). Hence (up to
+duality) the coherent sheaf T (F) encodes the variation of the k-linear spaces H0(A, F ⊗ Pα) as
+α varies in Pic0A. Therefore it is natural to expect that T (F) must be related somehow to the
+generation of the coherent sheaf F. It turns out that it is the torsion of the sheaf T (F) what really
+matters.
+Of course, in general it is not the case that the naive transform coincides (up to shift) with
+the transform in the derived category, namely that
+ΦA
+P−1(F∨) = RgΦA
+P−1(F∨)[−g].
+(1.5)
+In fact the sheaf F is said to be a GV sheaf (generic vanishing sheaf) precisely when (1.5) holds.
+
+4
+G.PARESCHI
+The precise relation of the FMP transform with the generation problem is stated in Corollaries
+3.2.1 and 3.2.4. It could be somewhat imprecisely summarized as follows
+Theorem C (Generation of sheaves and the FMP transform). (a) The surjectivity of the map
+(1.1) can be rephrased in terms of a condition involving both the FMP transform ΦA
+P−1(F∨) and
+the naive FMP transform RgΦA
+P−1(F∨);
+(b) if F is generated, then a certain subset of the set of irreducible components of the supports of the
+sheaves appearing in the torsion filtration of the naive FMP transform of F form an (irredundant)
+generating set for F .
+As mentioned above, the problem in applying condition (b) is that at present there is no
+general way of describing, in terms of F, the supports of the sheaves appearing in the torsion
+filtration of the naive FMP transform of F (only for GV sheaves there we have such a description,
+see Remark 3.2.5).
+1.4. Ampleness of naive FMP transforms (= generalized Picard bundles). The next
+result concerns a lucky class of coherent sheaves which turn out to be generated and ample, even
+when they are not GV sheaves: the naive FMP transforms themselves. More generally, the same
+holds for the naive FMP transforms of coherent sheaves on certain reduced schemes mapping to
+abelian varieties. These are defined similarly, with the difference that the FMP functor is not an
+equivalence anymore (it turns out that, for the specific result described in the present subsection,
+this is unnecessary).
+Specifically, given a equidimensional, reduced, Cohen-Macaulay projective scheme, equipped
+with a morphism to an abelian variety f : X → A, one defines PX = (f × id)∗P and considers the
+functor (not an equivalence anymore)
+ΦX
+P−1
+X : D(X) → D(Pic0A)
+and the (unshifted) dualization functor
+∆X : D(X) → D(X),
+∆X(F) = RHom(F, ωX)
+The naive FMP transform of a coherent sheaf F on X is defined as
+T (F) = RdΦX
+P−1
+X (∆X(F)),
+where d = dim X, and has the same meaning as discussed in the previous subsection.
+We will consider coherent sheaves F on a reduced scheme X as above with the property that
+all subsheaves of F have reduced scheme-theoretic support (equivalently, one can consider only the
+subsheaves appearing in the torsion filtration of F). We will adopt the following notation: given
+a sheaf F on X, we denote Z(F) = {Zi} the set of irreducible components of the supports of the
+sheaves on X appearing in the torsion filtration of F, and f(Z(F)) := {f(Zi)}.
+Theorem D (Generation and ampleness of naive FMP transforms). In the above setting, let F be a
+coherent sheaf on a scheme X as above, such that all subsheaves of F have reduced scheme-theoretic
+support (the simplest example are torsion free sheaves on X). Then the naive FMP transform T (F)
+is generated by a subset of the collection f(Z(F)).
+In particular, if each subvariety of the set Z(F) maps, via the morphism f, to a subvariety spanning
+the abelian variety A, then T (F) is an ample sheaf on Pic0A.
+
+GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES
+5
+The above theorem is in fact is a generalization of a result of Schnell, who proved that naive
+FM transforms of torsion free sheaves on abelian varieties are CGG ([Sch, Theorem 4.1]).4 The
+argument used in the proof is essentially Schnell’s.
+We remark that, although in a different language, a well known example of Theorem D is
+that of (dual) Picard bundles: here X is a smooth complex projective variety, equipped with its
+Albanese morphism a : X → Alb X, and F = L is a line bundle on X satisfying the following
+vanishing condition on the higher cohomology:
+Hi(X, L ⊗ Pα) = 0
+(1.6)
+for all α ∈ Pic0X and i > 0.5 By base change this ensures that T (L) is a vector bundle on Pic0X.
+Note that in this case the generating collection is given by a single subvariety, namely the Albanese
+image {a(X)}, which automatically spans. By Theorem D, T (L) is ample. The dual of T (L),
+namely R0ΦX
+PX(F), is called the Picard bundle associated to the line bundle L.6 Picard bundles
+were classically known to be be negative for dim X = 1 ([ACGH]). This result was subsequently
+generalized by Lazarsfeld to all smooth projective varieties ([L, §6.3.C]). Thus Theorem D can be
+seen as a vast generalization (with a completely different proof) of that result.
+It is worth to remark that in Theorem D we are not assuming condition (1.6), nor any other
+vanishing conditions. Interestingly, there are many examples of coherent sheaves not verifying (1.6)
+such that nevertheless their naive FMP transform is locally free (see Remark 4.2.2).
+1.5. Application to Brill-Noether theory of singular curves. One immediate application
+of Theorem D is to Brill-Noether theory of (complex) singular curves (even reducible and with
+non-planar singularities) equipped with a morphism to an abelian variety. In this setting we provide
+analogues of the existence and connectedness theorems for special divisors, Ghione and Segre-
+Nagata theorem. We refer to Subsection 5 for these results.
+1.6. A general inequality of Brill-Noether type. Another application of Theorem D is a
+general existence result of Brill-Noether type which, although somewhat weak, is optimal, and
+turns out to be useful in some applications. In this subsection we will assume that the ground field
+is C. Given a coherent sheaf F on a complex abelian variety A,7 one defines the cohomological
+support loci
+V i(F) = {α ∈ Pic0A | hi(F ⊗ Pα) > 0}
+(1.7)
+and
+V >0(F) =
+�
+i>0
+V i(F)
+(1.8)
+Theorem E. Let F be a coherent sheaf on a complex abelian variety A such that all its torsion
+subsheaves (if any) have reduced scheme-theoretic support, and each component of the support spans
+A (simplest example: a torsion free sheaf on A). Assume that condition (1.6) holds, i.e.
+V >0(F) = ∅.
+4Of course Schnell does not use this terminology. Moreover his result is stated in the language of the symmetric
+Fourier transform, introduced in the same paper.
+5When dealing with sheaves on abelian varieties this condition is usually referred to as the IT(0) condition (namely
+F satisfied the index theorem with index 0).
+6It is easily seen that this definition is equivalent to the one of [L, §6.3.C], where the Picard bundle is defined as
+a vector bundle on PicλX, where λ is the algebraic equivalence class of L.
+7also in this case one could extend the discussion of this topic to varieties equipped of a finite morphism to an
+abelian varieties, but for sake of simplicity we will stick to abelian varieties
+
+6
+G.PARESCHI
+Then
+χ(F) ≥ hd(F) + 1.
+Here hd(F) denotes the homological dimension. Easy examples showing that (at least the
+way it is stated), Theorem E is optimal, are:
+(a) locally free sheaves F on abelian varieties (of arbitrarily high rank) such that V >0(F) = ∅ and
+χ(F) = 1. (It is known that such sheaves are of the form F = ΦP−1(L∨), where L is any ample
+line bundle on A. Then χ(F) = 1 and rk(F) = χ(L).)
+(b) Let i : C → J(C) a Abel-Jacobi embedding of a curve in its Jacobian and let F = i∗L,
+where L is a line bundle on C. In this case V >0(F) = V 1(F), which is non-empty if and only if
+deg(L) ≤ 2g − 2, i.e. χ(F) ≤ g − 1 = hd(F).
+An example of application of Theorem E is as follows. Let J (D) be the multiplier ideal sheaf
+of an effective Q-divisor D on an abelian variety A, and let L be an ample line bundle on A such
+that L − D is nef and big. By Nadel’s vanishing the sheaf J (D) ⊗ L satisfies the assumptions of
+Theorem E. Letting Z the scheme of zeroes of J (D), it follows that
+χ(J (D) ⊗ L) ≥ codim Z
+(1.9)
+(meaning the maximal codimension of a component of Z).
+This is instrumental in the proof of a result of the author on singularities of divisors on
+complex simple abelian varieties ([P2], see also Corollary 5.2.2).
+1.7. The case of GV sheaves. GV sheaves are those sheaves such that their naive FMP transform
+T (F) coincides, up to shift, with the FMP transform in the derived category (see (1.5)).8 They
+are very special cases of naive FMP transforms since it follows from the inversion formula for the
+FMP equivalence that
+F = T (T (F))
+(1.10)
+(see Proposition 6.2.2). Therefore part (a) of the next result, on the generation and ampleness
+of such sheaves, follows from Theorems C and D. As M regular sheaves form a subclass of GV
+sheaves, this recovers as a particular case the M-regularity criterion (iii) of Subsection 1.2. Item
+(b) is a structure result for such GV sheaves, following from the analysis of the torsion filtration
+of the FMP transform. Loosely speaking, the content is that the building blocks for GV sheaves
+are either CGG (hence ample) sheaves, or sheaves of the form (p∗G) ⊗ Pα, where p : A → B is a
+surjective homomorphism with connected kernel onto a lower dimensional abelian variety and G is
+a CGG (hence ample) sheaf on B.
+Theorem F (see Theorem 6.3.1). Let F be a GV sheaf on a g-dimensional abelian variety A, and
+assume that all subsheaves of the FMP transform T (F) have reduced support. Then:
+(a) F is generated and an irredundant generating set can be explicitly described;
+(b) F has a cofiltration whose kernels have in turn a cofiltration whose kernels are dominated either
+by ample sheaves or by sheaves of the form (p∗G) ⊗ Pα as above.
+Concerning (a), let us recall that not all GV sheaves are generated. This is shown by the well
+known example of non-trivial unipotent vector bundles on abelian varieties (see Example 6.2.3).
+This is caused by the fact that, unless F is trivial, the FMP transform T (F) is a torsion sheaf
+supported at a non-reduced zero-dimensional scheme, set-theoretically supported at the origin of
+8If this is the case the sheaf T (F) is often denoted �
+F∨ in the literature
+
+GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES
+7
+Pic0A. It is worth to recall that a generation criterion for GV sheaves was already given by Popa
+and the author (WIT criterion, [PP2, Theorem 4.1]). Item (a) is a more precise version of that
+result.
+In turn, item (b) can be seen as a weak analog of the Chen-Jiang decomposition ([CJ]),
+a quite useful property satisfied by direct images of pluricanonical sheaves under morphisms to
+abelian varieties (see Subsection 6.3).
+2. Generated and ample coherent sheaves on abelian varieties
+In this section we provide some generalities on the notion of generation of coherent sheaves
+on abelian varieties introduced in Subsection 1.2, and prove Theorems A and B.
+2.1. Generalities. To begin with, we remark that, although the focus of this paper is on coherent
+sheaves on abelian varieties, one can extend the definition given in Subsection 1.2 to the following
+setting: let X be a projective variety, equipped with a morphism to an abelian variety
+f : X → A.
+Definition 2.1.1. Let Z = {Zi}i∈I be a finite set of irreducible subvarieties of Pic0A, such that no
+subvariety belonging to Z is contained in another. A coherent sheaf F on X is said to be generated
+by Z at a given point x ∈ X (with respect to te morphism f) if the map
+evU(x) :
+�
+α∈U
+H0(F ⊗ f ∗Pα) ⊗ f ∗P −1
+α
+→ F|x
+is surjective for all open sets U such that U meets Zi for all i. If this happens for all x ∈ X, namely
+the map
+evU :
+�
+α∈U
+H0(F ⊗ f ∗Pα) ⊗ f ∗P −1
+α
+→ F
+is surjective, the sheaf F is said to be generated by the set Z (with respect to the morphism f).
+If Z = {Pic0A} then F is said to be continuously globally generated (CGG). Moreover F is said to
+be generated (with respect to the morphism f) if there is some set of subvarieties Z generating F
+(with respect to the morphism f). This is equivalent to the surjectivity of the map
+evPic0A :
+�
+α∈Pic0A
+H0(F ⊗ f ∗Pα) ⊗ f ∗P −1
+α
+→ F.
+(2.1)
+Remark 2.1.2. A finite set of subvarieties Y = {Yj} as above is said to be covered by another
+Z = {Zi} if for all i there is a j such that Yj is contained in Zi. If a sheaf F is generated by a set
+Z then, trivially, it is generated by any set covered by Z, and, as it will be clear in a moment, it
+turns out that is useful to identify a maximal (with respect to the relation of being covered) set
+doing the job. Such a set of subvariety will be called an irredundant generating set for F.
+Remark 2.1.3. In practice in some arguments it may happen to find generating sets Z = {Zi}
+such that Zj is contained in Zk for some j and k. However this does not cause any problem because,
+if this is the case, to be generated by the set Z is the same thing of being generated by the set
+Z ∖ {Zk}. Hence, by taking the subset of minimal subvarieties belonging to set Z, one can always
+reduce to the assumption of Definition 2.1.1.
+
+8
+G.PARESCHI
+Remark 2.1.4. By noetherianity and quasi-compactness, Definition 4.1 can be equivalently for-
+mulated replacing the map evU of the Definition 2.1.1 with the sum of finite number of evaluation
+maps. The required condition is the existence of positive integer N0 and a collection of positive
+integers {Ni}i∈I such that the sum of twisted evaluation maps
+�
+i∈I∪{0},1≤j≤Ni
+H0(F ⊗ f ∗Pαi,j) ⊗ f ∗P −1
+αi,j → F
+is surjective for all sufficiently general (α0,1, . . . α0,N) ∈ (Pic0A)N and (αi,1, . . . , αi,Ni) ∈ (Zi)Ni for
+all i ∈ I. Therefore also in the sum (2.1) one can take a finite number of summands. In particular,
+the notion of generated coherent sheaf introduced here coincides with the the notion of algebraically
+generated coherent sheaf introduced in the recent paper [LY, Definition 3.2].
+It follows that a coherent generated (with respect to any morphism) sheaf is nef, as it is the
+quotient of the direct sum of numerically trivial line bundles.
+2.2. Proof of Theorem A. The main point is in the following easy lemma.
+Lemma 2.2.1. In the setting of Definition 2.1.1, let Z = {Zi}n
+i=1 and Y = {Yj}m
+j=1 be two finite
+sets of subvarieties of Pic0A. Let moreover F and G be coherent sheaves on X respectively generated
+by Z and Y (with respect to the morphism f). Then F ⊗ G is generated by the set of subvarieties
+(see Remark 2.1.3):
+Z + Y := {Zi + Yj}(n,m)
+(i,j)=(1,1).
+Here Zi + Yj denotes, as usual, the image of Zi × Yj via the group law of Pic0A.
+Proof. For open subsets of Pic0A, say U and V , respectively meeting all the Zi’s and all the Yi’s
+we have the surjective map
+�
+(α,β)∈U×V H0(F ⊗ f ∗Pα) ⊗ H0(G ⊗ f ∗Pβ) ⊗ f ∗P −1
+α
+⊗ f ∗P −1
+β
+∥
+��
+α∈U H0(A, F ⊗ f ∗Pα
+�
+⊗ f ∗P −1
+α ) ⊗
+��
+β∈V H0(A, G ⊗ f ∗Pβ) ⊗ f ∗P −1
+β
+�
+−→
+F ⊗ G
+.
+This map factors through the map
+�
+γ∈U+V
+H0(F ⊗ G ⊗ f ∗Pγ) ⊗ f ∗P −1
+γ
+→ F ⊗ G
+which is, therefore, surjective. Finally, any open subset of Pic0A meeting all the subvarieties Xi+Yj,
+for i = 1, . . . , n and j = 1, . . . , m, contains some open subset of the form U + V , with U and V as
+above.
+□
+The following result is Theorem A of the Introduction, in a slightly more general form. It is
+an extension of the above quoted result of Debarre asserting that a CGG sheaf with respect to a
+finite onto its image morphism to an abelian variety is ample.
+Theorem 2.2.2. In the setting of Definition 2.1.1 let us assume that the morphism f : X → A is
+finite onto its image. Let F be a coherent sheaf on X such that F is generated by a finite set of
+subvarieties Z = {Zi}i∈I strongly spanning Pic0A (see Subsection 1.2). Then F is ample.
+Proof. To begin with, we note that the fact that the set Z = {Zi}i∈I strongly spans Pic0A implies
+that there is a positive integer M such that Zi1 + · · · + ZiM = Pic0A for all (i1, . . . , iM) ∈ IM. By
+
+GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES
+9
+Lemma 2.2.1 we know that the sheaf F⊗M is generated by the set {Zi1 + · · · + ZiM }(i1,...,iM)∈IM .
+Hence F⊗M is CGG and therefore the same holds true for SMF. Therefore, by Debarre’s theorem,
+SMF is ample, hence F is ample.
+□
+As a particular case, a coherent sheaf on an abelian variety which is generated by a single
+irreducible subvariety Z spanning Pic0A is ample. Obviously this is in general false if F is generated
+by the set of components of a reducible subvariety spanning Pic0A, but some individual component
+do not (e.g. on a product of abelian varieties A = A1 × A2, let F = p∗
+1F1 ⊕ p∗
+2F2, with Fi CGG
+(hence ample) sheaves on Ai for i = 1, 2.
+The sheaf F is not ample, but is generated by the
+collection {Pic0A1 × {ˆ0}, {ˆ0} × Pic0A2}.)
+2.3. Proof of Theorem B. As mentioned in Subsection 1.1 (item (ii)), the CGG condition
+provides a useful criterion for global generation. The natural extension of this is the following9
+Proposition 2.3.1. In the setting of Definition 2.1.1, let F be a coherent sheaf on X, generated
+(with respect to the morphism f) by a set Z = {Zi}. Let Y be a subvariety of X and let L be a
+CGG (with respect to f) line bundle on Y . Then
+B(F ⊗ L) ⊂
+�
+i
+� �
+α∈Zi
+B(L ⊗ P −1
+α )
+�
+.
+Proof. Note that for a line bundle L, to be CGG means that the intersection of the base loci
+�
+α∈V B(L ⊗ Pα) is empty for all (non-empty) open subsets V ⊆ Pic0A. Let y ∈ Y . Since L is
+CGG the subset {α ∈ Pic0A | y ̸∈ B(L ⊗ P −1
+α )} contains an open subset Uy(L) ⊆ Pic0A. Assume
+that y ̸∈ �
+i(�
+α∈Zi B(L ⊗ P −1
+α )). Then the open set Uy(L) meets all irreducible subvarieties Zi.
+Therefore, given another open set U ⊂ Pic0A meeting all irreducible subvarieties Zi, also the open
+subset U ∩ Uy(L) meets all subvarieties Zi. Therefore, in the commutative diagram
+�
+α∈U∩Uy(L) H0(F ⊗ Pα) ⊗ H0(L ⊗ P −1
+α )
+�
+�
+H0(F ⊗ L)
+�
+�
+α∈U∩Uy(L) H0(F ⊗ Pα) ⊗ (L ⊗ P −1
+α )|y
+� (F ⊗ L)|y
+both the left arrow and the bottom arrow are surjective. Hence the right arrow is surjective.
+□
+3. Relationship with the FMP transform
+3.1. The basic relation. The relevance of the FMP functor in the study of the generation of
+coherent sheaves on abelian varieties stems from the following relation between the evaluation
+maps of a coherent sheaf on A and of its naive FMP transform on Pic0A. This is known to the
+experts (see Schnell’s paper [Sch], proof of Proposition 4.1). In what follows we will keep the setting
+and notation of the Introduction, Subsection 1.3.
+Given an open subset U of Pic0A, the continuous evaluation map at x ∈ A
+evU(x) :
+�
+α∈U
+H0(F ⊗ Pα) ⊗ P −1
+α
+→ F|x
+(3.1)
+9also this result is an extension of a result of Popa and the author in the context of the above mentioned notion
+of weak global generation ([PP2, Proposition 2.4(c)])
+
+10
+G.PARESCHI
+factors through the map
+�
+α∈U
+H0(F ⊗ Pα) ⊗ (P −1
+α )|x → F|x
+(3.2)
+and (3.1) is surjective if and only if (3.2) is.
+Let us consider the individual maps of (3.2), i.e.
+H0(F ⊗ Pα) ⊗ P −1
+α
+⊗ k(x) → F ⊗ k(x).
+(3.3)
+The next Proposition describes the image of the dual map of (3.3) via the (contravariant) equiva-
+lence
+F : D(A) → D(Pic0A),
+F( · ) = ΦA
+P−1(( · )∨)
+Proposition 3.1.1. The functor F identifies the Serre-Grothendieck dual of the linear map (3.3)
+to the linear map
+Hg(ΦA
+P−1(F∨) ⊗ Px) → (RgΦA
+P−1(F∨) ⊗ Px) ⊗ k(α)
+factoring as follows
+Hg(ΦP−1(F∨) ⊗ Px)
+edx �
+�❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+H0(RgΦP−1(F∨) ⊗ Px)
+evx(α)
+�
+(RgΦP−1(F∨) ⊗ Px)|α
+(3.4)
+where evx(α) is the evaluation at the point α ∈ �A of the coherent sheaf RgΦP−1(F∨) ⊗ Px.
+Proof. The proof is straightforward. We identify the map (3.3) to
+H0(F ⊗ Pα) ⊗ H0(P −1
+α
+⊗ k(x)) → H0(F ⊗ k(x)).
+(3.5)
+Applying Serre-Grothedieck duality, we write the dual map of (3.5) as
+HomD(A)(k(x), F∨[g]) = Extg(k(x), F∨) → Hom(H0(P −1
+α
+⊗ k(x)), Hg(F∨ ⊗ P −1
+α )).
+(3.6)
+Applying the functor ΦA
+P−1 to the the source of (3.6), we have the chain of isomorphisms
+HomD(A)(k(x), F∨[g]) ∼= HomD(Pic0A)(P −1
+x , ΦA
+P−1(F∨)[g]) ∼=
+∼= Extg(P −1
+x , ΦA
+P−1(F∨)) ∼= Hg(ΦA
+P−1(F∨) ⊗ Px).
+(3.7)
+Concerning the target of the map (3.6), there are the canonical identifications
+H0(P −1
+α
+⊗ k(x)) ∼= R0ΦA
+P−1
+�
+k(x)
+�
+⊗ k(α) = ΦA
+P−1(k(x)) ⊗ k(α) = P −1
+x
+⊗ k(α)
+and
+Hg(F∨ ⊗ P −1
+α ) ∼=
+�
+RgΦA
+P−1(F∨)
+�
+⊗ k(α).
+We conclude that the map (3.6) is identified, via the functor ΦA
+P−1, to a linear map
+Hg(ΦA
+P−1(F∨) ⊗ Px) →
+�
+RgΦA
+P−1(F∨) ⊗ Px
+�
+⊗ k(α).
+factorizing trough the evaluation map of the sheaf RgΦA
+P−1(F∨) ⊗ Px at the point α, as stated in
+(3.4).
+□
+
+GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES
+11
+Remark 3.1.2. The map
+edx : Hg(ΦA
+P−1(F∨) ⊗ Px) → H0(RgΦA
+P−1(F∨) ⊗ Px).
+(3.8)
+appearing in (3.4) is in fact the edge map in the hypercohomology spectral sequence
+Hp(RqΦA
+P−1(F∨) ⊗ Px) =⇒ Hp+q(ΦA
+P−1(F∨) ⊗ Px).
+3.2. Consequences. From this point we will adopt the notation of the Introduction (see 1.4),
+namely
+T (F) := RgΦA
+P−1(F∨).
+(3.9)
+This sheaf will be referred to as the naive FMP transform of F. Proposition 3.1.1 allows to express
+the surjectivity of the map evU(x) of (3.1) as follows (this is Theorem C(a) of the Introduction).
+Corollary 3.2.1. Let U be an open subset of Pic0A and let x ∈ A. The map evU(x) of (3.1) is
+surjective if and only if the following two conditions hold:
+(1) the map edx of (3.8) is injective;
+(2) the restriction of the simultaneous evaluation map
+evx(U) : H0(T (F) ⊗ Px) −→
+�
+α∈U
+(T (F) ⊗ Px)|α
+(3.10)
+to the image of the map edx is injective.
+Remark 3.2.2. Notice that the map edx depends only on x and not on the open subset U ⊂ Pic0A.
+Therefore if F is generated at x, i.e. the map evU(x) (see (3.1)) is surjective for some U, then edx
+is injective.
+Remark 3.2.3. In view of the previous corollary, it is useful to understand the kernel of the
+evaluation maps evx(U) of (3.10) for a non empty open subset U. A non-zero section s ∈ ker evx(U)
+must be a torsion section, i.e. the image of the map s : OPic0A → T (F) ⊗ Px must be a torsion
+sheaf. If the support of such image is reduced then it is the closure of the subset of α ∈ Pic0A
+such that s|α ∈ (T (F) ⊗ Px)|α is non zero. If this is the case then U must be contained in the
+complement of such support. Of course for all x ∈ A the torsion subsheaves of T (F) ⊗ Px coincide,
+after tensorization with P −1
+x , with the torsion subsheaves of T (F), hence they do not depend on
+x ∈ A. Moreover all the irreducible components of supports of torsion subsheaves of T (F) appear
+in the support of sheaves appearing in the torsion filtration of T (F).
+Summarizing, we have the following Corollary, which is Theorem C(b) of the Introduction.
+In the statement we denote
+�V 0(τ) = {x ∈ A | H0(τ ⊗ Px) ∩ Im(edx) ̸= 0}
+Corollary 3.2.4. Let F be a coherent sheaf on an abelian variety A. Assume that:
+(a) the maps edx of (3.8) are injective for all x ∈ A;
+(b) all sheaves τ appearing in the torsion filtration of the naive FMP transform T (F), and such
+that �V 0(τ) is non empty, have reduced scheme-theoretic support.
+Then F is generated by the set of minimal irreducible components of supports of all such sheaves τ.
+Remark 3.2.5. In general, it is not known how to identify in function of F the various sheaves
+τ appearing in the torsion filtration of the naive FMP transform T (F), let alone those such that
+�V0(τ) in non empty. Not surprisingly, in the case of GV sheaves this can be done. Recalling the
+
+12
+G.PARESCHI
+definition and notation for cohomological support loci given in (1.7), it is well known that F is a
+GV sheaf if and only if, for all i ≥ 0
+codimPic0A V i(F) ≥ i
+([PP6, Corollary 3.10] or [PP5, Theorem 2.3]). If this is the case, the components of supports of
+the torsion sheaves appearing in the torsion filtration of the sheaf T (F) are components W of the
+loci V i(F) of the minimalcodimension, namely codimPic0A W = i (this follows from a result of Popa
+and the author, see [P1, Theorem 1.10]). We refer also to [CLP, §9] where an even more precise
+description of the torsion filtration is given in the case of coherent sheaves admitting a Chen-Jiang
+decomposition, a special class of GV-sheaves which includes direct images of pluricanonical bundles
+with respect to morphism to an abelian variety. See Remark 6.2.5 below for more on these sheaves.
+4. Generation and ampleness of naive FMP transforms and generalized Picard
+bundles.
+A known class of examples of ample vector bundles on abelian varieties is the one of (dual)
+Picard bundles (see Subsection 1.4 of the Introduction). In the case where X is a smooth curve
+naturally embedded in its Jacobian, it was classically known that the projectivization of the dual
+of the Picard bundle of a line bundle of degree d ≥ 2g − 1 is the d-symmetric product of the
+curve. Based on this observation, it follows that the dual of Picard bundles of curves are ample,
+and, as a consequence, one gets the existence and connectedness theoremsn of Brill-Noether theory
+([ACGH, Chapter VII]). The ampleness of dual Picard bundles and some of its applications were
+subsequently generalized to all smooth complex projective varieties by Lazarsfeld ([L, §6.3.C and
+7.2.C]). In this section we show that the FMP methods of the previous section quickly provide
+(with a different argument) a vast generalization of such results.
+4.1. Naive FMP transforms of sheaves on abelian varieties. In order to give a quick idea
+of the result, we start from a particular case already met in the previous section, namely naive
+FMP transforms of coherent sheaves on abelian varieties. We keep the notation of Subsection 1.4,
+especially on the set of subvarieties Z(F) (irreducible components of supports of coherent sheaves
+appearing in the torsion filtration of F).
+Proposition 4.1.1. Let F be a coherent sheaf on an abelian variety A such that all of its subsheaves
+have reduced scheme-theoretic support. Then the naive FMP transform T (F) = RgΦA
+P−1(F∨) is
+generated by a subset of the set Z(F) (see Theorem D). In particular, T (F) is an ample sheaf on
+Pic0A as soon as all subvarieties appearing in the set Z(F) span A.
+As mentioned in the Introduction, this result, for torsion free coherent sheaves, is already
+present in Schnell’s paper [Sch, Theorem 4.1]. The present argument is borrowed from his.
+Proof. We use the notation on evaluation maps of (3.1) and (3.10). The arguments consists in
+considering diagram (3.4) and let x (rather than α, vary.
+Specifically, the generation of T (F)
+means the surjectivity of the map
+evA(α) :
+�
+x∈A
+: H0(RgΦA
+P−1(F∨) ⊗ Px) ⊗ (P −1
+x )|α −→ RgΦA
+P−1(F∨)|α
+for all α ∈ Pic0A. By Proposition 3.1.1 (and its proof) the map
+evα(A) : H0(F ⊗ Pα) −→
+�
+x∈A
+(F ⊗ Pα)|x
+
+GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES
+13
+is the dual of a map factorizing trough evA(α). Therefore asserted surjectivity follows from the
+injectivity of the map evα(A), which follows from Corollary 3.2.4 because we are assuming there
+are no subsheaves with non reduced scheme-theoretic support. In fact, also the map
+evα(V ) : H0(F ⊗ Pα) −→
+�
+x∈V
+(F ⊗ Pα)|x
+is injective, for all open subsets V of A meeting all irreducible components of the support of all
+subsheaves of F. In fact it is enough to consider the subsheaves appearing in the torsion filtration
+of F.
+□
+4.2. Generalization to transforms from reduced schemes mapping to abelian varieties.
+In this subsection we prove Theorem D of the Introduction, namely the generalization of Proposition
+4.1.1 to naive transforms from certain reduced schemes mapping to abelian varieties, rather than
+from abelian varieties. Again, we keep the notation and setting introduced in Subsection 1.4.
+Theorem 4.2.1. Let f : X → A be a reduced Cohen-Macaulay equidimensional (of dimension
+d) projective scheme equipped with a morphism to an abelian variety. Let F be a coherent sheaf
+on X such that all of its subsheaves have reduced scheme theoretic support. Then its naive FMP
+transform T (F) = RdΦX
+P−1
+X (∆X(F)) is generated by a subset of the set f(Z(F)). In particular,
+T (F) is ample sheaf on Pic0A as soon as, for each i, the image f(Zi) of the subvariety Zi ∈ Z(F)
+spans A (e.g. this happens when f(X) spans A and F is a torsion free sheaf on X.)
+Proof. The argument is similar to the one of Proposition 4.1.1 but needs to be adjusted because
+the functor ΦX
+P−1
+X
+is not anymore an equivalence, hence Proposition 3.1.1 does not hold anymore.
+We still consider the linear map
+H0(F ⊗ f ∗Pα) ⊗ H0(f ∗P −1
+α
+⊗ k(x)) → H0(F ⊗ k(x))
+Dualizing we get
+HomD(X)(k(x), ∆X(F)[d]) = Extd(k(x), ∆X(F)) → Hom(H0(f ∗P −1
+α ⊗k(x)), Hd(∆X(F)⊗f ∗P −1
+α )).
+(4.1)
+We still have that ΦX
+P−1
+X (k(x)) = R0ΦX
+P−1
+X (k(x)) = P −1
+f(x) and
+H0(f ∗P −1
+α
+⊗ k(x)) ∼= R0ΦP−1
+X (k(x)) ⊗ k(α) = ΦP−1
+X (k(x)) ⊗ k(α) = P −1
+f(x) ⊗ k(α)
+Therefore the target of the map (4.1) is still identified by the functor ΦP−1
+X to the fibre
+(RdΦP−1
+X (∆X(F)) ⊗ f ∗Pf(x))|α
+Even if the analog, in the present setting, of the first map in the chain of isomorphisms (3.7) is
+not an isomorphism anymore, still it follows that the map (4.1) factorizes, via the functor ΦX
+P−1
+X ,
+trough the evaluation map of global sections at the point α ∈ Pic0A of the naive FMP transform,
+twisted by the line bundle Pf(x):
+Extd(k(x), ∆X(F))
+�
+(4.1)
+�❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+❯
+H0(RdΦP−1
+X (∆X(F)) ⊗ f ∗Pf(x))
+evx(α)
+�
+(RdΦP−1
+X (∆X(F)) ⊗ f ∗Pf(x))|α
+(4.2)
+At this point the argument goes exactly as in Proposition 4.1.1.
+□
+
+14
+G.PARESCHI
+Example 4.2.2. Note that in Theorem 4.2.1 and Proposition 4.1.1 we are not assuming the
+vanishing condition (1.6), namely that Hi(X, F ⊗ f ∗Pα) = 0 for all α ∈ Pic0A and i > 0, which
+implies that the naive FMP transform (or Picard sheaf) T (F) is a locally free sheaf on Pic0A. It
+is worth to note that in fact there are many examples where such condition does not hold but still
+T (F) is locally free. Therefore, even if one is interested only in ample vector bundles the above
+results produce a wider class of examples.
+Let us outline a simple way to produce such sheaves, similar to Raynaud’s construction of
+stable vector bundles ”without theta divisor” on curves ([R]). The simplest example is as follows.
+Let C be a smooth curve embedded in its Jacobian J(C) := A. For odd coprime positive integers
+a, b let Wa,b be the semhomogenous vector bundles on a p.p.a.v., as J(C), considered by Mukai and
+Oprea ([Mu1], Theorem 7.1 and Remark 7.3, [O]). Denoting µb : A → A the multiplication by b,
+and L = OA(Θ), these are vector bundles on A such that
+µ∗
+aWa,b = (Lab)⊕ag.
+(4.3)
+We claim that if a and b are such that a
+b ∈ (0, 1) then hi(F ⊗ Pα) is non-zero and constant for
+α ∈ Pic0A for both for i = 0, 1. This can be shown as follows. By (4.3), we have that
+hi((Wa,b)|C ⊗ Pα) = ag
+a2g hi(a∗
+A(OC ⊗ Pα) ⊗ OA(abΘ)).
+for a general α ∈ Pic0A. This shows that, for i = 0, 1 and α general in Pic0A, by definition, the
+rational number
+1
+ag hi((Wa,b)|C ⊗Pα) equals the value of the cohomological rank functions hi
+OC(xθ),
+x = b
+a (see [JP]). These functions have been computed in loc cit, p. 837, proof of Theorem 7.5.
+Specifically, in the interval [0, 1], they are
+h0
+OC(xθ) = xg
+and
+h1
+OC(xθ) = xg − g + 1 − xg.
+(4.4)
+If there was a non empty jump locus for the function α �→ hi((Wa,b)|C ⊗ Pα) (α ∈ Pic0A) then as
+it is easy to see, this would cause a critical point of the functions hi
+OC(xθ) (see §5, loc cit) in the
+interval (0, 1) but (4.4) shows that there is none. Therefore the functions α �→ hi((Wa,b)|C ⊗ Pα)
+are constant. It follows, in particular, that T ((Wa,b)|C) is a locally free sheaf on A.
+Similar examples can be obtained starting from any subvariety X of a ppav and any coherent
+sheaf G on X, considering the sheaves F = G ⊗Wa,b for b
+a in a suitable range. Constructions of this
+type can be performed on abelian varieties with arbitrary polarizations (not necessarily principal).
+5. Applications to Brill-Noether theory
+5.1. Singular curves equipped with a morphism to an abelian variety. The interest of
+results as Theorem 4.2.1, besides providing examples of ample vector bundles, is in their appli-
+cation to Brill-Noether theory via the nowdays classical results of Kempf, Kleiman-Laksov, and
+Fulton-Lazarsfeld on non-emptyness and connectedness of degeneracy loci [ACGH, Chapter VII],
+[L, §7.2.C]. Such applications mainly concern locally free sheaves on smooth curves. Via Theorem
+D they can be generalized e.g. to the following setting: torsion free sheaves F on singular curves C
+(connected reduced 1-dimensional Cohen-Macaulay projective schemes), equipped with a morphism
+f : C → A to an abelian variety, such that the image of each component of spans the abelian variety
+A.
+In this sort of application we will mainly concerned with Brill-Noether loci
+V 0,r
+f
+(X, F) = {α ∈ Pic0A | h0(X, F ⊗ f ∗Pα) ≥ r + 1}
+
+GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES
+15
+According to the previous notation, the locus V 0,0
+f
+(X, F) is simply denoted V 0
+f (X, F). The loci
+V 0,r
+f
+(X, F) can be realized as degeneracy loci of a map of locally free sheaves in the usual way,
+namely considering the exact sequence
+0 → F → F(D) → F(D)|D → 0
+where D is a Cartier divisor avoiding the singular points of X, such that D is of degree high
+enough on all components of X so that Hi(X, F(D) ⊗ f ∗Pα) = 0 for all α ∈ Pic0A. It turns out
+that ΦX
+PX(F(D)) = R0ΦX
+PX(F(D)) is a locally free sheaf on Pic0A (dual to the naive FMP transform
+T (F)) whose fibre at α is identified to H0(X, F(D) ⊗ f ∗Pα). Therefore the locus V 0,r
+f
+(X, F) is the
+k’th degeneracy locus of the map of vector bundles
+ϕ : ΦX
+P−1
+X (F(D)) → ΦX
+P−1
+X (F(D)|D)
+where k = χ(F(D))−(r+1). The target is a locally free sheaf on Pic0A of rank equal to � νi deg Di,
+where (ν1, .., νh) is the multirank of F (the tuple of the ranks of F at the various components of
+X) and deg Di is the degree of D at the various components of X. The expected dimension of the
+locus V 0,r
+f
+(X, F) is
+(r + 1)(
+�
+νi deg Di) − (χ(F(D)) − (r + 1))) = (r + 1)(r + 1 − χ(F))
+By Theorem 4.2.1 the source of the map ϕ is negative. On the other hand it is well known that
+the target is a homogeneous vector bundle, namely admitting a filtration whose quotients are line
+bundles parametrized by Pic0(Pic0A)) = A. Therefore, by the Theorem of Fulton-Lazarsfeld, we
+have
+Theorem 5.1.1. In the above setting, the locus V 0,r
+f
+(X, F) is nonempty (resp. connected) as soon
+as
+(r + 1)(r + 1 − χ(F)) ≤ dim A
+(resp. (r + 1)(r + 1 − χ(F)) < dim A).
+In particular V 0
+f (F) is non empty as soon as
+χ(F) ≥ − dim A + 1.
+This statement recovers (for smooth curves embedded in their Jacobians) the classical exis-
+tence and connectedness theorems of Brill-Noether theory, as well as Ghione’s theorem ([L, Example
+7.2.13]). As a consequence, we have the following version of the theorem of Segre-Nagata. To sim-
+plfy, assume furthermore that, in the setting of Theorem 4.2.1, the torsion free sheaf F has uniform
+rank ν (i.e. ν1 = ... = νh = ν). Let us define
+deg F := χ(F) − ν(1 − pa(X))
+Corollary 5.1.2. The sheaf F has an invertible subsheaf B of degree
+deg B ≥ 1 − pa(X) + deg F + dim A − 1
+ν
+.
+Proof. By the last inequality of Theorem 5.1.1 it follows that the locus V 0
+f (F ⊗ B−1) is non-empty
+as soon as χ(F) − ν deg B ≥ − dim A + 1.
+□
+Similarly, one can extend to the setting of Theorem 4.2.1 the result of [L, Example 7.2.15].
+
+16
+G.PARESCHI
+5.2. A Brill-Noether inequality. Let F be a coherent sheaf on an abelian variety A. Recalling
+the notation of (1.7) and (1.8) about the cohomological support loci of F, assume that the loci
+V >0(F) is strictly contained in Pic0A. Then χ(F) ≥ 0 (since it is equal to the generic value of
+h0(F ⊗Pα)). It turns out that this easy inequality can be improved if the cohomological loci V i(F),
+i > 0, are suitably small. For example, the ”Castelnuovo - De Franchis inequality” of Popa and
+the author ([PP4, Theorem 3.3]) states that, if F is a GV sheaf and V >0(F) is non empty, then
+χ(F) ≥ gvα(F) := mini>0codimα{V i(F) − i | i > 0}
+(5.1)
+for all α ∈ Pic0A, where codimαV i(F) denotes the codimension of V i(F) in Pic0A, in the neigh-
+borhood of a given point α ∈ Pic0A.
+The result stated as Theorem E in the Introduction complements the inequality (5.1), dealing
+with the case where the locus V >0(F) is empty. The statement asserts that if this is the case, and
+the coherent sheaf F is torsion free, or, more generally, all of its subsheaves have all reduced support,
+and each irreducible component of the support spans A, then
+χ(F) ≥ hd(F) + 1.
+(5.2)
+Proof. (of Theorem E) By Theorem D the naive FMP transform of F, namely
+T (F) = RgΦA
+P−1(F∨)
+is an ample sheaf on Pic0A. Moreover the hypothesis that V >0(F) = ∅ yields, by base change, that
+T (F) is a locally free sheaf on Pic0A.
+We claim that
+Supp
+�
+Exti(F, OA)
+�
+⊆ V i(T (F))
+(5.3)
+The assertion of Theorem E follows from the claim. Indeed, by Le Potier vanishing, V j(T (F)) = ∅
+as soon as j ≥ rkT (F) = χ(F). Hence (5.3) yields that χ(F) > max {j | Extj(F, OA) ̸= 0}.
+To prove (5.3), note that the hypothesis V >0(F) = ∅ yields trivially that RiΦA
+P−1(F∨) = 0 for
+i ̸= g, and therefore T (F) = ΦA
+P−1(F∨)[g] (in other words F is a GV-sheaf (see (1.5)). Therefore,
+since the inverse of the FMP equivalence ΦA
+P−1 : D(A) → D(Pic0A) is the equivalence ΦPic0A
+P
+[g] :
+D(Pic0A) → D(A), it follows that
+ΦP(T (F)) = F∨ = RHom(F, OA)
+hence
+RiΦP(T (F)) = Exti(F, OA)
+(see also Proposition 6.2.2 below for a related statement). This yields (5.3) since, by base change,
+SuppRiΦP(T (F)) ⊆ V i(T (F)).
+□
+Remark 5.2.1. Interestingly, the main point of the proof of Theorem E is Le Potier vanishing
+theorem, while the essential ingredient of the proof of inequality (5.1) is the syzygy theorem of
+Evans-Griffith ([EG, Corollary 1.7]), which is in turn quite related to the theorem of Le Potier
+(as shown by Ein, [E]). Therefore it seems possible that the inequalities (5.1) and (5.2) are both
+manifestations of some more general phenomenon.
+Theorem E was already implicitly used by the author (in a special case) in the proof of the
+following result (conjectured in [DH, §6]).
+
+GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES
+17
+Corollary 5.2.2 ([P2] Theorem B(1)). Let A be a complex simple abelian variety and D an effective
+Q-divisor on A such that (A, D) is a non klt pair. If L is a line bundle on A such that L − D is
+nef and big then
+χ(L) ≥ dim A + 1.
+Proof. The hypothesis means that J (D), the multiplier ideal sheaf of D, is non trivial. By Nadel’s
+vanishing V >0(J (D) ⊗ L) = ∅. Let Z be the zero-scheme of J(D). By Theorem E, χ(J (D) ⊗
+L) ≥ codim Z, where codim Z (resp. dim Z) denotes the maximal codimension (resp. maximal
+dimension) of a component of Z.
+On the other hand, an inequality of Debarre-Hacon ([DH,
+Lemma 5(e)]) states that, if A is simple, χ(J (D) ⊗ L) ≤ χ(L) − dim Z − 1. Combining the two
+inequalities it follows that
+χ(L) ≥ codim Z + dim Z + 1 ≥ dim A + 1 .
+□
+6. GV sheaves
+6.1. Recap on GV and M-regular sheaves. We recall from (1.5) that a sheaf F on abelian
+variety A is said to be GV if ΦP−1(F∨) is a sheaf in cohomological degree g, so that ΦP−1(F∨)[g]
+coincides with the naive FMP transform T (F) = ΦP−1(F∨)[g]. Therefore for a GV sheaf F the
+naive FMP transform will be simply referred to as the FMP transform of F.10 Moreover a M-regular
+sheaf is a GV sheaf such that T (F) is torsion free.
+It is also worth to recall that to be GV (resp.
+M-regular) is equivalent to the follow-
+ing condition on the cohomological support loci of F (see (1.7)): codimPic0A V i(F) ≥ i (resp.
+codimPic0A V i(F) > i) for all i > 0 (see e.g. [PP5, Theorem 2.3, Proposition 2.8]).
+A result of Popa and the author states that a M-regular sheaf is CGG ((iii) of Subsection
+1.1). This is a particular case of Corollary 3.2.4. Indeed if F is GV then the source and target of
+the map edx of (3.8) simply coincide, and the map is the identity. Moreover, if F is, in addition,
+M-regular, the FMP transform T (F) is torsion free and therefore all maps evx(U) of (3.10) are
+injective for all (non-empty) open subsets U of Pic0A and for all x ∈ A. It also follows that a
+M-regular sheaf is ample by the mentioned result of Debarre ((i) of the Introduction).
+The purpose of this section is to extend this analysis from M-regular to GV sheaves.
+6.2. Generation of GV sheaves. The generation of GV sheaves follows in the same way (under
+the usual reducedness assumption) from Corollary 3.2.4. Therefore we have the following result,
+partly known by work of Popa and the author in [PP2, Theorem 4.2].
+Corollary 6.2.1. Let F be a GV sheaf on an abelian variety A. Assume that all subsheaves ap-
+pearing in the torsion filtration of the FMP transform T (F) have reduced scheme-theoretic support.
+Then F is generated by the set of minimal irreducible components of the supports of such sheaves.
+Alternatively, Corollary 6.2.1 follows from Theorem D (or Proposition 4.1.1) combined with
+the following inversion result
+10In the literature of GV sheaves, including papers of the author, this is usually denoted �
+F∨. In this paper we
+changed notation because we have been considering also non-GV sheaves, where the naive FMP transform does not
+coincide with T (F) = ΦP−1(F∨)[g].
+
+18
+G.PARESCHI
+Proposition 6.2.2. If F is a GV sheaf then also TA(F) is a GV sheaf (on Pic0A) and
+TPic0A(TA(F)) = F.
+Note that, in the the statement above, we have denoted TA(F) = RgΦA
+P−1(F∨) and, for a coherent
+sheaf G on Pic0A, TPic0A(G) = RgΦPic0A
+P−1 (G∨). However in the sequel the ambient abelian variety
+will be omitted from the notation, unless this will be cause of confusion.
+Proof. By definition a sheaf F is a GV sheaf if TA(F) = ΦA
+P−1(F∨)[g]. Therefore, since the inverse
+of the FMP equivalence ΦA
+P−1 : D(A) → D(Pic0A) is the equivalence ΦPic0A
+P
+[g] : D(Pic0A) → D(A),
+it follows that
+ΦPic0A
+P
+(TA(F)) = F∨ = RHom(F, OA)
+Therefore, by Grothendieck duality ([Mu2, (3.8)]), ΦPic0A
+P−1 (TA(F)∨)[g] = F.
+□
+Example 6.2.3 (Unipotent vector bundles). Concerning the assumption of Corollary 6.2.1, its
+necessity is shown by the well known example of unipotent vector bundles on abelian varieties.
+These are, by definition, vector bundles F admitting a filtration whose successive quotients are
+trivial line bundles.
+Since a trivial bundle on an abelian variety is evidently a GV sheaf, any
+unipotent bundle is a GV sheaf. For such vector bundles H0(F ⊗ Pα) ̸= 0 if and only if α = ˆ0
+(the identity point of Pic0A). Therefore the only possible set of subvarieties generating F is {ˆ0},
+and this happens if and only if F is trivial, because otherwise H0(F) < rk F. In conclusion, any
+non-trivial unipotent vector bundle is GV but not generated. This is explained by the fact that
+the FMP transform T (F) is supported at zero, but the scheme theoretic support is non-reduced,
+unless the sheaf F is trivial ([Mu1, Lemma 4.8]).
+Other examples obtained from this one are
+e.g. homogeneous vector bundles (direct sums of unipotent vector bundles twisted by line bundles
+parametrized by Pic0A), and, on a product of abelian varieties A = B × C, coherent sheaves of the
+form p∗
+BG ⊠ p∗
+CU, where G is GV on B and U is homogeneous on C.
+Example 6.2.4 (Any subset of subvarieties can be realized as a irredundant generating subset).
+Corollary 6.2.1, combined with Proposition 6.2.2, can be used to construct examples showing that
+any subset of subvarieties is the irredundant generating set of some (locally free) sheaf. Indeed let A
+be an abelian variety and let G be any coherent sheaf on Pic0A, such that all of its subsheaves have
+reduced scheme-theoretic support. Twisting with a sufficiently high power of an ample line bundle
+L on Pic0A we have that V >0(G⊗L) = ∅. Therefore, ΦPic0A
+P−1 ((G⊗L)∨) = RgΦPic0A
+P−1 ((G⊗L)∨)[−g] =
+TA(G ⊗ L)[−g]. Let
+F := TA(G ⊗ L)
+From Proposition 6.2.2 it follows that F is a (locally free) GV sheaf and TPic0A(F) = G ⊗ L.
+Therefore the assertion follows from Corollary 6.2.1, because it can be easily shown, with the help
+of Corollary 3.2.4, that the generating set of minimal irreducible components of support of the
+sheaves appearing in the torsion filtration of G ⊗ L, i.e. of G, is actually irredundant, and G is
+arbitrary.
+For the next example illustrating Corollary 6.2.1, it is useful to recall the compatibility
+property of the FMP functor with respect to homomorphisms of abelian varieties. Let pB : A → B
+be a surjective homomorphism and iB := b∗ : Pic0B → Pic0A the dual inclusion. Then
+ΦA
+P−1
+A ◦ p∗
+B = iB∗ ◦ p∗ΦB
+P−1
+B
+ΦPic0A
+P−1
+A
+◦ iB∗ = p∗
+B ◦ p∗ΦPic0B
+P−1
+B
+.
+(6.1)
+(This can be deduced e.g. from [Sch, Proposition 1.1]).
+
+GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES
+19
+Example 6.2.5 (Sheaves admitting a Chen-Jiang decomposition). A significant class of GV sheaves
+where the assumption of Corollary 6.2.1(a) is always verified is given by coherent sheaves having
+the Chen-Jiang decomposition property ([LPS, §B.3] and references therein). They were already
+mentioned in Remark 3.2.5. By definition, these sheaves admit a (essentially canonical) decompo-
+sition
+F =
+�
+i
+(p∗
+BiGi) ⊗ Pαi
+(6.2)
+where pBi : A → Bi are surjective homomorphisms, with connected kernel, of abelian varieties, Gi
+are M-regular sheaves (hence ample) on Bi, and αi ∈ Pic0A are torsion points. It follows that such
+sheaves are generated and semiample. This class is important because it contains higher direct
+images of canonical sheaves of smooth complex projective varieties mapping to abelian varieties
+([PPS, Theorem A]), as well as direct images of pluricanonical sheaves ([LPS, Theorem C]), and
+direct images of log-pluricanonical sheaves for klt pairs ([M1, Theorem 1.3]).
+It follows from (6.1) that each summand in (6.2) is a GV sheaf, and its FMP transform is
+T ((p∗
+BiGi) ⊗ Pαi) = t∗
+−αi(iB∗TB(Gi)).
+In particular the various FMP transforms of the summands are scheme-theoretically supported
+on the translated abelian subvarieties Pic0Bi − αi. Therefore such sheaves, as well as F, satisfy
+the assumption of Corollary 6.2.1. (In fact, in the proof of the above quoted results about direct
+images of pluricanonical sheaves, this is an essential point, whose proof revolves around the minimal
+extension property of a positively curves metrics on such direct images, whose existence was shown
+by Cao-Paun [CP]. See also the survey [HPS] on this matters.)
+Therefore, as noted in [CLP, §9], the torsion filtration of the FMP transform:
+T0 ⊂ · · · ⊂ Td ⊂ T (F),
+where d = dim T (F), is given by
+Tk =
+�
+dim Bi≤k
+t∗
+−αi(iB∗TB(Gi)).
+(Note that if d = dim A, among the summands of (6.2) there is one with Bi = A, hence Gi is already
+M-regular in A. Its transform is the torsion free sheaf T (F)/Tg−1.)
+6.3. Structure of GV sheaves. In the previous example, applying the inverse FMP transform
+ΦPic0A
+P
+: D(Pic0A) → D(A) to the torsion filtration of T (F) one gets back the cofiltration
+F = Fd ։ · · · ։ F0 → 0 ,
+(6.3)
+where Fk = �
+dim Bi≤k(p∗
+BiGi) ⊗ Pαi. (In particular, if d = dim A(= g) the kernel of F ։ Fg−1 is
+M-regular hence ample.) This is a sort of weak version of the decomposition (6.2)
+The following results says that, under the reducedness assumption of Corollary 6.2.1, the
+structure of GV sheaves can be described by a weak analog of (6.3).
+Theorem 6.3.1. Let F be a GV sheaf on an abelian variety A, satisfying the assumption of
+Corollary 6.2.1. Then F has a cofiltration
+F
+ϕ0
+։ F1
+ϕ1
+։ · · ·
+ϕs−1
+։ Fs
+ϕs
+→ 0
+such that: (a) The kernel of the surjection ϕ0 : F
+ϕ0
+։ F1 is either zero or ample, the latter case
+holding if and only if dim T (F) = dim A.
+
+20
+G.PARESCHI
+(b) For each i ≥ 1, the sheaf ker ϕi has, in turn, a cofiltration
+ker ϕi
+ϕ0i
+։ F1i
+ϕ1i
+։ · · ·
+ϕs−1 i
+։
+Fsi i
+ϕsi i
+→ 0
+such that for all (j, i), there is a surjection fj i : p∗
+BiFj i ⊗ Pαj i ։ ker ϕj i, where: pBj i : A → Bj i
+is a surjective homomorphism of abelian varieties with connected kernel, Fj i is an ample coherent
+sheaf on Bj i and αj i ∈ Pic0A
+Proof. We begin with the general, and well known to the experts (see e.g. [Sch, Propositions 1.1,
+1.2]),
+Lemma 6.3.2. Let p : A → B be a surjective homomorphism of abelian varieties, with connected
+kernel. i : Pic0B ֒→ A the dual inclusion, τ a coherent sheaf on Pic0B and α ∈ Pic0A. Then
+TPic0A(t∗
+−αi∗τ) = p∗TPic0B(τ) ⊗ Pα here tα denotes the translation by α on Pic0A).
+Proof.
+Recalling that, on an abelian variety C, the functor TC((·)) is defined as Rdim CΦP−1((·)∨)
+(see (1.3) for the dualizing functor), the assertion of the Lemma follows from: (i) the second identity
+in (6.1), (ii) the fact that RHomPic0A(i∗τ, OPic0A) = i∗RHomPic0B(τ, OB)[dim A − dim B], and,
+(iii) the well known identity ΦPic0A
+P−1 (t∗
+−α(·)) = ΦPic0A
+P−1 (·) ⊗ Pα. This proves Lemma 6.3.2.
+Going back to the Theorem, the initial step of the cofiltration is defined as follows. Let τ be
+the torsion part of TA(F). We apply the right exact contravariant functor TPic0A(·) to the exact
+sequence 0 → τ → TA(F) → TA(F)/τ → 0. By Proposition 6.2.2 we get the exact sequence
+TPic0A(TA(F)/τ) −→ F
+ϕ0
+−→ TPic0A(τ) → 0
+We define F1 := TPic0A(τ).
+Since the sheaf TA(F)/τ is either zero or torsion free, the sheaf
+TPic0A(TA(F)/τ) is either zero or ample (Proposition 4.1.1). Therefore the same is true for ker ϕ0.
+Hence we have accomplished the first step of the cofiltration. Next, let d = dim τ and 0 → τd−1 →
+τ → τ/τd−1 → 0 the first step of the torsion filtration of τ. Applying the functor TPic0A(·) we get
+the exact sequence
+TPic0A(τ/τd−1) → F1
+ϕ1
+→ F2 := TPic0A(τd−1) → 0
+We need to prove condition (b) on ker ϕ1. The sheaf σ := τ/τd−1 has pure dimension d. Let us
+pick an irreducible component V of the support of σ. We apply the functor TPic0A(·) to the exact
+sequence 0 → K → σ → σ|V → 0. We get
+TPic0A(σ|V ) → TPic0A(σ) → TPic0A(K) → 0
+By Proposition 4.1.1 if V spans Pic0A then the sheaf TPic0A(σ|V ) is ample.
+Otherwise V is a
+subvariety of a translate of an abelian subvariety C of Pic0A. In this case, by Lemma 6.3.2, the
+sheaf TPic0A(σ|V ) is of the form p∗TPic0B(G)⊗Pα, where B is the dual of C, and the sheaf TPic0B(G)
+is ample on B again by Proposition 4.1.1. This settles the first step of the cofiltration on ker ϕ1.
+Next, the sheaf K is either zero or of pure dimension d. Therefore we can apply the same
+procedure to the sheaf K, and so on. In this way, after a finite number of steps the cofiltration on
+ker ϕ1 is settled. We now proceed inductively applying the same procedure to the sheaves τk for
+k ≤ d − 1.
+□
+References
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+Email address: pareschi@mat.uniroma2.it
+

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+1
+Sparse Array Design for Dual-Function
+Radar-Communications System
+Huiping Huang, Student Member, IEEE, Linlong Wu, Member, IEEE, Bhavani Shankar, Senior Member, IEEE,
+and Abdelhak M. Zoubir, Fellow, IEEE
+Abstract—The problem of sparse array design for dual-
+function radar-communications is investigated. Our goal is to
+design a sparse array which can simultaneously shape desired
+beam responses and serve multiple downlink users with the
+required signal-to-interference-plus-noise ratio levels. Besides, we
+also take into account the limitation of the radiated power by
+each antenna. The problem is formulated as a quadratically
+constrained quadratic program with a joint-sparsity-promoting
+regularization, which is NP-hard. The resulting problem is solved
+by the consensus alternating direction method of multipliers,
+which enjoys parallel implementation. Numerical simulations
+exhibit the effectiveness and superiority of the proposed method
+which leads to a more power-efficient solution.
+Index Terms—Alternating direction method of multipliers,
+dual-funtion radar-communications, sparse array design
+I. INTRODUCTION
+D
+UAL-function radar-communications (DFRC) systems
+have been recently widely investigated [1]–[4]. They find
+applications in a wide range of areas, including vehicular net-
+works, indoor positioning and covert communications [5], [6].
+To achieve accurate sensing and high throughput, the DFRC
+systems probably requires a large number of antennas [7]–
+[10]. In practice, a base station usually equips with less radio-
+frequency (RF) chains than antennas given the hardware cost
+consideration. For such a configuration, it raises a question
+on how to adaptively switch the available RF chains to the
+corresponding subset of antennas [11]–[13], which can be
+interpreted as sparse array design.
+In this work, we consider the sparse array design for the
+DFRC system. Similar problems have been studied in [14]–
+[19]. The authors in [14] derived a Cram´er-Rao bound for the
+cooperative radar-communications system, where they focused
+on target parameter estimation. The work [15] developed an
+antenna selection strategy by using a learning approach. This
+method requires a training process, which might be unavailable
+in some practical applications. The authors in [16] introduced a
+realistic waveform constraint in the DFRC system, in order to
+improve the power efficiency. A surrogate subproblem instead
+of the original problem was solved, which might lead to a
+highly suboptimal solution. In [17] and [18], several types of
+DFRC systems were proposed which implement simultaneous
+beamformers associated with single and different sparse arrays
+with shared aperture. However, these two works consider a
+single-user case, and the corresponding methods cannot be
+The work of H. Huang is supported by the Graduate School CE within the
+Centre for Computational Engineering at Technische Universit¨at Darmstadt.
+H. Huang and A. M. Zoubir are with Technische Universit¨at Darmstadt,
+Darmstadt, Germany (e-mail: {h.huang; zoubir}@spg.tu-darmstadt.de).
+L. Wu and B. Shankar are with University of Luxembourg, Luxembourg
+City, Luxembourg (e-mail: {linlong.wu; Bhavani.Shankar}@uni.lu).
+applicable to the case of multi-users. A weighted ℓ1,q-norm
+optimization based approach was developed in [19]. Note that
+all the problems considered in [17]–[19] are convex and were
+solved by the existing toolbox such as CVX.
+Unlike in previous works, we propose a novel system model
+for DFRC systems, and formulate the corresponding sparse
+array design problem as a quadratically constrained quadratic
+program (QCQP) regularized by a joint-sparsity-promoting
+term. Besides the control on illumination beampattern and
+communication signal-to-interference-plus-noise ratio (SINR),
+we also take into account the limitation of the radiated power
+by each antenna. We propose an algorithm based on the con-
+sensus alternating direction method of multipliers (ADMM)
+to solve the resulting problem. Note that at each ADMM
+iteration, the primary variable has a closed-form solution, and
+the auxiliary variables can be solved efficiently in a parallel
+manner. Simulation results show its superior performance
+compared to other examined methods.
+The remainder of the paper is organized as follows. The
+system model is established in Section II. The proposed
+algorithm is presented in Section III. Simulation results are
+shown in Section IV, while Section V concludes the paper.
+Notations: Throughout this paper, bold-faced lower-case
+(upper-case) letters denote vectors (matrices). Superscripts ·T,
+·H, ·∗, ·−1 denote transpose, Hermitian transpose, conjugate,
+and inverse, respectively. E{·} denotes the expectation opera-
+tor. | · | and ∠· are the modulus and phase, respectively, both
+in an element-wise manner. C and R are the sets of complex
+and real numbers, respectively, and ȷ = √−1. ∥·∥0, ∥·∥1, and
+∥·∥2 are ℓ0-quasi-norm, ℓ1-norm, and ℓ2-norm, respectively. ⊗
+denotes the Kronecker product. ⊘ is the element-wise division
+operator. IN and OM×N are the N × N identity matrix and
+M ×N zero matrix, respectively. 1 and 0 are the all-ones and
+all-zeros vector of appropriate size, respectively.
+II. SYSTEM MODEL AND PROBLEM FORMULATION
+We consider a DFRC system, as indicated in Fig. 1. The
+transmitter, consisting of N antennas, emits M signals, each
+to a single user. The weight vector wm ∈ CN is designed to
+transfer the data symbol, sm(t), to user m, ∀m = 1, 2, · · ·, M.
+All the M transmitted signals are also received by a target.
+The total transmitted signal can then be formulated as
+x(t) = [w1, w2, · · ·, wM]
+�
+����
+s1(t)
+s2(t)
+...
+sM(t)
+�
+���� ,
+(1)
+arXiv:2301.00786v1  [eess.SY]  2 Jan 2023
+
+2
+…
+1
+2
+N
+User 1
+User M
+Target
+w1s1(t)
+wMsM(t)
+…
+x(t) =
+M
+∑
+m=1
+wmsm(t)
+…
+Fig. 1: Illustration of a DFRC system.
+where t is the time index. We assume that the data symbols,
+sm(t) ∀m = 1, 2, · · ·, M, are mutually uncorrelated, and each
+sm(t) is zero-mean, spatially white with unit variance, i.e.,
+E{|sm(t)|2} = 1, ∀m=1, 2, · · ·, M,
+(2a)
+E{|si(t)s∗
+j(t)|} = 0, ∀i, j =1, 2, · · ·, M, and i̸=j.
+(2b)
+For the radar system, we expect that the response within
+some desired angular region, say B, is not less than a preset
+threshold, while the response within ¯B is not higher than a
+small threshold. Mathematically,
+E
+�
+xH(t)a(θi)aH(θi)x(t)
+�
+≥ ϵp, ∀θi ∈ B,
+(3a)
+E
+�
+xH(t)a(θj)aH(θj)x(t)
+�
+≤ ϵs, ∀θj ∈ ¯B,
+(3b)
+where a(θ) denotes the steering vector of θ, ϵp and ϵs are two
+preset thresholds corresponding to the mainlobe and sidelobe,
+respectively. Substituting (1) and (2) into (3), we have
+M
+�
+m=1
+wH
+ma(θi)aH(θi)wm ≥ ϵp, ∀θi ∈ B,
+(4a)
+M
+�
+m=1
+wH
+ma(θj)aH(θj)wm ≤ ϵs, ∀θj ∈ ¯B.
+(4b)
+For the downlink communication systems, the received data
+by the m-th user is given as
+ym(t) = hH
+mwmsm(t)
+�
+��
+�
+communication signal
++
+M
+�
+j=1,j̸=m
+hH
+mwjsj(t)
+�
+��
+�
+interference signals
++ nm(t)
+� �� �
+noise
+, (5)
+where nm(t) is the additive white Gaussian noise with mean
+zero and known variance σ2
+m, ∀m = 1, 2, · · ·, M. Besides,
+hm ∈ CN denotes the channel state information (CSI) vector,
+which models the propagation loss and phase shift of the
+frequency-flat quasi-static channel from the transmitter to the
+m-th user. The CSI, {hm}m, is assumed to be perfectly
+available at the transmitter. The maximum power radiated by
+each antenna n is given by Pn. Therefore, we have [7]
+M
+�
+m=1
+wH
+mEnwm ≤ Pn, ∀n = 1, 2, · · ·, N,
+(6)
+where En is the N×N all-zero matrix except the n-th diagonal
+entry being 1. The received SINR of user m is defined below,
+and a certain minimum SINR needs to be guaranteed, i.e.,
+SINRm ≜
+��hH
+mwm
+��2
+�
+j̸=m |hH
+mwj|2+σ2m
+≥γm, ∀m = 1, · · ·, M, (7)
+where �
+j̸=m is short notation for �M
+j=1,j̸=m.
+Now we consider the scenario where only K ≤ N RF
+chains are available, and thus only K antennas can be simul-
+taneously utilized in the DFRC system. Therefore, wm, ∀m,
+are sparse, and moreover, they share the same sparsity pattern.
+To fulfill this requirement, we need:
+∥�w∥0 ≤ K,
+(8)
+where �w ≜ [ �w1, �w2, · · ·, �wN]T with its component defined as
+�wn ≜ ∥[w1(n), w2(n), · · ·, wM(n)]∥2 ∀n = 1, 2, · · · , N, and
+wm(n) being the n-th entry of vector wm, ∀m = 1, 2, · · ·, M.
+The quality of service (QoS) problem aims at minimizing
+the total transmit power (TxPower), with several constraints
+described above. To this end, the QoS problem is given as
+min
+{wm} TxPower ≜
+M
+�
+m=1
+∥wm∥2
+2
+(9a)
+s.t.
+(4), (6), (7), and (8).
+(9b)
+By replacing the non-convex ℓ0-quasi-norm in (8) with the ℓ1-
+norm, and incorporating it into Problem (9) as a regularization
+term, we have
+min
+{wm}
+M
+�
+m=1
+∥wm∥2
+2 + η∥�w∥1
+s.t. (4), (6), and (7),
+(10)
+where η is a parameter controlling the sparsity of the solution.
+III. PROPOSED METHOD
+The difficulties of solving Problem (10) are twofold. First,
+the discorporated sparsity has a joint structure among all wm
+[20], which is different from the classical sparsity and con-
+sequently, the existing solutions cannot be directly extended.
+Second, the constraints (4a) and (7) are nonconvex.
+In what follows, we first deal with the joint-sparsity struc-
+ture of the solution by reformulating the problem, and then,
+we develop an algorithm based on the consensus ADMM.
+A. Problem Reformulation
+First of all, we define a long column vector w ∈ CMN
+as w ≜ [wT
+1 , wT
+2 , · · · , wT
+M]T and define M matrices Φm ∈
+RN×MN as Φm ≜ [ON×(m−1)N, IN, ON×(M−m)N]. Then,
+Problem (10) can be reformulated as
+min
+w
+∥w∥2
+2 + η∥w∥2,1
+(11a)
+s.t. wHAθiw ≥ ϵp, ∀θi ∈ B,
+(11b)
+wHAθjw ≤ ϵs, ∀θj ∈ ¯B,
+(11c)
+wHBnw ≤ Pn, ∀n = 1, 2, · · ·, N,
+(11d)
+wHCmw
+wHC ¯mw + σ2m
+≥ γm, ∀m = 1, 2, · · ·, M.
+(11e)
+where we have utilized �M
+m=1 ∥wm∥2
+2 = ∥w∥2
+2, wm = Φmw,
+and ∥w∥2,1 ≜ �N
+n=1
+��M
+m=1 |wm(n)|2. Further, since the
+denominators, wHC ¯mw + σ2
+m, are always positive, constraint
+(11e) can be rewritten as
+wHDmw ≥ γmσ2
+m, ∀m = 1, 2, · · ·, M,
+(12)
+
+3
+where Dm ≜ Cm − γmC ¯m, ∀m = 1, 2, · · ·, M. We observe
+that (11b), (11c), (11d), and (12) take the form: wHFw ≤ f.
+Therefore, Problem (11) can be reformulated as
+min
+w
+∥w∥2
+2 + η∥w∥2,1
+(13a)
+s.t. wHFlw ≤ fl, ∀l = 1, 2, · · ·, L,
+(13b)
+where the subscript ·l is used to indicate the l-th constraint
+in Problem (11), and L is the total number of constraints.
+It is worth noting that Fl corresponding to −Aθi is neg-
+ative semidefinite, and Fl corresponding to −Dm could be
+indefinite. Therefore, in general, Problem (13) is non-convex
+and thus NP-hard [21]. To solve this problem, we propose an
+algorithm based on the consensus ADMM, which is capable
+of handling all the constraints in parallel.
+B. Proposed Algorithm
+Firstly, we formulate Problem (13) by introducing L auxil-
+iary variables {vl ∈ CMN}L
+l=1, and settle the original variable
+w and the auxiliary variables {vl}l in a separable fashion, as
+min
+w,{vl} ∥w∥2
+2 + η
+L
+L
+�
+i=1
+∥vl∥2,1
+(14a)
+s.t.
+vH
+l Flvl ≤ fl, vl = w, ∀l = 1, 2, · · ·, L.
+(14b)
+Next, we form the scaled-form augmented Lagrangian func-
+tion related to the above problem, as: L(w, {vl}, {ul}) =
+∥w∥2
+2 + η
+L
+�L
+l=1∥vl∥2,1 + ρ
+2
+�L
+l=1
+�
+∥vl−w + ul∥2
+2 − ∥ul∥2
+2
+�
+,
+where ρ > 0 stands for the augmented Lagrangian parameter,
+and ul is the scaled dual variable corresponding to the equality
+constraint vl = w in Problem (14).
+Finally, the consensus ADMM updating equations can be
+written down as
+w ← arg min
+w
+∥w∥2
+2 + ρ
+2
+L
+�
+l=1
+∥vl − w + ul∥2
+2,
+(15a)
+vl ←
+�
+arg min
+vl
+η
+L∥vl∥2,1 + ρ
+2∥vl − w + ul∥2
+2
+s.t.
+vH
+l Flvl ≤ fl,
+(15b)
+ul ← ul + vl − w.
+(15c)
+In what follows, we show how to solve w and vl from (15a)
+and (15b), respectively. We start by solving w from (15a). It
+is straightforward to see that, by calculating the derivative of
+the objective function of (15a) with respect to (w.r.t.) w and
+setting it to 0, we obtain the solution to (15a) as
+�w =
+ρ
+2 + ρL
+L
+�
+l=1
+(vl + ul).
+(16)
+On the other hand, to solve vl from (15b), we firstly consider
+the unconstrained minimization problem:
+min
+vl
+f(vl) ≜ η
+L∥vl∥2,1 + ρ
+2∥vl − w + ul∥2
+2.
+(17)
+The derivative of f(vl) w.r.t. vl is calculated as
+∇vlf(vl) =
+� η
+L (IM ⊗ G) + ρIMN
+�
+vl − ρ(w − ul),
+where G∈RN×N is a diagonal matrix with diagonal being
+1⊘
+�
+�
+�
+�
+�
+�
+M
+�
+m=1
+|vl(1+(m−1)N)|2, · · ·,
+�
+�
+�
+�
+M
+�
+m=1
+|vl(N +(m−1)N)|2
+�
+�
+T
+.
+By setting the derivative to 0, we obtain
+� η
+L (IM ⊗ G) + ρIMN
+�
+vl = ρ(w − ul).
+(18)
+Since
+� η
+L (IM ⊗G)+ρIMN
+�
+is real-valued, ∠vl = ∠(w−ul).
+Thus, we only need to calculate the modulus of vl, using
+� η
+L (IM ⊗ G) + ρIMN
+�
+|vl| = ρ|w − ul|.
+(19)
+By exploring the structure of IM ⊗G, we observe that there
+are N blocks each containing M equal entries. Extracting the
+rows of the equal entries yields
+�
+η
+L∥vl(n)∥2
++ ρ
+�
+|vl(n)| = ρ|cl(n)|,
+(20)
+∀n = 1, 2, · · ·, N, where vl(n) ≜ [vl(n), vl(n+N), · · ·, vl(n+
+(M−1)N)]T ∈ CM, and cl(n) ∈ CM contains the correspond-
+ing entries of (w − ul). From (20), we have
+|vl(n)| =
+ρL∥vl(n)∥2
+η + ρL∥vl(n)∥2
+|cl(n)|.
+(21)
+Performing the element-wise square operation, we have
+|vl(n)|2 =
+ρ2L2∥vl(n)∥2
+2
+η2 + ρ2L2∥vl(n)∥2
+2 + 2ηρL∥vl(n)∥2
+|cl(n)|2. (22)
+Hence, we further have
+∥vl(n)∥2
+2 =1T|vl(n)|2
+(23a)
+=
+ρ2L2∥vl(n)∥2
+2
+η2+ρ2L2∥vl(n)∥2
+2+2ηρL∥vl(n)∥2
+1T|cl(n)|2. (23b)
+The above equation leads to
+ρ2L2∥vl(n)∥2
+2+2ηρL∥vl(n)∥2+η2−ρ2L21T|cl(n)|2 = 0, (24)
+the left-hand side of which is a simple quadratic function w.r.t.
+∥vl(n)∥2, and its unique1 root is given as
+∥vl(n)∥2 =
+ρL
+�
+1T|cl(n)|2 − η
+ρL
+.
+(25)
+Then, by substituting (25) into (21), we obtain
+|vl(n)| =
+ρL
+�
+1T|cl(n)|2 − η
+ρL
+�
+1T|cl(n)|2
+|cl(n)|.
+(26)
+In (26), we define |vl(n)| = 0 if |cl(n)| = 0. The solution for
+(18), referred to as ¯vl, is finally obtained by combining vl(n)
+(which can be calculated as |vl(n)|eȷ∠vl(n)). Then, the solution
+to (15b), denoted by �vl, is found via the following theorem.
+Theorem. If ρ satisfies ρ
+2 ≫ η
+L, then �vl can be solved via:
+�vl ← arg min
+vl
+∥vl − ¯vl∥2
+2
+s.t. vH
+l Flvl ≤ fl.
+(27)
+Proof. See Appendix A.
+1Note that the quadratic function in (24) has two roots, one positive and one
+negative. In our case, the negative one is omitted, since its root ∥vl(n)∥2 ≥ 0.
+
+4
+Algorithm 1 Consensus ADMM for solving Problem (13)
+Input: η, ρ, kmax, Fl ∈CMN×MN, fl, ∀l = 1, 2, · · ·, L
+Output: �w ∈ CMN
+Initialize: �v(0)
+l
+←v(init)
+l
+, �u(0)
+l
+←u(init)
+l
+, k←0
+1: while k < kmax do
+2:
+�w(k+1) ←
+ρ
+2+ρL
+L
+�
+l=1
+�
+�v(k)
+l
++ �u(k)
+l
+�
+3:
+for each l = 1, 2, · · ·, L do
+4:
+cl ← �w(k+1) − �u(k)
+l
+5:
+∠¯v(k+1)
+l
+← ∠cl
+6:
+for each n = 1, 2, · · ·, N do
+7:
+|¯v(k+1)
+l(n) | ←
+ρL√
+1T|cl(n)|2−η
+ρL√
+1T|cl(n)|2 |cl(n)|
+8:
+¯v(k+1)
+l(n)
+← |¯v(k+1)
+l(n) |eȷ∠¯v(k+1)
+l(n)
+9:
+end for
+10:
+Construct ¯v(k+1)
+l
+using ¯v(k+1)
+l(n) , ∀n
+11:
+if ¯v(k+1)H
+l
+Fl¯v(k+1)
+l
+≤fl then �v(k+1)
+l
+←¯v(k+1)
+l
+12:
+else �v(k+1)
+l
+←
+�
+arg min
+vl
+∥vl − ¯v(k+1)
+l
+∥2
+2
+s.t.
+vH
+l Flvl ≤ fl
+13:
+end if
+14:
+�u(k+1)
+l
+← �u(k)
+l
++ �v(k+1)
+l
+− �w(k+1)
+15:
+end for
+16:
+k ← k + 1
+17: end while
+18: �w ← �w(k)
+Remark 1. Note that η is related to the sparsity of the solution
+of Problem (10), and L is the total number of constraints in
+Problem (13). Both of them are known for a specific problem.
+Hence, it is easy to choose a ρ such that ρ
+2 ≫ η
+L.
+Remark 2. If ¯vl satisfies the constraint of (27), i.e., ¯vH
+l Fl¯vl ≤
+fl, it is easy to have �vl = ¯vl. Otherwise, notice that Problem
+(27) is a QCQP with one constraint, which can be solved
+optimally despite that Fl may be indefinite [21].
+So far, we have presented how to solve w and vl from (15a)
+and (15b), respectively. Note that {vl}L
+l=1 and {ul}L
+l=1 can be
+calculated in parallel. The complete consensus ADMM for
+solving Problem (13) is summarized in Algorithm 1, in which
+kmax is used to terminate the iteration, and the superscript ·(k)
+denotes the corresponding variable at the k-th iteration.
+In Parallel
+In Parallel
+IV. SIMULATION RESULTS
+In this section, we evaluate the performance of the proposed
+algorithm compared with the feasible point pursuit successive
+convex approximation (FPP-SCA) method [22]. FPP-SCA was
+proposed for general QCQP and it is adapted to our problem.
+Note that unlike our solution, no closed-form solution of FPP-
+SCA is given. The following two metrics are adopted: the Tx-
+Power defined in (9a) and the mainlobe-to-sidelobe response
+ratio (MSRR) defined as MSRR =
+�
+θi∈B
+�M
+m=1|wH
+ma(θi)|
+2
+�
+θj ∈ ¯
+B
+�M
+m=1|wH
+ma(θj)|2 .
+We first consider a transmit system with a uniform linear
+array of N = 10 antennas and K = 8 or 10 RF chains.
+There are M = 2 users. The mainlobe and sidelobe are B =
+[−5◦, 5◦] and ¯B = [−90◦, −20◦]∪[20◦, 90◦], respectively. The
+-90
+-60
+-30
+0
+30
+60
+90
+Angle (Degree)
+0
+2
+4
+6
+8
+10
+12
+14
+Beampattern
+Fig. 2: Beampattern comparison.
+4
+6
+8
+10
+Number of Selected Sensors
+2
+4
+6
+8
+10
+12
+14
+TxPower (dB)
+4
+6
+8
+10
+-5
+0
+5
+10
+MSRR (dB)
+Fig. 3: TxPower (left) and MSRR (right) versus K.
+thresholds for the mainlobe and sidelobe response are ϵp = 10
+and ϵs = 0.1. The maximum power radiated by each antenna
+is Pn = 40 dBm, ∀n, the same as [7]. The noise variance is set
+to σ2
+m = 1, ∀m. The threshold for the received SINR is γm =
+10 dB, ∀m. The number of constraints in Problem (13) is
+L = 53, the tuning parameter is2 η = 0.1, and the augmented
+Lagrangian parameter is ρ = 50 (ρ/2 ≫ η/L is satisfied). The
+value of v(init)
+l
+is given by any feasible point, while u(init)
+l
+= 0
+and kmax = 100. The beampatterns are drawn in Fig. 2, which
+indicates that the proposed method has nearly perfect response
+controls in both mainlobe and sidelobe while the FPP-SCA has
+lower mainlobe and higher sidelobe correspondingly.
+Next, we examine the TxPower and MSRR versus the
+number of selected sensors, i.e., K. We also consider a strategy
+of randomly selecting K sensors. The parameters are the
+same as those in the last example. 500 Monte-Carlo trials are
+performed. The results are plotted in Fig. 3. It is seen that
+our proposed method has the lowest transmit power and the
+highest MSRR, among all tested methods.
+Finally, we test the TxPower and MSRR versus the number
+of users, i.e., M. The number of selected antennas is fixed as
+2We do not study the relationship between η and the sparsity of the solution,
+because of the space limitation of the paper. Instead, when we obtain �w, we
+choose K antennas corresponding to the largest (in an ℓ2,1-norm sense) K
+components. We have good results when the tuning parameter is η = 0.1.
+
+5
+2
+4
+6
+Number of Users
+4
+6
+8
+10
+12
+14
+16
+18
+TxPower (dB)
+2
+4
+6
+-5
+0
+5
+10
+15
+MSRR (dB)
+Fig. 4: TxPower (left) and MSRR (right) versus M.
+K = 8, and the other parameters are unchanged as in the last
+example. The results are depicted in Fig. 4, which again show
+that our algorithm leads to a more power-efficient solution.
+V. CONCLUSION
+We studied the problem of sparse array design for dual-
+function radar-communications. Our design aimed at main-
+taining good control in both mainlobe and sidelobe, and also
+to keep the signal-to-interference-plus-noise ratio for the users
+larger than a certain level. In addition, we considered the limi-
+tation of the radiated power by each antenna. The problem was
+formulated as a quadratically constrained quadratic program,
+and solved by consensus alternating direction method of multi-
+pliers. The proposed algorithm was able to be implemented in
+parallel. Simulation results demonstrated better performance
+of the proposed algorithm than the other tested methods.
+APPENDIX A
+PROOF OF THEOREM
+Solving (27) is equal to finding a point in {vl :vH
+l Flvl ≤fl},
+such that it is closest (in an ℓ2-norm sense) to ¯vl. Hence,
+the theorem equivalently states that the solution to Problem
+(15b) is the point closest (in an ℓ2-norm sense) to ¯vl, provided
+that ρ/2 ≫ η/L. To show this, we denote �vl as the point in
+{vl : vH
+l Flvl ≤ fl}, such that
+∥�vl − ¯vl∥2 ≤ ∥vl − ¯vl∥2
+(28)
+holds for any vl ∈ {vl : vH
+l Flvl ≤ fl}. Our goal is to show
+f(�vl) ≤ f(vl), for any vl ∈ {vl : vH
+l Flvl ≤ fl}.
+The Lagrangian parameter ρ is chosen as ρ = Cη/L, where
+C is a constant. Then, |g(vl) − f(vl)| → 0 as C → ∞ (i.e.,
+ρ/2 ≫ η/L), where g(vl) ≜ ρ
+2∥vl − w + ul∥2
+2. Moreover,
+f(vl) = g(vl) = ρ
+2∥vl − ¯vl∥2
+2,
+(29)
+as long as C → ∞. Note that, in the second equality above,
+we used the fact that ¯vl = w − ul as C → ∞.
+Suppose that there exists a point ˘vl ∈ {vl : vH
+l Flvl ≤ fl},
+such that f(˘vl) < f(�vl). Thus, by using (29), we obtain that
+ρ
+2∥˘vl − ¯vl∥2
+2 <
+ρ
+2∥�vl − ¯vl∥2
+2, which contradicts (28). This
+implies that f(�vl) ≤ f(vl) holds for all feasible vl, that is,
+�vl is the solution to Problem (15b). This completes the proof.
+REFERENCES
+[1] K. V. Mishra, B. Shankar, V. Koivunen, B. Ottersten, and S. A. Vorobyov,
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+abs/2208.12313
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+804–808, Jul. 2015.
+

CdAyT4oBgHgl3EQf4fpA/content/tmp_files/load_file.txt

@@ -0,0 +1,502 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf,len=501
+page_content='1  Sparse Array Design for Dual-Function  Radar-Communications System  Huiping Huang, Student Member, IEEE, Linlong Wu, Member, IEEE, Bhavani Shankar, Senior Member, IEEE,  and Abdelhak M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Zoubir, Fellow, IEEE  Abstract—The problem of sparse array design for dual-  function radar-communications is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Our goal is to  design a sparse array which can simultaneously shape desired  beam responses and serve multiple downlink users with the  required signal-to-interference-plus-noise ratio levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Besides, we  also take into account the limitation of the radiated power by  each antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The problem is formulated as a quadratically  constrained quadratic program with a joint-sparsity-promoting  regularization, which is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The resulting problem is solved  by the consensus alternating direction method of multipliers,  which enjoys parallel implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Numerical simulations  exhibit the effectiveness and superiority of the proposed method  which leads to a more power-efficient solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  Index Terms—Alternating direction method of multipliers,  dual-funtion radar-communications, sparse array design  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' INTRODUCTION  D  UAL-function radar-communications (DFRC) systems  have been recently widely investigated [1]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' They find  applications in a wide range of areas, including vehicular net-  works, indoor positioning and covert communications [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  To achieve accurate sensing and high throughput, the DFRC  systems probably requires a large number of antennas [7]–  [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' In practice, a base station usually equips with less radio-  frequency (RF) chains than antennas given the hardware cost  consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' For such a configuration, it raises a question  on how to adaptively switch the available RF chains to the  corresponding subset of antennas [11]–[13], which can be  interpreted as sparse array design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  In this work, we consider the sparse array design for the  DFRC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Similar problems have been studied in [14]–  [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The authors in [14] derived a Cram´er-Rao bound for the  cooperative radar-communications system, where they focused  on target parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The work [15] developed an  antenna selection strategy by using a learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' This  method requires a training process, which might be unavailable  in some practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The authors in [16] introduced a  realistic waveform constraint in the DFRC system, in order to  improve the power efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' A surrogate subproblem instead  of the original problem was solved, which might lead to a  highly suboptimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' In [17] and [18], several types of  DFRC systems were proposed which implement simultaneous  beamformers associated with single and different sparse arrays  with shared aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' However, these two works consider a  single-user case, and the corresponding methods cannot be  The work of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Huang is supported by the Graduate School CE within the  Centre for Computational Engineering at Technische Universit¨at Darmstadt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Huang and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Zoubir are with Technische Universit¨at Darmstadt,  Darmstadt, Germany (e-mail: {h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='huang;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' zoubir}@spg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='tu-darmstadt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Wu and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Shankar are with University of Luxembourg, Luxembourg  City, Luxembourg (e-mail: {linlong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='wu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Bhavani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='Shankar}@uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='lu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  applicable to the case of multi-users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' A weighted ℓ1,q-norm  optimization based approach was developed in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Note that  all the problems considered in [17]–[19] are convex and were  solved by the existing toolbox such as CVX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  Unlike in previous works, we propose a novel system model  for DFRC systems, and formulate the corresponding sparse  array design problem as a quadratically constrained quadratic  program (QCQP) regularized by a joint-sparsity-promoting  term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Besides the control on illumination beampattern and  communication signal-to-interference-plus-noise ratio (SINR),  we also take into account the limitation of the radiated power  by each antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' We propose an algorithm based on the con-  sensus alternating direction method of multipliers (ADMM)  to solve the resulting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Note that at each ADMM  iteration, the primary variable has a closed-form solution, and  the auxiliary variables can be solved efficiently in a parallel  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Simulation results show its superior performance  compared to other examined methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The  system model is established in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The proposed  algorithm is presented in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Simulation results are  shown in Section IV, while Section V concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  Notations: Throughout this paper, bold-faced lower-case  (upper-case) letters denote vectors (matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Superscripts ·T,  H, ·∗, ·−1 denote transpose, Hermitian transpose, conjugate,  and inverse, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' E{·} denotes the expectation opera-  tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' | · | and ∠· are the modulus and phase, respectively, both  in an element-wise manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' C and R are the sets of complex  and real numbers, respectively, and ȷ = √−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' ∥·∥0, ∥·∥1, and  ∥·∥2 are ℓ0-quasi-norm, ℓ1-norm, and ℓ2-norm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' ⊗  denotes the Kronecker product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' ⊘ is the element-wise division  operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' IN and OM×N are the N × N identity matrix and  M ×N zero matrix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' 1 and 0 are the all-ones and  all-zeros vector of appropriate size, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' SYSTEM MODEL AND PROBLEM FORMULATION  We consider a DFRC system, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The  transmitter, consisting of N antennas, emits M signals, each  to a single user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The weight vector wm ∈ CN is designed to  transfer the data symbol, sm(t), to user m, ∀m = 1, 2, · · ·, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  All the M transmitted signals are also received by a target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  The total transmitted signal can then be formulated as  x(t) = [w1, w2, · · ·, wM]  �  ����  s1(t)  s2(t)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  sM(t)  �  ���� ,  (1)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='00786v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='SY]  2 Jan 2023  2  …  1  2  N  User 1  User M  Target  w1s1(t)  wMsM(t)  …  x(t) =  M  ∑  m=1  wmsm(t)  …  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' 1: Illustration of a DFRC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  where t is the time index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' We assume that the data symbols,  sm(t) ∀m = 1, 2, · · ·, M, are mutually uncorrelated, and each  sm(t) is zero-mean, spatially white with unit variance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=',  E{|sm(t)|2} = 1, ∀m=1, 2, · · ·, M,  (2a)  E{|si(t)s∗  j(t)|} = 0, ∀i, j =1, 2, · · ·, M, and i̸=j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  (2b)  For the radar system, we expect that the response within  some desired angular region, say B, is not less than a preset  threshold, while the response within ¯B is not higher than a  small threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Mathematically,  E  �  xH(t)a(θi)aH(θi)x(t)  �  ≥ ϵp, ∀θi ∈ B,  (3a)  E  �  xH(t)a(θj)aH(θj)x(t)  �  ≤ ϵs, ∀θj ∈ ¯B,  (3b)  where a(θ) denotes the steering vector of θ, ϵp and ϵs are two  preset thresholds corresponding to the mainlobe and sidelobe,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Substituting (1) and (2) into (3), we have  M  �  m=1  wH  ma(θi)aH(θi)wm ≥ ϵp, ∀θi ∈ B,  (4a)  M  �  m=1  wH  ma(θj)aH(θj)wm ≤ ϵs, ∀θj ∈ ¯B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  (4b)  For the downlink communication systems, the received data  by the m-th user is given as  ym(t) = hH  mwmsm(t)  �  ��  �  communication signal  +  M  �  j=1,j̸=m  hH  mwjsj(t)  �  ��  �  interference signals  + nm(t)  � �� �  noise  , (5)  where nm(t) is the additive white Gaussian noise with mean  zero and known variance σ2  m, ∀m = 1, 2, · · ·, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Besides,  hm ∈ CN denotes the channel state information (CSI) vector,  which models the propagation loss and phase shift of the  frequency-flat quasi-static channel from the transmitter to the  m-th user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The CSI, {hm}m, is assumed to be perfectly  available at the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The maximum power radiated by  each antenna n is given by Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Therefore, we have [7]  M  �  m=1  wH  mEnwm ≤ Pn, ∀n = 1, 2, · · ·, N,  (6)  where En is the N×N all-zero matrix except the n-th diagonal  entry being 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The received SINR of user m is defined below,  and a certain minimum SINR needs to be guaranteed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=',  SINRm ≜  ��hH  mwm  ��2  �  j̸=m |hH  mwj|2+σ2m  ≥γm, ∀m = 1, · · ·, M, (7)  where �  j̸=m is short notation for �M  j=1,j̸=m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  Now we consider the scenario where only K ≤ N RF  chains are available, and thus only K antennas can be simul-  taneously utilized in the DFRC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Therefore, wm, ∀m,  are sparse, and moreover, they share the same sparsity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  To fulfill this requirement, we need:  ∥�w∥0 ≤ K,  (8)  where �w ≜ [ �w1, �w2, · · ·, �wN]T with its component defined as  �wn ≜ ∥[w1(n), w2(n), · · ·, wM(n)]∥2 ∀n = 1, 2, · · · , N, and  wm(n) being the n-th entry of vector wm, ∀m = 1, 2, · · ·, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  The quality of service (QoS) problem aims at minimizing  the total transmit power (TxPower), with several constraints  described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' To this end, the QoS problem is given as  min  {wm} TxPower ≜  M  �  m=1  ∥wm∥2  2  (9a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  (4), (6), (7), and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  (9b)  By replacing the non-convex ℓ0-quasi-norm in (8) with the ℓ1-  norm, and incorporating it into Problem (9) as a regularization  term, we have  min  {wm}  M  �  m=1  ∥wm∥2  2 + η∥�w∥1  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' (4), (6), and (7),  (10)  where η is a parameter controlling the sparsity of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' PROPOSED METHOD  The difficulties of solving Problem (10) are twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' First,  the discorporated sparsity has a joint structure among all wm  [20], which is different from the classical sparsity and con-  sequently, the existing solutions cannot be directly extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  Second, the constraints (4a) and (7) are nonconvex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  In what follows, we first deal with the joint-sparsity struc-  ture of the solution by reformulating the problem, and then,  we develop an algorithm based on the consensus ADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Problem Reformulation  First of all, we define a long column vector w ∈ CMN  as w ≜ [wT  1 , wT  2 , · · · , wT  M]T and define M matrices Φm ∈  RN×MN as Φm ≜ [ON×(m−1)N, IN, ON×(M−m)N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Then,  Problem (10) can be reformulated as  min  w  ∥w∥2  2 + η∥w∥2,1  (11a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' wHAθiw ≥ ϵp, ∀θi ∈ B,  (11b)  wHAθjw ≤ ϵs, ∀θj ∈ ¯B,  (11c)  wHBnw ≤ Pn, ∀n = 1, 2, · · ·, N,  (11d)  wHCmw  wHC ¯mw + σ2m  ≥ γm, ∀m = 1, 2, · · ·, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  (11e)  where we have utilized �M  m=1 ∥wm∥2  2 = ∥w∥2  2, wm = Φmw,  and ∥w∥2,1 ≜ �N  n=1  ��M  m=1 |wm(n)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Further, since the  denominators, wHC ¯mw + σ2  m, are always positive, constraint  (11e) can be rewritten as  wHDmw ≥ γmσ2  m, ∀m = 1, 2, · · ·, M,  (12)  3  where Dm ≜ Cm − γmC ¯m, ∀m = 1, 2, · · ·, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' We observe  that (11b), (11c), (11d), and (12) take the form: wHFw ≤ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  Therefore, Problem (11) can be reformulated as  min  w  ∥w∥2  2 + η∥w∥2,1  (13a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' wHFlw ≤ fl, ∀l = 1, 2, · · ·, L,  (13b)  where the subscript ·l is used to indicate the l-th constraint  in Problem (11), and L is the total number of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  It is worth noting that Fl corresponding to −Aθi is neg-  ative semidefinite, and Fl corresponding to −Dm could be  indefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Therefore, in general, Problem (13) is non-convex  and thus NP-hard [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' To solve this problem, we propose an  algorithm based on the consensus ADMM, which is capable  of handling all the constraints in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Proposed Algorithm  Firstly, we formulate Problem (13) by introducing L auxil-  iary variables {vl ∈ CMN}L  l=1, and settle the original variable  w and the auxiliary variables {vl}l in a separable fashion, as  min  w,{vl} ∥w∥2  2 + η  L  L  �  i=1  ∥vl∥2,1  (14a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  vH  l Flvl ≤ fl, vl = w, ∀l = 1, 2, · · ·, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  (14b)  Next, we form the scaled-form augmented Lagrangian func-  tion related to the above problem, as: L(w, {vl}, {ul}) =  ∥w∥2  2 + η  L  �L  l=1∥vl∥2,1 + ρ  2  �L  l=1  �  ∥vl−w + ul∥2  2 − ∥ul∥2  2  �  ,  where ρ > 0 stands for the augmented Lagrangian parameter,  and ul is the scaled dual variable corresponding to the equality  constraint vl = w in Problem (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  Finally, the consensus ADMM updating equations can be  written down as  w ← arg min  w  ∥w∥2  2 + ρ  2  L  �  l=1  ∥vl − w + ul∥2  2,  (15a)  vl ←  �  arg min  vl  η  L∥vl∥2,1 + ρ  2∥vl − w + ul∥2  2  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  vH  l Flvl ≤ fl,  (15b)  ul ← ul + vl − w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  (15c)  In what follows, we show how to solve w and vl from (15a)  and (15b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' We start by solving w from (15a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' It  is straightforward to see that, by calculating the derivative of  the objective function of (15a) with respect to (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=') w and  setting it to 0, we obtain the solution to (15a) as  �w =  ρ  2 + ρL  L  �  l=1  (vl + ul).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  (16)  On the other hand, to solve vl from (15b), we firstly consider  the unconstrained minimization problem:  min  vl  f(vl) ≜ η  L∥vl∥2,1 + ρ  2∥vl − w + ul∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  (17)  The derivative of f(vl) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' vl is calculated as  ∇vlf(vl) =  � η  L (IM ⊗ G) + ρIMN  �  vl − ρ(w − ul),  where G∈RN×N is a diagonal matrix with diagonal being  1⊘  �  �  �  �  �  �  M  �  m=1  |vl(1+(m−1)N)|2, · · ·,  �  �  �  �  M  �  m=1  |vl(N +(m−1)N)|2  �  �  T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  By setting the derivative to 0, we obtain  � η  L (IM ⊗ G) + ρIMN  �  vl = ρ(w − ul).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  (18)  Since  � η  L (IM ⊗G)+ρIMN  �  is real-valued, ∠vl = ∠(w−ul).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  Thus, we only need to calculate the modulus of vl, using  � η  L (IM ⊗ G) + ρIMN  �  |vl| = ρ|w − ul|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  (19)  By exploring the structure of IM ⊗G, we observe that there  are N blocks each containing M equal entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Extracting the  rows of the equal entries yields  �  η  L∥vl(n)∥2  + ρ  �  |vl(n)| = ρ|cl(n)|,  (20)  ∀n = 1, 2, · · ·, N, where vl(n) ≜ [vl(n), vl(n+N), · · ·, vl(n+  (M−1)N)]T ∈ CM, and cl(n) ∈ CM contains the correspond-  ing entries of (w − ul).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' From (20), we have  |vl(n)| =  ρL∥vl(n)∥2  η + ρL∥vl(n)∥2  |cl(n)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  (21)  Performing the element-wise square operation, we have  |vl(n)|2 =  ρ2L2∥vl(n)∥2  2  η2 + ρ2L2∥vl(n)∥2  2 + 2ηρL∥vl(n)∥2  |cl(n)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' (22)  Hence, we further have  ∥vl(n)∥2  2 =1T|vl(n)|2  (23a)  =  ρ2L2∥vl(n)∥2  2  η2+ρ2L2∥vl(n)∥2  2+2ηρL∥vl(n)∥2  1T|cl(n)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' (23b)  The above equation leads to  ρ2L2∥vl(n)∥2  2+2ηρL∥vl(n)∥2+η2−ρ2L21T|cl(n)|2 = 0, (24)  the left-hand side of which is a simple quadratic function w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  ∥vl(n)∥2, and its unique1 root is given as  ∥vl(n)∥2 =  ρL  �  1T|cl(n)|2 − η  ρL  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  (25)  Then, by substituting (25) into (21), we obtain  |vl(n)| =  ρL  �  1T|cl(n)|2 − η  ρL  �  1T|cl(n)|2  |cl(n)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  (26)  In (26), we define |vl(n)| = 0 if |cl(n)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The solution for  (18), referred to as ¯vl, is finally obtained by combining vl(n)  (which can be calculated as |vl(n)|eȷ∠vl(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Then, the solution  to (15b), denoted by �vl, is found via the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' If ρ satisfies ρ  2 ≫ η  L, then �vl can be solved via:  �vl ← arg min  vl  ∥vl − ¯vl∥2  2  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' vH  l Flvl ≤ fl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  (27)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  1Note that the quadratic function in (24) has two roots, one positive and one  negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' In our case, the negative one is omitted, since its root ∥vl(n)∥2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  4  Algorithm 1 Consensus ADMM for solving Problem (13)  Input: η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' kmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Fl ∈CMN×MN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' ∀l = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' L  Output: �w ∈ CMN  Initialize: �v(0)  l  ←v(init)  l  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' �u(0)  l  ←u(init)  l  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' k←0  1: while k < kmax do  2:  �w(k+1) ←  ρ  2+ρL  L  �  l=1  �  �v(k)  l  + �u(k)  l  �  3:  for each l = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' L do  4:  cl ← �w(k+1) − �u(k)  l  5:  ∠¯v(k+1)  l  ← ∠cl  6:  for each n = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' N do  7:  |¯v(k+1)  l(n) | ←  ρL√  1T|cl(n)|2−η  ρL√  1T|cl(n)|2 |cl(n)|  8:  ¯v(k+1)  l(n)  ← |¯v(k+1)  l(n) |eȷ∠¯v(k+1)  l(n)  9:  end for  10:  Construct ¯v(k+1)  l  using ¯v(k+1)  l(n) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' ∀n  11:  if ¯v(k+1)H  l  Fl¯v(k+1)  l  ≤fl then �v(k+1)  l  ←¯v(k+1)  l  12:  else �v(k+1)  l  ←  �  arg min  vl  ∥vl − ¯v(k+1)  l  ∥2  2  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  vH  l Flvl ≤ fl  13:  end if  14:  �u(k+1)  l  ← �u(k)  l  + �v(k+1)  l  − �w(k+1)  15:  end for  16:  k ← k + 1  17: end while  18: �w ← �w(k)  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Note that η is related to the sparsity of the solution  of Problem (10), and L is the total number of constraints in  Problem (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Both of them are known for a specific problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  Hence, it is easy to choose a ρ such that ρ  2 ≫ η  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' If ¯vl satisfies the constraint of (27), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=', ¯vH  l Fl¯vl ≤  fl, it is easy to have �vl = ¯vl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Otherwise, notice that Problem  (27) is a QCQP with one constraint, which can be solved  optimally despite that Fl may be indefinite [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  So far, we have presented how to solve w and vl from (15a)  and (15b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Note that {vl}L  l=1 and {ul}L  l=1 can be  calculated in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The complete consensus ADMM for  solving Problem (13) is summarized in Algorithm 1, in which  kmax is used to terminate the iteration, and the superscript ·(k)  denotes the corresponding variable at the k-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  In Parallel  In Parallel  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' SIMULATION RESULTS  In this section, we evaluate the performance of the proposed  algorithm compared with the feasible point pursuit successive  convex approximation (FPP-SCA) method [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' FPP-SCA was  proposed for general QCQP and it is adapted to our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  Note that unlike our solution, no closed-form solution of FPP-  SCA is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The following two metrics are adopted: the Tx-  Power defined in (9a) and the mainlobe-to-sidelobe response  ratio (MSRR) defined as MSRR =  �  θi∈B  �M  m=1|wH  ma(θi)|  2  �  θj ∈ ¯  B  �M  m=1|wH  ma(θj)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  We first consider a transmit system with a uniform linear  array of N = 10 antennas and K = 8 or 10 RF chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  There are M = 2 users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The mainlobe and sidelobe are B =  [−5◦, 5◦] and ¯B = [−90◦, −20◦]∪[20◦, 90◦], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The  90  60  30  0  30  60  90  Angle (Degree)  0  2  4  6  8  10  12  14  Beampattern  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' 2: Beampattern comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  4  6  8  10  Number of Selected Sensors  2  4  6  8  10  12  14  TxPower (dB)  4  6  8  10  5  0  5  10  MSRR (dB)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' 3: TxPower (left) and MSRR (right) versus K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  thresholds for the mainlobe and sidelobe response are ϵp = 10  and ϵs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The maximum power radiated by each antenna  is Pn = 40 dBm, ∀n, the same as [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The noise variance is set  to σ2  m = 1, ∀m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The threshold for the received SINR is γm =  10 dB, ∀m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The number of constraints in Problem (13) is  L = 53, the tuning parameter is2 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='1, and the augmented  Lagrangian parameter is ρ = 50 (ρ/2 ≫ η/L is satisfied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The  value of v(init)  l  is given by any feasible point, while u(init)  l  = 0  and kmax = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The beampatterns are drawn in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' 2, which  indicates that the proposed method has nearly perfect response  controls in both mainlobe and sidelobe while the FPP-SCA has  lower mainlobe and higher sidelobe correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  Next, we examine the TxPower and MSRR versus the  number of selected sensors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' We also consider a strategy  of randomly selecting K sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The parameters are the  same as those in the last example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' 500 Monte-Carlo trials are  performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The results are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' It is seen that  our proposed method has the lowest transmit power and the  highest MSRR, among all tested methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  Finally, we test the TxPower and MSRR versus the number  of users, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The number of selected antennas is fixed as  2We do not study the relationship between η and the sparsity of the solution,  because of the space limitation of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Instead, when we obtain �w, we  choose K antennas corresponding to the largest (in an ℓ2,1-norm sense) K  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' We have good results when the tuning parameter is η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  5  2  4  6  Number of Users  4  6  8  10  12  14  16  18  TxPower (dB)  2  4  6  5  0  5  10  15  MSRR (dB)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' 4: TxPower (left) and MSRR (right) versus M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  K = 8, and the other parameters are unchanged as in the last  example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The results are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' 4, which again show  that our algorithm leads to a more power-efficient solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' CONCLUSION  We studied the problem of sparse array design for dual-  function radar-communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Our design aimed at main-  taining good control in both mainlobe and sidelobe, and also  to keep the signal-to-interference-plus-noise ratio for the users  larger than a certain level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' In addition, we considered the limi-  tation of the radiated power by each antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The problem was  formulated as a quadratically constrained quadratic program,  and solved by consensus alternating direction method of multi-  pliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' The proposed algorithm was able to be implemented in  parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Simulation results demonstrated better performance  of the proposed algorithm than the other tested methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  APPENDIX A  PROOF OF THEOREM  Solving (27) is equal to finding a point in {vl :vH  l Flvl ≤fl},  such that it is closest (in an ℓ2-norm sense) to ¯vl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Hence,  the theorem equivalently states that the solution to Problem  (15b) is the point closest (in an ℓ2-norm sense) to ¯vl, provided  that ρ/2 ≫ η/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' To show this, we denote �vl as the point in  {vl : vH  l Flvl ≤ fl}, such that  ∥�vl − ¯vl∥2 ≤ ∥vl − ¯vl∥2  (28)  holds for any vl ∈ {vl : vH  l Flvl ≤ fl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Our goal is to show  f(�vl) ≤ f(vl), for any vl ∈ {vl : vH  l Flvl ≤ fl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  The Lagrangian parameter ρ is chosen as ρ = Cη/L, where  C is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Then, |g(vl) − f(vl)| → 0 as C → ∞ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=',  ρ/2 ≫ η/L), where g(vl) ≜ ρ  2∥vl − w + ul∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Moreover,  f(vl) = g(vl) = ρ  2∥vl − ¯vl∥2  2,  (29)  as long as C → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Note that, in the second equality above,  we used the fact that ¯vl = w − ul as C → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content='  Suppose that there exists a point ˘vl ∈ {vl : vH  l Flvl ≤ fl},  such that f(˘vl) < f(�vl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Thus, by using (29), we obtain that  ρ  2∥˘vl − ¯vl∥2  2 <  ρ  2∥�vl − ¯vl∥2  2, which contradicts (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' This  implies that f(�vl) ≤ f(vl) holds for all feasible vl, that is,  �vl is the solution to Problem (15b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
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+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
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+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
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+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf'}
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@@ -0,0 +1,1280 @@
+1 
+ 
+TITLE: 
+Joint k-TE Space Image Reconstruction and Data Fitting for T2 Mapping 
+ 
+RUNNING TITLE: 
+     Joint k-TE Reconstructed T2 Mapping 
+ 
+AUTHORS: 
+Yan Dai B.S. 1, Xun Jia Ph.D.2, Yen-Peng Liao, PhD1, Jiaen Liu, PhD3, Jie Deng Ph.D.1 
+ 
+1 Department of Radiation Oncology, University of Texas Southwestern Medical Centre, TX, USA 
+2 Department of Radiation Oncology and Molecular Radiation Sciences, Johns Hopkins 
+University, MD, USA 
+3 Advanced Imaging Research Center, University of Texas Southwestern Medical Centre, TX, 
+USA 
+ 
+CORRESPONDENCE: 
+Jie Deng, Ph.D. 
+214-645-5140 
+Department of Radiation Oncology,  
+University of Texas Southwestern Medical Centre, 2280 Inwood Rd, Dallas. 
+Email: Jie.Deng@UTSouthwestern.edu 
+ 
+ACKNOWLEDGEMENT 
+This work was supported in part by the Cancer Prevention and Research Institute of Texas 
+(grant \#RP200573) and the National Cancer Institute (grant \#R01CA227289). 
+ 
+
+2 
+ 
+ABSTRACT 
+Objectives: To develop a joint k-TE reconstruction algorithm to reconstruct the T2-weighted 
+(T2W) images and T2 map simultaneously. 
+Materials and Methods: The joint k-TE reconstruction model was formulated as an 
+optimization problem subject to a self-consistency condition of the exponential decay relationship 
+between the T2W images and T2 map. The objective function included a data fidelity term 
+enforcing the agreement between the solution and the measured k-space data, together with a 
+spatial regularization term on image properties of the T2W images. The optimization problem was 
+solved using Alternating-Direction Method of Multipliers (ADMM). We tested the joint k-TE 
+method in phantom data and healthy volunteer scans with fully-sampled and under-sampled k-
+space lines. Image quality of the reconstructed T2W images and T2 map, and the accuracy of T2 
+measurements derived by the joint k- TE and the conventional signal fitting method were 
+compared. 
+Results: The proposed method improved image quality with reduced noise and less artifacts on 
+both T2W images and T2 map, and increased measurement consistency in T2 relaxation time 
+measurements compared with the conventional method in all data sets. 
+Conclusions: The proposed reconstruction method outperformed the conventional magnitude 
+image-based signal fitting method in image quality and stability of quantitative T2 measurements. 
+Key words: Quantitative MRI, joint reconstruction, under-sampled reconstruction, T2 
+relaxation, optimization, denoising 
+ 
+ 
+
+3 
+ 
+1. INTRODUCTION  
+T2-weighted (T2W) MR images provide an important image contrast mechanism resulting 
+from the difference of tissue transverse relaxation time for anatomical delineation, disease 
+diagnosis, and tissue characterization1,2. The T2 relaxation time measurement derived from a series 
+of T2W images acquired at different echo-times (TEs) based on a mono-exponential signal decay 
+model provided a quantitative method for tissue characterization, which has been used for cancer 
+diagnosis and treatment response assessment 3-5.  Pixel-by-pixel T2 measurements, called T2 
+mapping, has been integrated into various clinical MRI protocols such as cardiovascular MRI for 
+the diagnosis of myocardial inflammation and edema6-9, and neuroimaging for brain maturation 
+evaluation 10-14. The conventional T2 mapping reconstruction method includes a two-step process, 
+in which the magnitude T2W image acquired at each TE is reconstructed from its own k-space 
+data, e.g., by Fast Fourier Transform (FFT), followed by a pixel-by-pixel  fitting of the T2W MR 
+signal decay to derive the T2 map. The accuracy of T2 measurement using the conventional 2-step 
+method is limited by low signal-to-noise-ratio (SNR), low spatial resolution, motion artifacts,  as 
+well as image quality degradation5,15,16. Acquiring a full set of T2W image based on spin echo 
+(SE) or fast spin echo (FSE) pulse sequences takes a relatively long imaging time, resulting in 
+pixel misregistration and thus inaccurate signal fitting. Other types of MRI pulse sequences have 
+also been used for T2 mapping such as T2-prepared balanced steady-state free precession (bSSFP), 
+and Gradient And Spin Echo (GraSE)17, a hybrid technique that acquires a series of gradient echoes 
+and spin echoes from a train of 180 radiofrequency pulses, were used for fast myocardial T2 
+mapping. In addition, various types of post-processing algorithms for deriving T2 map based on 
+reconstructed magnitude T2W images have been used but with the problems of noisy data fitting 
+and reproducibility155.  
+
+4 
+ 
+Parallel imaging techniques were introduced to accelerate MRI acquisition by estimating 
+missing data through coil-based calibration. The generalized auto-calibrating partially parallel 
+acquisitions(GRAPPA)18 is one of the parallel imaging methods that uses the linear relationship 
+between the acquired k-space lines and adjacent missing lines as a constraint in the block-wise 
+calibrations across all coil elements to estimate the missing data. This constraint was further 
+developed to calibrate between every single k-space data point and its neighboring data points 
+across all coil elements using a matrix called SPiRIT (iterative self-consistent parallel imaging 
+reconstruction) operator19. Similarly, spatial sparsity of an MR image was considered as prior 
+knowledge in image domain to reconstruct under-sampled data, namely compressed sensing (CS). 
+The most common form of CS-based reconstruction is to minimize a loss function consisting of a 
+data fidelity term and a specifically designed regularization term that penalizes violations the 
+sparsity assumption20-22. Essentially, MRI reconstruction can be generally viewed as an inverse 
+problem that aims to restore data from non-ideal measurements. Prior knowledge is useful in data 
+correction and constraining solutions as used in the above-mentioned parallel imaging and CS 
+techniques. More recently, neural network demonstrated good performance in MR image 
+reconstruction23-25. A U-net was proposed to reconstruct under-sampled T2W images given the 
+corresponding T1-weighted (T1W) images26. With the flexibility of neural network, the sparsity 
+between T2W images at different TEs and that between T2W image and T1W image can be well 
+integrated into the reconstruction of T2W images27.  
+In this study, we propose a joint k-TE space reconstruction method that exploits structural 
+correlations between different T2W images acquired at different TEs and the mono-exponential 
+decay relationship between T2W images and the corresponding T2 map as the regularization 
+terms28. Furthermore, the joint k-TE reconstruction algorithm can be applied to under-sampled 
+
+5 
+ 
+T2W images, in which CS was not only used in each T2W image itself but also between different 
+T2W images27,29-31. This work aims to reconstruct T2W images and T2 mapping simultaneously 
+by solving an optimization problem.  The error in an T2W image was iteratively corrected by the 
+information obtained from other T2W images at different TEs as well as from the corresponding 
+T2 map by enforcing the exponential decay constraint, which in turn also reduced the noise in T2 
+map. In addition, an image denoising filter algorithm was applied to T2W images as a prior to 
+restore image smoothness via the plug-and-play approach. We tested this joint k-TE reconstruction 
+method in both phantom data and healthy volunteer images, and compared the image quality and 
+accuracy of T2 measurements with those obtained by the conventional method for fully sampled 
+data and CS-based method for under-sampled data. 
+ 
+2. MATERIALS AND METHODS 
+2.A. Model for joint reconstruction and data fitting 
+Let us denote the measured k-space data at 𝑇𝐸𝑖 as 𝒈(𝑧, 𝑇𝐸𝑖) and the magnitude image to be 
+reconstructed as 𝒇(𝑥, 𝑇𝐸𝑖), with 𝑥 being the spatial coordinate. There exists a Fourier transform 
+relationship between them: 
+𝒈(𝑧, 𝑇𝐸𝑖) = 𝑺𝑭𝑨(𝑥, 𝑇𝐸𝑖)𝒇(𝑥, 𝑇𝐸𝑖) + 𝒏 
+eq. 1 
+where 𝒏 denotes noise signal in data acquisition. 𝑺 is the sampling matrix, 𝑭 is the Fourier 
+transform operator. 𝑨(𝑥, 𝑇𝐸𝑖) is the phase image. In this study, we assumed this is known and it 
+was estimated from the image obtained by inverse Fourier transform of 𝒈(𝑧, 𝑇𝐸𝑖) by taking its 
+phase factors. The T2W MR magnitude images at different TEs are related by the T2 relaxation 
+model: 
+
+6 
+ 
+𝒇(𝑥, 𝑇𝐸𝑖) = 𝒇(𝑥, 𝑇𝐸0)𝑒
+−𝑇𝐸𝑖−𝑇𝐸0
+𝑻𝟐(𝑥)  
+eq. 2 
+where 𝑻𝟐(𝑥) is the T2 map. 
+We proposed to solve the following optimization model to jointly estimate the image 𝒇(𝑥, 𝑇𝐸) 
+and the T2 map 𝑻𝟐(𝑥):  
+{𝒇∗, 𝑻𝟐∗} = argmin
+𝒇,𝐓𝟐
+∑|𝑺𝑭𝑨(𝑥, 𝑇𝐸𝑖)𝒇(𝑥, 𝑇𝐸𝑖) − 𝒈(𝑧, 𝑇𝐸𝑖)|2 + 𝑅[𝒇(𝑥, 𝑇𝐸𝑖), 𝜆]
+𝑖
+, 
+𝑠. 𝑡.⁡ 𝒇(𝑥, 𝑇𝐸𝑖) = 𝒇(𝑥, 0)𝑒
+−𝑇𝐸𝑖−𝑇𝐸0
+𝑻𝟐(𝑥) , for 𝑖 > 0 
+eq. 3 
+There are two terms in the objective function. The first one is a data fidelity term that ensures the 
+agreement between the reconstructed magnitude images 𝒇(𝑥, 𝑇𝐸𝑖)  and the corresponding 
+measurements 𝒈(𝑧, 𝑇𝐸𝑖). The second term is employed to provide regulation on MR image in the 
+spatial domain, where 𝜆 is the parameter in the regularization function. In this study, we used the 
+Block-matching and 3D filtering (BM3D) method for regularization32, which was employed via 
+the plug-and-play (PnP) approach presented in the next subsection. The constraint posts the 
+connection among MR images 𝒇(𝑥, 𝑇𝐸𝑖) and the T2 map 𝑻𝟐(𝑥).     
+2.B. Numerical algorithm and implementation 
+The Alternating Direction Method of Multipliers was employed to solve this optimization 
+problem33. As such, we first considered the optimization problem equivalent to eq. 3 by 
+introducing a variable 𝒗: 
+
+7 
+ 
+{𝒇, 𝑻𝟐, 𝒗} = argmin
+���,𝑻𝟐,𝒗
+1
+2 ∑|𝑺𝑭𝑨(𝑥, 𝑇𝐸𝑖)𝒇(𝑥, 𝑇𝐸𝑖) − 𝒈(𝑧, 𝑇𝐸𝑖)|2 + 𝑅[𝒗(𝑥, 𝑇𝐸𝑖), 𝜆]
+𝑖
+, 
+𝑠. 𝑡 . 𝒇(𝑥, 𝑇𝐸𝑖) = 𝒇(𝑥, 𝑇𝐸0)𝑒
+−⁡𝑇𝐸𝑖−𝑇𝐸0
+𝑻𝟐(𝑥) , for 𝑖 > 0 ; ⁡𝒗(𝑥, 𝑇𝐸𝑖) = ⁡𝒇(𝑥, 𝑇𝐸𝑖). 
+eq. 4 
+The augmented Lagrangian of this problem is 
+𝐿𝜌 = 1
+2 ∑|𝑺𝑭𝑨(𝑥, 𝑇𝐸𝑖)𝒇(𝑥, 𝑇𝐸𝑖) − 𝒈(𝑧, 𝑇𝐸𝑖)|2 + 𝑅[𝒗(𝑥, 𝑇𝐸𝑖), 𝜆]
+𝑖
++ ∑ 𝒚𝑖
+𝑇 [𝒇(𝑥, 𝑇𝐸𝑖) − 𝒇(𝑥, 𝑇𝐸0)𝑒
+−⁡𝑇𝐸𝑖−𝑇𝐸0
+𝑻𝟐(𝑥) ]
+𝑖>0
++ ∑ 𝜌
+2 |𝒇(𝑥, 𝑇𝐸𝑖) − 𝒇(𝑥, 𝑇𝐸0)𝑒
+−⁡𝑇𝐸𝑖−𝑇𝐸0
+𝑻𝟐(𝑥) |
+2
+𝑖>0
++ ∑ 𝑧𝑖
+𝑇[𝒗(𝑥, 𝑇𝐸𝑖) − 𝒇(𝑥, 𝑇𝐸𝑖)]
+𝑖
++ ∑ ⁡𝜌
+2 |𝒗(𝑥, 𝑇𝐸𝑖) − 𝒇(𝑥, 𝑇𝐸𝑖)|2
+𝑖
+ 
+eq. 5 
+where 𝒚𝑖, 𝒛𝑖 and 𝜌 are variables introduced in the algorithm. The ADMM solved the optimization 
+problem by iteratively performing the following steps with 𝑘 being the index of iteration: 
+𝒇(𝑘+1)(𝑥, 𝑇𝐸𝑖) = argmin
+𝒇(𝑥,𝑇𝐸𝑖)
+1
+2 |𝑺𝑭𝑨(𝑥, 𝑇𝐸𝑖)𝒇(𝑥, 𝑇𝐸𝑖) − 𝒈(𝑧, 𝑇𝐸𝑖)|2
++ 𝒚𝒊
+𝑇 [𝒇(𝑥, 𝑇𝐸𝑖) − 𝒇(𝑘)(𝑥, 𝑇𝐸0)𝑒
+−⁡𝑇𝐸𝑖−𝑇𝐸0
+𝑻𝟐(𝑘)(𝑥)]
++ 𝜌
+2 |𝒇(𝑥, 𝑇𝐸𝑖) − 𝒇(𝑘)(𝑥, 𝑇𝐸0)𝑒
+−⁡𝑇𝐸𝑖−𝑇𝐸0
+𝑻𝟐(𝑘)(𝑥)|
+2
++ 𝒛𝑖
+𝑇[𝒗(𝑘)(𝑥, 𝑇𝐸𝑖) − 𝒇(𝑥, 𝑇𝐸𝑖)] + 𝜌
+2 |𝒗(𝑘)(𝑥, 𝑇𝐸𝑖) − 𝒇(𝑥, 𝑇𝐸𝑖)|
+2,
+for⁡𝑖 > 0 
+eq. 6 
+
+8 
+ 
+ 
+𝒇(𝑘+1)(𝑥, 𝑇𝐸0) = argmin
+𝒇(𝑥,𝑇𝐸0)
+1
+2 |𝑺𝑭𝑨(𝑥, 𝑇𝐸0)𝒇(𝑥, 𝑇𝐸0) − 𝒈(𝑧, 𝑇𝐸0)|2
++ ∑ 𝒚𝒊
+𝑇 [𝒇(𝑘+1)(𝑥, 𝑇𝐸𝑖) − 𝒇(𝑥, 𝑇𝐸0)𝑒
+−⁡𝑇𝐸𝑖−𝑇𝐸0
+𝑻𝟐(𝑘)(𝑥)]
+𝑖>0
++ ∑ 𝜌
+2 |𝒇(𝑘+1)(𝑥, 𝑇𝐸𝑖) − 𝒇(𝑥, 𝑇𝐸0)𝑒
+−⁡𝑇𝐸𝑖−𝑇𝐸0
+𝑻𝟐(𝑘)(𝑥)|
+2
+𝑖>0
++ 𝒛𝒊
+𝑇[𝒗(𝑘)(𝑥, 𝑇𝐸0)
+− 𝒇(𝑥, 𝑇𝐸0)] + 𝜌
+2 |𝒗(𝑘)(𝑥, 𝑇𝐸0) − 𝒇(𝑥, 𝑇𝐸0)|
+2 
+eq. 7 
+ 
+1
+𝑻𝟐(𝑘+1)(𝑥) = argmin
+1
+𝑻𝟐(𝑥)
+∑ 𝒚𝑖
+𝑇 [𝒇(𝑘+1)(𝑥, 𝑇𝐸𝑖) − 𝒇(𝑘+1)(𝑥, 𝑇𝐸0)𝑒
+−⁡𝑇𝐸𝑖−𝑇𝐸0
+𝑻𝟐(𝑘)(𝑥)]
+𝑖>0
++ ∑ 𝜌
+2 |𝒇(𝑘+1)(𝑥, 𝑇𝐸𝑖) − 𝒇(𝑘+1)(𝑥, 𝑇𝐸0)𝑒
+−⁡𝑇𝐸𝑖−𝑇𝐸0
+𝑻𝟐(𝑘)(𝑥)|
+2
+𝑖>0
+ 
+eq. 8 
+ 
+𝒗(𝑘+1)(𝑥, 𝑇𝐸𝑖) = argmin
+𝒗(x,𝑇𝐸𝑖) ⁡𝑅[𝒗(𝑥, 𝑇𝐸𝑖), 𝜆] + 𝒛𝑖
+𝑇[𝒗(𝑥, 𝑇𝐸𝑖) − 𝒇(𝑘+1)(𝑥, 𝑇𝐸𝑖)]
++ 𝜌
+2 |𝒗(𝑥, 𝑇𝐸𝑖) − 𝒇(𝑘+1)(𝑥, 𝑇𝐸𝑖)|
+2 
+eq. 9 
+ 
+𝒚𝑖
+(𝑘+1) = 𝒚𝑖
+(𝑘) + 𝜌[𝒇(𝑘+1)(𝑥, 𝑇𝐸𝑖) − 𝒇(𝑘+1)(𝑥, 𝑇𝐸0)𝑒
+−⁡ 𝑇𝐸𝑖−𝑇𝐸0
+𝑻𝟐(𝑘+1)(𝑥)] 
+eq. 10 
+ 
+
+9 
+ 
+𝒛𝒊
+(𝑘+1) = 𝒛𝒊
+(𝑘) + 𝜌[𝒗(𝑘+1)(𝑥, 𝑇𝐸𝑖) − 𝒇(𝑘+1)(𝑥, 𝑇𝐸𝑖)] 
+eq. 11 
+ 
+The first two subproblems in eq. 6 and eq. 7 with respect to the images 𝒇(𝑥, 𝑇𝐸𝑖), 𝑖 = 0,1, …⁡, 
+are quadratic optimization problems, and the solutions are computed by solving the linear equation 
+corresponding to the optimality condition.  
+The subproblem in eq. 8 with respect to T2 map 𝑻𝟐(𝑥) is essentially data fitting to determine 
+the T2 map 𝑻𝟐(𝑥) by fitting data 𝒇(𝑘+1)(𝑥, 𝑇𝐸𝑖) pixelwise in an exponential decay function form. 
+This was achieved by using a weighted least square (WLS) fitting (eq. 13) of a linear function (eq. 
+12),  where the weight 𝑤𝑏 for each term should be inversely proportional to the measurement 
+uncertainty of  
+log
+𝒇(𝑘+1)(𝑥,𝑇𝐸𝑖)
+𝒇(𝑘+1)(𝑥,𝑇𝐸0)
+𝑇𝐸𝑖−⁡𝑇𝐸0
+. Generally, as this is only a subproblem of the iterative process, it is 
+not necessary to solve this subproblem accurately at each iteration. Hence, a first order 
+approximation of the measurement uncertainty was used as shown in eq. 14, where 𝜎 is the 
+variance of noise in the measurement of 𝒇(𝑘+1)(𝑥, 𝑇𝐸𝑖) and its measurement is described later.  
+log[𝒇(𝑘+1)(𝑥, 𝑇𝐸𝑖)] = log[𝒇(𝑘+1)(𝑥, 𝑇𝐸0)] − 𝑇𝐸𝑖 − 𝑇𝐸0
+𝑻𝟐(𝑥)
+ 
+eq. 12 
+1
+𝑻𝟐(𝑘+1)(𝑥)⁡= argmin
+1
+𝑻𝟐(𝑥)
+∑ 𝑤𝑖 |log 𝒇(𝑘+1)(𝑥, 𝑇𝐸𝑖)
+𝒇(𝑘+1)(𝑥, 𝑇𝐸0) + 𝑇𝐸𝑖 − 𝑇𝐸0
+𝑻𝟐(𝑥)
+|
+𝑖
+2
+ 
+eq. 13 
+𝑤𝑖 ∝ (−
+1
+𝑇𝐸𝑖 − 𝑇𝐸0
+log(𝒇(𝑘+1)(𝑥, 𝑇𝐸𝑖) + 𝜎)
++
+1
+𝑇𝐸𝑖 − 𝑇𝐸0
+log(𝒇(𝑘+1)(𝑥, 𝑇𝐸𝑖) − 𝜎))
+−1
+ 
+eq. 14 
+
+10 
+ 
+As for the subproblem defined in eq. 9, which is a essentially image domain processing 
+problem, the PnP approach34 was employed. The PnP method enabled plugging in a powerful 
+image denoising algorithm and its validity has been empirically demonstrated in previous studies 
+in terms of producing high-quality results in various applications35-37.  
+For processing of image 𝒗(𝑥, 𝑇𝐸𝑖), assuming that the noise in k-space is Gaussian, the noise in 
+the magnitude MR image will still follow a Gaussian distribution after phase correction, or can be 
+approximated as Gaussian distribution when 𝑆𝑁𝑅 > 338. In this study, BM3D method32 was used 
+in the PnP framework for Gaussian noise removal, and this step is denoted as:  
+𝒗(𝑘+1)(𝑥, 𝑇𝐸𝑖) = 𝐵𝑀3𝐷[𝒇(𝑘+1)(𝑥, 𝑇𝐸𝑖), 𝜎] 
+eq. 15 
+where 𝐵𝑀3D(. ) stands for the BM3D denoising operation. It needs to be mentioned that the 
+BM3D algorithm requires a hyper-parameter 𝜎 specifying the Gaussian noise standard deviation. 
+To estimate the 𝜎 in MR images, which was also used as the measurement uncertainty in weighted 
+linear regression for T2 map fitting, we first removed the background with a threshold-based 
+segmentation algorithm and then estimated the local noise standard deviation of patches with a 
+size of 5 × 5 pixels. The mode value of these local noise variance values was computed as an 
+approximation of the noise standard deviation. Then this value was either directly used as 
+parameter 𝜎 or multiplied with 0.5 as a conservative estimation to preserve more details in the 
+image.  
+The iterative process in eq. 6-eq. 11 continued until convergence, when the mean relative 
+intensity change of 𝒇(𝑥, 𝑇𝐸𝑖) in two successive iteration steps is less than a threshold 𝜖. Note that 
+due to the modifications to the ADMM algorithm and the application of PnP framework, 
+theoretical convergence of this iterative process was not guaranteed. 𝜖 was set as 1% in this study. 
+
+11 
+ 
+The algorithm used to solve the joint reconstruction and data fitting problem is summarized in 
+Algorithm 1. Solving the problem in eq. 3. The algorithm was implemented in Python 3.8, and 
+computation was performed on a workstation with an Intel(R) Xeon(R) Gold 6230R CPU of 
+2.1GHz frequency. The other adjustable parameter in this algorithm, 𝜌, was quite robust and was 
+set as 0.5 for all the applications.  
+ Algorithm 1. Solving the problem in eq. 3. 
+Input:  
+K-space data 𝒈(𝑧, 𝑇𝐸𝑖) with at least 3 different 𝑇𝐸⁡values, sampling 
+matrix 𝑺 and phase matrix 𝑨  
+Parameters: 
+𝜌, 𝜖 
+Initialize: 
+ 
+ 
+𝒗(0)(𝑥, 𝑇𝐸𝑖) =⁡𝒇(0)(𝑥, 𝑇𝐸𝑖) =⁡𝑨−1𝑭−1𝑺𝒈(𝑧, 𝑇𝐸𝑖), 
+ 
+1
+𝑻𝟐(0)(𝑥) = argmin
+1
+𝑻𝟐(𝑥)
+⁡ ∑ 𝑤𝑖 |log 𝒇(0)(𝑥, 𝑇𝐸𝑖)
+𝒇(0)(𝑥, 𝑇𝐸0) + 𝑇𝐸𝑖 − 𝑇𝐸0
+𝑻𝟐(𝑥)
+|
+2
+,
+𝑖
+⁡ 
+ 
+𝒚𝑖
+(0) = 𝒛𝑖
+(0) = 0, and 𝑘⁡ = ⁡0 
+While: 
+ 
+ 
+mean (
+|𝒇(𝑘+1)(𝑥,𝑇𝐸𝑖)−𝒇(𝑘)(𝑥,𝑇𝐸𝑖)|
+𝒇(𝑘)(𝑥,𝑇𝐸𝑖)
+)⁡> 𝜖⁡or 𝑘 = 0 
+Do: 
+ 
+ 
+𝑘⁡ = ⁡𝑘⁡ + ⁡1 
+ 
+Solve problems in eq. 6 and eq. 7 using CGLS algorithm 
+ 
+Update 𝑻𝟐 by pixelwise data fitting eq. 13 
+ 
+Update 𝒗 using BM3D algorithm in eq. 15 
+ 
+Update 𝒚𝑖, 𝒛𝑖 by eq. 10 and eq. 10 
+End while 
+ 
+Return:  
+𝒇∗ = 𝒇(𝑘+1) and 𝑫∗ = 𝑫(𝑘+1) 
+ 
+ 
+ 
+2.C. Evaluation 
+
+12 
+ 
+The joint reconstruction algorithm was evaluated in phantom and patient studies. The study 
+protocol was approved by our Institutional Review Board and written informed consent was 
+waived from each subject. T2W imaging data sets were acquired on a phantom (Essential System 
+Phantom for Relaxometry, Model 106, CaliberMRI, Boulder, CO) consisting of 14 vials with 
+different concentration of MnCl2 solution providing a T2 range from 8 to 850 𝑚𝑠.  measured on 
+1.5T at 20C. MR images were acquired on a 1.5T clinical MRI scanner (Ingenia Ambition X, 
+Philips Healthcare, Netherlands) with a 20-channel brain coil. In phantom study, T2W imaging 
+was acquired using a multi-echo FSE sequence following the vendor-provided protocol: field-of-
+view (FOV) = ⁡250 × ⁡250⁡𝑚𝑚2, slice thickness/gap = 6/0 mm, matrix size = 256 × ⁡256, 
+repetition time (TR) = 5000 ms, eleven TEs from 11 to 176 ms with 11 ms step,  bandwidth = 170 
+Hz/pixel, number of excitation (NEX) = 1, echo train length = 16, acquisition time = 21 min 25 
+𝑠𝑒𝑐. In subject scan, T2W imaging was acquired using a GraSE sequence with the following 
+parameters: field-of-view (FOV) = ⁡256 × ⁡207⁡𝑚𝑚2, slice thickness/gap = 3/9 mm, matrix size 
+= 400 × ⁡400, repetition time (TR) = 1892 ms, 32 TEs from 14.4 to 461.6 ms with 14.4 ms step,  
+bandwidth = 691 Hz/pixel, number of excitation (NEX) = 2, echo train length = 32, echo planar 
+factor = 5, acquisition time =2 min 42 sec. 
+For both phantom and subject scans, each T2W image was converted to the corresponding k-
+space data through FFT. Furthermore, we tested the effectiveness of the proposed joint k-TE 
+reconstruction method in reconstructing under-sampled k-space data, in which the central 10% k-
+space data was kept while 25% and 33% of the rest 90% k-space data was randomly discarded. A 
+CS-based reconstruction algorithm39 was used as the conventional reconstruction method to 
+reconstruct the T2W image from the under-sampled data. Each MR image was reconstructed under 
+the regularization of 1D Total Variation (TV) as shown in eq.16, where  𝑭𝑝ℎ𝑎𝑠𝑒
+1𝐷
+, 𝑭𝑓𝑟𝑒𝑞
+1𝐷 ⁡and⁡𝑇𝑉𝑝ℎ𝑎𝑠𝑒
+1𝐷
+ 
+
+13 
+ 
+denoted 1D Fourier transform applied along phase direction, 1D Fourier transform applied along 
+frequency direction, and 1D TV applied along phase encoding direction, respectively. 
+𝒇∗ = argmin
+𝒇
+∑ |𝑺𝑭𝑓𝑟𝑒𝑞
+1𝐷 𝑭𝑝ℎ𝑎𝑠𝑒
+1𝐷
+𝑨𝒇(𝑥, 𝑇𝐸𝑖) − 𝒈(𝑧, 𝑇𝐸𝑖)|
+𝑖
++ 𝑇𝑉𝑝ℎ𝑎𝑠𝑒
+1𝐷
+[𝑭𝑝ℎ𝑎𝑠𝑒
+1𝐷
+𝑨𝒇(𝑥, 𝑇𝐸𝑖)] 
+eq. 16 
+To numerically evaluate the performance of the proposed joint k- TE reconstruction method on 
+both fully sampled and under-sampled phantom data, mean and standard deviation of T2 
+measurements were calculated in each vial of the phantom with ground truth T2 values, as well as 
+in three regions-of-interests (ROIs) including white matter (WM), gray matter (GM), and ventricle 
+on subject images. The joint reconstruction results were compared with those computed using the 
+conventional 2-step WLS fitting method on the fully sampled data. 
+3. RESULTS 
+In all of the fully sampled and under-sampled datasets of phantom and subject scans, the 
+proposed algorithm converged within 10 iterations in fully sampled data, as shown in Figure. 1. It 
+took slightly more iterations for the algorithm to converge on the under-sampled data due to the 
+ill-defined problem. The mean relative change (MRC) of the image intensity increased at the 
+beginning of iterations and we started to check it from the 4th iteration. 
+
+14 
+ 
+` 
+   
+Figure. 1. The convergence plot of the proposed algorithm, shown as the mean relative change (MRC) 
+along with the number of iterations in fully sampled phantom data (A), 25% under-sampled phantom data 
+(B), 33% under-sampled phantom data (C), fully sampled patient data (D), 25% under-sampled patient 
+data (E), and 33% under-sampled patient data (F). The horizontal dashed red line in each subplot shows 
+the stopping criteria in each data set.  
+ 
+3.A. Image Quality Evaluation 
+Image quality comparison between fully sampled, under-sampled joint reconstruction methods, 
+and full sampled conventional method were shown in Fig. 2 (phantom) and Fig. 3 (subject). The 
+joint reconstruction method demonstrated improved image quality with less noise on T2W images 
+at low, medium, and high TEs and the corresponding T2 map in both fully sampled and under-
+sampled data sets. The proposed algorithm outperformed the conventional CS method in under-
+sampled image reconstruction by removing the aliasing artifacts more effectively.  
+
+A
+B
+c
+109
+MRimage
+MR image
+MR image
+R images(%) 
+: of MR images(%)
+ of MR images(%)
+107
+105
+MRC of MR 
+103
+MRC
+100
+MRC
+100
+101
+2
+3
+4
+5
+8
+6
+10 11
+Numberofiteration
+Number
+ofiteration
+Number
+ofiteration
+D
+E
+F
+109
+MRimage
+MR image
+MR image
+MRC of MR images(%)
+MRC of MR images(%)
+MRC of MR images(%)
+107
+101
+101
+105
+103
+101
+100
+100
+2
+6
+6
+10 11
+91011121314
+3
+6
+111315171921
+Numberofiteration
+Numberofiteration
+Numberofiteration15 
+ 
+ 
+   
+
+A
+TE=11.000ms
+TE=88.000ms
+TE=176.000ms
+T2
+Recon
+2-step
+e
+Recon
+Joint
+B
+a
+Recon
+CS
+e
+Recon
+Joint
+C
+a
+Recon
+e
+Recon
+Joint
+0
+500
+1000
+1500
+2000
+500
+1000
+15000
+400
+800
+1200100
+101
+102
+10316 
+ 
+ Figure 2. The reconstructed phantom images by the conventional 2-step and proposed joint 
+reconstruction method in the fully sampled (A), 25% under-sampled (B) and 33% under-sampled (C) 
+phantom data set. In all data sets, T2W MR images and the corresponding T2 map reconstructed by the 
+k-TE joint reconstruction (Joint Recon) method improved image quality compared with those 
+reconstructed by the conventional 2-step (2-step Recon) method for fully sample data and by the 
+compressed-sensing (CS Recon) based method for under-sampled data.  
+ 
+
+17 
+ 
+   
+ 
+
+TE=14.426ms
+TE=100.982ms
+TE=216.390ms
+T2
+a
+b
+2-step Recon
+e
+g
+h
+Joint Recon
+B
+a
+.
+d
+CS Recon
+e
+9
+h
+JointRecon
+c
+a
+b
+d
+CS Recon
+9
+JointRecon
+500
+10001500
+2000
+2500
+500
+1000
+15000
+250
+500
+750
+0
+150
+300>=35018 
+ 
+Figure 3. The reconstructed brain images by the conventional and proposed method in the fully 
+sampled (A), 25% under-sampled (B) and 33% under-sampled (C) subject brain data. In all data sets, 
+T2W MR images and the corresponding T2 map reconstructed by the k-TE joint reconstruction (Joint 
+Recon) method improved image quality compared with those reconstructed by the conventional 2-step (2-
+step Recon) method for fully sampled data and by the compressed-sensing (CS Recon) based method for 
+under-sampled data. 
+ 
+  3.B. Quantitative T2 Measurement Evaluation 
+The accuracy of T2 measurements in the Relaxometry phantom by the conventional and 
+proposed algorithms on both fully sampled and under-sampled data was shown in Table. 1. The 
+vendor provided T2 values for each vial in the phantom were used as the ground truth. The mean 
+and standard deviation of the pixel-wise T2 values were calculated within a ROI manually placed 
+in each vial of the phantom, theoretically all pixels within each ROI should have the same T2 
+value.  
+In the phantom study, the joint reconstruction method applied to both fully sampled and under-
+sampled data provided comparable mean T2 measurements as the conventional method in fully 
+sampled dataset, and they were consistent with the gold standard T2 values. In addition, the 
+variation in pixel-by-pixel T2 measurements was greatly reduced (i.e., less standard deviations) 
+using the joint reconstruction method in the full sample data set comparison. The mean T2 values 
+of the under-sampled reconstruction by the proposed algorithm showed minimal differences 
+compared with those reconstructed from the fully sampled data, however, the variations in T2 
+measurements of under-sampled data were higher than those of fully sampled reconstruction. 
+ 
+Table 1. Comparison of mean and standard deviation of T2 values reconstructed by the conventional 2-
+step reconstruction on fully sampled phantom data (Full 2-step Recon) and the proposed joint reconstruction 
+method on fully sampled phantom data (Full Joint Recon), 25% under-sampled phantom data (25% Joint 
+Recon) and 33% under-sampled phantom data (33% Joint Recon). 
+
+19 
+ 
+Grou
+nd Truth  
+(𝑚𝑠) 
+ 
+8.75 
+12.8 
+17.9 
+26.1 
+34.3 
+53.0 
+82.2 
+Full 
+2-step 
+Recon   
+mean 
+8.44 
+12.4 
+16.1 
+24.0 
+31.7 
+48.6 
+75.7 
+ 
+std 
+1.67 
+1.23 
+0.92
+4 
+1.01 
+0.77
+5 
+0.91
+6 
+0.86
+2 
+Full 
+Joint 
+Recon 
+mean 
+8.40 
+12.4 
+16.0 
+23.8 
+31.2 
+47.6 
+73.8 
+ 
+std 
+1.20 
+0.91
+8 
+0.75
+6 
+0.79
+8 
+0.57
+8 
+0.94
+5 
+0.91
+1 
+25% 
+Joint 
+Recon 
+mean 
+7.45 
+10.6 
+14.5 
+21.3 
+26.6 
+46.4 
+73.1 
+ 
+std 
+2.55 
+3.45 
+2.24 
+1.77 
+4.35 
+3.22 
+1.06 
+33% 
+Joint 
+Recon 
+mean 
+11.7 
+13.5 
+15.5 
+20.4 
+26.3 
+44.5 
+72.8 
+ 
+std 
+7.98 
+9.53 
+6.86 
+3.44 
+6.29 
+6.78 
+1.73 
+*The unit is 𝑚𝑠 for all mean and std of ROI T2 
+ 
+Table 2 continued.  
+Ground Truth  
+(𝑚𝑠) 
+ 
+116 
+167 
+194 
+323 
+479 
+692 
+853 
+Full 2-step 
+Recon   
+mean 
+109 
+147 
+187 
+278 
+421 
+641 
+954 
+ 
+std 
+1.53 
+2.03 
+3.28 
+6.38 
+13.1 
+29.8 
+85.3 
+Full Joint 
+Recon 
+mean 
+105 
+141 
+179 
+263 
+385 
+556 
+797 
+ 
+std 
+1.34 
+1.18 
+2.33 
+4.60 
+5.31 
+15.1 
+37.1 
+25% Joint 
+Recon 
+mean 
+104 
+140 
+179 
+264 
+376 
+539 
+781 
+ 
+std 
+1.33 
+2.43 
+4.15 
+8.23 
+11.5 
+28.9 
+69.3 
+33% Joint 
+Recon 
+mean 
+104 
+139 
+179 
+262 
+376 
+535 
+786 
+ 
+std 
+1.46 
+2.54 
+4.57 
+9.37 
+12.3 
+35.7 
+79.7 
+*The unit is 𝑚𝑠 for all mean and std of ROI T2 
+ 
+
+20 
+ 
+In human brain data set, the mean and standard deviation of ADC values were calculated within 
+three selected ROIs (white matter, gray matter, and CSF: cerebrospinal fluid) as shown in Table. 2.  
+Since the noise distribution in MR images can be approximated as Gaussian distribution, it had little 
+impact on the mean ADC value reconstructed by the conventional 2-step algorithm and thus we used 
+the conventional reconstructed ADCs as ground truth.  In all three ROIs, the joint reconstructed ADC 
+values were comparable to the ground truth while showing higher consistency (i.e., less standard 
+deviation). 
+ 
+Table 2. Comparison of mean and standard deviation of T2 values reconstructed by the conventional 2-
+step reconstruction on fully sampled (Full 2-step Recon) and the proposed joint reconstruction method on 
+fully sampled (Full Joint Recon), 25% under-sampled (25% Joint Recon), and 33% under-sampled data 
+(33% Joint Recon) of a brain scan. 
+*The unit is 𝑚𝑠 for all mean and std of ROI T2 
+ 
+4. DISCUSSION 
+In this work, we developed a novel optimization method for joint reconstruction of T2W MR 
+images and T2 map by exploiting constraints simultaneously from k-space and TE-space that used 
+regularizations in image domain and self-consistency condition of the T2 weighted exponential 
+ 
+ 
+CSF 
+White Matter 
+Gray Matter 
+Full 2-step Recon   
+mean 
+1716 
+94.96 
+117.7 
+ 
+std 
+177.6 
+4.266 
+9.220 
+Full Joint Recon 
+mean 
+1733 
+104.1 
+124.8 
+ 
+std 
+139.5 
+2.305 
+7.974 
+25% Joint Recon 
+mean 
+1729 
+102.1 
+124.8 
+ 
+std 
+146.4 
+2.156 
+7.156 
+33% Joint Recon 
+mean 
+1731 
+101.6 
+128.3 
+ 
+std 
+140.4 
+1.550 
+8.367 
+
+21 
+ 
+signal decay form. This proposed algorithm improved image quality in T2W images and T2 maps, 
+and reduced variations in pixel-by-pixel T2 measurements. 
+Technical highlights of the proposed algorithm reside in the synergy of k-TE space constraints 
+in MR image reconstruction and T2 fitting through multiple iterations. Considering the mono-
+exponential decay relationship between T2W signals and T2 value, we incorporated a constraint 
+in TE space to the optimization problem, which enforced the mono-exponential decay calibration 
+through MR signals acquired at different TEs with the signal decay coefficient being 
+1
+𝑇2. This TE-
+space constraint was based on that all k-space data acquired at different TEs rather than single k-
+space data acquired at single TE. The detailed updating formulas through each iteration to 
+reconstruct a T2W MR image (𝑖 = 0 and 𝑖 > 0) were shown in Eq. 17 and Eq. 18.  
+The numerator contained terms for data fidelity (i.e., k-space constraint), TE-space constraint, 
+and the smoothness constraint, aside from the Lagrange multiplier terms about 𝒛 and 𝒚. The 
+denominator was the normalization term. The meanings of these equations are interpretable. For 
+MR image at 𝑇𝐸0, each MR image with 𝑇𝐸𝑖, 𝑖 > 0 from the last iteration estimated an expected 
+MR image at 𝑇𝐸0 based on the exponential decay relationship. Then these expected images were 
+averaged together with the image generated based on data fidelity, the image after applying 
+smoothing constraints and those from multipliers. Similarly, for updating an MR image at 𝑇𝐸𝑖, 𝑖 >
+0, the expected MR image given by 𝒇0 was involved. Overall, as the iterations continued, each 
+𝒇0 =
+𝒈𝑇𝑺𝑭∗𝑨∗ + 𝒈𝐻𝑺𝑭𝑨
+2
++ ∑
+(𝒚𝑖
+𝑇𝑒−𝑇𝐸𝑖−𝑇𝐸0
+𝑻𝟐
++ 𝜌𝒇𝑏
+𝑇𝑒−𝑇𝐸𝑖−𝑇𝐸0
+𝑻𝟐
+)
+𝑖>0
++ 𝒛0
+𝑇 + 𝜌𝒗0
+𝑇
+𝑨𝐻𝑭∗𝑺𝑭𝑨 + 𝑨𝑭𝑺𝑭∗𝑨∗
+2
++ 𝜌 + 𝜌 ∗ ∑
+𝑒−2𝑇𝐸𝑖−𝑇𝐸0
+𝑻𝟐
+𝐼>0
+ 
+Eq. 17 
+𝒇𝑖,𝑖>0 ⁡=
+𝒈𝑇𝑺𝑭∗𝑨∗ + 𝒈𝐻𝑺𝑭𝑨
+2
+⁡−⁡𝒚𝑖
+𝑇 + 𝜌𝒇0
+𝑇𝑒−𝑇𝐸𝑖−𝑇𝐸0
+𝑻𝟐
++ 𝒛𝑖
+𝑇 + 𝜌𝒗𝒊
+𝑇
+𝑨𝐻𝑭∗𝑺𝑭𝑨 + 𝑨𝑇𝑭𝑺𝑭∗𝑨∗
+2
++ 𝜌 + 𝜌
+ 
+Eq. 18 
+
+22 
+ 
+T2W MR image was estimated by calibrating data from all k-space data at different TEs rather 
+than relying on its only k-space data. The iterative process effectively removed noise that was 
+randomly distributed in each MR image, and thus improve the image quality and the accuracy of 
+T2 measurements.  
+In addition to the k-TE space constraints, the smoothness constraint was also considered in MR 
+images. BM3D was incorporated in our algorithm using a PnP method as the smoothness 
+regularization for MR images to remove noise with Gaussian distribution. We also tested Rician-
+based BM3D method instead assuming the noise in MR images follows Rician distribution but 
+found out both BM3D and Rician-based BM3D gave similar results. This could be because the 
+Rician-distributed noise can be approximated as Gaussian noise in most of our data sets. We also 
+tested apply the BM3D spatial regularizations in T2 map, hoping to further improve the jointly 
+reconstructed image quality. However, this approach tended to over-smooth the reconstructed 
+images, which may be due to the unknown noise distribution in T2 map.  
+It is not able to mention that the joint reconstruction algorithm was more effective in removing 
+the aliasing artifacts on the under-sampled data compared with CS-based algorithm, mainly 
+because of the combined application of TE-space constraints, rather than the application of the 
+denoising tool BM3D, but. To validate the effectiveness of combining both k-space and TE-space 
+constraints in the proposed algorithm, we tested only applying the k-space constraint regularized 
+with BM3D solely to reconstruct the fully sampled and under-sampled phantom data by 
+minimizing eq. 19: 
+{𝒇∗} = argmin
+𝒇
+∑|𝑺𝑭𝑨(𝑥, 𝑇𝐸𝑖)𝒇(𝑥, 𝑇𝐸𝑖) − 𝒈(𝑧, 𝑇𝐸𝑖)|2 + 𝑅[𝒇(𝑥, 𝑇𝐸𝑖), 𝜆]
+𝑖
+ 
+eq. 19 
+
+23 
+ 
+The optimization process still exploited the ADMM and PnP algorithms, which was similar to that 
+of the proposed joint k-TE reconstruction algorithm. In this test, MRC failed to converge in all 
+phantom data sets and thus the iteration was manually stopped at the 4th iteration. As shown in Fig. 
+4. the approach cannot remove noise in the fully sampled data set, especially for T2W MR images 
+at high TE. The results became worse in under-sampled data set, where the aliasing patterns with 
+high spatial correlation were more obvious. This test demonstrated the effectiveness and necessity 
+of joint application of the constraints from both k-space and TE-space.  
+ 
+ 
+Figure 4. Images reconstruction by only applying BM3D regularized k-space constraint in the fully 
+sampled (a-f), 25% under-sampled (e-f) and 33% under-sampled (i-l) phantom data.  
+ 
+
+b=11s/mm2
+b=88s/mm2
+b=176s/mm2
+T2
+a
+b
+Under-sampled Fully sampled
+e
+9
+25%
+Under-sampled
+33%
+0
+500
+1000
+1500
+2000
+500
+1000
+1500
+250
+500
+750
+1000
+12500
+250
+500
+750
+1000
+125024 
+ 
+The adjustable parameters in the joint reconstruction method included the model parameter 𝜌 
+in eq. 5 brought by augmented Lagrange method and the threshold 𝜖 for convergence judgement. 
+Among them, the model parameter 𝜌 did not impact the final convergence result but only affected 
+the rate and stability of convergence. It can be set to a larger value when the convergence process 
+was not stable at the expense of having more iterations before reaching convergence. The threshold 
+𝜖 was used to stop the algorithm at a proper iteration and needed to be fine-tuned based on different 
+data sets. In this study, the threshold 𝜖 was set as 1% for both phantom and subject data sets. 
+There are several limitations in our work. First, the threshold of MRC for stopping the 
+iterations may vary in different data sets. Currently a conservative way to find the best threshold 
+was to first run the algorithm with a small threshold and then select a proper threshold that 
+preserves as much fine structures as possible while having the noise suppressed to overcome the 
+over smoothing issue. Second, the reconstruction time for the proposed algorithm with weighted 
+least square fitting is long (5 mins each iteration for each slice with 400 × 400 pixels and 32 TEs) 
+compared with the conventional FFT method. Over half the time was spent on noise variance 
+estimation, which can be accelerated using high performance GPU computing.  
+Future work will include extending the joint k-TE reconstruction method to more complex 
+models such as 3-parameter T2 model fitting model that takes into account the effect of imaging 
+pluses40. Also, a more robust algorithm will be implemented to estimate the parameter 𝜎 for BM3D 
+in order to avoid fine tuning the threshold 𝜖. Taking one step further, BM3D might not be the 
+optimal choice to be plugged in as the denoising regularization tool, more flexible tools, such as a 
+neural network, can be integrated to achieve better denoising effect and image reconstruction. 
+ 
+5. CONCLUSION 
+
+25 
+ 
+We developed a novel joint k-TE space optimization algorithm for simultaneous T2W MR 
+image and T2 map reconstruction. In this method, the k-space constraint enforced data fidelity, 
+and TE-space constraint enabled information to be shared among MR images at different TEs and 
+the corresponding T2 map. Our algorithm improved the image quality of T2W MR images and T2 
+maps with better SNR, increased the stability of pixelwise T2 measurements compared with the 
+conventional magnitude image-based 2-step signal fitting method. 
+ 
+ 
+ 
+
+26 
+ 
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+ 
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+arXiv:2301.02215v1  [math.CV]  5 Jan 2023
+ON 1/2 ESTIMATE FOR GLOBAL NEWLANDER-NIRENBERG THEOREM
+ZIMING SHI
+Dedicated to Prof. Xianghong Gong for his 60th Birthday
+Abstract. Given a formally integrable almost complex structure X defined on the closure of a
+bounded domain D ⊂ Cn, and provided that X is sufficiently close to the standard complex struc-
+ture, the global Newlander-Nirenberg problem asks whether there exists a global diffeomorphism
+defined on D that transform X into the standard complex structure, under certain geometric and
+regularity assumptions on D. In this paper we prove a regularity result of this problem. Assuming
+D is a strictly pseudoconvex domain in Cn with C2 boundary, and that the almost structure X is
+of the H¨older-Zygmund class Λr(D) for r > 1, we prove the existence of a global diffeomorphism
+(independent of r) in the class Λr+ 1
+2 −ε(D), for any ε > 0.
+Contents
+1.
+Introduction
+1
+2.
+Preliminaries
+3
+2.1.
+Function space and stability of constant
+3
+2.2.
+Smoothing operator on bounded Lipschitz domains
+7
+2.3.
+Paraproduct estimate
+10
+2.4.
+∂ equation in H¨older space of negative smooth index
+12
+3.
+Norm estimates for the error term
+14
+4.
+Iteration Scheme and convergence of maps
+18
+References
+22
+1. Introduction
+The main goal of the paper is the following result:
+Theorem 1.1. Let D be a domain in Cn with C2 boundary that is strictly pseudoconvex with respect
+to the standard complex structure in Cn, for n ≥ 2. Let X be a Λr(D) formally integrable complex
+structure on D, for 1 < r ≤ ∞, and and let Xst be the standard complex structure in Cn. For each
+ε > 0, there is δ = δ(ε) > 0 such that if |X − Xst|D, 3
+2 +ε < δ(ε), then there exists an embedding
+F of D into Cn such that F transforms X into Xst. Furthermore, F ∈ Λr+ 1
+2
+−
+(D) if r < ∞ and
+F ∈ C∞(D) if r = ∞. If bD is C3, the hypothesis can be weakened to |X − Xst|D,1+ε < δ(ε).
+2020 Mathematics Subject Classification. 32T15, 32Q60, 32Q40.
+1
+
+2
+An almost complex structure on a real differentiable manifold M is a tensor field J, which is,
+at every point x of M, an endomorphism of the tangent space Tx(M) such that J2 = −I, where I
+denotes the identity transformation of Tx(M). If M is a complex manifold with local holomorphic
+coordinate chart {Uα, zα}α, then one can define J
+�
+∂
+∂zk
+�
+= i
+� ∂
+∂zk
+�
+and J
+�
+∂
+∂zk
+�
+= −i
+�
+∂
+∂zk
+�
+, which
+is a globally well-defined almost complex structure on M. Consequently we have the decomposition
+of the complexified tangent bundle into the +i and −i eigenspaces of J: TCM = T +M ⊕ T −M.
+Conversely, one wants to know when an almost complex structure J is induced by some complex
+structure. In other words, the question is whether there exists a diffeomorphism so that in the
+new coordinate the vector fields making up the i-eigenspace of J defines a holomorphic coordinate
+system. In this case we say that X is integrable, and the underlying complex structure is said to
+be compatible with X.
+We call an almost complex structure X formally integrable if T +M is closed under the bracket
+[·, ·]. The classical Newlander-Nirenberg theorem states that an almost complex structure defined in
+a neighborhood of an interior point of M is locally integrable if and only if it is formally integrable.
+For real analytic X, the Newlander-Nirenberg theorem follows easily from the analytic Frobenius
+theorem. However if X is only C∞ or less regular, the proof becomes much more difficult. Here
+we refer to the work of Newlander-Nirenberg [NN57], Nijenhuis-Woolf [NW63], Malgrange [Mal69]
+and Webster [Web89]. We point out that Webster’s proof yields the sharp regularity result in the
+H¨older space Cr, namely, if X ∈ Ck+α in a neighborhood of a point p for k ≥ 1 and α ∈ (0, 1),
+then there exists a local diffeomorphism near p of the class Ck+1+α transforming X to the complex
+structure.
+We now consider the analogous global problem.
+Suppose an almost complex structure X is
+defined on the closure of a subset D in a complex manifold, such that X is a small perturbation of
+the given complex structure as measured by certain norms (e.g. H¨older-Zygmund). The problem
+asks whether there exists a global holomorphic coordinate system on D which is compatible with
+X.
+Under the assumption that both boundary and almost complex structures are C∞, Hamilton
+[Ham77] proved a more general version of Theorem 1.1. More specifically, he showed that global
+Newlander-Nirenberg theorem holds for a domain D which is a relatively compact subset with C∞
+boundary in a complex manifold Y , such that H1(D, T D) = 0, where T stands for the holomorphic
+tangent bundle of D, and the Levi form on bD has either n − 1 positive eigenvalues or 1 negative
+eigenvalue. Catlin [Cat88] proved a local Newlander-Nirenberg theorem with smooth pseudoconvex
+boundary. Most recently, Gan-Gong studied the problem when D is a strictly pseudoconvex domain
+in Cn with C2 boundary. Assuming X ∈ Λr(D), r > 5, they proved the existence of a diffeomor-
+phism in the class Λr−1(D). However for this result they had to assume that |X − Xst|Λr(D) < δ(r)
+for some sufficiently small δ(r).
+We now describe our proof, which is based on the rapid convergence scheme of [Web89] and [GG].
+Given D0 with an initial almost complex structure X0, we look for a succession of diffeomorphisms
+{Fi}∞
+i=1 that maps Di−1 with structure Xi−1 to a new domain Di with structure Xi.
+During
+the iteration, the error |Ai| = |Xi − Xst|C0(Di) converges to 0 while each Di remains strictly
+pseudoconvex.
+In the end we have the convergence of �Fi = Fi ◦ · · · ◦ F1 to a diffeomorphism
+F : D0 → D∞ with F∗X0 = Xst. The map Fi is constructed by using a ∂ homotopy formula
+Ai = ∂PAi + Q∂Ai and setting Fi = I − PAi, where I is the identity map on Di. In Webster’s
+proof for the classical Newlander-Nirenberg theorem, the operator P gains one derivative in the
+interior and there is no loss of derivative at the iteration step. Hence he is able to prove the rapid
+convergence of the error term for all derivatives: |Ai+1|k+α ≤ |Ai|2
+k+α for any positive integer k
+and α ∈ (0, 1). For the global problem this is no longer true since the operator P gains only 1/2
+derivative up to boundary, and there is loss of 1/2 derivative at each step, meaning the iteration
+will break down after finite iteration. To address this issue, Gan-Gong [GG] applied a smoothing
+
+3
+operator at each step. The cost for doing this is that one needs to estimate both the lower and higher
+order derivatives and use the convexity of norms to estimate the intermediate derivatives. In this
+case one still have the rapid convergence of the lower-order norms, but not for higher-order norms.
+In the iteration scheme of Gan-Gong, the higher-order norms blow up like |Ai+1|r ≤ Crt− 1
+2 |Ai|r,
+r > 5, where t is the parameter in the smoothing operator that tends to 0. In our case, we use a
+different solution operator Fi which allows us to prove the estimate
+(1.1)
+|Ai+1|r ≤ Cr|Ai|r,
+r > 1.
+The key is the construction of smoothing operator on bounded Lipschitz domains. Consequently
+we are able to obtain a gain of 1/2 − ε estimate for any ε > 0. This is the best possible estimate
+using current method, due to the convex interpolation of norms.
+The paper is organized as follows. In Section 2, we recall some basic properties of the H¨older
+Zygmund norm.
+In particular we show it has the Littlewood-Paley characterization using the
+Besov norm Bs
+∞,∞. In Section 2.2 we construct the important Moser-type smoothing operator on
+bounded Lipschitz domains. We also prove a commutator estimate which is crucial for bounding
+the norms of the error in the iteration. In Section 2.3 we prove a paraproduct estimate. The idea
+is that if both u, v are Λs for s > − 1
+2, we can define the paraproduct uv in the space Λ− 1
+2 by using
+the equivalent Littlewood-Paley characterization of Zygmund norm. This result will be used in the
+case of C3 boundary to optimize the assumption on the original almost complex structure. To this
+end, we also need a ∂ homotopy operator which gains 1/2 derivative in negative Zygmund space.
+We prove this result in section 2.4, using the same method we developed in [SY21a] for Sobolev
+space. In section 3 we derive the estimates for the lower and higher order norms of the new error
+term in term of the norms of old error term, for each iteration. In section 4 we set up the induction
+scheme and establish the rapid convergence of the lower order norms and estimate (1.1) for higher
+order norm. Finally we use the convexity of norms to show that the composition of the iterating
+maps converge to the limiting diffeomorphism in the appropriate norm.
+Acknowledgment. The author would like to thank Liding Yao and Xianghong Gong for many helpful
+discussions.
+2. Preliminaries
+2.1. Function space and stability of constant. In this section, we recall some basic results for
+the standard H¨older space Cr(D) with norm ∥ · ∥D,r, 0 ≤ r < ∞, and the H¨older-Zygmund space
+Λr with norm | · |D,r, 0 < r < ∞.
+For a bounded Lipschitz domain D, the following H¨older estimates for interpolation, product
+rule and chain rule are well-known.
+∥u∥D,(1−θ)a+θb ≤ Ca,b∥u∥1−θ
+D,a∥u∥θ
+D,b,
+0 ≤ θ ≤ 1;
+∥uv∥D,a ≤ Ca(∥u∥D,a∥v∥D,0 + ∥u∥D,0∥v∥D,a)
+a ≥ 0;
+∥u ◦ ϕ∥D,a ≤ Ca
+�
+∥u∥D′,a∥ϕ∥a
+D,1 + ∥u∥D′,1∥ϕ∥D,a + ∥u∥D′,0
+�
+,
+a ≥ 0.
+Here ϕ : D → D′ is a C1 map between two bounded Lipschitz domains D, D′.
+Definition 2.1 (H¨older-Zygmund). Let U ⊆ Rd be an open subset. We define the H¨older-Zygmund
+space Λs(U) for s ∈ R by the following:
+• For 0 < s < 1, Λs(U) consists of all f ∈ C0(U) such that ∥f∥Λs(U) := sup
+U
+|f|+ sup
+x,y∈U
+|f(x)−f(y)|
+|x−y|s
+<
+∞.
+• Λ1(U) consists of all f ∈ C0(U) such that ∥f∥Λ1(U) := sup
+U
+|f|+
+sup
+x,y∈U; x+y
+2 ∈U
+|f(x)+f(y)−2f( x+y
+2 )|
+|x−y|
+<
+∞.
+
+4
+• For s > 1 recursively, Λs(U) consists of all f ∈ Λs−1(U) such that ∇f ∈ Λs−1(U; CN). We
+define ∥f∥Λs(U) := ∥f∥Λs−1(U) + �N
+j=1 ∥Djf∥Λs−1(U).
+• For s ≤ 0 recursively, Λs(U) consists of all distributions that have the form g0 + �N
+j=1 ∂jgj
+where g0, . . . , gN ∈ Λs+1(U).
+We define ∥f∥Λs(U) := inf{�N
+j=0 ∥gj∥Λs+1(U) : f = g0 +
+�N
+j=1 ∂jgj ∈ D′(U)}.
+• We define C∞(U) := �
+s>0 Λs(U) be the space of bounded smooth functions.
+Like in the case of H¨older norm, we have the following estimates for the H¨older-Zygmund norm:
+Lemma 2.2. [GG, Lemma 3.2] Let D, D′ be connected bounded Lipschitz domains and let ϕ maps
+D into D′. Suppose that ∥ϕ∥D,1 < C. Then we have
+|u|D,(1−θ)a+θb ≤ Ca,b(D)|u|1−θ
+D,a|u|θ
+D,b,
+0 ≤ θ ≤ 1,
+a, b > 0.
+(2.1)
+|uv|D,a ≤ Ca(|u|D,a∥v∥D,0 + ∥u∥D,0|v|D,a),
+a > 0;
+(2.2)
+|u ◦ ϕ|D,1 ≤ C(D, D′)|u|D′,1(1 + C1/ε∥ϕ∥
+1
+1+ε
+D,1+ε);
+|u ◦ ϕ|D,a ≤ C(a, D, D′)(C1/ε|u|D′,a∥ϕ∥
+1+2ε
+1+ε
+D,1+ε + ∥u∥D′,1|ϕ|D,a + ∥u∥D′,0),
+a > 1.
+(2.3)
+|u ◦ ϕ|D,a ≤ |u|D′,a∥ϕ∥a
+D,1,
+0 < a < 1.
+(2.4)
+Here C1/ε is a positive constant depending on ε that tends to ∞ as ε → 0.
+We also need the following more general chain rule estimate. The proof for H¨older norms can be
+found in the appendix of [Gon20] and the estimate for Zygmund norms can be done similarly. We
+leave the details to the reader.
+Lemma 2.3. Let D be a sequence of Lipschitz domains in Rd of which the Lipschitz constants are
+bounded. Let Fi = I + fi map Di into Di+1, with ∥fi∥1 ≤ C0. Then
+∥u ◦ Fm ◦ · · · ◦ F1∥D0,r ≤ Cm
+r
+�
+∥u∥r +
+�
+i
+∥u∥1∥fi∥r + ∥u∥r∥fi∥1
+�
+,
+r ≥ 0;
+|u ◦ Fm ◦ · · · ◦ F1|D0,r ≤ Cm
+r
+�
+|u|r +
+�
+i
+∥u∥1|fi|r + C1/ε|u|r∥fi∥
+1
+1+ε
+1+ε
+�
+,
+r > 1.
+We now recall the definition of Besov space, which includes the H¨older-Zygmund space as a
+special case.
+In what follows we denote by S ′(Rd) the space of tempered distributions, and for an arbitrary
+open subset U ⊂ Rd, we denote by S ′(U) := { �f|U : �f ∈ S ′(Rd)} the space of distributions in U
+which can be extended to tempered distributions in Rd.
+Definition 2.4. A classical Littlewood-Paley family λ is a collection of Schwartz functions λj such
+that the Fourier transform �λj(ξ) =
+�
+Rd λj(x)e−2πix·ξ satisfies
+• �λ0 ∈ C∞
+c (B2(0)) and �λ0 ≡ 1 in B1(0);
+• �λj(ξ) = �λ0(2−jξ) − �λ0(2−(j−1)ξ) for j ≥ 1 and ξ ∈ Rd.
+We denote by C = C(Rd) the set of all such families λ.
+Definition 2.5. We use S0(Rd) to denote the space of all infinite order moment vanishing Schwartz
+functions, that is, all f ∈ S (Rd) such that
+�
+Rd xαf(x) dx = 0 for all α ∈ Nd, or equivalently,
+f ∈ S (Rd) such that �f(ξ) = O(|ξ|∞) as ξ → 0.
+Definition 2.6. A generalized Littlewood-Paley family η = (ηj)∞
+j=1 is a collection of Schwartz
+functions depending only on η1, such that
+
+5
+• η1 ∈ S (Rd) and all its moments vanish.
+• ηj = 2(j−1)dη1(2j−1x) for j ≥ 2, x ∈ Rd.
+We denote by G = G(Rd) the set of all such families η.
+Notation 2.7. In Rd, we use the xd-directional cone K := {(x′, xd) : xd > |x′|} and its reflection
+−K := {(x′, xd) : xd < −|x′|}.
+Definition 2.8. A K-Littlewood-Paley pair is a collection of Schwartz functions (φj, ψj)∞
+j=0 such
+that
+• φ = (φj)∞
+j=1 and ψ = (ψj)∞
+j=1 ∈ G.
+• suppφj, supp ψj ⊂ −K ∩ {xd < −2−j} for all j ≥ 0.
+• �∞
+j=0 φj = �∞
+j=0 ψj ∗ φj = δ0 is the Direc delta measure at 0 ∈ Rd.
+Proposition 2.9. Let η0, θ0 ∈ S0(Rd) and define ηj(x) := 2jdη0(2jx) and θj(x) := 2jdθ0(2jx) for
+j ∈ Z+. Then for any M, N ≥ 0, there is C = C(η, θ, M, N) > 0 such that
+�
+Rd |ηj ∗ θk(x)|(1 + 2max(j,k)|x|)N dx ≤ C2−M|j−k|,
+∀j, k ∈ N.
+Let s ∈ R and 1 ≤ p, q ≤ ∞. Let λ ∈ C. The nonhomogeneous Besov norm Bs
+p,q(λ) of u ∈ S ′(Rd)
+is defined by
+∥u∥Bsp,q(λ) =
+���
+�
+2js|λj ∗ u|Lp�
+j∈N
+���
+ℓq(N) < ∞.
+The norm topology is independent of the choice of λ. In other words, for any λ, λ′ ∈ C, 1 ≤ p, q ≤ ∞
+and s ∈ R, there is a C = Cλ,λ′,p,q,s > 0 such that for every f ∈ S ′(Rd),
+∥f∥Bp,q(λ) ≤ C∥f∥Bp,q(λ′).
+The reader may refer to [Tri10, Prop.2.3.2] for the proof of this fact.
+For this reason we will
+henceforth drop λ in the definition of the norm and write simply ∥ · ∥Bsp,q.
+Definition 2.10. The nonhomogeneous Besov space Bs
+p,q(Rd) is defined by
+Bs
+p,q(Rd) = {u ∈ S ′(Rd) : ∥u∥Bsp,q < ∞}.
+Let Ω ⊂ Rd be an arbitrary open subset. We define Bs
+p,q(Ω) := { �f|Ω : �f ∈ Bs
+p,q(Rd)}.
+Let Ω ⊂ Rd. We denote by ˚
+Bs
+p,q(Ω) the norm closure of C∞
+c (Ω) in Bs
+p,q(Rd).
+The following result establishes the equivalence between the various spaces:
+Proposition 2.11. Let Cs, Λs and Bs
+∞,∞ be defined as above and let Ω ⊂ Rd is either a bounded
+Lipschitz domain or the whole space. Then the following statements are true.
+(i) Cs(Ω) = Λs(Ω), for all positive non-integer s.
+(ii) Λs(Ω) = Bs
+∞,∞(Ω), for s ∈ R.
+(iii) ˚
+Bs
+p,q(Ω) = {f ∈ ˚
+Bs
+p,q(Ω) : f|Ω
+c ≡ 0}.
+(iv) B−s
+p,q(Ω) = [ ˚
+Bs
+p′,q′(Ω)]′, for 1 ≤ p, q ≤ ∞, 1
+p + 1
+p′ = 1, 1
+q + 1
+q′ = 1.
+We shall frequently use the following extension operator due to Rychkov.
+Proposition 2.12. [Ryc99] Let Ω be a bounded Lipschitz domain in Rd.
+Then there exists a
+extension operator EΩ such that
+• EΩ defines a bounded map Bs
+p,q(Ω) → Bs
+p,q(Rd), for any 1 ≤ p, q ≤ ∞ and s ∈ R.
+• EΩf|Ω = f, for f ∈ S ′(Ω).
+
+6
+We refer the reader to [SY21b] for some more detailed properties of the Rychkov extension
+operator. The following anti-derivative operator, which we introduced in [SY21a], is used in the
+construction of our ∂ solution operator.
+Proposition 2.13 (Anti-derivative operator with support condition). Let Ω ⊂ Rd be a bounded
+Lipschitz domain. Then for any positive integer m, there exist linear operators Sm,α
+Ω
+: S ′(Rd) →
+S ′(Rd), |α| ≤ m, such that
+(i) Sm,α
+Ω
+: Λs(Rd) → Λs+m(Rd), for all s ∈ R.
+(ii) g = �
+|α|≤m ∂α(Sm,α
+Ω
+g) for all g ∈ S ′(Rd).
+(iii) If g ∈ S ′(Rd) satisfies g|Ω ≡ 0, then (Sm,α
+Ω
+g)|Ω ≡ 0 for all |α| ≤ m.
+Lemma 2.14. Let F = I + f be a C1 map from Br = {x ∈ Rd : ∥x∥ ≤ r} ⊂ Rd into Rd with
+f(0) = 0,
+∥Df∥Br,0 ≤ θ < 1
+2
+Let r′ = (1 − θ)r. Then the range of F contains Br′ and there exists a C1 inverse map G = I + g
+which maps Br′ injectively into Br, with
+g(0) = 0,
+∥Dg∥Br′,0 ≤ 2∥Df∥Br,0.
+Assume further that f ∈ Λa+1(Br). Then g ∈ Λa+1(Br′) and
+∥Dg∥Br′,a ≤ Ca∥Df∥Br,a,
+a ≥ 0;
+|Dg|Br′,a ≤ Ca|Df|Br,a(1 + C1/ε∥f∥
+1+2ε
+1+ε
+1+ε )
+a > 1.
+In practice our f will have compact support in Br and we can take r = r′.
+Lemma 2.15. Let Xα = ∂α + Aβ
+α∂β be a formally integrable almost complex structure, then
+∂A = [A, ∂A].
+Proof. Let Xβ = ∂β + Aα
+β∂α, Xγ = ∂γ + Aη
+γ∂η. The integrability condition says that [Xβ, Xγ] ∈
+span Xβ. By an easy computation we obtain
+[Xβ, Xγ] = XβXγ − XγXβ
+=
+�
+∂Aα
+γ
+∂zβ
+−
+∂Aα
+β
+∂zγ
++ Aη
+β
+∂Aα
+γ
+∂zη
+− Aη
+γ
+∂Aα
+β
+∂zη
+�
+∂
+∂zα
+.
+If [Xβ, Xγ] = �
+ν cν
+βγXν, then cν
+βγ = 0 for all ν. Hence for each α, β, γ, we have
+(2.5)
+∂Aα
+γ
+∂zβ
+−
+∂Aα
+β
+∂zγ
+= Aη
+γ
+∂Aα
+β
+∂zη
+− Aη
+β
+∂Aα
+γ
+∂zη
+.
+Now for each α, we can identify Aα as a (0, 1)-form: Aα = �
+β Aα
+βdzβ, then
+∂Aα =
+�
+β<γ
+(∂βAα
+γ − ∂γAα
+β)dzβ ∧ dzγ
+∂Aα =
+�
+η
+(∂ηAα
+β)dzη ∧ dzβ.
+If we view ∂Aα as a matrix whose (β, γ) -entry is ∂βAα
+γ − ∂γAα
+β, and ∂Aα as a matrix whose
+(β, η)-entry is ∂ηAα
+β, then (2.5) implies that ∂A = (∂A)A − A(∂A)T .
+□
+The following result shows how an almost complex structure changes under transformation of
+the form F = I + f, where I is the identity map.
+
+7
+Lemma 2.16. Let F = I +f be a C1 map with f(0) = 0 and Df is small. The associated complex
+structure {dF(Xα} has a basis {X′
+α} such that X′
+α + A′β
+α ∂β, where A′ is given by
+A(z) + ∂zf + A(z)∂zf = (I + ∂zf(z) + A(z)∂zf(z))A′ ◦ F(z).
+This is proved in [Web89]. For a more detailed proof the reader may also refer to [GG, Lemma
+2.1].
+The integrability condition is invariant under diffeomorphism. The proof is almost trivial but
+we include here for completeness.
+Lemma 2.17. Let F be a diffeomorphism defined on a domain D ⊂ Cn, and let {Xα} be a
+formally integrable almost complex structure on D. Then {F∗Xα} defines a formally integrable
+almost complex structure on F(D).
+Proof. This follows from the fact that if [Xα, Xβ] = cγ
+αβXγ, then [F∗(Xα), F∗(Xβ)] = (cγ
+αβ ◦
+F −1)F∗(Xγ).
+□
+2.2. Smoothing operator on bounded Lipschitz domains.
+Proposition 2.18. Let Ω be a bounded Lipschitz domain in Rd. Then there exists operator St :
+S ′(Ω) → C∞(Ω) such that
+(i) ∥Stu∥Ω,r ≲ ts−r∥u∥Ω,s,
+s ≤ r.
+(ii) ∥(I − St)u∥Ω,r ≲ ts−r∥u∥Ω,s,
+s ≥ r.
+Proof. First we prove the statements when the domain is a special Lipschitz domain of the form
+ω = {(x′, xd) ∈ Rd : xd > ρ(x′)}. In particular, we have ω + K = ω. We also note that it suffices to
+assume t = 2−N, N ∈ N. Let (φj, ψj)∞
+j=0 be a K dyadic pair. We define the following smoothing
+operator St on S ′(ω):
+(2.6)
+SK
+t u :=
+N
+�
+k=0
+ψk ∗ φk ∗ u,
+t = 2−N.
+Using the equivalence of the H¨older-Zygmund Λs norm and the Besov Bs
+∞,∞-norm, it suffices to
+prove that
+sup
+j≥0
+2jr|λj ∗ (SK
+t u)|L���(Ω) ≲ ts−r sup
+j≥0
+2js|φj ∗ u|L∞(Ω).
+We have
+sup
+j≥0
+2jr|λj ∗ (SK
+t u)|L∞(ω) = sup
+j≥0
+2jr
+�����λj ∗
+N
+�
+k=0
+ψk ∗ φk ∗ u
+�����
+L∞(ω)
+≤ sup
+j≥0
+2jr
+N
+�
+k=0
+|λj ∗ ψk ∗ φk ∗ u|L∞(ω)
+≤ sup
+j≥0
+2j(r−s)2js
+N
+�
+k=0
+|λj ∗ ψk|L1(Ω)|φk ∗ u|L∞(ω)
+where in the last inequality we used Young’s inequality. By Proposition 2.9, the last expression is
+bounded up to a constant multiple C = C(M) by
+sup
+j≥0
+2j(r−s)2js
+N
+�
+k=0
+2−M|j−k||φk ∗ u|L∞(ω) = 2N(r−s) sup
+j≥0
+2(j−N)(r−s)2js
+N
+�
+k=0
+2−M|j−k||φk ∗ u|L∞(ω).
+
+8
+If j < N, then 2(j−N)(r−s)2− M
+2 |j−k| < 1. If j ≥ N, then |j − k| = j − k ≥ j − N for k ≤ N,
+so 2− M
+2 |j−k| ≤ 2− M
+2 (j−N). It follows that 2(j−N)(r−s)2− M
+2 |j−k| ≤ 2(j−N)(r−s− M
+2 ) < 1 if we choose
+M > 2(r − s). In any case, the above estimate leads to
+(2.7)
+sup
+j≥0
+2jr|λj ∗ (SK
+t u)|L∞(ω) ≤ 2N(r−s) sup
+j≥0
+2js
+N
+�
+k=0
+2− M
+2 |j−k||φk ∗ u|L∞(ω)
+≤ 2N(r−s) sup
+j≥0
+N
+�
+k=0
+2(s− M
+2 )|j−k|2ks|φk ∗ u|L∞(ω)
+Here we assume that M > 2s. Let
+u[j] := 2j(s− M
+2 ),
+v[j] = 2js|φj ∗ f|L∞(ω).
+Then
+sup
+j≥0
+N
+�
+k=0
+2(s− M
+2 )|j−k|2ks|φk ∗ u|L∞(ω) = |u ∗ v|l∞ ≤ |u|l1|v|l∞ ≲ |v|l∞ = sup
+j≥0
+2js|φj ∗ f|L∞(ω).
+Thus we get from (2.7)
+sup
+j≥0
+2jr|λj ∗ (SK
+t u)|L∞(ω) ≤ ts−r sup
+j≥0
+2js|φj ∗ f|L∞(ω).
+In other words, we have shown that |SK
+t u|Λr(ω) ≤ ts−r|u|Λs(ω).
+(ii) Assume now that s ≥ r. From (2.6) we have
+(I − SK
+t )u =
+�
+k>N
+ψk ∗ φk ∗ u.
+It suffices to show that
+sup
+j≥0
+2jr|λj ∗ [(I − SK
+t )u]|L∞(ω) ≲ ts−r sup
+j≥0
+2js|φj ∗ u|L∞(ω).
+We have
+sup
+j≥0
+2jr|λj ∗ [(I − SK
+t )u]|L∞(ω) = sup
+j≥0
+2jr
+�����λj ∗
+∞
+�
+k=N+1
+ψk ∗ φk ∗ u
+�����
+L∞(ω)
+≤ 2jr sup
+j≥0
+∞
+�
+k=N+1
+|λj ∗ ψk|L1(ω)|φk ∗ u|L∞(ω)
+By Proposition 2.9, the last expression is bounded up to a constant multiple C = C(M) by
+sup
+j≥0
+2j(r−s)2js
+∞
+�
+k≥N+1
+2−M|j−k||φk ∗ u|L∞(ω) = 2N(r−s) sup
+j≥0
+2(j−N)(r−s)2js
+N
+�
+k=0
+2−M|j−k||φk ∗ u|L∞(ω).
+If j ≥ N, then 2(j−N)(r−s) ≤ 1.
+If j < N, then −(j − N) = |j − N| ≤ |j − k|.
+Hence
+2(j−N)(r−s)− M
+2 |j−k| ≤ 2−(j−N)(s−r)− M
+2 |j−k| ≤ 2|j−k|(s−r− M
+2 ) < 1, where we choose M > 2(s − r). In
+all cases, we get from the above estimates that
+sup
+j≥0
+2jr|λj ∗ [(I − SK
+t )u]|L∞(ω) ≲ 2N(r−s) sup
+j≥0
+2js
+∞
+�
+k=0
+2−M|j−k||φk ∗ u|L∞(ω).
+The rest of the estimates follow identically as that in (i), and consequently we prove that |(I −
+SK
+t )u|Λr(ω) ≤ ts−r|u|Λs(ω) for s ≥ r ≥ 0.
+
+9
+Finally we prove both (i) and (ii) for general bounded Lipschitz domains.
+For this we use
+partition of unity. Take an open covering {Uν}M
+ν=0 of Ω such that
+U0 ⊂⊂ Ω,
+bΩ ⊆
+M
+�
+ν=1
+Uν,
+Uν ∩ Ω = Uν ∩ Φν(ων),
+ν = 1, . . . , M.
+Here each ων is special Lipschitz domain of the form ων = {xN > ρν(x′)}, and Φν, 1 ≤ ν ≤ M are
+invertible affine linear transformations.
+If f has compact support in Ω, we define the smoothing operator S0
+t by
+S0
+t f =
+N
+�
+k=0
+ηk ∗ θk ∗ f,
+t = 2−N.
+Here we can choose any Littlewood-Paley pair (θj, ηj)∞
+j=0 with no cone support condition. Then
+the same proof as above shows that ∥S0
+t f∥Ω,r ≲ ts−r∥f∥Ω,s for 0 ≤ s ≤ r, and ∥(I − St)f∥Ω,r ≲
+ts−r∥f∥Ω,s for 0 ≤ r ≤ s.
+Let χν be a partition of unity associated with {Uν}, i.e. χν ∈ C∞
+c (Uν) and χ0 + � χ2
+ν = 1. For
+each 1 ≤ ν ≤ M, we have the property ων + K = ων, where K := {x ∈ Rd : xd > |x′|}. Let SK
+t be
+given as above, we define
+(2.8)
+Stu := S0
+t (χ0u) +
+M
+�
+ν=1
+χνSν
+t (χνu),
+where Sν
+t g := [SK
+t (g ◦ Φν)] ◦ Φ−1
+ν , 1 ≤ ν ≤ M.
+Applying the estimates for S0
+t and SK
+t , we get
+∥Stu∥Ω,k ≲ ts−r (∥χ0u∥Ω,s + ∥χνu∥Ω,s) ≲ ts−r∥u∥Ω,s,
+0 < s ≤ r
+On the other hand, since u = χ0u + �M
+ν=1 χ2
+νu we have
+∥(I − St)u∥Ω,r ≤ ∥(I − S0
+t )(χ0u)∥Ω,r +
+M
+�
+ν=1
+∥χν(I − Sν
+t )(χνu)∥Ω,r
+≲ ts−r (∥χ0u∥Ω,s + ∥χνu∥Ω,j) ≲ ts−r∥u∥Ω,s.
+s ≥ r.
+□
+Lemma 2.19. Let Ω be a bounded Lipschitz domain and {Sν
+t }M
+ν=1 be the smoothing operator con-
+structed in the proof of Proposition 2.18. Then [D, Sν
+t ](χνf) ≡ 0 for all f ∈ Λr(Ω), r ≥ 1.
+Proof. Recall that Sν
+t g := [�Sν
+t (g ◦ Φν)] ◦ Φ−1
+ν , where �Sν
+t is the smoothing operator defined on ων
+and �Sν
+t (g ◦ Φν) := (g ◦ Φv) ∗ φt. Hence we have
+[D, Sν
+t ](χνf) = D(Sν
+t (χνf)) − Sν
+t (D(χνf))
+= D(�Sν
+t [(χνf) ◦ Φν] ◦ Φ−1
+ν ) − �Sν
+t [D(χvf) ◦ Φν] ◦ Φ−1
+ν
+= (D �Sν
+t [(χνf) ◦ Φν]) ◦ Φ−1
+ν
+· DΦ−1
+ν
+− �Sν
+t [D((χνf) ◦ Φν)] ◦ Φ−1
+ν
+· DΦ−1
+ν
++ �Sν
+t [D((χνf) ◦ Φν)] ◦ Φ−1
+ν
+· DΦ−1
+ν
+− �Sν
+t [D(χνf) ◦ Φν] ◦ Φ−1
+ν
+= [D, �Sν
+t ]((χνf) ◦ Φν) ◦ Φ−1
+ν
+· DΦ−1
+ν .
+where in the last step we used the fact that Φν is a linear transformation so that
+�Sν
+t [D((χνf) ◦ Φν)] ◦ Φ−1
+ν
+· DΦ−1
+ν
+= �Sν
+t [D(χνf) ◦ Φν] ◦ Φ−1
+ν
+· DΦν · DΦ−1
+ν
+= �Sν
+t [D(χνf) ◦ Φν] ◦ Φ−1
+ν .
+Now since �Sν
+t is a convolution operator, we have [D, �Sν
+t ] ≡ 0 on ων. Thus [D, Sν
+t ](χνf) ≡ 0.
+□
+
+10
+Lemma 2.20. Let Ω be a bounded Lipschitz domain and let St be the smoothing operator constructed
+in the proof of Proposition 2.18. Then for all u ∈ Λs(Ω) with s ≥ 1, the following holds
+(2.9)
+|[D, St]u|r ≲ ts−r∥u∥s,
+0 < r ≤ s.
+Proof. By the formula for St (2.8), we can write
+[D, St]u = DStu − StDu
+= DS0
+t (χ0u) +
+M
+�
+ν=1
+D(χνSν
+t (χνu) − S0
+t (χ0(Du)) −
+M
+�
+ν=1
+χνSν
+t (χν(Du))
+= {DS0
+t (χ0u) − S0
+t D(χ0u)} + S0
+t ((Dχ0)u) +
+M
+�
+ν=1
+(Dχν)Sν
+t (χνu)
++
+M
+�
+ν=1
+χνDSν
+t (χνu) − χνSν
+t D(χνu) +
+M
+�
+ν=1
+χνSν
+t ((Dχν)u)
+= S0
+t ((Dχ0)u) +
+M
+�
+ν=1
+(Dχν)Sν
+t (χνu) +
+M
+�
+ν=1
+χνSν
+t ((Dχν)u).
+Here to get the last line we use [D, S0
+t ](χ0u) ≡ 0 and Lemma 2.19. Since 0 = D(χ0 + �M
+ν=1 χ2
+ν) =
+Dχ0 + 2 �M
+ν=1 χνD(χnu), we can write
+[D, St]u = (S0
+t − I)((Dχ0)u) +
+M
+�
+ν=1
+(Dχν)(Sν
+t − I)(χνu) +
+M
+�
+ν=1
+χν(Sν
+t − I)((Dχν)u).
+Applying Proposition 2.18 to the right-hand side above we get (2.9).
+□
+2.3. Paraproduct estimate.
+Proposition 2.21. Let A be the annulus {ξ ∈ Rd : 3
+4 ≤ |ξ| ≤ 8
+3}. There exist radial functions χ
+and ϕ, valued in the interval [0, 1], belonging respectively to C∞
+c (B(0, 4
+3) and C∞
+c (A), and such that
+∀ξ ∈ Rd, χ(ξ) +
+�
+j≥0
+ϕ(2−jξ) = 1,
+|j − j′| ≥ 2 =⇒ suppϕ(2−j·) ∩ suppϕ(2−j′·) = ∅,
+j ≥ 1 =⇒ supp χ ∩ suppϕ(2−j·) = ∅,
+j ≥ 1.
+Write h = F−1ϕ and �h = F−1χ. The non-homogeneous dyadic blocks ∆j are defined by
+∆ju = 0,
+if j ≤ −2,
+∆−1u = χ(D)u =
+�
+Rd
+�h(y)u(x − y) dy,
+and
+∆ju = ϕ(2−jD)u = 2jd
+�
+Rd h(2jy)u(x − y) dy
+if j ≥ 0.
+The nonhomogeneous low frequency cut-off operator Sj is defined by
+Sju =
+�
+ℓ≤j−1
+∆ℓu.
+Let u, v ∈ S ′. The nonhomogeneous paraproduct of v by u is defined by
+(2.10)
+Tuv =
+�
+j
+Sj−1u ∆jv.
+
+11
+The nonhomogeneous remainder of u and v is defined by
+R(u, v) =
+�
+|i−j|≤1
+∆iu ∆jv.
+Formally we have the following Bony decomposition:
+uv = Tuv + Tvu + R(u, v).
+Proposition 2.22. [BCD11, Theorem 2.82] For any couple of real numbers (s, t) with t negative
+and any (p, r1, r2) ∈ [1, ∞]3:
+∥T∥L(L∞×Bsp,r;Bsp,r) ≤ C|s|+1;
+∥T∥L(Bt∞,r1×Bs∞,r2;Bs+t
+p,r ) ≤ C|s+t|+1
+−t
+,
+with 1
+r := min
+�
+1, 1
+r1
++ 1
+r2
+�
+.
+In particular, by taking p = r1 = r2 = ∞, we have
+∥T∥L(Λt×Λs;Λt+s) ≤ C|s+t|+1
+−t
+.
+Proposition 2.23. [BCD11, Theorem 2.85] Let (s1, s2) ∈ R2 and (p1, p2, r1, r2) ∈ [1, ∞]4. Suppose
+that
+1
+p := 1
+p1
++ 1
+p2
+≤ 1
+and
+1
+r := 1
+r1
++ 1
+r2
+≤ 1.
+If s1 + s2 > 0, then there exists a constant C > 0 such that, for any (u, v) in Bs1
+p1,r1 × Bs1
+p2,r2,
+∥R(u, v)∥Bs1+s2
+p,r
+≤ C|s1+s2|+1
+s1 + s2
+∥u∥Bs1
+p1,r1∥v∥Bs2
+p2,r2.
+In particular, by taking p = p1 = p2 = r1 = r2 = ∞, we have
+∥R(u, v)∥Λs1+s2 ≤ C|s1+s2|+1
+s1 + s2
+∥u∥Λs1∥v∥Λs2.
+Proposition 2.24. Let ε > 0. Then for any m ∈ (1
+2, 1),
+|[∂A, A]|m−1−ε ≤ Cε,m|A|2
+m,
+where Cε,m tends to ∞ as ε → 0 or m → 1.
+Proof. First we apply Rychkov’s extension operator to A and denote the extended function still by
+A. Hence we can assume that A ∈ Λm(Rd). By the Bony decomposition, we can write
+[∂A, A] = T∂AA + TA(∂A) + R(A, ∂A).
+By Proposition 2.22, we get for t < 0,
+|T∂AA|D,m−1 ≤ C|s+t|+1
+−t
+|∂A|t|A|s,
+where t + s = m − 1 < 0. Choosing t = m − 1 and s = 0, we get
+(2.11)
+|T∂AA|D,m−1 ≤
+Cm
+1 − m|∂A|m−1|A|0 ≤
+Cm
+1 − m|A|2
+m.
+By Proposition 2.22 again, we get for t < 0,
+|TA(∂A)|D,m−1 ≤ C|s+t|+1
+−t
+|A|t|∂A|s.
+Choosing t = −ε and s = m − 1 + ε for ε > 0, we get
+|TA(∂A)|D,m−1 ≤ Cm
+ε |A|−ε|∂A|m−1+ε ≤ Cm
+ε |A|−ε|A|m+ε ≤ Cm
+ε |A|2
+m+ε.
+
+12
+Replacing m by m − ε in the above inequality we get
+(2.12)
+|TA(∂A)|D,m−1−ε ≤ Cm
+ε |A|2
+m.
+Finally applying Proposition 2.23 we get for any r > 0
+|R(∂A, A)|−r+2ε ≤ |R(∂A, A)|2ε ≤ |∂A|ε−r|A|ε+r ≤ |A|ε−r+1|A|ε+r
+If r ≤ 1
+2, then ε − r + 1 ≥ ε + r, and the above implies
+(2.13)
+|R(∂A, A)|−r+2ε ≤ |A|2
+−r+ε+1.
+Set m − 1 = −r + 2ε. Then −r + ε + 1 = m − ε, and (2.13) gives
+(2.14)
+|R(∂A, A)|m−1 ≤ |A|2
+m−ε ≤ |A|2
+m,
+where m = −r + 1 + 2ε ≥ 1
+2 + 2ε. Putting together estimates (2.11), (2.12) and (2.14) we obtain
+the desired conclusion.
+□
+2.4. ∂ equation in H¨older space of negative smooth index.
+Proposition 2.25. Let Ω be a bounded Lipschitz domain and let δ(x) denote the distance function
+to the boundary bΩ.
+Let m ∈ N and r ∈ R be such that r < m.
+Suppose u ∈ Λm
+loc(Ω) and
+|δm−rDmu|L∞(Ω) < ∞, then there exists a constant C = C(Ω, r) such that
+(2.15)
+|u|Ω,r ≤ C
+�
+|β|≤m
+∥δm−rDβu∥L∞(Ω).
+Proof. By the property of the H¨older-Zygmund norm (see for example [Ste70, Chapter V] or [SY21b,
+Theorem 1.1]), we have the following equivalence:
+(2.16)
+|u|Ω,r ≈
+�
+|α|≤m
+|Dαu|Ω,r−m,
+for every m ∈ N.
+We now claim that
+|u|Ω,−s ≤ C∥δsu∥L∞(Ω),
+s > 0,
+for all u ∈ L∞
+loc(Ω). Assume the claim is true. Then by choosing m > r in (2.16) we obtain (2.15).
+It remains to prove the claim. By Proposition 2.11, we have Λ−s(Ω) = B−s
+∞,∞(Ω) = [ ˚
+Bs
+1,1(Ω)]′.
+Notice that
+�����sup
+j∈N
+2js|λj ∗ f|
+�����
+L1(Ω)
+≤ C(Ω)
+��2js|λj ∗ f|L1(Ω)
+��
+l1(N) .
+The left-hand side can be identified as the Tribel-Lizorkin norm ∥ · ∥F s
+1,∞(Ω). By [Yao22, Corollary
+5.5], we have the following embedding of Banach spaces:
+˚
+F s
+1,∞(Ω) ֒→ L1(Ω, δ−s).
+Hence for every f ∈ L∞(Ω, δs),
+|f|Ω,−s =
+sup
+g∈ ˚
+B1,1(Ω), ∥g∥Bs
+1,1(Ω)≤1
+⟨f, g⟩ ≤
+sup
+g∈L1(Ω,δ−s dx), ∥δ−sg∥L1(Ω)≤Cs
+�
+Ω
+|fg|
+≤
+sup
+∥δ−sg∥L1(Ω)≤Cs
+�
+Ω
+|δsf||δ−sg| ≤ Cs∥δsf∥L∞(Ω).
+□
+Theorem 2.26. Let Ω be a strictly pseudoconvex domain with Ck+2 boundary, where k is a non-
+negative integer. Then there exists a linear operator Hq (depending on k) such that for all r > −k,
+
+13
+(i) Hq : Λr
+(0,q)(Ω) → Λ
+r+ 1
+2
+(0,q−1)(Ω);
+(ii) u = Hqϕ is a solution to the equation ∂u = ϕ for any ϕ ∈ Λr(Ω) which is ∂-closed.
+Proof. We shall use the homotopy operator Hq constructed in [SY21a], given as follows
+(2.17)
+Hqϕ(z) =
+�
+U
+K0
+0,q−1(z, ·) ∧ Eϕ + (−1)|γ| �
+|γ|≤l
+�
+U\Ω
+Dγ(K01
+0,q−1(z, ·)) ∧ Sl,γ
+Ω [∂, E]ϕ
+:= H0
+qϕ(z) + H1
+qϕ(z).
+Here E is the extension operator of Rychkov (Proposition 2.12), and S is the anti-derivative oper-
+ator in Proposition 2.13. We multiply E and Sl,γ
+Ω
+by suitable cut-off functions so that supp(Eϕ),
+supp Sl,γ
+Ω ϕ ⊂⊂ U. The proof of (i) follows straight from Proposition 2.25 and Proposition 2.28
+below. Using (i) and arguing in the same way as the proof of [SY21a, Corollary 4.3], we can show
+that the following homotopy formula holds in the sense of distributions:
+ϕ = ∂Hqϕ + Hq+1∂ϕ.
+In particular for ϕ in Ω with Λr(Ω) which is ∂-closed, we have ∂Hϕ = ϕ.
+□
+In what follows we recall that the first integral and the second integral in (2.17) are denoted by
+H0
+q and H1
+q, respectively.
+Proposition 2.27. Let r ∈ R. Suppose ϕ ∈ Λr
+(0,q)(U). Then H0
+qϕ ∈ Λr+1
+(0,q−1)(Cn)
+Proposition 2.28. Let Ω ⊂ Cn be a bounded strictly pseudoconvex domain with Ck+2 boundary,
+where k is a non-negative integer. For q ≥ 1, let H1
+qϕ be given by (2.17), where we choose l to
+be any positive number greater or equal to k + 1. Let r > −k. Then for any non-negative integer
+m > r + k + 1
+2, there exists a constant C = C(Ω, k, m, r, p) such that for all ϕ ∈ Λr
+(0,q)(Ω),
+(2.18)
+sup
+z∈Ω
+δ(z)m−r− 1
+2Dm
+z H1
+q(z) ≤ C∥ϕ∥Λr(Ω).
+Proof. We need to estimate
+(2.19)
+δ(z)(m−r− 1
+2)
+������
+�
+|γ|≤l
+�
+U\Ω
+Dm
+z Dγ
+ζ K(z, ·) ∧ Sl,γ([∂, E]ϕ) dV (z)
+������
+.
+The idea is to choose l large enough so that Sl,γ[∂, E]ϕ lies in some positive index space. Let f
+be a coefficient function of [∂, E]ϕ so that f ∈ Λr−1(U \ Ω). By Sl,γf we mean each component of
+Sl,γ([∂, E]ϕ) . In the computation below we shall fix a multi-index γ and write simply Sl for Sl,γ
+without causing ambiguity. By Proposition 2.13, Slf ∈ Λr+l−1(U \Ω). Since l ≥ k+1, and r > −k,
+we have l > −r + 1, or r − 1 + l > 0. Since f ≡ 0 in Ω, by Proposition 2.13 (iii) we have Slf ≡ 0
+in U \ Ω. Denote the above integral by Kf(z). Arguing as in the proof of [SY21a, Proposition 4.9],
+we can show that
+(2.20)
+|Kf(z)| ≲ |K0f(z)| + |K1f(z)|.
+where
+K0f(z) :=
+�
+U\Ω
+Slf(ζ)
+DζW(z, ζ)
+Φm+l+1(z, ζ)|ζ − z|2n−3 dV (ζ),
+K1f(z) :=
+�
+U\Ω
+Slf(ζ)
+Dl+1
+ζ
+W(z, ζ)
+Φm+1(z, ζ)|ζ − z|2n−3 dV (ζ).
+
+14
+We shall denote the kernels of K0 and K1 by B0(z, ζ) and B1(z, ζ), respectively. By estimates from
+the proof of [SY21a, Proposition 4.9], we have
+|B0(z)| ≲
+1
+|Φ(z, ζ)|m+l+1|ζ − z|2n−3 ,
+|B1(z)| ≲
+δ(ζ)k−l
+|Φ(z, ζ)|m+1|ζ − z|2n−3 .
+(2.21)
+For the K0f integral, we have
+|K0f(z)| =
+�����
+�
+U\Ω
+Slf(ζ)B0(z, ζ) dV (ζ)
+�����
+=
+�
+U\Ω
+�
+δ(ζ)−(r−1+l)Slf(ζ)
+� �
+δ(ζ)r−1+lB0(z, ζ)
+�
+dV (ζ)
+≤ ∥δ−(r−1+l)Slf∥L∞(U\Ω)
+�
+U\Ω
+δ(ζ)r−1+l
+|Φ(z, ζ)|m+l+1|ζ − z|2n−3 dV (ζ).
+If r − 1 + l < m + l − 1 − 1
+2, or m > r + 1
+2, then we can apply [SY21a, Lemma 4.8] to obtain
+|K0f(z)| ≲ ∥δ−(r−1+l)Slf∥L∞(U\Ω)δ(z)(r−1+l)−(m+l−1)+ 1
+2 ≲ ∥f∥Λr−1(U\Ω)δ(z)r−m+ 1
+2.
+For the K1f integral, we have using (2.21)
+|K1f(z)| =
+�����
+�
+U\Ω
+Slf(ζ)B1(z, ζ) dV (ζ)
+�����
+=
+�
+U\Ω
+�
+δ(ζ)−(r−1+l)Slf(ζ)
+� �
+δ(ζ)r−1+lB1(z, ζ)
+�
+dV (ζ)
+≤ ∥δ−(r−1+l)Slf∥L∞(U\Ω)
+�
+U\Ω
+δ(ζ)r−1+l+k−l
+|Φ(z, ζ)|m+1|ζ − z|2n−3 dV (ζ).
+If r − 1 + k > −1 and r − 1 + k < m − 1 − 1
+2, then by [SY21a, Lemma 4.8] we get
+|K1f(z)| ≲ ∥δ−(r−1+l)Slf∥L∞(U\Ω)δ(z)(r−1+k)−(m−1)+ 1
+2 ≲ ∥f∥Λr−1(U\Ω)δ(z)r+k−m+ 1
+2 .
+Putting together estimates for K0f and K1f and using (2.20), we get
+δ(z)m−r− 1
+2|Kf(z)| ≲ ∥f∥Λr−1(U\Ω)δ(z)m−r− 1
+2
+�
+δ(z)r−m+ 1
+2 + δ(z)r+k−m+ 1
+2
+�
+≲ ∥f∥Λr−1(U\Ω).
+Since ∥f∥Λr−1(U\Ω) ≲ ∥ϕ∥Λr(Ω), we obtain (2.18).
+□
+3. Norm estimates for the error term
+Let D0 be a strictly pseudoconvex domain in Cn. Given the initial integrable almost complex
+structure on D0, we want to find a transformation F defined on D0 to transform the structure to a
+new structure closer to the standard complex structure while D0 is transformed to a new domain
+that is still strictly pseudoconvex. We shall assume the following initial condition
+(3.1)
+t− 1
+2 |A|D0,s ≤ 1
+C∗s
+.
+which will later be verified to hold at each induction step. Here t is the parameter of the smoothing
+operator which will converge to 0 in the iteration.
+We take the map in the form F = I + f.
+Applying the extension operator E to f we can assume that f is defined with compact support on
+some large ball B0 , where
+D0 ⊂ U ⊂ B0.
+
+15
+Below we will take f = −StPA, where St is the smoothing operator constructed in Proposition 2.18.
+By (2.18) (i) and Theorem 2.26, we have the following estimates for f:
+|f|D0,m = |StPA|D0,m ≤ Cm|PA|D0,m ≤ Cm|A|D0,m− 1
+2 ,
+m > 0.
+(3.2)
+In particular we have
+|f|D0, 3
+2 ≤ C2|A|D0,1.
+In view of (3.2) and the initial condition with C∗
+s chosen sufficiently large, we have
+|Df|D0,0 ≤ |f|D0,1 ≤ C|A|D0, 1
+2 ≤ 1
+2.
+By Lemma 2.14, there exists an inverse map G = I + g of F, which takes B0 to B0 and satisfies
+the estimate
+|Dg|Br,a ≤ Ca|Df|Br,a(1 + |f|
+4
+3
+3
+2 ) ≤ Ca|Df|Br,a,
+a > 0
+which together with (3.2) implies
+(3.3)
+|g|Br,m ≤ Cm|f|m ≤ Cm|A|D0,m− 1
+2 ,
+m > 1.
+In particular g satisfies
+(3.4)
+|g|Br, 3
+2 ≤ C|f|Br, 3
+2 ≤ C|A|D0,1.
+By Lemma 2.16, the new structure takes the form
+A′ ◦ F = (I + ∂f + A∂f)−1(A + ∂f + A∂f).
+As in the proof of Lemma 2.15, we regard Aα
+β as the coefficients of the (0, 1) form Aα := Aα
+βdzβ.
+We then apply the homotopy formula component-wise to A = (A1, . . . , An) on D0 so that
+A = ∂PA + Q∂A.
+Set f = −StPA. Then we have
+(3.5)
+A + ∂f + A∂f = A − ∂(StPA) + A∂f
+= A − St∂PA + [St, ∂]PA + A∂f
+= A − St(A − Q∂A) + [St, ∂]PA + A∂f
+= (I − St)A + StQ∂A + [St, ∂]PA + A∂f.
+We shall use the following notation:
+K : ∂f + A∂f,
+I1 = (I − St)A,
+I2 = StQ∂A,
+I3 = [St, ∂]PA,
+I4 = A∂f,
+and consequently we can rewrite (3.5) as
+�A = (I + K0)−1(I1 + I2 + I3 + I4),
+�A = A′ ◦ F.
+We first estimate the Ij-s. By Proposition 2.18, we get
+(3.6)
+|I1|D0,m = |(I − St)A|D0,m ≤ Crtr−m|A|D0,r,
+0 ≤ m ≤ r.
+Substituting s for m in (3.8) we have
+(3.7)
+|I1|D0,s ≤ Crtr−s|A|D0,r,
+0 ≤ s ≤ r.
+Substituting m for r in (3.8) we have
+(3.8)
+|I1|D0,m ≤ Cm|A|D0,m,
+m ≥ 0
+
+16
+For I2, we consider two cases.
+Case 1: m ≥ 1. We apply Proposition 2.18 and use the integrability condition ∂A = [∂A, A] and
+the estimate for Q to get
+|I2|D0,m = |StQ∂A|D0,m ≲ t− 1
+2 |Q∂A|D0,m− 1
+2
+≲ Cmt− 1
+2|∂A|D0,m−1 = Cmt− 1
+2 |[∂A, A]|D0,m−1
+≲ Cmt− 1
+2 (|A|D0,m−1|∂A|D0,0 + |∂A|D0,m−1|A|D0,0)
+≤ Cmt− 1
+2|A|D0,s|A|D0,m,
+m ≥ 1, s ≥ 0.
+Substituting s for m in the above inequality we get
+(3.9)
+|I2|D0,s ≤ Cst− 1
+2|A|2
+D0,s,
+s ≥ 1.
+Alternatively, if we use the initial condition (3.1) we get
+(3.10)
+|I2|D0,m ≤ Cm|A|D0,m,
+m ≥ 1.
+Case 2: 1
+2 < m < 1.
+For every ε > 0, we have
+|I2|D0,m = |StQ∂A|D0,m ≲m t− 1
+2−ε|Q∂A|D0,m− 1
+2−ε ≲ t− 1
+2−ε|∂A|D0,m−1−ε.
+Here we assumed bD0 ∈ C3 and therefore Proposition 2.26 applies with r = m − 1 − ε > −1. By
+the integrability condition ∂A = [∂A, A] and Proposition 2.24, we get
+(3.11)
+|I2|D0,m ≲ Cε,mt− 1
+2−ε|A|2
+D0,m,
+1
+2 < m < 1,
+where Cε,m → ∞ as ε �� 0 or m → 1.
+For I3, we apply Lemma 2.20 to get
+|I3|D0,m = |[St, ∂]PA|D0,m ≲ tr+ 1
+2−m|PA|D0,r+ 1
+2 ≲ tr+ 1
+2 −m|A|D0,r,
+r ≥ 1
+2,
+r + 1
+2 ≥ m.
+(3.12)
+Substituting s for m we have
+(3.13)
+|I3|D0,s ≤ Crtr+ 1
+2−s|A|D0,r,
+r + 1
+2 ≥ s,
+r ≥ 1
+2,
+s ≥ 0.
+Substituting m for r in (3.12) we have
+(3.14)
+|I3|D0,m ≤ Cm|A|D0,m.
+To estimate I4, we recall that f = −StPA, so
+|I4|D0,m = |A∂f|D0,m ≤ Cm (|A|D0,m|∂f|D0,0 + |A|D0,0|∂f|D0,m)
+≤ Cm (|A|D0,m|f|D0,1 + |A|D0,0|f|D0,m+1)
+≤ Cm
+�
+|A|D0,mt− 1
+2 |A|D0,0 + |A|D0,0t− 1
+2|A|D0,m
+�
+≤ Cmt− 1
+2 |A|D0,s|A|D0,m,
+m, s ≥ 0
+where we used that |f|D0,1 = |StPA|D0,1 ≲ t− 1
+2 |PA|D0, 1
+2 ≲ t− 1
+2|A|D0,0, and similarly |f|D0,m+1 ≲
+Cmt− 1
+2 |A|D0,m for any m ≥ 0. Note that if bΩ is merely C2, then |I4|D0,m ≤ Cmt− 1
+2 |A|D0,s|A|D0,m,
+for m, s > 0. Applying the above estimate with s in place of m we get
+(3.15)
+|I4|D0,s ≤ Cst− 1
+2|A|2
+D0,s,
+s ≥ 0.
+Alternatively, by using the initial condition (3.1), we have
+(3.16)
+|I4|D0,m ≤ Cm|A|D0,m,
+m ≥ 0
+
+17
+Next, we estimate the low and high-order norms of (I + K)−1, where K := ∂f + A∂f. By using
+f = −StPA and the initial condition (3.1), we have
+|K|D0,m ≤ |∂StPA|D0,m + |A∂StPA|D0,m
+≤ |StPA|D0,m+1 + |A|D0,m|∂StPA|D0,0 + |A|D0,0|∂StPA|D0,m
+≲ Cmt− 1
+2 |A|D0,m,
+m ≥ 0.
+Applying the above estimate with m = s and using the initial condition (3.1) we get
+(3.17)
+|K|D0,s ≤ 1/C∗
+s ,
+0 < s < 1.
+We now consider (I + K)−1. Using the formula (I + K)−1 = [det(I + K)]−1B, where B is the
+adjugate matrix of I + K, we see that every entry in (I + K)−1 is a polynomial in [det(I + K)]−1
+and entries of I5. By using the product estimate (2.2) and (3.17), we get
+(3.18)
+|(I + K)−1|D0,m ≤ Cm(1 + |K|D0,m) ≤ Cm(1 + t− 1
+2 |A|D0,m),
+m > 0.
+In particular by the initial condition (3.1), we have
+(3.19)
+|(I + K)−1|D0,s ≤ Cs(1 + |K|D0,s) ≲ Cs,
+0 ≤ s ≤ 1.
+We now estimate the s-norm of �A = (I + K)−1(�4
+j=1 Ij). Applying the product estimate (2.2) and
+(3.7), (3.8), (3.18), (3.19), we get
+(3.20)
+|(I + K)−1I1|m ≤ |(I + K)−1|m|I1|0 + |(I + K)−1|0|I1|m
+≲ Cm(1 + t− 1
+2 |A|D0,m)(t
+1
+2|A|D0, 1
+2) + Cm|A|D0,m
+≲ Cm|A|D0,m + Cm|A|D0, 1
+2 |A|D0,m ≲ Cm|A|D0,m,
+m ≥ 0,
+where in the last inequality we used the initial condition (3.1). If 0 ≤ s < 1, we can use estimate
+(3.7) to get
+(3.21)
+|(I + K)−1I1|s ≤ |(I + K)−1|s|I1|0 + |(I + K)−1|0|I1|s
+≲ |(I + K)−1|s|I1|s ≤ Crtr−s|A|r,
+s ≤ r.
+Using estimates (3.9), (3.10), (3.18) and (3.19) we get
+(3.22)
+|(I + K)−1I2|m ≤ |(I + K)−1|m|I2|0 + |(I + K)−1|0|I2|m
+≲ Cm(1 + t− 1
+2|A|D0,m)(t− 1
+2 |A|2
+D0,s) + Cm|A|D0,m
+≲ Cmt−1|A|2
+D0,s|A|D0,m + Cm|A|D0,m ≲ Cm|A|D0,m,
+m ≥ 1.
+If 1
+2 < s < 1, we can use estimate (3.11) to get
+(3.23)
+|(I + K)−1I2|s ≲ |(I + K)−1|s|I2|s ≤ Cs,εt− 1
+2 −ε|A|2
+s.
+Here Cs,ε → ∞ as ε → 0 or s → 1.
+In a similar way, by using estimates (3.13), (3.14), (3.15) and (3.16), we can show that
+(3.24)
+|(I + K)−1I3|D0,m, |(I + K)−1I4|D0,m ≲ Cm|A|D0,m,
+m ≥ 1,
+and for 1
+2 < s < 1,
+(3.25)
+|(I + K)−1I3|D0,s ≲ Crtr−s|A|r, s ≤ r,
+|(I + K)−1I4|D0,s ≲ Cst− 1
+2|A|2
+s.
+Combining estimates (3.20), (3.22), (3.24), we obtain for �A = (I + K)−1(�4
+j=1 Ij) the following
+m-norm estimate:
+(3.26)
+| �A|D0,m ≤ Cm|A|D0,m,
+m ≥ 1.
+
+18
+By using (3.21), (3.23) and (3.25), we obtain the following estimate for the s-norm of �A:
+| �A|D0,s ≲ Crtr−s|A|D0,r + Cs,εt− 1
+2−ε|A|2
+D0,s,
+1
+2 < s < 1,
+s ≤ r.
+(3.27)
+Finally, we estimate the norms of A′ = �A ◦ G, where G = I + g = F −1. Using (3.4) and the initial
+condition, we may assume that |g| 3
+2 < 1
+2. Let D1 = F(D0). Applying the chain rule estimate (2.3)
+with ε = 3
+2 to A′ on D1 and also (3.3) we obtain
+|A′|D1,m = | �A ◦ G|D1,m ≤ C(a, D)(| �
+A|D0,m|G|
+4
+3
+D1, 3
+2 + ∥ �A∥D0,1|G|D1,m + ∥ �A∥D0,0)
+≤ C(m, D)| �A|D0,m, m > 1.
+If 0 < s < 1, we can apply (2.4) to get
+|A′|D1,s = | �A ◦ G|D1,s ≤ | �A|D0,s∥G∥α
+D1,1 ≲ |A|D0,s.
+Using estimates (3.26) and (3.26) for �A, we obtain
+|A′|D1,m ≤ C(m, D)|A|D0,m,
+m > 1
+(3.28)
+|A′|D0,s ≲ Crtr−s|A|D0,r + Cs,εt− 1
+2−ε|A|2
+D0,s,
+1
+2 < s < 1,
+s ≤ r.
+(3.29)
+4. Iteration Scheme and convergence of maps
+Proposition 4.1. Let 1
+2 < s < 1 and r > 1. Let α, β, d, λ, γ be positive numbers satisfying
+r − s − λ − γ > αd + β,
+α(2 − d) > 1
+2 + ε + λ,
+β(d − 1) > λ,
+1 < d < 2.
+Note that the second and fourth conditions imply that α > 1
+2. Let D0 be a strictly pseudoconvex
+domain with a C3 defining function ρ0 on U and X(0) = ∂ + A0∂ ∈ Λr(D0) be a formally integrable
+almost complex structure. There exists a constant
+ˆt0 = ˆt0(C∗
+s , Cr, Cs,ε),
+ε = ε(r)
+such that if 0 < t0 ≤ ˆt0 and
+|A0|D0,s ≤ tα
+0 ,
+|A0|D0,r ≤ t−γ
+0 ,
+then the following statements are true for i = 0, 1, 2, . . .
+(i) There exists a C∞ diffeomorphism Fi = I + fi from B0 onto itself with F −1
+i
+= I + gi such
+that fi, gi satisfy
+|gi|B0, ≤
+(ii) Set ρi+1 = ρi ◦ F −1
+i
+. Then Di+1 := Fi(Di) = {z ∈ U : ρi+1 < 0} and
+∥ρi+1 − ρ0∥U,2 ≤ ε(D0),
+(4.1)
+dist(Di+1, ∂U) ≥ dist(D0, ∂U) − Cε.
+(4.2)
+(iii) (Fi|Di)∗(X(i)) is in the span of Xi+1 := ∂ + Ai+1∂ on Di+1.
+Moreover, ai = |Ai|Di,s,
+Li = |Ai|Di,r satisfy
+ai ≤ tα
+i ,
+Li ≤ L0t−β
+i
+.
+(iv) Suppose A0 ∈ C∞(D0). There exist some η(d) > 0 independent of m and N = N(m, d) ∈ N
+such that
+Mi ≤ MNt−η
+i
+,
+i > N.
+
+19
+Proof. We do induction on i. First we prove (ii)-(iv) for i = 0. Fix 1
+2 < s < 1, r ≥ s, and set
+ai = |Ai|Di,s, Li = |Ai|Di,r. We choose
+(4.3)
+ˆt0 ≤
+� 1
+C∗s
+�
+2
+2α−1
+.
+Then
+t
+− 1
+2
+0
+|A0|D0,s ≤ t
+α− 1
+2
+0
+≤ 1
+C∗s
+,
+t0 < ˆt0.
+By (3.28)-(3.29) we have
+a1 ≤ Crtr−s
+0
+L0 + Cs,εt− 1
+2 −εa2
+0
+L1 ≤ CrL0.
+For some fixed λ > 0, we further choose ˆt0 satisfying
+(4.4)
+ˆt0 ≤ min
+�� 1
+2Cr
+� 1
+λ
+,
+�
+1
+2Cs,ε
+� 1
+λ
+�
+.
+Then for all 0 < t0 < ˆt0, we have Cr, Cs,ε ≤ 1
+2t−λ
+0 . Hence
+a1 ≤ 1
+2(tr−s
+0
+t−γ−λ
+0
++ t2α
+0 t
+− 1
+2 −ε
+0
+t−λ
+0 ) ≤ tdα
+0
+= tα
+1
+L1 ≤ t−λ
+0 L0 ≤ t−βd
+0
+L0 = t−β
+1 L0.
+where we have assumed the following constraints:
+(4.5)
+αd < r − s − γ − λ
+α(2 − d) > 1
+2 + ε + λ
+βd > λ.
+Thus we have verified for i = 0 assuming the intersection of the above constraints is nonempty. We
+will see in the induction step that this is true provided r > s + 1
+2.
+Now assume that the induction hypotheses hold for some i − 1 ∈ N, i ≥ 1. Since ti < t0, by
+condition (4.3) we have
+(4.6)
+t
+− 1
+2
+i
+|Ai|Di,s ≤ t
+α− 1
+2
+i
+≤ 1
+C∗s
+,
+ti < t0 < ˆt0.
+Therefore estimates (3.28)-(3.29) hold
+ai+1 ≤ Crtr−s
+i
+Li + Cs,εt
+− 1
+2 −ε
+i
+a2
+i ,
+Li+1 ≤ CrLi.
+Notice that by the condition (4.4), we still have Cr, Cs,ε ≤
+1
+2t−λ
+i
+since ti < t0.
+Also we have
+Li ≤ L0t−β
+i
+≤ t−γ−β
+i
+. Hence
+ai+1 ≤ 1
+2(tr−s
+i
+t−λ−γ−β
+i
++ t−λ
+i
+t
+− 1
+2−ε
+i
+t2α
+i ) ≤ tdα
+i
+= tα
+i+1,
+Li+1 ≤ t−λ
+i
+Li ≤ t−λ−β
+i
+L0 ≤ t−dβ
+i
+L0 = t−β
+i+1L0.
+
+20
+where we have assumed
+(4.7)
+αd + β < r − s − λ − γ,
+α(2 − d) > 1
+2 + ε + λ,
+β >
+λ
+d − 1.
+Notice that the above constraint is more strict than (4.5). Let ξ := r − s and D(ξ, d, λ, γ, ε) be
+the set of (α, β) such that (4.7) is satisfied.
+We now determine the values of ξ, d, λ such that
+D(ξ, d, λ, γ, ε) is non-empty. Consider the limiting domain for fixed ξ, d and λ, γ, ε = 0:
+D(ξ, d, 0) =
+�
+(α, β) ∈ (0, ∞)2 : αd + β < ξ,
+α(2 − d) > 1
+2,
+α > 1
+2
+�
+.
+By the defining inequalities of D(ξ, d, 0), it is non-empty if and only if
+ξ > p(d),
+p(d) :=
+d
+2(2 − d).
+On the interval (1, 2), p is a strictly increasing function with infimum value p(1) = 1
+2. This implies
+that
+(4.8)
+r − s > p(1) = 1
+2,
+r > s + 1
+2 > 1
+(since s > 1
+2).
+Notice that under the above condition for r, s, D(ξ, d, λ, γ, ε) is still non-empty for sufficiently small
+λ, γ, ε. In summary, given r = 1 + δ0 for sufficiently small δ0 > 0, we first choose s > 1
+2 + δ0
+2 such
+that (4.8) is satisfied. This is possible by choosing d ∈ (1, 2) sufficiently close to 1. We then choose
+(α, β) ∈ D(ξ, d, λ, γ, ε) with λ, γ, ε chosen sufficiently small, such that (4.7) holds.
+(iii) First, notice that since all previous assumptions still hold, we have (i),(ii),(iii). Denote Mi :=
+|Ai|m.
+Now, since we have condition (4.6), we get from estimate (3.28) that
+Mi+1 ≤ CmMi.
+Fix λ > 0. There exists N = N(m, d) ∈ N such that
+(4.9)
+Cm ≤ t−λ
+i
+,
+for all i ≥ N.
+We would like to show that there exists η such that for all i ≥ N, the following holds
+Mi ≤ MNt−η
+i
+,
+i ≥ N.
+For i = N, the above inequality is obvious. Assume it holds for some i ≥ N. Then
+Mi+1 ≤ CmMi ≤ t−λ
+i
+t−η
+i
+MN ≤ t−dη
+i
+MN = t−η
+i+1,
+i ≥ N,
+where we have chosen η > 0 to satisfy
+η(d − 1) > λ.
+□
+Proposition 4.2. Let r > 1 and 1
+2 < s < 1. Let D0 be a C3 strictly pseudoconvex domain in Cn
+and let Xα = ∂α + Aβ
+α∂β ∈ Λr(D0), α = 1, . . . , n be a formally integrable almost complex structure
+on D0. There exist constants α > 1
+2, γ ∈ (0, 1) and ˆt0 > 0 such that if
+|A|D0,s ≤ tα
+0 ,
+|A|D0,r ≤ t−γ
+0 .
+where 0 < t0 ≤ ˆt0, then the following statements are true.
+
+21
+(i) Suppose A ∈ Λl+ε(D0), with l > 1 and ε > 0. Then there exists a sequence of mappings �Fj
+converging to an embedding F : D0 → Cn in Λl+ 1
+2 (D0).
+(ii) If A ∈ C∞(D0), then F ∈ C∞(D0).
+(iii) F∗(Xα) are in the span of ∂1, . . . , ∂n and F(D0) is strictly pseudoconvex.
+Proof. Write l = (1 − θ)s + θm, where s < l < m + ℓ + ε and θ ∈ (0, 1) is to be chosen. By
+Proposition 4.1, there exist d ∈ (1, 2) and α > 1
+2 such that ai := |Ai|Di,s and Mi := |Ai|Di,m satisfy
+ai ≤ tα
+i ,
+Mi ≤ MNt−η
+i
+,
+ti+1 = td
+i ,
+i ≥ N.
+Here η satisfies the condition
+(4.10)
+η >
+λ
+d − 1.
+Here λ > 0 is some fixed constant for which
+(4.11)
+Cm ≤ t−λ
+i
+,
+i ≥ N = N(m, d).
+Using convexity of H¨older-Zygmund norm (2.1), we have for all i ≥ N,
+|fi|Di,ℓ+ 1
+2 ≤ Cm|fi|1−θ
+Di,s+ 1
+2 |fi|θ
+Di,m+ 1
+2 ≲ |Ai|1−θ
+Di,s|Ai|θ
+Di,m ≤ MNt(1−θ)α
+i
+t−θη
+i
+≤ MNt(1−θ)α−θη
+i
+.
+Consider the composition �Fi+1 = Fi◦Fi−1◦· · ·◦F0, where Fi = I +fi for i ≥ 0. By using Lemma 2.3
+and above estimate for fi, we obtain for all i ≥ N,
+(4.12)
+| �Fi+1 − �Fi|D0,ℓ+ 1
+2 = |fi ◦ Fi−1 ◦ · · · F0|D0,ℓ
+≤ Cj
+ℓ
+�
+|fi|ℓ+ 1
+2 +
+�
+i
+∥fi∥1|fi|ℓ+ 1
+2 + |fi|ℓ+ 1
+2∥fi∥1
+�
+≲ Cj
+ℓ |fi|ℓ+ 1
+2 ≲ Cj
+ℓ MNt(1−θ)α−θη
+i
+Hence �Fi is a Cauchy sequence in Λℓ+ 1
+2 (D0), provided that (1 − θ)α − θη > 0, which can be
+achieved by choosing 0 < θ <
+α
+α+η < 1. In view of (4.10), we can choose η to be arbitrarily close
+to 0 by choosing λ small while fixing d > 1 (Notice that (4.11) then requires N = N(m, d) to be
+chosen large.) Hence θ can be chosen to be arbitrarily close to 1. In other words, m can be chosen
+arbitrarily close to l = s+θ(m−s) and ε can be arbitrarily close to 0. This shows that �Fi converges
+to some limit F ∈ Λℓ+ 1
+2 (D0), provided that A0 ∈ Cl+ε(D0) for any ε > 0.
+(ii) Fix any ℓ > 1 and we have A ∈ Λℓ(D0). We can then choose arbitrarily large m > ℓ and
+set ℓ = (1 − θ)s + θm. Then the same argument as in (i) shows that �Fi converges to some limit
+F ∈ Cℓ+ 1
+2 (D0). Since ℓ is arbitrary, we have F ∈ C∞(D0).
+Finally we show that F is a diffeomorphism. By the inverse function theorem, it suffices to check
+that the Jacobian of F(x) is invertible at every x0 ∈ D. Write DF = I − (DF − I), then DF is
+invertible with inverse (I − (I − DF))−1 if |DF − I| < 1. Write
+DF − I = lim
+j→∞D �Fj+1 = lim
+j→∞[D �Fj+1 − D �Fj] + [D �Fj − D �Fj−1] + · · · [D �F2 − D �F1].
+where we set �F1 to be the identity map. Then
+|DF − I|D0,0 ≤
+∞
+�
+j=1
+|D �Fj+1 − D �Fj|D0,0 ≤
+∞
+�
+j=1
+| �Fj+1 − �Fj|D0,1.
+
+22
+By the estimates in (4.12), we have for some s ∈ (1
+2, 1):
+∞
+�
+j=1
+| �Fj+1 − �Fj|D0,1 ≤
+∞
+�
+j=1
+Cj|fj|Dj,s+ 1
+2 ≤
+∞
+�
+j=1
+Cj
+s|Aj|Dj,s ≤
+∞
+�
+j=1
+Cj
+stα
+j =
+∞
+�
+j=1
+Cj
+stdjα
+0
+which is less than 1 if we choose t0 < ε0 for some ε0 > 0.
+(iii) This follows from the fact that |Ai|Di,s ≤ tα
+i which converges to 0 as i → ∞.
+□
+References
+[BCD11] Hajer Bahouri, Jean-Yves Chemin, and Rapha¨el Danchin, Fourier analysis and nonlinear partial differen-
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+Email address: zs327@rutgers.edu
+

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+arXiv:2301.00577v1  [hep-th]  2 Jan 2023
+January, 2023
+Weyl invariance, non-compact duality and conformal
+higher-derivative sigma models
+Darren T. Grasso, Sergei M. Kuzenko and Joshua R. Pinelli
+Department of Physics M013, The University of Western Australia
+35 Stirling Highway, Crawley W.A. 6009, Australia
+Email: darren.grasso@uwa.edu.au, sergei.kuzenko@uwa.edu.au,
+joshua.pinelli@research.uwa.edu.au
+Abstract
+We study a system of n Abelian vector fields coupled to 1
+2n(n + 1) complex
+scalars parametrising the Hermitian symmetric space Sp(2n, R)/U(n). This model
+is Weyl invariant and possesses the maximal non-compact duality group Sp(2n, R).
+Although both symmetries are anomalous in the quantum theory, they should be
+respected by the logarithmic divergent term (the “induced action”) of the effec-
+tive action obtained by integrating out the vector fields. We compute this induced
+action and demonstrate its Weyl and Sp(2n, R) invariance. The resulting confor-
+mal higher-derivative σ-model on Sp(2n, R)/U(n) is generalised to the cases where
+the fields take their values in (i) an arbitrary K¨ahler space; and (ii) an arbitrary
+Riemannian manifold. In both cases, the σ-model Lagrangian generates a Weyl
+anomaly satisfying the Wess-Zumino consistency condition.
+
+Contents
+1
+Introduction
+1
+2
+Computing the induced action
+6
+2.1
+Quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.2
+Heat kernel calculations
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+2.3
+Geometric expression for the induced action . . . . . . . . . . . . . . . . .
+10
+3
+Generalisations and open problems
+12
+A Hermitian symmetric space Sp(2n, R)/U(n)
+15
+B Alternative field redefinition
+19
+C Curved space basis structures
+20
+1
+Introduction
+A unique feature of the Weyl multiplet of N = 4 conformal supergravity [1] is the
+presence of a dimensionless complex scalar field φ that parametrises the Hermitian sym-
+metric space SL(2, R)/SO(2).1 The most general family of invariant actions for N = 4
+conformal supergravity was derived only a few years ago by Butter, Ciceri, de Wit and
+Sahoo, [2, 3]. Such an action is uniquely determined by a holomorphic function H(φ)
+which accompanies the terms quadratic in the Weyl tensor in the Lagrangian.
+For the special choice H = const, in which case the N = 4 conformal supergravity
+action proves to be invariant under rigid SL(2, R) transformations, the corresponding
+action was constructed in 2015 by Ciceri and Sahoo [4] to second order in fermions. The
+1This coset space can equivalently be realised as SU(1, 1)/U(1), since the groups SL(2, R) and SU(1, 1)
+are isomorphic. Refs. [2,3] make use of the latter group.
+1
+
+bosonic sector of the latter action had been computed in 2012 by Buchbinder, Pletnev and
+Tseytlin [5] as an “induced action”, obtained by integrating out an Abelian N = 4 vector
+multiplet coupled to external N = 4 conformal supergravity.2 The purely φ-dependent
+part of the Lagrangian is a higher-derivative σ-model of the form [6]:
+L(φ, ¯φ) =
+1
+(Im φ)2
+�
+D2φD2 ¯φ − 2(Rmn − 1
+3gmnR)∇mφ∇n ¯φ
+�
++
+1
+12(Im φ)4
+�
+α∇mφ∇mφ∇n ¯φ∇n ¯φ + β∇mφ∇m ¯φ∇nφ∇n ¯φ
+�
+,
+(1.1)
+where
+D2φ := ∇m∇mφ +
+i
+Im φ∇mφ∇mφ ,
+(1.2)
+and α and β are numerical parameters. In the case of N = 4 conformal supergravity,
+these coefficients are [5]: α = 1
+2β = 1. The Lagrangian (1.1) is invariant under SL(2, R)
+transformations
+φ → φ′ = aφ + b
+cφ + d ,
+
+ a
+b
+c
+d
+
+ ∈ SL(2, R)
+(1.3)
+acting on the upper half-plane Im φ > 0 with metric
+ds2 = dφ d¯φ
+(Im φ)2
+=⇒
+Γφ
+φφ =
+i
+Im φ ,
+Γ
+¯φ
+¯φ¯φ = −
+i
+Im φ .
+(1.4)
+The functional
+�
+d4x √−g L proves to be invariant under Weyl transformations
+gmn(x) → e2σ(x)gmn(x) ,
+(1.5)
+since the scalar field φ is inert under such transformations. The higher-derivative σ-model
+(1.1) possesses the N = 1 supersymmetric extension [7] which relates the parameters α
+and β. Both parameters are completely fixed if N = 2 supersymmetry is required [7–9].
+The conformal higher-derivative σ-model (1.1) admits a nontrivial generalisation that
+is obtained by replacing the Hermitian symmetric space SL(2, R)/SO(2) with an arbitrary
+n-dimensional K¨ahler manifold Mn, with n the complex dimension. We assume that Mn
+is parametrised by n local complex coordinates φI and their conjugates ¯φ¯I. Let K(φ, ¯φ)
+2Some of the relevant terms were missed in [5].
+2
+
+be the corresponding K¨ahler potential such that the K¨ahler metric gI ¯J(φ, ¯φ) is given by
+gI ¯J = ∂I∂ ¯JK. Associated with Mn is a higher-derivative sigma model of the form
+S =
+�
+d4x √−g
+�
+gI ¯J(φ, ¯φ)
+�
+D2φID2 ¯φ
+¯J − 2
+�
+Rmn − 1
+3Rgmn�
+∇mφI∇n ¯φ
+¯J�
++ FIJ ¯
+K ¯L(φ, ¯φ)∇mφI∇mφJ∇n ¯φ
+¯
+K∇n ¯φ
+¯L
++
+�
+GIJ ¯
+K ¯L(φ, ¯φ)∇mφI∇nφJ∇m ¯φ
+¯
+K∇n ¯φ
+¯L + c.c.
+��
+,
+(1.6)
+where Rmn is the spacetime Ricci tensor,
+D2φI := ∇m∇mφI + ΓI
+JK(φ, ¯φ)∇mφJ∇mφK ,
+(1.7)
+with ΓI
+JK being the Christoffel symbols for the K¨ahler metric gI ¯J. Finally, FIJ ¯
+K ¯L and
+GIJ ¯
+K ¯L are tensor fields on the target space, which are constructed from the K¨ahler metric
+gI ¯J, Riemann tensor RI ¯JK ¯L and, in general, its covariant derivatives. We recall that the
+Christoffel symbols ΓI
+JK and the curvature tensor RI ¯JK ¯L are given by the expressions3
+ΓI
+JK = gI ¯L∂J∂K∂¯LK ,
+RI ¯JK ¯L = ∂I∂K∂ ¯J∂¯LK − gM ¯
+N∂I∂K∂ ¯
+NK∂ ¯J∂¯L∂MK .
+(1.8)
+A typical expression for FIJ ¯
+K ¯L is
+FIJ ¯
+K ¯L = α1R(I ¯
+KJ)¯L + α2g(I ¯
+KgJ)¯L + . . .
+(1.9)
+The possible structure of GIJ ¯
+K ¯L is analogous. It should be pointed out that actions of
+the form (1.6) naturally emerge at the component level in N = 2 superconformal higher-
+derivative σ-models [8] (see also [9]), and in N = 1 ones [7].
+By construction, the action (1.6) is invariant under arbitrary holomorphic isometries
+of Mn.
+A nontrivial observation is that (1.6) is also invariant under arbitrary Weyl
+transformations of spacetime provided the scalars φI are inert under these transformations.
+Choosing Mn = Cn and gI ¯J(φ, ¯φ) = δI ¯J in (1.6) and integrating by parts, one obtains the
+Fradkin-Tseytlin (FT) operator [13]
+∆0 = (∇m∇m)2 + 2∇m�
+Rmn ∇n − 1
+3R ∇m
+�
+,
+(1.10)
+3The reader is referred, e.g., to [10–12] for a pedagogical review of K¨ahler geometry in the framework
+of nonlinear sigma models.
+3
+
+which is conformal when acting on dimensionless scalar fields.4 Given a Weyl inert scalar
+field ϕ, the Weyl transformation (1.5) acts on ∆0ϕ as
+∆0ϕ → ∆0ϕ = e−4σ∆0ϕ .
+(1.11)
+An action of the form
+�
+d4x √−g L(φ, ¯φ), with L given by (1.1), naturally emerges as
+an induced action in Maxwell’s electrodynamics coupled to a dilaton ϕ and an axion a
+with Lagrangian
+L(F; φ, ¯φ) = −1
+4e−ϕF mnFmn − 1
+4a ˜F mnFmn
+= i
+2φF αβFαβ + c.c. ,
+φ = a + ie−ϕ .
+(1.12)
+Here ˜F mn =
+1
+2εmnrsFrs is the Hodge dual of the electromagnetic field strength Fmn =
+2∇[mAn] = 2∂[mAn], with εmnrs the Levi-Civita tensor. The second form of the Lagrangian
+(1.12) is written using two-component spinor notation, where the field strength Fmn =
+−Fnm is replaced with a symmetric rank-2 spinor Fαβ = Fβα and its conjugate ¯F ˙α ˙β.
+More precisely, if one considers the effective action, Γ[φ, ¯φ], obtained by integrating out
+the quantum gauge field in the model (1.12), then the logarithmically divergent part of
+Γ[φ, ¯φ] is given by
+�
+d4x √−g L(φ, ¯φ), as demonstrated by Osborn [6].
+An important
+question arises: why is the induced action Weyl and SL(2, R) invariant?
+We recall that the group of electromagnetic duality rotations of free Maxwell’s equa-
+tions is U(1). More than forty years ago, it was shown by Gaillard and Zumino [16, 17]
+that the non-compact group Sp(2n, R) is the maximal duality group of n Abelian vec-
+tor field strengths Fmn = (Fmn,i), with i = 1, . . . , n, in the presence of a collection of
+complex scalars φij = φji parametrising the homogeneous space Sp(2n, R)/U(n), with
+i, j = 1, . . . , n. In the absence of such scalars, the largest duality group proves to be U(n),
+the maximal compact subgroup of Sp(2n, R). These results admit a natural extension to
+the case when the pure vector field part L(F) of the Lagrangian L(F; φ, ¯φ) is a nonlinear
+U(1) duality invariant theory [18–22] (see [23–25] for reviews), for instance Born-Infeld
+theory. However, in the case that L(F) is quadratic, the F-dependent part of L(F; φ, ¯φ)
+is also invariant under the Weyl transformations in curved space. Then, computing the
+4This operator was re-discovered by Paneitz in 1983 [14] and Riegert in 1984 [15].
+4
+
+path integral over the gauge fields leads to an effective action, Γ[φ, ¯φ], such that its log-
+arithmically divergent part is invariant under Weyl and rigid Sp(2n, R) transformations,
+see, e.g., [26,27] for formal arguments. Both symmetries are anomalous at the quantum
+level, but the logarithmically divergent part of the one-loop effective action is invariant
+under these transformations.
+In this paper we demonstrate that an action of the type (1.6) emerges as an induced
+action in a model for n Abelian gauge fields Am = (Am,i), i = 1, . . . , n, coupled to a
+complex field φ = (φij) and its conjugate ¯φ = (¯φ¯i¯j) parametrising the homogeneous space
+Sp(2n, R)/U(n),
+φ = φT ∈ Mat(n, C) ,
+i(¯φ − φ) > 0 .
+(1.13)
+The corresponding Lagrangian is
+L(n)(F; φ, ¯φ) = −1
+4
+�
+(F mn)T ΞFmn + (F mn)T Υ ˜Fmn
+�
+= i
+2
+�
+F αβ�T φFαβ + c.c. ,
+(1.14)
+where we have also introduced the real matrices Ξ and Υ defined by
+φ = Υ + i Ξ ,
+(1.15)
+with Ξ being positive definite.
+The model described by (1.14) has two fundamental
+properties: (i) its duality group is Sp(2n, R) (see, e.g. [23] for the technical details); and
+(ii) it is Weyl invariant. The induced action must respect these properties.
+This paper is organised as follows. In section 2 we compute the logarithmically diver-
+gent part of the effective action obtained by integrating out the vector fields in the model
+(1.14). Generalisations of our analysis and open problems are briefly discussed in section
+3. The main body of the paper is accompanied by three technical appendices. In appendix
+A we collect necessary facts about the Hermitian symmetric space Sp(2n, R)/U(n). Ap-
+pendix B provides an alternative calculation of the induced action compared with that
+given in subsection 2.2. Appendix C provides a complete list of the structures introduced
+in (2.22).
+5
+
+2
+Computing the induced action
+In this section we compute the logarithmically divergent part of the effective action,
+Γ[φ, ¯φ], defined by
+eiΓ[φ,¯φ] =
+�
+[DA] δ
+�
+η − χ(A)
+�
+Det (∆gh) eiS[A;φ,¯φ] .
+(2.1)
+Here S[A; φ, ¯φ] is the classical action corresponding to (1.14),
+S[A; φ, ¯φ] =
+�
+d4x √−g L(n)(F; φ, ¯φ) ,
+(2.2)
+χ(A) denotes a gauge fixing condition, ∆gh the corresponding Faddeev-Popov operator
+[28], and η an arbitrary background field. Since the effective action is independent of η,
+this field can be integrated out with some weight that we choose to be
+exp
+�
+− i
+2
+�
+d4x √−g ηTΞη
+�
+.
+(2.3)
+In general, the logarithmically divergent part of the effective action has the form
+Γ∞ = − ln Λ
+(4π)2
+�
+d4x √−g (a2)total ,
+(2.4)
+where (a2)total denotes the appropriate sum of diagonal DeWitt coefficients. We identify
+the induced action with
+�
+d4x √−g (a2)total, modulo an overall numerical coefficient.
+2.1
+Quantisation
+We choose the simplest gauge-fixing condition
+χ(A) = ∇mAm ,
+(2.5)
+which leads to the ghost operator
+∆gh := ✷1 ,
+(2.6)
+with 1 the n × n unit matrix. Integrating the right-hand side of (2.1) with the weight
+functional (2.3) leads to the gauge-fixing term
+SG.F.[A; φ, ¯φ] = −1
+2
+�
+d4x √−g (∇mAm)TΞ(∇nAn) .
+(2.7)
+6
+
+As a result, the gauge-fixed action becomes
+Squadratic[A; φ, ¯φ] = 1
+2
+�
+d4x √−g A ˆm∆ ˆmˆnAˆn ,
+(2.8)
+where here we have introduced hatted indices corresponding to a pair of spacetime and
+internal indices A ˆm := (Ami), ∆ ˆmˆn := (∆mi,nj). Contractions over hatted indices encode
+summations over both indices, however the position of the hatted indices (up or down)
+indicates only the position of the spacetime indices, internal indices are always understood
+as matrix multiplication. The non-minimal operator ∆ ˆmˆn is defined as:
+∆mi,nj := Ξijgmn✷ + V mi,p,nj∇p − RmnΞij ,
+✷ := ∇m∇m ,
+(2.9a)
+V mi,p,nj := (∇pΞij)gmn − (∇nΞij)gmp + (∇mΞij)gpn − (∇qΥij)εmpnq .
+(2.9b)
+From here onward matrix indices will be suppressed, unless there may be ambiguity or
+confusion. The one-loop effective action is specified by
+Γ(1)[φ, ¯φ] = i
+2Tr ln ∆ − iTr ln ∆gh .
+(2.10)
+Since Ξ is symmetric and positive definite, due to (1.13), its inverse Ξ−1, square root Ξ1/2
+and inverse square root Ξ−1/2 are well-defined. We perform a local field redefinition in
+the path integral:
+Am → Ξ−1/2Am ,
+(2.11)
+so that the operator which appeared in (2.8) becomes5
+˜∆ ˆmˆn = Ξ−1/2∆mnΞ−1/2 .
+(2.12)
+Inserting the explicit form of ∆ ˆmˆn from (2.9a) and (2.9b), the ˜∆ ˆm
+ˆn operator is now
+minimal:
+˜∆ ˆm
+ˆn = 1 δm
+n✷ + Q ˆm
+pˆn∇p + T ˆm
+ˆn ,
+(2.13a)
+Q ˆm
+pˆn := −2
+�
+∇pΞ1/2�
+Ξ−1/2δm
+n + Ξ−1/2 V m
+pn Ξ−1/2 ,
+(2.13b)
+T ˆm
+ˆn := −
+�
+✷Ξ1/2�
+δm
+n + 2
+�
+∇pΞ1/2�
+Ξ−1/2 �
+∇pΞ1/2�
+Ξ−1/2δm
+n
+− Ξ−1/2 V m
+pnΞ−1/2 �
+∇pΞ1/2�
+Ξ−1/2 − Rm
+n1 .
+(2.13c)
+After our field redefinition the one-loop effective action is given by
+Γ(1)[φ, ¯φ] = i
+2Tr ln ˜∆ − iTr ln ∆gh .
+(2.14)
+5In appendix B, we provide the results for an alternative field redefinition which leads to an equivalent
+logarithmic divergence up to total derivative.
+7
+
+2.2
+Heat kernel calculations
+Since the operator ˜∆ ˆm
+ˆn defined by (2.13a) is minimal, we can proceed with the stan-
+dard heat kernel technique in curved space, by bringing it to the form:
+˜∆ ˆm
+ˆn =
+� ˆ∇p ˆ∇p
+� ˆm
+ˆn + ˆP ˆm
+ˆn ,
+(2.15a)
+ˆP ˆm
+ˆn := −1
+2∇pQ ˆm
+pˆn − 1
+4 Q ˆm
+qˆpQˆpq
+ˆn + T ˆm
+ˆn .
+(2.15b)
+The generalised covariant derivative ˆ∇m introduced above is defined to act on a column
+matrix A ˆm = (Am
+i) as
+� ˆ∇pA
+� ˆm := ∇pA ˆm + 1
+2 Q ˆm
+pˆnAˆn .
+(2.16)
+The generalised covariant derivatives have no torsion, meaning
+� ˆ∇p, ˆ∇q
+�
+A ˆm = ˆR ˆm
+ˆnpqAˆn ,
+(2.17)
+with ˆR ˆm
+ˆnpq some generalised curvature anti-symmetric in p, q. Explicitly it has the form
+ˆR ˆm
+ˆnpq = Rm
+npq1 + 1
+2∇pQ ˆm
+qˆn − 1
+2∇qQ ˆm
+pˆn + 1
+4 Q ˆm
+pˆrQˆr
+qˆn − 1
+4 Q ˆm
+qˆrQˆr
+pˆn .
+(2.18)
+Using the standard Schwinger-DeWitt formalism [29–33] in curved spacetime for an oper-
+ator of the form (2.15a), in the coincidence limit the DeWitt coefficient traced over matrix
+indices, (a2) ˜∆(x, x), is given by
+(a2)
+˜∆(x, x) =
+� 1
+45RmnpqRmnpq − 1
+45RmnRmn + 1
+18R2 + 2
+15✷R
+�
+Tr1
++ 1
+12
+ˆR ˆmˆnpq ˆRˆn ˆmpq + 1
+6
+� ˆ∇p ˆ∇p ˆP
+� ˆm
+ˆm + 1
+2
+� ˆP 2� ˆm
+ˆm + 1
+6R ˆP ˆm
+ˆm ,
+(2.19)
+where ‘Tr’ denotes the matrix trace. Similarly for the ghost operator (2.6), the corre-
+sponding traced DeWitt coefficient (a2)∆gh(x, x) (noting that the generalised curvature
+vanishes) is
+(a2)∆gh(x, x) =
+� 1
+180RmnpqRmnpq −
+1
+180RmnRmn + 1
+72R2 + 1
+30✷R
+�
+Tr1 ,
+(2.20)
+which contains purely gravitational components. Armed with the set of equations (2.15a
+– 2.18), we expand out (a2) ˜∆(x, x) (2.19) explicitly in terms of Q ˆm
+pˆn (2.13b) and T ˆm
+ˆn
+(2.13c)
+(a2)
+˜∆(x, x) =
+�
+− 11
+180RmnpqRmnpq − 1
+45RmnRmn + 1
+18R2 + 2
+15✷R
+�
+Tr1
+8
+
++ 1
+6Rp
+qr
+m
+�
+∇qQmi
+r,pi
+�
++ 1
+12Rp
+qr
+m Qmi
+qˆsQˆs
+r,pi − 1
+12R
+�
+∇pQ ˆm
+p ˆm
+�
+− 1
+24R Q ˆm
+qˆpQˆpq
+ˆm + 1
+6R T ˆm
+ˆm − 1
+12✷∇pQ ˆm
+p ˆm − 1
+12
+�
+✷Q ˆm
+qˆp
+�
+Qˆpq
+ˆm
+− 1
+24
+�
+∇qQ ˆm
+rˆp
+� �
+∇qQˆpr
+ˆm
+�
+− 1
+24
+�
+∇qQ ˆm
+rˆp
+� �
+∇rQˆp
+q ˆm
+�
++ 1
+8
+�
+∇rQ ˆm
+rˆp
+� �
+∇sQˆp
+s ˆm
+�
++ 1
+24
+�
+∇qQ ˆm
+rˆp
+�
+Qˆp
+qˆsQˆsr
+ˆm − 1
+24
+�
+∇qQ ˆm
+rˆp
+�
+Qˆpr
+ˆsQˆs
+q ˆm
++ 1
+8
+�
+∇rQ ˆm
+rˆp
+�
+Qˆp
+tˆsQˆst
+ˆm + 1
+96 Q ˆm
+qˆsQˆs
+rˆpQˆpq
+ˆtQˆtr
+ˆm + 1
+48 Q ˆm
+qˆpQˆpq
+ˆrQˆr
+sˆtQˆts
+ˆm
+− 1
+2
+�
+∇pQ ˆm
+pˆq
+�
+T ˆq
+ˆm − 1
+4T ˆm
+ˆrQˆr
+pˆqQˆqp
+ˆm + 1
+6✷T ˆm
+ˆm + 1
+2(T 2) ˆm
+ˆm .
+(2.21)
+Using the definitions of of Q ˆm
+pˆn (2.13b) and T ˆm
+ˆn (2.13c), we perform the laborious task
+of expanding (a2) ˜∆(x, x) in terms of the matrices Ξ and Υ. It reduces to the following
+form:
+(a2)
+˜∆(x, x) = Tr
+� 7
+24T1 + 1
+48T2 − 5
+12T3 + 1
+12T4 + 7
+12T5 + 5
+24T6 − 1
+24T7 + 5
+24T8 + 7
+24T9
++ 1
+48T10 + 1
+4T11 + 1
+4T12 − 1
+2T13 + 1
+2T14 − 1
+2T15 − 1
+2T16 − 1
+2T17 − 1
+2T18
++ 1
+6T19 + 1
+6T20 + X 1 + ∇mYm + ✷Z
+�
+,
+(2.22)
+where the contributions T1, . . . , T20 are listed in appendix C. We have also introduced:
+X := − 11
+180RmnpqRmnpq + 43
+90RmnRmn − 1
+9R2 − 1
+30✷R ,
+(2.23a)
+Ym := − 1
+4Ξ−1(∇mΞ)Ξ−1(∇nΞ)Ξ−1(∇nΞ) + 1
+4Ξ−1(∇mΞ)Ξ−1(∇nΥ)Ξ−1(∇nΥ)
++ 1
+12Ξ−1(∇nΞ)Ξ−1(∇mΥ)Ξ−1(∇nΥ) + 1
+12Ξ−1(∇nΞ)Ξ−1(∇nΥ)Ξ−1(∇mΥ)
++ 1
+6Ξ−1(∇mΞ)Ξ−1(✷Ξ) − 1
+6Ξ−1(∇mΥ)Ξ−1(✷Υ)
+− 1
+3
+�
+Rmn − 1
+2Rgmn�
+Ξ−1(∇nΞ) ,
+(2.23b)
+Z := − 1
+6Ξ−1(∇mΥ)Ξ−1(∇mΥ) − 1
+3Ξ−1(✷Ξ) + 4
+3Ξ−1/2(✷Ξ1/2)
++ 4
+3Ξ−1(∇nΞ)Ξ−1/2(∇nΞ1/2) .
+(2.23c)
+The total DeWitt coefficient corresponding to the logarithmic divergence of the effective
+action (2.14) is given by
+(a2)total = (a2)
+˜∆(x, x) − 2(a2)∆gh(x, x) ,
+(2.24)
+9
+
+where (a2)∆gh(x, x) was given in (2.20). Recalling the expression for Ξ and Υ in terms of
+the original fields φ and its conjugate ¯φ (1.15), and defining
+D2φ := ✷φ + i(∇mφ)Ξ−1(∇mφ) ,
+D2 ¯φ := ✷¯φ − i(∇m ¯φ)Ξ−1(∇m ¯φ) ,
+(2.25)
+the total DeWitt coefficient is given by
+(a2)total = n
+� 1
+10F − 31
+180G − 1
+10✷R
+�
++ 1
+4Tr
+�
+Ξ−1(D2φ)Ξ−1(D2 ¯φ) − 2
+�
+Rmn − 1
+3Rgmn�
+Ξ−1(∇mφ)Ξ−1(∇n ¯φ)
+�
++ 1
+24Tr
+�
+Ξ−1(∇mφ)Ξ−1(∇m ¯φ)Ξ−1(∇nφ)Ξ−1(∇n ¯φ)
+�
++ 1
+48Tr
+�
+Ξ−1(∇mφ)Ξ−1(∇n ¯φ)Ξ−1(∇mφ)Ξ−1(∇n ¯φ)
+�
+,
+(2.26)
+where F is the square of the Weyl tensor, G is the Euler density,
+F = RmnpqRmnpq − 2RmnRmn + 1
+3R2 ,
+G = RmnpqRmnpq − 4RmnRmn + R2 , (2.27)
+and we have removed the total derivative pieces Tr
+�
+∇mYm�
+and Tr
+�
+✷Z
+�
+since they do
+not contribute to the induced action
+�
+d4x √−g (a2)total. The ✷R in (2.26) is also a total
+derivative and can be omitted.
+Setting n = 1 in (2.26) yields the expected result derived in [5,6]
+(a2)total = 1
+10F − 31
+180G − 1
+10✷R
++
+1
+4(Imφ)2
+�
+D2φD2 ¯φ − 2
+�
+Rmn − 1
+3Rgmn�
+∇mφ∇n ¯φ
+�
++
+1
+48(Imφ)4
+�
+∇mφ∇mφ∇n ¯φ∇n ¯φ + 2∇mφ∇m ¯φ∇nφ∇n ¯φ
+�
+.
+(2.28)
+2.3
+Geometric expression for the induced action
+To recast (2.26) in terms of geometric objects defined on the Hermitian symmetric
+space Sp(2n, R)/U(n), here we analyse the dependence of (2.26) on the symmetric matrix
+φ = (φij) = (φji) ≡ (φI) and its conjugate ¯φ =
+�¯φ¯i¯j�
+=
+�¯φ ¯j¯i�
+≡ (¯φ¯I).
+We make the standard choice for K¨ahler potential K(φij, ¯φ¯i¯j) on Sp(2n, R)/U(n)
+K(φ, ¯φ) := −4Tr ln Ξ ,
+(2.29)
+10
+
+which is well defined since Ξ is a positive definite matrix. The group Sp(2n, R) acts on
+Sp(2n, R)/U(n) by fractional linear transformations (A.18). Given such a transformation,
+the K¨ahler potential changes as
+K(φ, ¯φ) → K(φ, ¯φ) + Λ(φ) + ¯Λ(¯φ) ,
+(2.30)
+in accordance with (A.20).
+Therefore, the K¨ahler metric is invariant under arbitrary
+Sp(2n, R) transformations.
+The K¨ahler metric is given by6
+gij,¯k¯l = g(ij),(¯k¯l) =
+∂2K
+∂φij∂ ¯φ ¯k¯l = (Ξ−1)i(¯k(Ξ−1)¯l)j ,
+(2.31)
+where (i1 · · · in) denotes symmetrisation in indices i1, . . . , in. Note that pairs of indices
+are symmetrised over due to φ being symmetric (1.13). Here and in what follows, we use
+the notation
+Ki1i2,...,i2p−1i2p,¯i1¯i2,...,¯i2q−1¯i2q =
+∂p+qK
+∂φi1i2 · · · ∂φi2p−1i2p∂ ¯φ¯i1¯i2 · · ·∂ ¯φ¯i2q−1¯i2q .
+(2.32)
+In accordance with (1.8), the Christoffel symbols are given by
+Γi1i2
+i3i4,i5i6 = gi1i2,¯i7¯i8Ki3i4,i5i6,¯i7¯i8 ,
+Γ
+¯i1¯i2
+¯i3¯i4,¯i5¯i6 = (Γi1i2
+i3i4,i5i6)∗ ,
+(2.33)
+and the Riemann curvature tensor is
+Ri1i2,¯i3¯i4,i5i6,¯i7¯i8 = Ki1i2,i5i6,¯i3¯i4,¯i7¯i8 − gi9i10,¯i11¯i12Ki1i2,i5i6,¯i11¯i12Ki9i10,¯i3¯i4,¯i7¯i8 .
+(2.34)
+Noting that the inverse K¨ahler metric of (2.31) is
+gij,¯k¯l = Ξi(¯k Ξ
+¯l)j ,
+(2.35)
+one can calculate the Christoffel symbols, Riemann curvature tensor and Ricci tensor for
+the metric considered in (2.31) and we find:
+Γi1i2
+i3i4,i5i6 = iδ(i1
+(i3(Ξ−1)i4)(i5δi2)
+i6) ,
+(2.36a)
+6The partial derivatives with respect to symmetric matrices φ = (φij) and ¯φ = (¯φ¯i¯j) are defined by
+dK(φ, ¯φ) = dφij ∂K(φ, ¯φ)
+∂φij
++ d¯φ¯i¯j ∂K(φ, ¯φ)
+∂φ¯i¯j
+, and therefore ∂φkl
+∂φij = δk
+(iδl
+j). Symmetrisation of n indices includes
+a 1/n! factor. Vertical bars are notation to exclude indices contained between them from a separate
+symmetrisation, for example, (i1|(i2i3)|i4).
+11
+
+Γ
+¯i1¯i2
+¯i3¯i4,¯i5¯i6 = −iδ(¯i1
+(¯i3(Ξ−1)¯i4)(¯i5δ
+¯i2)
+¯i6) ,
+(2.36b)
+Ri1i2,¯i3¯i4,i5i6,¯i7¯i8 = 1
+2(Ξ−1)(¯i8|(i1(Ξ−1)i2)(¯i3(Ξ−1)¯i4)(i5(Ξ−1)i6)|¯i7) ,
+(2.36c)
+Ri1i2,¯i3¯i4 = −n
+2(Ξ−1)i1(¯i3(Ξ−1)¯i4)i2 = −n
+2gi1i2,¯i3¯i4 .
+(2.36d)
+The latter relation means that Sp(2n, R)/U(n) is an Einstein space.
+As pointed out at the beginning of this subsection, the complex variables φ and their
+conjugates ¯φ can be viewed either as symmetric matrices φ = (φij) and ¯φ = (¯φ¯i¯j) or as
+vector columns φ = (φI) and ¯φ = (¯φ¯I), with I, ¯I = 1, . . . , 1
+2n(n + 1). Resorting to the
+latter notation, the geometric structures (2.31) and (2.36a – 2.36c) can be used to recast
+(2.26) in the form:
+(a2)total = n
+� 1
+10F − 31
+180G − 1
+10✷R
+�
++ 1
+4gI ¯J(φ, ¯φ)
+�
+D2φID2 ¯φ
+¯J − 2
+�
+Rmn − 1
+3Rgmn�
+∇mφI∇n ¯φ
+¯J�
+(2.37a)
++ 1
+24RI ¯JK ¯L(φ, ¯φ)
+�
+2∇mφI∇m ¯φ
+¯J∇nφK∇n ¯φ
+¯L + ∇mφI∇n ¯φ
+¯J∇mφK∇n ¯φ
+¯L�
+,
+where
+D2φI = ✷φI + ΓI
+JK∇mφJ∇mφK ,
+D2 ¯φ
+¯I = ✷¯φ
+¯I + Γ
+¯I
+¯J ¯
+K∇m ¯φ
+¯J∇m ¯φ
+¯
+K .
+(2.37b)
+Every isometry transformation (A.18) acts on ∇φ and D2φ as follows:
+∇mφ′ =
+�
+(Cφ + D)−1�T(∇mφ)(Cφ + D)−1 ,
+(2.38a)
+D2φ′ =
+�
+(Cφ + D)−1�T(D2φ)(Cφ + D)−1 .
+(2.38b)
+It is now seen that the induced action defined by (2.37) is invariant under the isometry
+transformations on Sp(2n, R)/U(n).
+3
+Generalisations and open problems
+Relation (2.37), which constitutes the induced action, is our main result. The same
+structure also determines the Weyl anomaly of the effective action
+δσΓ ∝
+1
+(4π)2
+�
+d4x √−g σ (a2)total .
+(3.1)
+12
+
+It is well known that the purely gravitational part of this variation satisfies the Wess-
+Zumino consistency condition [34]
+[δσ2, δσ1]Γ = 0 ,
+(3.2)
+see, e.g., [35–37] for a review.7 The φ-dependent part of the Weyl anomaly will be dis-
+cussed below.
+As a generalisation of (1.6), we can introduce a conformal higher-derivative σ-model
+associated with a Riemannian manifold (Wd, g) parametrised by local coordinates ϕµ.
+The action is
+S =
+�
+d4x √−g
+�
+gµν(ϕ)
+�
+D2ϕµD2ϕν − 2
+�
+Rmn − 1
+3Rgmn�
+∇mϕµ∇nϕν�
++ Fµνσρ(ϕ)∇mϕµ∇mϕν∇nϕσ∇nϕρ
+�
+,
+(3.3a)
+where
+D2ϕµ := ✷ϕµ + Γµ
+νσ∇mϕν∇mϕσ ,
+✷ = ∇m∇m ,
+(3.3b)
+and Fµνσρ(ϕ) is a tensor field of rank (0, 4) on Wd. The Weyl invariance of the above
+action follows from
+δσ
+�√−g gµν(ϕ)
+�
+D2ϕµD2ϕν − 2
+�
+Rmn − 1
+3Rgmn�
+∇mϕµ∇nϕν��
+= 4√−g∇m
+�
+gµν(ϕ)
+�
+∇nσ∇mϕµ∇nϕν − 1
+2∇mσ∇nϕµ∇nϕν��
+.
+(3.4)
+In the case that the target space is K¨ahler, eq. (1.6), the relation (3.4) takes the form
+δσ
+�√−g gI ¯J(φ, ¯φ)
+�
+D2φID2 ¯φ
+¯J − 2
+�
+Rmn − 1
+3Rgmn�
+∇mφI∇n ¯φ
+¯J��
+= 2√−g ∇m
+�
+gI ¯J(φ, ¯φ)
+�
+∇nσ
+�
+∇mφI∇n ¯φ
+¯J + ∇nφI∇m ¯φ
+¯J�
+− ∇mσ∇nφI∇n ¯φ
+¯J��
+. (3.5)
+Choosing Wd to be R, specifying gµν(ϕ) and Fµνσρ(ϕ) to be constant, respectively,
+and restricting the background spacetime to be flat, the action (3.3) turns into
+S =
+�
+d4x L ,
+L = (∂2ϕ)2 + f(∂mϕ∂mϕ)2 ,
+(3.6)
+7The ✷R term, which contributes to (a2)total in (3.1), can be removed since it is generated by a local
+counterterm
+�
+d4x √−g R2.
+13
+
+with f a coupling constant, which is the model studied recently by Tseytlin [38].
+Assuming the model (3.3) originates from an induced action of some theory, the ϕ-
+dependent part of the Weyl anomaly should have the form
+δσΓ ∝
+�
+d4x √−g σ
+�
+gµν(ϕ)
+�
+D2ϕµD2ϕν − 2
+�
+Rmn − 1
+3Rgmn�
+∇mϕµ∇nϕν�
++ Fµνσρ(ϕ)∇mϕµ∇mϕν∇nϕσ∇nϕρ
+�
+.
+(3.7)
+The anomaly satisfies the Wess-Zumino consistency condition (3.2) as a consequence of
+the relation (3.4).
+It would be interesting to study renormalisation properties of a higher-derivative the-
+ory in Minkowski space with Lagrangian of the form
+L(F,φ, ¯φ) = L(n)(F; φ, ¯φ) + f1gI ¯J(φ, ¯φ)D2φID2 ¯φ
+¯J
++ RI ¯JK ¯L(φ, ¯φ)
+�
+f2∂mφI∂m ¯φ
+¯J∂nφK∂n ¯φ
+¯L + f3∂mφI∂n ¯φ
+¯J∂mφK∂n ¯φ
+¯L�
++ . . . ,
+(3.8)
+where L(n)(F; φ, ¯φ) is given by (1.14), the complex scalar fields φI and their conjugates
+¯φ¯I parametrise Sp(2n, R)/U(n), and f1, f2 and f3 are dimensionless coupling constants.
+All the structures in the F-independent part of (3.8) appear in the induced action (2.37).
+The ellipsis in (3.8) denotes other Sp(2n, R) terms that are quartic in ∂φ and ∂ ¯φ, such
+as the K¨ahler metric squared structure in (1.9).
+Such terms are possible for n > 1.
+The renormalisation of the most general fourth-order sigma models with dimensionless
+couplings in four dimensions was studied in [39,40]. All couplings constants in (3.8) are
+dimensionless, and the freedom to choose them is dictated by Sp(2n, R). This implies
+that the theory with classical Lagrangian (3.8) is renormalisable at the quantum level.
+It was discovered two years ago that Maxwell’s theory possesses a one-parameter
+conformal and U(1) duality invariant deformation [41, 42]; it was called the ModMax
+theory in [41]. Using the methods developed in [19–21], it can be coupled to the dilaton-
+axion field (1.12) to result in a conformal and SL(2, R) duality invariant model described
+by the Lagrangian [43]
+Lγ(F; φ, ¯φ) = −1
+2e−ϕ�
+ω + ¯ω
+�
+(cosh γ − 1) + e−ϕ√
+ω¯ω sinh γ + i
+2
+�
+φ ω − ¯φ ¯ω
+�
+, (3.9a)
+where
+ω = α + iβ = F αβFαβ ,
+α = 1
+4 F abFab ,
+β = 1
+4 F ab ˜Fab ,
+(3.9b)
+14
+
+and γ is a non-negative coupling constant [41]. For γ = 0 the model (3.9) reduces to
+(1.12). A challenging problem is to compute an induced action generated by (3.9).
+Acknowledgements:
+The work of SK is supported in part by the Australian Research Council, project No.
+DP200101944. The work of JP is supported by the Australian Government Research
+Training Program Scholarship.
+A
+Hermitian symmetric space Sp(2n, R)/U(n)
+In this appendix we collect necessary facts about the Hermitian symmetric space
+Sp(2n, R)/U(n). Here the symplectic group is defined by
+Sp(2n, R) =
+
+
+g ∈ GL(2n, R),
+gTJg = J ,
+J =
+
+
+0
+1n
+−1n
+0
+
+
+
+
+ .
+(A.1)
+Its maximal compact subgroup, H = Sp(2n, R)∩SO(2n), proves to be isomorphic to U(n).
+One way to see this is to make use of the isomorphism
+Sp(2n, R) ∼= Sp(2n, C) ∩ SU(n, n) ≡ G ,
+(A.2)
+which is obtained by considering the bijective map
+ϕ : g → g = A−1gA ,
+A = 1
+√
+2
+
+
+1n
+1n
+−i
+1n
+i1n
+
+ ,
+(A.3)
+for any g ∈ Sp(2n, R). Each complex matrix g = ϕ(g) is characterised by the properties
+gTJg = J ,
+g†In,ng = In,n ,
+In,n =
+
+
+1n
+0
+0
+−1n
+
+ ,
+(A.4)
+and thus g ∈ Sp(2n, C) ∩ SU(n, n).
+The inverse map ϕ−1 takes every group element
+g ∈ Sp(2n, C) ∩ SU(n, n) to some g ∈ GL(2n, R). For every element h from the maximal
+compact subgroup, h ∈ H = Sp(2n, R)∩SO(2n), its image h = ϕ(h) is unitary, h†h =
+12n.
+Simple calculations show that every h ∈ H = Sp(2n, R) ∩ SO(2n) has the form
+h =
+
+
+A
+B
+−B
+A
+
+ ,
+AAT + BBT =
+1n ,
+ABT = BAT ,
+(A.5)
+15
+
+and for its image h = ϕ(h) we obtain
+ϕ(h) =
+
+ A − iB
+0
+0
+A + iB
+
+ ,
+(A + iB)(A + iB)† =
+1n .
+(A.6)
+The group Sp(2n, R) naturally acts on C2n.
+This action is extended to that on a
+complex Grassmannian Grm,2n(C), with 0 < m < 2n, the space of m-planes through
+the origin in C2n. Of special interest is the Grassmannian Grn,2n(C). Given an n-plane
+P ∈ Grn,2n(C), it can be identified with a 2n × n matrix of rank n
+P =
+
+ M
+N
+
+ ,
+(A.7a)
+defined modulo equivalence transformations
+
+ M
+N
+
+ ∼
+
+ MR
+NR
+
+ ,
+R ∈ GL(n, C) .
+(A.7b)
+We denote by X ⊂ Grn,2n(C) the collection of all n-planes satisfying the two conditions:
+PTJP = 0 ;
+(A.8a)
+P†(iJ)P > 0 .
+(A.8b)
+Condition (A.8b) means that the Hermitian matrix P†(iJ)P is positive definite. Condition
+(A.8a) means that P is a Lagrangian subspace of C2n with respect to the symplectic
+structure J.
+By construction, the group Sp(2n, R) naturally acts on X.
+It turns out that this
+action is transitive, and X can be identified with Sp(2n, R)/U(n). The simplest way to
+see this is to make use of the realisation G of Sp(2n, R), eq. (A.2). The picture changing
+transformation (A.3) is accompanied with
+P → P = A−1P =
+
+ M
+N
+
+ .
+(A.9)
+For the transformed n-plane P the conditions (A.8) take the form
+PTJP = 0 ;
+(A.10a)
+P†In,nP > 0 .
+(A.10b)
+16
+
+The latter condition tells us that the matrix M in (A.9) is nonsingular, and therefore
+
+ M
+N
+
+ ∼
+
+
+1n
+NM −1
+
+ ≡
+
+
+1n
+ψ
+
+ .
+(A.11)
+In terms of the n × n matrix ψ, the conditions (A.10) are equivalent to
+ψT = ψ ,
+1n > ψ†ψ .
+(A.12)
+The complex n × n matrix ψ constrained by (A.12) and its conjugate ¯ψ provide a global
+coordinate system for X. Associated with ψ and ¯ψ is the group element
+S(ψ, ¯ψ) =
+
+ s
+¯ψ¯s
+ψs
+¯s
+
+ ∈ Sp(2n, C) ∩ SU(n, n) ,
+s =
+�
+1n − ¯ψψ
+�− 1
+2 .
+(A.13)
+Its important property is
+S(ψ, ¯ψ)P0 =
+
+ s
+ψs
+
+ ∼
+
+
+1n
+ψ
+
+ ,
+P0 :=
+
+
+1n
+0
+
+ .
+(A.14)
+Thus S(ψ, ¯ψ) maps the “origin” P0 to the point (A.11) of X, and therefore the group
+Sp(2n, C) ∩ SU(n, n) acts transitively on X. The isotropy subgroup of P0 can be seen to
+consist of the group elements (A.6), which span U(n). We conclude that
+X = Sp(2n, C) ∩ SU(n, n)
+U(n)
+= Sp(2n, R)
+U(n)
+.
+(A.15)
+Making use of (A.9) – (A.11), we can reconstruct a generic element of X in the original
+real realisation (A.1).
+P = A
+
+
+1n
+ψ
+
+ ∼
+
+
+1n + ψ
+−i(1n − ψ)
+
+ .
+(A.16)
+Since the matrices
+1n ± ψ are non-singular, P is equivalent to
+P ∼
+
+ φ
+1n
+
+ ,
+φ = i
+1n + ψ
+1n − ψ .
+(A.17a)
+The properties of φ follow from (A.8):
+φT = φ ,
+i
+�¯φ − φ
+�
+> 0 .
+(A.17b)
+17
+
+In this paper we make use of the φ-parametrisation (A.17) of the coset space (A.15).
+The group Sp(2n, R) acts on Sp(2n, R)/U(n) by fractional linear transformations
+φ → φ′ = (Aφ + B)(Cφ + D)−1 ,
+
+A
+B
+C
+D
+
+ ∈ Sp(2n, R) ,
+(A.18)
+and therefore
+dφ′ =
+�
+(Cφ + D)−1�Tdφ (Cφ + D)−1 .
+(A.19)
+Using the definition of symplectic matrices, eq. (A.1), one can show that the positive-
+definite matrix Ξ, which is defined by (1.15), transforms as follows:
+(Ξ′)−1 = (Cφ + D)Ξ−1(C ¯φ + D)T = (C ¯φ + D)Ξ−1(Cφ + D)T .
+(A.20)
+Let P1 and P2 be two points in Sp(2n, R)/U(n). We associate with them the following
+two-point function:
+s2(P1, P2) = −4Tr
+��
+P†
+1J ¯P2
+��
+PT
+2 J ¯P2
+�−1�
+PT
+2 J ¯P1
+��
+P†
+1J ¯P1
+�−1�
+.
+(A.21)
+By construction, it is invariant under arbitrary Sp(2n, R) transformations
+P1,2 → gP1,2 ,
+g ∈ Sp(2n, R) .
+(A.22)
+It is also invariant under arbitrary equivalence transformations (A.7),
+P1,2 → P1,2R1,2 ,
+R1,2 ∈ GL(n, C) .
+(A.23)
+Therefore, s2(P1, P2) is a two-point function of Sp(2n, R)/U(n) which is invariant under
+the isometry group Sp(2n, R). Due to the invariance of s2(P1, P2) under arbitrary right
+shifts (A.23), both P1 and P2 can be chosen to have the form (A.17). In the case that P1
+and P2 are infinitesimally separated, s2(P1, P2) becomes
+ds2 = Tr
+�
+d¯φ Ξ−1dφ Ξ−1�
+,
+(A.24)
+which is the K¨ahler metric on Sp(2n, R)/U(n), eq. (2.31).
+18
+
+B
+Alternative field redefinition
+Here we describe an alternative calculation of Tr ln ∆ compared with that given in
+section 2.2. Consider a path integral over two vector fields
+Det ∆−1 = N
+�
+[DB][DA] ei �d4x √−g B ˆ
+m∆ ˆ
+mˆnAˆn
+(B.1)
+with the operator ∆ ˆmˆn given in (2.9a). We perform an alternative local field definition in
+the path integral
+A ˆm → A ˆm,
+B ˆm → Ξ−1Bm ,
+(B.2)
+which was not possible for the original quadratic action (2.8). Now the operator which
+appeared in (B.1) has the form
+˜∆ ˆm
+ˆn := Ξ−1∆m
+n ,
+(B.3)
+which, once expanded explicitly from (2.9a) and (2.9b), is minimal:
+˜∆ ˆm
+ˆn = 1 δm
+n✷ + Qm
+pn∇p + T m
+n ,
+(B.4a)
+Q ˆm
+pˆn := Ξ−1 (∇pΞ) δm
+n − Ξ−1 (∇nΞ) δm
+p + Ξ−1 (∇mΞ) gpn − Ξ−1 (∇qΥ) ǫm
+pnq ,
+(B.4b)
+T ˆm
+ˆn := −Rm
+n1 .
+(B.4c)
+Although both approaches of obtaining a minimal operator will lead to equivalent log-
+arithmic divergences up to total derivative, the alternative field definition proves to be
+much easier to manage computationally, since it solely involves derivatives of Ξ, Ξ−1 and
+Υ, rather than Ξ1/2 and Ξ−1/2. Following the same procedure as section 2.2, the final
+result for (a2)total (including total derivative contributions) is
+(a2)total = n
+� 1
+10F − 31
+180G − 1
+10✷R
+�
++ 1
+4Tr
+�
+Ξ−1(D2φ)Ξ−1(D2 ¯φ) − 2
+�
+Rmn − 1
+3Rgmn�
+Ξ−1(∇mφ)Ξ−1(∇n ¯φ)
+�
++ 1
+24Tr
+�
+Ξ−1(∇mφ)Ξ−1(∇m ¯φ)Ξ−1(∇nφ)Ξ−1(∇n ¯φ)
+�
++ 1
+48Tr
+�
+Ξ−1(∇mφ)Ξ−1(∇n ¯φ)Ξ−1(∇mφ)Ξ−1(∇n ¯φ)
+�
++ Tr
+�
+∇mYm�
++ Tr
+�
+✷ ˜Z
+�
+.
+(B.5)
+19
+
+Compared to the original field redefinition, the total derivative contributions are the same
+for Ym (2.23b) and differ from Z (2.23c)
+˜Z := − 1
+6Ξ−1(∇mΥ)Ξ−1(∇mΥ) − 1
+3Ξ−1(✷Ξ) + 1
+3Ξ−1(∇mΞ)Ξ−1(∇mΞ) .
+(B.6a)
+Indeed the two results for (a2)total (2.26) and (B.5) differ only by a total derivative, which
+does not contribute to the induced action
+�
+d4x √−g (a2)total.
+C
+Curved space basis structures
+Included below is a complete list of the basis structures introduced in (2.22). Note that
+under the trace over matrix indices some of these structures are equivalent to one another
+(via their transpose), however, since these structures are generated directly during the
+computation we have left them distinct for ease of computational reproducibility.
+T1 := Ξ−1(∇mΞ)Ξ−1(∇mΞ)Ξ−1(∇nΞ)Ξ−1(∇nΞ) ,
+T2 := Ξ−1(∇mΞ)Ξ−1(∇nΞ)Ξ−1(∇mΞ)Ξ−1(∇nΞ) ,
+T3 := Ξ−1(∇mΞ)Ξ−1(∇mΞ)Ξ−1(∇nΥ)Ξ−1(∇nΥ) ,
+T4 := Ξ−1(∇mΞ)Ξ−1(∇nΞ)Ξ−1(∇mΥ)Ξ−1(∇nΥ) ,
+T5 := Ξ−1(∇mΞ)Ξ−1(∇nΞ)Ξ−1(∇nΥ)Ξ−1(∇mΥ) ,
+T6 := Ξ−1(∇mΞ)Ξ−1(∇mΥ)Ξ−1(∇nΞ)Ξ−1(∇nΥ) ,
+T7 := Ξ−1(∇mΞ)Ξ−1(∇nΥ)Ξ−1(∇mΞ)Ξ−1(∇nΥ) ,
+T8 := Ξ−1(∇mΞ)Ξ−1(∇nΥ)Ξ−1(∇nΞ)Ξ−1(∇mΥ) ,
+T9 := Ξ−1(∇mΥ)Ξ−1(∇mΥ)Ξ−1(∇nΥ)Ξ−1(∇nΥ) ,
+T10 := Ξ−1(∇mΥ)Ξ−1(∇nΥ)Ξ−1(∇mΥ)Ξ−1(∇nΥ) ,
+T11 := Ξ−1(✷Ξ)Ξ−1(✷Ξ) ,
+T12 := Ξ−1(✷Υ)Ξ−1(✷Υ) ,
+T13 := Ξ−1(✷Ξ)Ξ−1(∇mΞ)Ξ−1(∇mΞ) ,
+T14 := Ξ−1(✷Ξ)Ξ−1(∇mΥ)Ξ−1(∇mΥ) ,
+T15 := Ξ−1(✷Υ)Ξ−1(∇mΞ)Ξ−1(∇mΥ) ,
+T16 := Ξ−1(✷Υ)Ξ−1(∇mΥ)Ξ−1(∇mΞ) ,
+20
+
+T17 := Rmn Ξ−1(∇mΞ)Ξ−1(∇nΞ) ,
+T18 := Rmn Ξ−1(∇mΥ)Ξ−1(∇nΥ) ,
+T19 := R Ξ−1(∇mΞ)Ξ−1(∇mΞ) ,
+T20 := R Ξ−1(∇mΥ)Ξ−1(∇mΥ) .
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+23
+

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@@ -0,0 +1,1056 @@
+SFP: Providing System Call Flow Protection against Software
+and Fault Attacks
+Robert Schilling
+robert.schilling@iaik.tugraz.at
+Graz University of Technology
+Graz, Austria
+Pascal Nasahl
+pascal.nasahl@iaik.tugraz.at
+Graz University of Technology
+Graz, Austria
+Martin Unterguggenberger
+martin.unterguggenberger@lamarr.at
+Graz University of Technology
+Lamarr Security Research
+Graz, Austria
+Stefan Mangard
+stefan.mangard@iaik.tugraz.at
+Graz University of Technology
+Lamarr Security Research
+Graz, Austria
+Abstract
+With the improvements in computing technologies, edge devices
+in the Internet-of-Things or the automotive area have become more
+complex. The enabler technology for these complex systems are
+powerful application core processors with operating system sup-
+port, such as Linux, replacing simpler bare-metal systems. While
+the isolation of applications through the operating system increases
+the security, the interface to the kernel poses a new threat. Different
+attack vectors, including fault attacks and memory vulnerabilities,
+exploit the kernel interface to escalate privileges and take over the
+system.
+In this work, we present SFP, a mechanism to protect the exe-
+cution of system calls against software and fault attacks providing
+integrity to user-kernel transitions. SFP provides system call flow
+integrity by a two-step linking approach, which links the system call
+and its origin to the state of control-flow integrity. A second linking
+step within the kernel ensures that the right system call is executed
+in the kernel. Combining both linking steps ensures that only the
+correct system call is executed at the right location in the program
+and cannot be skipped. Furthermore, SFP provides dynamic CFI
+instrumentation and a new CFI checking policy at the edge of the
+kernel to verify the control-flow state of user programs before en-
+tering the kernel. We integrated SFP into FIPAC, a CFI protection
+scheme exploiting ARM pointer authentication. Our prototype is
+based on a custom LLVM-based toolchain with an instrumented
+runtime library combined with a custom Linux kernel to protect sys-
+tem calls. The evaluation of micro- and macrobenchmarks based on
+SPEC 2017 show an average runtime overhead of 1.9 % and 20.6 %,
+which is only an increase of 1.8 % over plain control-flow protec-
+tion. This small impact on the performance shows the efficiency of
+SFP for protecting all system calls and providing integrity for the
+user-kernel transitions.
+Permission to make digital or hard copies of part or all of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for third-party components of this work must be honored.
+For all other uses, contact the owner/author(s).
+HASP ’22, October 1, 2022, Chicago, IL, USA
+© 2022 Copyright held by the owner/author(s).
+ACM ISBN 978-1-4503-9871-8/22/10.
+https://doi.org/10.1145/3569562.3569565
+CCS Concepts
+• Security and privacy → Tamper-proof and tamper-resistant
+designs; Side-channel analysis and countermeasures.
+Keywords
+System Call Flow Protection, Control-Flow Integrity, Fault Attacks.
+ACM Reference Format:
+Robert Schilling, Pascal Nasahl, Martin Unterguggenberger, and Stefan
+Mangard. 2022. SFP: Providing System Call Flow Protection against Software
+and Fault Attacks. In Hardware and Architectural Support for Security and
+Privacy (HASP ’22), October 1, 2022, Chicago, IL, USA. ACM, New York, NY,
+USA, 8 pages. https://doi.org/10.1145/3569562.3569565
+1
+Introduction
+Devices in the Internet-of-Things, automotive area, or indus-
+trial computers are getting more complex and powerful. While in
+the past, those systems used deeply embedded processing units
+with bare-metal applications, they now are based on powerful
+application-grade processors with the support for operating sys-
+tems [40]. Off-the-shelf operating systems, e.g., Linux, build the
+foundation for complex software [10]. They isolate different pro-
+grams, manage privileges, or restrict access to particular memory
+regions. User programs can only access kernel features via a small
+but well-defined interface, the system call (syscall) interface. For
+this reason, this interface to the kernel is a prominent target for
+attackers to escalate privileges and gain access to the system [48].
+One way of manipulating the system call interface is control-flow
+hijacking, which can be conducted with different methodologies.
+Classical control-flow attacks performed in software exploit a mem-
+ory vulnerability to modify a code-pointer or return address on the
+stack to redirect the execution of the program. When fault attacks
+are considered in the threat model, the attack surface in the kernel
+interface increases even more. While faults can manipulate the
+control-flow on a much finer granularity, e.g., they can manipulate
+direct branches, they can also manipulate the system call being
+executed. A control flow hijack can skip or change which system
+call gets executed, possibly with a critical security impact. Further-
+more, precise faults can directly manipulate which system call gets
+executed by manipulating the system call register containing the
+system call number.
+One way to counteract control-flow attacks is a generic mecha-
+nism called control-flow integrity (CFI) [1]. CFI exists at different
+arXiv:2301.02915v1  [cs.CR]  7 Jan 2023
+
+HASP ’22, October 1, 2022, Chicago, IL, USA
+Schilling et al.
+granularities, depending on which threat model is considered. In a
+classical software setting, only indirect branches are protected since
+those are the only ones an attacker can manipulate. Faults pose
+a more severe threat, thus requiring even more robust protection.
+Fine-grained instrumentation [2, 22, 36] protects the control-flow
+of a program on basic-block or even instruction-level [13, 53]. As a
+result, these countermeasures protect direct or indirect branches
+or even the whole instruction sequence. Instruction-granular pro-
+tection requires intrusive hardware changes to deal with the per-
+formance penalty, which is unsuitable for commodity devices.
+CFI can be enforced in different security domains. While tra-
+ditionally, CFI was only used to protect user-space applications,
+different CFI protection schemes can also protect the kernel [16, 21].
+However, currently, there are no CFI protection schemes available
+providing protection between different security domains, i.e., the
+transitions between the user-space program and the kernel. Thus,
+the large attack surface, the transitions between user programs
+to the kernel remain unprotected. Hence, there is a need for new
+countermeasures that protect the software interface to the kernel
+and provides system call flow integrity for commodity devices.
+Contribution
+In this work, we solve the problem of the unprotected system
+call interface and provide system call flow protection on top of
+CFI, protecting the interface to the kernel against both software
+and fault attacks. SFP cryptographically links the system call itself
+and its origin to a global CFI state that is verified at runtime in the
+operating system. A second-stage linking mechanism within the
+kernel dynamically applies a second link to ensure that the correct
+system call was selected and executed.
+To automatically protect arbitrary programs, we develop an
+LLVM-based toolchain to provide CFI and instrument all system
+calls. We provide an instrumented standard library, where all system
+calls are instrumented with our system call protection. Furthermore,
+we modify the Linux kernel to dynamically verify at runtime that
+the correct system call was executed.
+We implement SFP on top of FIPAC, a software-based CFI scheme
+exploiting ARM pointer authentication. We evaluate the perfor-
+mance of SFP based on a microbenchmark to measure the impact
+of SFP on the system call latency, leading to an overhead of 1.9 %.
+To show the applicability to real-world programs, we perform mac-
+robenchmarks using the SPEC 2017 application benchmark. On
+average, we measure a runtime overhead of 20.6 % for protected
+applications. Summarized, we make the following contributions:
+• We provide system call flow protection by linking the syscall
+and its origin to a global CFI state and verifying it at runtime.
+• We provide a prototype implementation comprising an LLVM-
+based toolchain, an instrumented C-standard library, and a
+modified Linux kernel.
+• We evaluate the performance based on a microbenchmark
+and on the application-grade SPEC 2017 benchmark.
+2
+Background
+This section provides background to fault attacks, pointer au-
+thentication, and control-flow integrity.
+2.1
+Fault Attacks
+Injecting faults into a digital circuit is a powerful threat allowing
+adversaries to break the security of a system entirely. The effect of
+an induced fault at the electrical level includes timing violations
+and transient voltage and current changes [42]. Typically, the effect
+of a fault is modeled at the bit-level with transient bit-flips and
+permanent stuck-at effects [52].
+Common fault injection approaches include voltage or clock
+glitching, laser fault injection (LFI), and electromagnetic fault injec-
+tion (EMFI) [54]. While these methodologies require physical access
+to the device, recently, new techniques relaxing this constraint have
+been released [11, 32, 39, 46]. E.g., in Plundervolt [32], the attacker
+utilizes the dynamic voltage scaling interface of the CPU to induce
+faults remotely in software.
+Independently of the injection technique, an attacker can exploit
+the effects of faults in various ways. E.g., fault attacks on encryp-
+tion primitives enable the attacker to leak secret keys [4, 12, 18].
+Despite dedicated attacks on encryption, fault attacks are also ac-
+tively used to bypass security features, such as secure boot, on
+embedded systems [17, 23, 35, 37, 49]. By inducing targeted faults
+into the program counter of a processor, faults enable an adversary
+to arbitrarily hijack the control-flow of a program [34, 47, 48].
+2.2
+Control-Flow Integrity
+The control-flow of a program can be hijacked using software
+attacks, fault attacks, or combined software-fault attacks. Therefore,
+various countermeasures targeting different attacker models were
+proposed to protect programs from these attack vectors.
+Software CFI Schemes. Software-based control-flow attacks are
+typically performed by exploiting a memory vulnerability. By over-
+writing control-flow-related data, e.g., return addresses or function
+pointers, the adversary can arbitrarily manipulate the execution of
+the program [5, 24, 45]. To mitigate these attacks, software control-
+flow integrity (SCFI) schemes [15, 25, 27, 31] aim to provide pointer
+integrity using different mechanisms. E.g., PARTS [27] uses ARM
+pointer authentication (PA) to cryptographically seal and verify
+security-sensitive pointers to protect them while stored in memory.
+Fault CFI Schemes. Software CFI schemes only protect control-
+flow transfers the adversary can also manipulate in the software
+threat model, i.e., return addresses and function pointers. Faults also
+allow the attacker to tamper with static control-flow data stored in
+the program or even skip instructions. Therefore, fault control-flow
+integrity (FCFI) schemes enforce their protection at a finer granular-
+ity, e.g., at the instruction level [13, 53]. However, as these schemes
+usually require custom hardware changes to avoid tremendous
+runtime overheads, software-based FCFI schemes typically operate
+at the function or basic block level [36, 41]. These schemes track
+the execution of the program using a signature and compare this
+running signature with a precomputed signature during runtime.
+Software Fault CFI Schemes. As most FCFI schemes [36, 41] do
+not consider a software attacker in their threat model, software
+attacks allow the adversary to bypass most FCFI schemes. Here, the
+adversary uses a memory bug to overwrite the state maintained in
+software and arbitrarily hijacks the control-flow. Hence, mitigating
+software, fault, and software-fault combined attacks require even
+stronger countermeasures, i.e., software fault CFI (SFCFI) schemes.
+
+SFP: System Call Flow Protection
+HASP ’22, October 1, 2022, Chicago, IL, USA
+2.3
+FIPAC
+FIPAC [44] is a software fault CFI (SFCFI) scheme mitigating soft-
+ware and fault-based control-flow attacks exploiting ARM pointer
+authentication. Internally, FIPAC maintains a global state through
+the entire program execution. When entering a basic block, i.e., a
+block of consecutive instructions without a control-flow transfer,
+FIPAC cryptographically updates the state. Depending on FIPAC’s
+configured checking policy, the value of the state is compared to the
+expected value determined during the compilation of the program
+at the end of each basic block, function, or program. On control-flow
+merges, i.e., indirect calls, the state is updated using a justification
+signature to ensure that different valid control-flow paths yield an
+identical state. To prevent a software adversary from predicting and
+overwriting the state using a memory bug, a MAC is utilized for
+the state update. Moreover, the state update and check functions
+cryptographically derive and verify the running signature on pro-
+gram execution. FIPAC uses the pointer authentication instructions
+of modern ARMv8.6A architectures for the MAC computation.
+ARM Pointer Authentication. ARM pointer authentication is a
+hardware feature introduced with ARMv8.3A [29] and updated in
+ARMv8.6A [30]. This extension provides new instructions to cryp-
+tographically sign and authenticate data. These instructions derive
+a message authentication code (MAC) using a secret key, a 64-bit
+modifier, and the value of a provided register, e.g., an address stored
+in a pointer. A fraction of this MAC, called the pointer authenti-
+cation code (PAC), is then stored in the upper bits of the provided
+register. By using the authentication instructions, the authenticity
+of the MAC and the data in the register can then be verified.
+2.4
+Linux and the System Call Interface
+Linux [50] is a monolithic kernel used in billions of devices [51]
+and embedded systems. To retrieve a particular service or get a
+specific resource, e.g., reading and writing a file, or to get dynamic
+memory, the user program needs to request this from the kernel,
+i.e., via a system call. A system call changes the privilege and trans-
+fers the execution from the user-space program to the kernel of
+the operating system, which then grants or denies the requested
+service. A user-space program aiming to execute a certain sys-
+tem call invokes the corresponding system call wrapper routine
+provided by a library. This wrapper then initiates a control-flow
+and privilege transfer into the kernel space by using a dedicated
+instruction, i.e., the svc instruction for AArch64. The system call
+instruction requires the system call number of the requested service
+and additional optional parameters as arguments.
+3
+Threat Model and Attack Scenario
+Our threat model considers a powerful adversary capable of per-
+forming software attacks, fault attacks, or combined software and
+fault attacks. This attacker can exploit memory vulnerabilities to ar-
+bitrarily read or modify data in memory. However, we assume that
+the code segment of the program cannot be modified by a software
+adversary by, for example, exploiting memory vulnerabilities. Nev-
+ertheless, by inducing faults, the attacker can flip bits in memory,
+the registers, the code segment, or the instruction pipeline of the
+processor. We assume that the control-flow of executed programs
+and the kernel is protected using an SFCFI scheme, such as FIPAC.
+Note that faults on the data, except the syscall register, are out
+of the scope of this work. It requires orthogonal schemes, e.g.,
+User mode
+Application
+...
+syscall_C()
+...
+libc syscall wrapper
+syscall_C()
+{
+...
+syscall args
+syscall numb
+svc
+...
+}
+Kernel mode
+Syscall handler
+syscall_C service routine
+...
+syscall_B service routine
+...
+sys_exit
+�
+Figure 1: Redirecting a system call using fault attacks.
+1 basic_block:
+2
+...
+3
+ldr w8, memAddress
+; load data from memAddress to w8
+4
+...
+5
+mov x0, #...
+; arguments for C system call
+6
+mov w8, #syscall_C
+; system call number for C
+7
+svc #0
+Listing 1: Invoking system call C on AArch64.
+redundancy encoding schemes for data [6], for their protection. We
+assume ARM PA to be cryptographically secure, and the attacker
+does not have access to the encryption keys. Furthermore, the
+operating system is assumed to be secure, providing isolation of
+the kernel task structure to the user program.
+3.1
+Attack Scenario
+Within this threat model, the adversary aims to hijack the pro-
+gram’s interface to the Linux kernel. In the example shown in
+Figure 1, the user program invokes the system call C using the
+Linux system call interface. However, by using a fault attack or a
+software-fault combined attack, the adversary can either (i) redirect
+the system call to B or (ii) entirely skip the system call.
+Listing 1 shows the instruction sequence to invoke the system
+call C on AArch64. The system call number is stored in register w8,
+and the system call arguments are stored in the remaining registers.
+By flipping bits in register w8 using faults, the adversary can redirect
+(i) the execution to a different system call.
+Moreover, the syscall gadget in Listing 1 is susceptible to com-
+bined attacks. A software-fault combined attacker utilizes a memory
+vulnerability to overwrite data at address memAddress. Afterward,
+in Line 4, the adversary hijacks the execution of the program by
+flipping bits in the program counter to redirect the control-flow
+to the svc instruction in Line 7, responsible for switching to the
+kernel. This attack enables the adversary to invoke arbitrary system
+calls. In addition to these attacks, a fault attacker can also corrupt
+the svc instruction to skip (ii) the execution of the entire syscall.
+SCFI schemes, such as FIPAC, currently cannot mitigate these
+attacks as these countermeasures do not consider transitions be-
+tween user-space and kernel space in their threat model. While
+they only protect the user-space application, they fail to provide
+protection for the kernel interface, posing a large threat surface
+for critical vulnerabilities. Furthermore, current SCFI protection
+schemes use static control-flow instrumentation, which is the same
+for subsequent calls to the program. As a result, an attacker with
+access to the code segment or to general-purpose registers can learn
+from subsequent program executions. Thus, it would be possible
+for an attacker to attempt multiple control-flow attacks until the
+hijack succeeds.
+3.2
+FIPAC Intra Basic Block Protection
+The authors of FIPAC describe a mechanism to extend the pro-
+tection guarantees of FIPAC from inter to intra-basic block secu-
+rity [44]. By applying a state update after every instruction within a
+
+HASP ’22, October 1, 2022, Chicago, IL, USA
+Schilling et al.
+Check(S, S𝐵)
+...
+Compute Sig𝐵2
+Patch(S, Sig𝐵2)
+Check(S, S𝐵2)
+Syscall exit
+Check(S, S𝐶)
+...
+Compute Sig𝐶2
+Patch(S, Sig𝐶2)
+Check(S, S𝐶2)
+Syscall exit
+Update(S, A)
+A
+Patch(S, Sig𝐶1)
+Syscall C
+...
+Application
+Kernel
+Syscall B
+Syscall C
+�
+✗
+✓
+Figure 2: System Call protection in SFP. Before a syscall, we
+cryptographically bind the syscall to the CFI state for later
+verification and second-stage linking in the kernel.
+basic block, the subsequently also update the CFI state continuously.
+Although this mechanism can be applied around syscalls, it does
+not add any protection. With a state update before and after the
+system call, an attacker can still fault the syscall number or manip-
+ulate the svc instruction to perform a nop instruction. Although
+this attack manipulates the execution of the system call, FIPAC’s
+extended intra-basic protection does not detect these attacks. Con-
+sequently, it requires a different protection scheme to provide call
+flow protection for system calls.
+4
+Design of SFP
+In this section, we present SFP, a mechanism that provides sys-
+tem call flow protection by exploiting a stateful CFI protection
+scheme. While SFP is generic and compatible with different CFI
+protection schemes, our design exploits FIPAC as the underlying
+CFI protection scheme. Section 7.3 discusses the compatibility as-
+pects and how SFP can be applied to different CFI schemes.
+4.1
+Requirements for System Call Protection
+The goal of SFP is to protect the system call interface to the
+kernel against software, fault, and combined attacks. Based on the
+attack scenario from Section 3, the protection of SFP must fulfill
+the following requirements.
+R1 System Call Number. Prevent an attacker from manipulating
+the system call number to a different system call.
+R2 System Call Execution. Ensure that a syscall cannot be skipped.
+R3 System Call Protection. Ensure the system call dispatcher in
+the kernel executes the correct system call function.
+R4 Dynamic CFI Instrumentation. Provide a dynamic CFI in-
+strumentation to ensure protection between consecutive
+program executions.
+4.2
+System Call Protection
+To fulfill requirements R1 to R3, SFP introduces a two-step ap-
+proach cryptographically linking the syscall to the state of the
+deployed SCFI scheme. First, at the system call caller site, we cryp-
+tographically link the system call origin and which system call we
+want to execute to the cryptographic CFI state. Second, at runtime,
+we perform a second-stage linking operation during the system call
+operation, confirming that the correct syscall gets executed.
+First-Stage System Call Linking. We statically identify at compile-
+time which system call is getting executed for all locations in the
+program. To protect the system call, SFP binds the syscall to the CFI
+state, i.e., to perform a CFI state update with the system call number.
+The system call number is a monotonically increasing number, thus
+not providing a significant Hamming distance between different
+system calls. A single bit-flip on the system call number changes the
+system call to a different one. As a result, the system call number
+cannot safely be used to bind it to the CFI state since faults can
+easily manipulate the system call to a different one.
+To overcome this limitation and perform a safe and secure state
+update, we need to compute a system call-dependent update value
+with a sufficiently large Hamming distance. In SFP, we exploit the
+cryptographic properties of ARM PA for this purpose. We use com-
+putation of a PACIA operation, with the system call and a random
+modifier as input, and compute a cryptographic 15-bit patch value
+for the particular system call. Due to the cryptographic MAC op-
+erations of ARM PA, the patch values for subsequent system call
+numbers have a large Hamming distance and cannot be computed
+without having access to the secret ARM PA key. The computa-
+tion of those patch values occurs at compile-time or load time and
+replaces the empty patch values in the binary.
+Before executing a system call and jumping to the kernel, we
+patch the CFI state with the statically computed system call patch,
+thus performing the first-stage linking. At this point in time, we
+bind the future execution of the particular system call to the CFI
+state ahead of executing it. Performing first-stage linking already
+provides protection for requirements R1 and partly R2.
+Second-Stage System Call Linking. After linking the system call
+to the CFI state in the user-space of the program, the system call
+is executed, and the execution switches into the kernel. Via dis-
+patching code and the selected system call in the general-purpose
+register w8, the kernel selects the correct system call function and
+executes it. At the end of each system call function, we apply a sec-
+ond patch, i.e., the second-stage linking to the CFI state, confirming
+that the previously selected system call was really executed. This
+patch value is computed dynamically during the execution of the
+syscall. The second linking step ensures that both requirements R2
+and R3 are fulfilled.
+In Figure 2, we summarize SFP’s system call protection. A user
+program performs the first-stage linking and patches the CFI state
+with a statically computed syscall patch to link the execution of a
+system call. The execution transitions to the kernel, which executes
+the desired system call function. At the end of the system call, the
+kernel performs the second-stage linking operation, followed by a
+CFI check operation. The later second-stage linking operation only
+succeeds when the correct system call is linked to the CFI state. As a
+result, SFP’s approach translates system call errors, independent of
+how they occur, to CFI state errors, which eventually are detected
+through the checking policy of the selected CFI protection scheme.
+Note, Figure 2 includes CFI checks at the beginning and end of the
+syscall to immediately detect a wrong syscall when entering the
+kernel and after the syscall’s execution.
+4.3
+Dynamic Instrumentation
+Existing SFCFI protection schemes [13, 33, 44, 53] use a static
+post-processing or encryption phase. A dedicated post-processing
+tool recovers the control-flow, computes the patch and check val-
+ues, and modifies the program. The static approach with a single
+
+SFP: System Call Flow Protection
+HASP ’22, October 1, 2022, Chicago, IL, USA
+encryption key leads to the fact that all executions of the same
+program use the same CFI values, e.g., patches, updates, or checks.
+By observing the used CFI-related values, attackers can more easily
+craft valid CFI states to bypass the control-flow protection.
+In SFP, we overcome this limitation by splitting up the toolchain
+and integrating the CFI instrumentation into the kernel. When
+starting a program, the ELF loader of the OS identifies a CFI in-
+strumented program. It generates a random ARM PA encryption
+key and stores it in the process task structure. The ELF loader then
+performs the per-program call unique CFI instrumentation and
+computes the expected CFI state and all patch values needed to
+handle the control-flow. The CFI states are stored along with the
+process task structure within the kernel. With this mechanism, sub-
+sequent calls to the same program create different encryption keys.
+As a result, it guarantees that different CFI values are generated on
+each new program start, i.e., fulfilling requirement R4.
+Kernel Checking Policy. In SFP, we develop a novel CFI checking
+policy at the edge of the operating system. Due to dynamically
+instrumenting the program when starting it, the operating system
+exactly knows the expected CFI state for every location of the
+program. When a user program now enters the kernel, e.g., due to
+a system call instruction, the kernel, which has access to both the
+user program state and the expected CFI states, can verify them. If
+the current CFI state matches the expected state, the system call
+continues. However, if the CFI state deviates from the expected
+state, a CFI error is detected, and the operating system aborts the
+program execution. A CFI check at the end of the syscall confirms
+the execution of the right syscall. Apart from system calls, a user
+program can enter the operating system also via different execution
+paths. We include the same checking policy when a timer interrupt
+is raised, and the kernel is entered.
+5
+Implementation
+The prototype implementation of SFP consists of two parts. First,
+we develop a toolchain to automatically compile and instrument
+arbitrary C-programs with CFI, including a custom runtime library.
+Second, we modify the kernel of the Linux operating system to
+include the system call verification, the new checking policy, and
+the dynamic instrumentation on the program start.
+5.1
+Toolchain
+Compiler. We base the toolchain on the modified compiler of
+FIPAC [43], which is based on the LLVM [26] compiler framework.
+We adapt the AArch64 backend of the compiler to instrument the
+control-flow and embed control-flow meta information in a custom
+section of the ELF binary. The compiler inserts the updates for
+every basic block, inserts patches for control-flow merges, and also
+deals with call instructions. Our modified compiler emits a running
+ELF binary but leaves all patch values for control-flow merges and
+system calls to be zero. The necessary post-processing step is shifted
+to the operating system, which computes all patches at the program
+start. Note that the instrumented program does not contain any
+check instructions as they are part of the transition to the operating
+system and are performed in the kernel.
+C Standard Library. System calls are typically invoked via wrap-
+per functions provided by the standard library of the programming
+language. This prototype toolchain uses a CFI-instrumented version
+of the musl [19] C standard library. The standard library provides
+1 basic_block:
+2
+...
+3
+mov x0, #...
+; arguments for B system call
+4
+mov x15, #0
+; Zero system call patch
+5
+eor x28, x28, x15
+; Perform a CFI state update
+6
+mov w8, #syscall_B
+; system call number for B
+7
+svc #0
+; Jump to kernel
+Listing 2: Patched system call in the musl standard library.
+wrapper functions for all system calls or uses system calls directly
+in different library functions. We identify every system call in the
+musl standard library and insert the necessary patch sequence con-
+taining an immediate load and the xor-based state update ahead
+of executing the system call. Listing 2 summarizes the first-stage
+linking, where the immediate value for the mov instruction is zero.
+When starting the binary, the operating system computes the actual
+patch value for this system call and fills out the correct load value.
+5.2
+Kernel Support
+SFP requires minor modifications to the operating system. We
+base the prototype of SFP on the Linux kernel in version 5.15.32 [3].
+Dynamic Instrumentation on Program Start. On program start,
+when an instrumented ELF binary is started, SFP performs the per-
+program instrumentation of the program. First, the kernel generates
+a random encryption key used for the PA instrumentation. With the
+help of control-flow metadata, which is stored along with the ELF
+binary in a metadata section, we compute the CFI state throughout
+the program and fill the necessary patch values for justifying sig-
+natures. Furthermore, we compute the syscall- and key-dependent
+patch values that are used to protect the system call interface. For
+every system call in the program, we compute its PAC based on
+the system call number and user-space program unique modifier.
+The resulting PAC value, which is not guessable by the attacker, is
+filling out the immediate patch value before the syscall.
+As discussed, the instrumented program does not contain dedi-
+cated CFI check operations as they are performed when entering the
+kernel. Instead, we store the expected CFI state for each program
+location in the task’s kernel structure. To reduce the storage over-
+head, we use a RangeMap, to only have one entry for a contiguous
+range of states, where it does not change.
+System Call Verification. During the system call, the user program
+updates the CFI state with a statically computed cryptographic
+patch value that depends on the system call number. The verification
+that the correct system call gets executed happens in the kernel.
+After the system call jumps into the kernel, a dispatcher code selects
+the correct system call function to be executed. At the end of every
+system call function in the kernel, we perform the second-stage
+linking. Based on the system call number, we dynamically compute
+a second patch value dependent on the currently executed system
+call. In Listing 3, we summarize this operation sequence, where we
+perform the second-stage linking within the kernel. To retrieve a
+cryptographically secure patch value, we exploit ARM PA’s PACIA
+instruction, which takes the system call and a modifier as input
+operands. Note that the modifier used for the kernel update of the
+CFI state is different from the one used for the first-stage linking
+in the user program. This property is essential to avoid attackers
+being able to skip system calls entirely since patching the CFI state
+twice with the same value would cancel out and has no permanent
+
+HASP ’22, October 1, 2022, Chicago, IL, USA
+Schilling et al.
+1 syscall_A:
+2
+...
+3
+mov x16, #1
+; Load kernel modifier
+4
+pacia x8, x16
+; Compute system call patch
+5
+eor x28, x28, x15
+; Perform 2nd stage linking
+6
+and x28, x28, #0xffffffff00000000 ; Clear syscall
+7
+ret
+; number from CFI state
+Listing 3: Dynamically computing the system call patch and
+removing it from the CFI state at the system call end.
+effect on the CFI state. We finally apply the computed patch to the
+CFI state and clear the lower bits from the system call.
+Checking Policy at the Kernel Boundary. Whenever a user pro-
+gram enters the kernel, SFP performs a CFI check to validate if the
+current CFI state still matches the expected state. We perform CFI
+checks on two entering points: During a system call and when a
+timer interrupt is raised. With the help of the CFI states stored in a
+RangeMap within the process structure and the knowledge of the
+program’s current program counter, we look up the expected CFI
+state for the program location. If the current CFI state, stored in the
+register x28 of the user program state, diverges from the expected
+state, a CFI error is raised, and SFP stops the program execution.
+For syscalls, we perform a second CFI check at the end of the syscall
+function in the kernel to ensure the syscall was really executed.
+6
+Evaluation
+In this section, we first evaluate the security of SFP and show
+how it provides protection and the defined threat model. Second,
+we evaluate the functionality and the performance overhead of the
+prototype implementation.
+6.1
+Security Evaluation
+We analyze the security guarantees of SFP and show how differ-
+ent attacks within the threat model are mitigated.
+Control-Flow Hijacks in the User-Space or Kernel. SFP provides CFI
+protection for the user-space application based on the selected un-
+derlying CFI protection scheme. The prototype uses FIPAC, a basic-
+block granular CFI scheme, protecting all direct/indirect branches
+as well as direct/indirect calls. The protection domain includes the
+C standard library, which is fully CFI instrumented. Consequently,
+an attacker cannot redirect syscalls in the user-space application
+by redirecting the control-flow to a different wrapper function of
+the standard library. Control-flow attacks in the kernel are detected
+via the kernel internal CFI protection scheme.
+Skipping a System Call. When skipping a system call instruction,
+i.e., the svc instruction, the first-stage linking already occurred. Sub-
+sequently, the skipped system call misses the second-stage linking
+from the kernel, which yields a wrong CFI state, which is detectable
+through the CFI checking policy. However, if the entire system call
+instruction sequence is skipped, i.e., first-stage patching and the
+syscall instruction are omitted, the hijack is still detectable. As both
+patch operations are missing on the CFI state, the state is wrong
+again, and a subsequent CFI check, e.g., when the program gets
+scheduled, detects the invalid state. In both cases, SFP transforms
+the skipped system call into a CFI error, which manifests itself in a
+wrong CFI state, which is detectable.
+Changing a System Call. A fault on the register containing the
+system call number, or a combined attack, in which the attacker
+controls the register used to execute the system call, redirects the
+system call to a different one. SFP protects against both attacks.
+By applying the first-stage linking to the CFI state, the correct
+system call is already bound to its future execution. Manipulating
+the system call register, e.g., due to a fault or software vulnerability,
+leads to applying the wrong system call patch to the CFI state. When
+the system call is executed, the CFI state for that program differs
+from the expected state, and the CFI check in the kernel detects the
+problem and aborts the program.
+To bypass a system call, the attacker only has a single chance
+to change the system call number and manipulate the previous
+system call patch to correct one for this location. However, the
+system call patch is protected via the secret ARM PA key, which
+the attacker cannot access. Guessing the PAC leads to a probability
+of 𝑝 =
+1
+215 = 0.0031 % for getting the correct patch value, where
+15 is the configured PAC size of our prototype implementation.
+Furthermore, due to the dynamic instrumentation on the program
+startup, the system call patches always differ between subsequent
+calls of the same program. As a result, the attacker cannot learn
+new patch information between subsequent program calls.
+6.2
+Functional Evaluation
+To validate the functional correctness of SFP, we emulate the exe-
+cution on the functional simulator QEMU [38] in version 7.0.0. Since
+this simulator currently only supports ARM PA from ARMv8.3-A,
+we extend it to include ARM PA of ARMv8.6-A to support the CFI
+protection. The functional evaluation runs the modified Linux ker-
+nel from the prototype and can start and execute instrumented
+programs, where all system calls are protected. Within the kernel,
+the functional simulator performs the second-stage linking and a
+CFI check to verify the execution of the correct syscall.
+To verify the functionality of the countermeasure, we emulated
+skipping a system call and modifying the system call number. In
+both cases, SFP detects the attack through the next CFI check since
+the CFI state became invalid and stops the program execution.
+6.3
+Performance Evaluation
+At the time of evaluation, there is no publicly available system
+supporting ARMv8.6-A needed to run FIPAC. However, to conduct
+the performance evaluation and to measure the performance im-
+pact of SFP, we emulate the runtime overhead of PA instructions.
+Therefore, we base the performance evaluation on a Raspberry Pi
+4 Model B [20] with 8 GB RAM configured with a fixed CPU fre-
+quency of 1.5 GHz. The Raspberry Pi contains an ARM Cortex-A72
+CPU based on ARMv8-A but without Pointer Authentication. To
+emulate the overhead of PA instructions, we replace them with
+a PA-analogue instruction sequence, i.e., four consecutive XORs.
+Related work [27, 28] evaluated this instruction sequence to mimic
+the timing behavior of a PA instruction.
+Microbenchmark. To evaluate the overhead of SFP executing sys-
+tem calls, we perform a simple microbenchmark. Our benchmark
+measures the syscall latency of the getpid system call, which is
+a side-effect-free syscall and is used in related works to bench-
+mark the syscall execution path [7, 8]. We execute the system call
+10 million times and measure the system call latency via the pro-
+cessor’s inbuilt cycle counter. Figure 3 summarizes our evaluation
+results, showing the syscall latency in different kernel configura-
+tions. On the plain unmodified Linux kernel, we measure an average
+
+SFP: System Call Flow Protection
+HASP ’22, October 1, 2022, Chicago, IL, USA
+Plain
+Syscall Check
+CFI Check
+Syscall + CFI Check
+2000
+2050
+2100
+2150
+2200
+Latency [cycles]
+2131.27
+2144.28
+2175.62
+2185.92
+Syscall Latency for getppid
+Figure 3: The microbenchmark shows the system call la-
+tency of the getpid system call for different kernel config-
+urations. SFP increases the system call latency by 1.9 %.
+lbm
+gcc
+perlbench
+xz
+x264
+mcf
+imagick
+nab
+0
+50
+100
+150
+Runtime Overhead [%]
+0.4
+60.2
+70.0
+14.4
+37.8
+30.5
+61.0
+9.1
+0.5
+61.8
+70.9
+17.2
+38.8
+32.4
+61.8
+11.2
+Basic CFI
+SFP
+Figure 4: Macrobenchmark shows the performance impact
+of SFP on the SPEC 2017 benchmark. We evaluate the impact
+of CFI only and SFP, including the system call protection.
+system call latency of 2131 cycles. When integrating the system
+call verification alone, the latency rises to 2144 cycles. Furthermore,
+with the CFI checks alone enabled, the latency increases to 2175 cy-
+cles. When both are active, we measure a system call latency of
+only 2185 cycles, impacting the system call latency by only 1.9 %.
+Macrobenchmark. To demonstrate the applicability of SFP on a
+larger scale, we perform a macrobenchmark on real-world applica-
+tions. We compiled the SPECspeed 2017 [14] benchmark with our
+toolchain, including only C-based programs. In Figure 4, we plot
+the runtime overheads in two different configurations compared to
+the plain uninstrumented code. First, we only include the dynamic
+verification, including the new CFI checking policy, that verifies
+the CFI state of user programs when entering the kernel. Second,
+we include the syscall protection based on the two-stage linking ap-
+proach together with the previously evaluated CFI checking policy.
+During the evaluation, we measure a geometric mean overhead
+of 18.8 % for the new CFI checking policy and 20.6 % with the system
+call protection and CFI checking policy in place. Based on the evalu-
+ation of the SPEC 2017 benchmark, we only measure a difference in
+the overhead of 1.8 % between the pure CFI protection and the full
+system call protection of SFP. This result shows that the dominating
+part of the overhead comes from the CFI instrumentation, not from
+the system call protection. Thus, reducing the overheads of the CFI
+protection directly influences the performance of SFP.
+7
+Discussion
+This section discusses prototype limitations and shows how SFP
+is compatible with other CFI protection schemes.
+7.1
+Dynamic System Call Instrumentation
+In our prototype, we manually instrument all syscalls of the C
+standard library with the necessary patch instructions, consisting
+of a load of an immediate patch value followed by applying the
+patch value to the CFI state. The immediate value is zero and is
+set to its concrete value during the dynamic instrumentation of
+the startup phase of the program. In a future version of SFP, we
+could instrument the compiler to detect syscall instructions, i.e., svc,
+and then automatically insert the necessary patch sequence. This
+enhancement would also include cases where syscalls are invoked
+manually without the wrapper functions of the standard library.
+7.2
+CFI Checking Policy Extension
+SFP currently performs CFI checks when entering the kernel
+through a syscall or a timer interrupt. A future version of this work
+can extend the CFI checking policy to include all interrupts of the
+system. Our microbenchmark shows adding new CFI checks adds
+minimal overhead to the syscall latency. Thus, adding additional
+CFI checks for all interrupt handlers are expected to have minimal
+impact on the system performance.
+7.3
+Compatibility
+Although SFP uses FIPAC as the underlying CFI protection scheme,
+the design or the protection mechanism of SFP is generic and com-
+patible with different CFI schemes. To apply the protection of SFP to
+a different protection scheme, two things are required. First, the CFI
+protection scheme must be stateful, and there must be a possibility
+to manipulate the state, e.g., via standard or custom instructions,
+to inject the system call patch. Second, it is necessary to be able to
+dynamically compute a second system call patch required for the
+second-stage linking in the kernel. With these requirements, SFP
+is compatible with existing CFI protection schemes such as SCFP,
+SOFIA, or any other state-based CFI protection scheme.
+8
+Related Work
+SCFP [53] and SOFIA [13] are hardware-assisted control-flow
+integrity schemes on the instruction level. They encrypt the pro-
+gram’s instruction stream at compile-time, and perform a fine-
+granular decryption during runtime to retrieve the correct instruc-
+tion sequence. In order to deal with the performance penalty, both
+protection schemes require intrusive hardware changes. This limits
+their applicability to small custom embedded processing cores but
+does not provide protection on a larger scale.
+FIPAC [44] is a software-based FCFI protection scheme that ex-
+ploits the architectural features of recent ARM processors. This
+protection scheme instruments all basic blocks of a user program
+with a running CFI signature, thus providing control-flow integrity
+at that granularity. They present three checking policies, i.e., where
+to check whether the running CFI signature still matches the ex-
+pected one. However, FIPAC only protects the control-flow of the
+user-space part of the program. Although FIPAC is developed for
+being used with operating systems, they miss the protection of the
+system call interface to the kernel.
+SFIP [9] implements coarse-grained syscall flow protection for
+user-space applications. They statically identify the possible transi-
+tions between different syscalls at compile-time and then enforce
+that at runtime. Since SFIP only considers software attackers in
+their threat model, they fail to protect against fault attacks.
+9
+Conclusion
+In this work, we presented SFP, a protection mechanism that
+provides system call flow protection on top of ordinary CFI, pro-
+tecting the interface to the kernel against both software and fault
+attacks. We show that an already employed CFI protection scheme
+
+HASP ’22, October 1, 2022, Chicago, IL, USA
+Schilling et al.
+can be used as a versatile tool to protect the system call interface
+to the kernel. Furthermore, we present a new CFI checking policy
+at the edge of the kernel to verify the CFI state for all transitions to
+the kernel. Combined with a dynamic CFI instrumentation on pro-
+gram startup, the attacker cannot learn CFI or system call-related
+information from subsequent program executions. We showed a
+prototype implementation comprising an LLVM-based toolchain to
+automatically instrument arbitrary programs and protect all system
+calls. A modified Linux kernel running on a Raspberry Pi evaluation
+setup is used to show the applicability of SFP to real-world pro-
+grams. Our evaluation based on a microbenchmark and on the SPEC
+2017 application benchmark shows an average runtime overhead
+of 20.6 %, which is only an increase of 1.8 % compared to plain CFI
+protection. This slight increase in the performance impact shows
+the effectiveness of SFP for protecting all system calls of a program.
+Acknowledgments
+This work has been supported by the Austrian Research Promo-
+tion Agency (FFG) under grant number 888087 (SEIZE).
+References
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+

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+Stochastic and Mixed Density Functional Theory within the projector augmented
+wave formalism for the simulation of warm dense matter
+Vidushi Sharma,1, 2 Lee A. Collins,1 and Alexander J. White1, ∗
+1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
+2Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
+(Dated: January 31, 2023)
+Stochastic and mixed stochastic-deterministic density functional theory (DFT) are promising new
+approaches for the calculation of the equation-of-state and transport properties in materials under
+extreme conditions.
+In the intermediate warm dense matter regime, a state between correlated
+condensed matter and kinetic plasma, electrons can range from being highly localized around nuclei
+to delocalized over the whole simulation cell. The plane-wave basis pseudo-potential approach is
+thus the typical tool of choice for modeling such systems at the DFT level.
+Unfortunately, the
+stochastic DFT methods scale as the square of the maximum plane-wave energy in this basis. To
+reduce the effect of this scaling, and improve the overall description of the electrons within the
+pseudo-potential approximation, we present stochastic and mixed DFT developed and implemented
+within the projector augmented wave formalism. We compare results between the different DFT
+approaches for both single-point and molecular dynamics trajectories and present calculations of
+self-diffusion coefficients of solid density carbon from 1 to 50 eV.
+The warm, dense matter (WDM) regime encompasses
+a wide variety of extreme environments and provides an
+excellent testing ground for methods that determine the
+basic properties of matter. These environments include,
+for example, planetary interiors [1–3], stellar systems
+such as brown and white dwarfs [4], and the capsule com-
+pression stage in inertial confinement fusion (ICF) [5–7].
+In the ice giant planets, the interiors may support a supe-
+rionic phase in which hydrogen remains fluid within the
+lattices of the heavier constituents such as oxygen, car-
+bon, nitrogen, and silicon [8–10], which may help explain
+the anomalous planetary magnetic fields of Neptune and
+Uranus. In addition, the difference between an exother-
+mic Neptune and an endothermic Uranus may originate
+in the nucleation of diamonds from hydrocarbon mixtures
+[11, 12]. Finally, properties of various hydrocarbons such
+as equations-of-state and thermal conductivities deter-
+mine the performance of ICF capsules irradiated by laser
+pulses [5], and stopping power characterizes the cooling
+effects of deposition of the capsule material into the hy-
+drogen fuel [13, 14]. The activation of the James Webb
+Space Telescope (JWST) presages an explosion in dis-
+coveries [15] of exoplanets representing a vast range of
+physical conditions, distributions, and dynamics involv-
+ing, to list just a few, surface-atmosphere couplings [16],
+interfaces between solid and liquid components of interi-
+ors [17], and formation pathways [18]. In another area,
+the breakthrough fusion milestone [19] at the National
+Ignition Facility emphasizes the role played in model-
+ing by ever-improved basic physical attributes. Both of
+these developments signal a pressing need for more accu-
+rate static and dynamical microscopic properties over a
+broad range of WDM conditions.
+Many methods exist to determine the basic structure
+and dynamics of WDM; the most accurate arise from
+∗ alwhite@lanl.gov
+first-principles (FP) techniques such as density functional
+theory (DFT) [20–22] and path integral Monte Carlo
+[23, 24], which supply a consistent set of basic material
+properties as equation-of-state, opacities, mass transport,
+electrical and thermal conduction. Recently, results from
+DFT simulations have provided training information to
+determine model potentials from machine learning tech-
+niques (MLP) [10, 11, 25, 26]. Kohn-Sham (KS) DFT
+combined with the plane-wave pseudo-potential (PWPP)
+method is the theory of choice for studying the electronic
+structure of numerous materials, ranging from solid-state
+condensed matter to hot dense plasmas.
+The success
+of DFT stems from the balance between computational
+complexity and useful accuracy, achieved by replacing
+the quantum-mechanical wavefunction by a much simpler
+quantity: the KS density matrix, typically constructed
+from the KS Hamiltonian eigenstates.
+The cubic scaling of the computational complexity of
+KS-DFT with respect to system size and temperature is
+a major limitation [27]. For WDM systems, orbital-free
+DFT has been a particularly useful alternative, but it is
+based on an approximate treatment of the electron non-
+interacting kinetic energy [28–30]. Linear scaling meth-
+ods, such as stochastic DFT (sDFT) [31], provide a full
+KS accuracy alternative for large or hot systems [32].
+The more general mixed stochastic-deterministic DFT
+(mDFT) shows great promise for providing full KS-DFT
+accuracy for calculations at any temperature [33]. How-
+ever, when combined with PWPP method, sDFT, and
+by extension mDFT, has a quadratic dependence of the
+computational cost on the maximum plane-wave energy
+(Ecut), i.e., on the grid resolutions, compared to stan-
+dard deterministic DFT’s linear dependence. Moreover
+it has only been formulated in combination with norm-
+conserving pseudopotentials, which typically show either
+low accuracy or require higher Ecut. Maintaining high
+accuracy and low Ecut, soft pseudopotentials, requires
+utilization of a non-orthogonal basis, as first developed
+arXiv:2301.12018v1  [physics.comp-ph]  27 Jan 2023
+
+2
+by Vanderbilt [34].
+The projector augmented wave (PAW) approach, first
+introduced by Bl¨ochl [35] and then reformulated by
+Kresse [36], generalizes the soft pseudopotentials to a for-
+mally “all-electron” formalism. The PAW method pro-
+vides a realistic description of core electrons, has a long
+and continued history of development, and yields accu-
+racy comparable to more expensive “all-electron” meth-
+ods [37].
+Moreover, it allows for very low Ecut, suit-
+able for sDFT calculations.
+In this letter, we develop
+mDFT, and sDFT by limitation, within the PAW for-
+malism and present isochoric calculations and analysis
+for warm dense carbon spanning the WDM regime, 1 to
+50 eV.
+While DFT was initially developed for electrons in
+their ground-state, i.e., zero temperature (T → 0), the
+temperatures in WDM systems are of the order of the
+Fermi energy and thus require the application of a finite-
+temperature formulation of KS-DFT. Mermin’s formula-
+tion of DFT within the grand canonical ensemble is the
+most common approach used in WDM [38]. In this for-
+mulation, the single-particle KS eigenstates are partially
+occupied according to the Fermi-Dirac distribution func-
+tion (here assuming paired spin):
+f(ε) =
+2
+1 + e(ε−µ)/kBT ,
+(1)
+where µ is the chemical potential, and kB is the Boltz-
+mann constant. Thus the thermal density matrix, �ρ =
+f( �HKS), is constructed as:
+�ρ =
+�
+b
+f(εb)|ψb⟩⟨ψb| ,
+(2)
+where ψb is an eigenvector of the KS Hamiltonian, �HKS,
+with eigenenergy εb.
+In finite-temperature metals and
+plasmas (where electrons populate the conduction band)
+the number of states required to resolve all the electrons
+grows as V T 3/2, where V is the system size, leading to
+cubic computational scaling in both size and tempera-
+ture.
+To address this drawback of traditional KS-DFT, Baer
+et al. proposed an alternative algorithmic approach to
+DFT that is stochastic in nature (sDFT) [31, 32, 39–41].
+In contrast to traditional KS-DFT, sDFT scales as V/T
+and the operations on the stochastic vectors are trivial
+to parallelize. sDFT is based on Hutchinson’s stochastic
+trace estimation (STE) [42] and the stochastic projec-
+tion for matrices, i.e., it is the application of STE to the
+Kohn-Sham density matrix. In sDFT, the thermal den-
+sity matrix, �ρ = f( �HKS), is projected onto Nχ stochastic
+vectors (χa):
+�ρ =
+�
+a∈Nχ
+f
+1
+2 ( �HKS)|χa⟩⟨χa|f
+1
+2 ( �HKS) ,
+(3)
+rather than on KS eigenstates. A converged calculation,
+with respect to Nχ has the same exact accuracy as that of
+FIG. 1.
+Disordered 64 carbon atoms system at (ρ, T) =
+(3.52 g/cc, 10 eV): Density of states (DOS), and (inset) Occu-
+pied DOS, obtained with Kohn-Sham (KS-DFT), stochastic
+(sDFT), and mixed (mDFT) methods. The chemical poten-
+tial of the system is µ = 7.92 eV. The pink and orange–shaded
+regions in the inset indicate the splitting due to deterministic
+(Nψ) and stochastic (Nχ) treatments in mDFT.
+a traditional KS-DFT calculation based on finding eigen-
+states.
+Recently, White and Collins [33] proposed the mDFT
+approach that generalizes stochastic and deterministic
+KS-DFT approaches and improves the computational
+complexity over a wide range of temperatures. It is based
+on partitioning the full eigenspectrum of �HDFT into low-
+energy and high-energy segments such that the maxi-
+mally occupied low-energy eigenstates (ψ) are explicitly
+resolved while the higher-energy states are spanned by
+random stochastic vectors (χ′). That is:
+|χ′
+a⟩ =
+�
+�I −
+�
+b∈Nψ
+|ψb⟩⟨ψb|
+�
+|χa⟩
+(4)
+where χ = ei2π⃗θ/N and θ ∈ {0, 1} is a set of uncorre-
+lated random numbers for each basis function and N is a
+normalization constant. We define “occupied” stochastic
+vectors, X′ = f
+1
+2 ( �HDFT)χ′, to obtain the mixed density
+matrix as,
+�ρ =
+�
+a∈Nχ
+|X′
+a⟩⟨X′
+a| +
+�
+b∈Nψ
+|ψb⟩f(εb)⟨ψb| .
+(5)
+All observables can be expressed as traces over appro-
+priate operators with this density matrix. See [33] for a
+detailed description.
+Figure 1 shows the density of states (DOS) and occu-
+pied DOS for a disordered carbon system, obtained with
+KS-DFT (Nψ = 1024), sDFT (Nχ = 256) and mDFT
+(Nψ/Nχ = 128/16) methods. The low-energy determin-
+istic component of mDFT is shown in pink and there is
+an overall good agreement across the three methods. The
+overlap of the components is due to the finite width of
+Gaussian functions used to define the continuous DOS.
+
+100
+KS-DFT
+mDFT(128/16)
+10
+SDFT
+mDFT (Nμ)
+80
+mDFT
+mDFT (Nx)
+5
+DOS (eV-1)
+60
+0
+50
+40
+20
+0
+0
+100
+200
+300
+400
+Energy (eV)3
+The state-of-the-art for balancing accuracy, computa-
+tional complexity, and grid resolution/plane-wave count
+is the pseudo-augmented wave (PAW) method. In princi-
+ple, PAW is an all-electron method, assuming a complete
+set of partial waves and projectors. However, only a finite
+set is used in practice. The PAW method is based on a
+linear transformation matrix (�τ) connecting the smooth
+pseudo density matrix (�ρ ) to an all-electron density ma-
+trix (�ρ ),
+�ρ = �τ �ρ �τ †
+(6)
+�τ = �I +
+�
+i
+�
+|φi⟩ − |˜φi⟩
+�
+⟨pi| ,
+(7)
+with ⟨pi| ˜φj⟩ = δij. Here, |φi⟩ is a ‘true’ all-electron par-
+tial wave and | ˜φi⟩ is a pseudo partial wave dual to the
+projector |pi⟩. This transformation is exact in the limit of
+a complete set of partial waves/projectors. These func-
+tions are defined in an “augmentation sphere” around
+an atom. Expectation values are preserved by defining
+pseudized operators, �O as:
+E[ �O] = Tr[�ρ �O] = Tr[�ρ �O] , with �O = �τ † �O�τ
+(8)
+The
+transformed
+identity
+operator
+gives
+an
+S-
+orthogonality condition for the transformed wavefunc-
+tions:
+�S = �τ †�τ = �I +
+�
+i,j
+|pi⟩
+�
+⟨φi|φj⟩ − ⟨˜φi|˜φj⟩
+�
+⟨pj| ,
+(9)
+⟨ψa|ψb⟩ = ⟨ ˜ψa|�S| ˜ψb⟩ = δab .
+(10)
+Therefore, ˜ψb is a solution to the generalized eigenvalue
+problem, �HKS ˜ψ = ε�S ˜ψ. This S-orthogonality condition
+complicates the generation of transformed stochastic vec-
+tors.
+Our approximate projection via all-electron norm-
+conserving and transformed stochastic vectors is given
+by:
+�I ≈
+�
+a
+|χa⟩⟨χa| =
+�
+a
+�τ|˜χa⟩⟨˜χa|�τ † .
+(11)
+From the transformation of the stochastic vectors and
+the identity operator, Eq. (11), and the �S operator, Eq.
+(9), we find that
+�
+a
+�τ †�τ|˜χa⟩⟨˜χa|�τ †�τ ≈ �S ,
+�
+a
+|˜χa⟩⟨˜χa| ≈
+�
+b
+| ˜ψb⟩⟨ ˜ψb| = �S−1 .
+(12)
+We now define a set of stochastic vectors |¯χa⟩ such that
+�I ≈
+�
+a
+|¯χa⟩⟨¯χa| , i.e. δ(⃗r,⃗r ′) =
+�
+a
+ei2π(⃗θa(⃗r)−⃗θa(⃗r ′))/N 2 ,
+which has the same form as the all-electron stochas-
+tic vectors, but with ⃗r and ⃗r ′ from the coarser grid
+of the transformed functions [32].
+Using the identity
+�I = �τ �S−1�τ † (see Supplementary Information [43], Eq.
+S8) and Eq. (11), we obtain
+�I ≈
+�
+a
+�τ �S− 1
+2 |¯χa⟩⟨¯χa|�S− 1
+2 �τ † =
+�
+a
+�τ|˜χa⟩⟨˜χa|�τ † ,
+where
+|˜χa⟩ = �S− 1
+2 |¯χa⟩ .
+(13)
+An efficient and sufficiently accurate procedure for ap-
+plying �S− 1
+2 to vectors was formulated recently by Li &
+Neuhauser [44]. Using the same identity, the pseudized
+density matrix can be written as (see Supplementary In-
+formation [43]):
+�ρ = f(�S−1 �HKS)�S−1 =
+(14)
+f
+1
+2 (�S−1 �HKS)�S−1f
+1
+2 ( �HKS �S−1) .
+The procedure for PAW mDFT is similar to the all-
+electron or norm-conserving case [33] with three modifi-
+cations: (i) the orthogonal stochastic vectors, ¯χ, are gen-
+erated and rotated to the standard PAW frame, ˜χ, (ii)
+the generalized eigenvalue problem is iteratively solved
+to obtain ˜ψ, (iii) the projection of the eigenstates from
+the stochastic vectors is performed via:
+|˜χ′
+a⟩ =
+�
+�S−1 −
+�
+b∈Nψ
+|ψb⟩⟨ψb|
+�
+�S|˜χa⟩, giving
+(15)
+�S−1 ≈
+�
+a∈Nχ
+|˜χ′
+a⟩⟨˜χ′
+a| +
+�
+b∈Nψ
+| ˜ψb⟩⟨ ˜ψb| and
+(16)
+�ρ =
+�
+a∈Nχ
+| ˜X′
+a⟩⟨ ˜X′
+a| +
+�
+b∈Nψ
+| ˜ψb⟩f(εb)⟨ ˜ψb| , with
+(17)
+| ˜X′
+a⟩ = f
+1
+2 (�S−1 �HKS)|˜χ′
+a⟩.
+(18)
+In Eq. (18), the S−1 can be applied via the Woodbury
+formula [45]. From Eq. (17), all observables necessary
+to complete the PAW formalism, e.g., the on-site density
+matrix and the compensation charge density, can be cal-
+culated [46]. The generalization of PAW force and stress
+tensor contributions for mDFT/sDFT are presented in
+Supplementary Information S2 [43].
+To test our method, we first perform single-point
+ground-state energy calculations on a 64-atom disordered
+carbon system at several temperatures and a solid-state
+density of 3.52 g/cc, using the 4e− PAW potential. All
+the computations are using our plane-wave DFT code,
+SHRED (Stochastic and Hybrid Representation Electronic
+Structure by Density Functional Theory) that relies on a
+modified version of a portable PAW library LibPAW [48],
+developed under the ABINIT [49] project, and LibXC [50]
+for the exchange-correlation energy functionals.
+A ki-
+netic energy cutoff Ecut of 426 eV (Ngrid = 483) is used
+for the transformed functions and 758 eV (Ngrid = 643)
+for the densities and the spherical PAW grid around each
+atom.
+Figure 2 shows a comparison of the three DFT algo-
+rithms in terms of computational time per self consistent
+
+4
+FIG. 2. Disordered carbon system comprising 64 atoms at
+ρ = 3.52 g/cc. Comparison of (a) SCF times per cycle, and
+(b) relative pressure with reference to SESAME 7833 (Psesame)
+[47] obtained for deterministic (Kohn-Sham), stochastic, and
+mixed DFT calculations performed using a Cray compilation
+of SHRED on 128 cores.
+field (SCF) cycle on 128 CPUs, and the accuracy and pre-
+cision of pressure (for this single-point calculation) refer-
+enced to SESAME 7833 value [47]. The free energy, pres-
+sure and chemical potential for each temperature, along
+with the combination of orbitals Nψ/Nχ (Nψ) used in
+mDFT are listed in Supplementary Information, Table
+S1 [43]. The combinations range from 136/4 at kBT = 1
+eV to 64/40 at kBT = 50 eV; whereas sDFT times are
+computed with Nχ = 128 orbitals for all temperatures.
+The pressures in sDFT and mDFT, are computed as av-
+erages over 10 independent SCF runs using a different
+set of Nχ stochastic orbitals; with the statistical error
+(the error bars) expressed as the standard deviation of
+the sample. The nonlinear nature of the SCF cycle leads
+to a potential bias in sDFT/mDFT given by the differ-
+ence between the expected value and the KS-DFT result.
+Mixed DFT yields energies to within 0.2% (standard de-
+viation of 0.3%) of the reference KS-DFT values with a
+42× speedup as compared to KS-DFT at T = 20 eV. Ad-
+ditionally, the chemical potential and pressure are con-
+verged to 0.13 eV (standard deviation of 0.18 eV) and
+7.27 GPa (standard deviation of 8.29 GPa) relative to
+their respective reference KS-DFT results.
+Local quantities such as the electronic forces on nu-
+clei depend on the electronic density and hence do not
+benefit from the self-averaging effect of stochastic DFT
+[39]. We examine the stochastic and mixed DFT forces
+for several Nψ/Nχ at different temperatures. A compar-
+ison is presented in Fig. 3, where F i,ψχ
+α,n
+indicates mixed
+or stochastic forces and Fi,ψ
+α
+indicates deterministic KS-
+forces, such that i = {x, y, z}, α = 1, . . . , Nat indexes the
+atoms, and n = 1, . . . , 10 indexes the stochastic/mixed
+run. The absence of an i index indicates the magnitude
+of the force, lack of n indicates the value is averaged
+FIG. 3. Comparison of mixed (Fi,ψχ
+α
+) vs Kohn-Sham (Fi,ψ
+α )
+DFT components of forces on all atoms obtained for vari-
+ous Nψ/Nχ. The agreement between stochastic (0/256) and
+deterministic forces improves at higher temperatures.
+The
+data points shown in magenta represent the chosen Nψ/Nχ
+for mixed DFT calculations at a given temperature (T). At
+higher temperatures, the area of the force plots is zoomed in
+to keep a constant scale. The order of lines at each T matches
+the key.
+over the 10 independent stochastic/mixed runs, and the
+absence of α indicates average over atoms.
+The stan-
+dard deviation over n independent runs, averaged over
+the atoms, is given by:
+σψχ =
+Nat
+�
+α=1
+N −1
+at
+�
+�
+�
+�
+10
+�
+n=1
+(Fψχ
+α,n − Fψ
+α)2
+10
+;
+(19)
+also see Supplementary information, Table S2 [43]. Upon
+comparing the bias in mixed forces (|Fψχ −Fψ|) with the
+statistical error (σψχ), one finds that the largest magni-
+tude of the force bias across temperatures is 1.280 eV/˚A,
+which is smaller than the largest magnitude of the sta-
+tistical error, 10.471 eV/˚A. Recently, a similar trend be-
+tween the errors in stochastic forces was seen for the case
+of an aqueous-solvated peptide system [52].
+Figure 3 shows a comparison of the components of
+mixed and stochastic DFT forces (Fi,ψχ
+α
+) obtained for
+several Nψ/Nχ with the deterministic Kohn-Sham forces
+(Fi,ψ
+α ). The displacement in the mixed forces about the
+linear curve indicates a statistical error that diminishes
+with an increasing number of stochastic orbitals at higher
+temperatures. At a given temperature, the data points
+shown in magenta indicate the mixed forces for Nψ/Nχ
+employed in other results presented in this work. The
+range of plotted forces is kept constant to compare the
+spread across temperatures. The forces at T = (30, 50)
+eV, are in good agreement between mixed and KS- forces
+with a standard deviation of 8.647 eV/˚A and 10.364
+eV/˚A respectively.
+These can be viewed in conjunc-
+tion with the purely stochastic forces shown in blue. At
+lower temperatures, increasing the number of determin-
+istic KS-orbitals reduces the fluctuations in forces, e.g.,
+
+(a)
+SCF time/step (s)
+10
+KS-DFT
+sesame
+SDFT
+sesame
+mDFT
+0.0
+P
+P
+-0.2
+0
+10
+20
+30
+40
+50
+Temperature (eV)60
+T=lev
+T= 10 eV
+(eVIA)
+40
+20
+Xm
+0
+0/256
+128/16
+0/256
+α
+20
+1024/8
+128/16
+256/16
+-40
+136/4
+96/24
+60
+= 30 eV
+T = 50 eV
+(eVIA)
+40
+%o
+20
+Xh
+0
+0/256
+0/256
+96/32
+64/40
+α
+-20
+128/16
+4000/400
+80/40
+128/16
+-40
+64/40
+16/128
+O
+-40
+-20
+0
+20
+40
+60
+-40
+-20
+0
+20
+40
+60
+Fi, 
+(eVIA)
+Fi, 
+(eVIA)
+α
+a5
+FIG. 4.
+Disordered 64 carbon atoms system at ρ = 3.52
+g/cc: average magnitude of force on atoms ⟨Fα⟩ obtained
+with Kohn-Sham and mixed DFT for a single snapshot. For
+KS-DFT, Nψ ranges from 256 at T = 1 eV to 6400 at T =
+50 eV. The error bars indicate the statistical error over mixed
+DFT runs, and the shaded region represents a Langevin-type
+friction term at the respective temperature, ⟨Fα⟩ ± 2ς with
+γα = 0.04 fs−1 [51]. A comparison of (b) velocity autocor-
+relation function (VACF) for KS (solid line), mixed (dash-
+dot line) and stochastic (dotted line) DFT methods at T=5
+eV, and (c) VACF/T at different temperatures (T). The self-
+diffusion coefficients (D, integral of VACF) are given in the
+key with units of 10−3 cm2/s.
+at T = 10 eV, increasing Nψ from 128/16 (magenta) to
+256/16 (yellow) improves the distribution of forces signif-
+icantly. However, at moderate to high temperatures one
+would require a drastically large Nψ to obtain accurate
+forces, see T = 50 eV in Fig. 3. Hence it is advantageous
+to increase Nχ’s and decrease Nψ as the temperature in-
+creases [33], as evidenced from the forces obtained with
+Nψ/Nχ = 128/16 vs. 16/128 at T = 50 eV.
+The magnitude of force averaged over all atoms ⟨Fα⟩ is
+computed with mixed and KS-DFT methods, as shown
+in Fig. 4(a). The error bars denote the standard devia-
+tion in the mixed DFT forces, σψχ, and the blue-shaded
+band indicates a thermal fluctuation region described by
+a Langevin-type fluctuation ς, such that
+mα¨qα = fα − γαpα + ςηα(t)
+(20)
+where ς ≡ √2mαγαkBT, (qα, pα) are the coordinates and
+momenta of the atoms, fα is the force on the atom, γα
+is the damping constant, and ηα(t) is a Gaussian process
+such that ⟨ηα(t)⟩ = 0 and ⟨ηα(t)ηα′(t′)⟩ = δαα′δ(t − t′).
+Langevin molecular dynamics was successfully used in
+previous studies to investigate forces from stochastic
+DFT–based simulations [51, 52]. At a given temperature,
+⟨Fα⟩±2ς is explicitly computed and then interpolated to
+yield the thermal-fluctuation band in Fig. 4(a). The av-
+eraged mixed DFT forces along with the error bars are
+contained within the thermal band. This serves to show
+that the statistical error as captured quantitatively by
+σψχ, and qualitatively in Fig. 3, can be absorbed by the
+thermal fluctuations in dynamical simulations. While the
+statistical fluctuations are contained within 2ς, the bias
+in the forces lies within 1ς indicating that the accuracy
+converges faster than precision [32].
+In order to investigate the effect of these relatively
+small biases on observable quantities, we apply mDFT
+to compute transport properties via molecular dynamics
+(MD) simulations. We employ an isokinetic [53], rather
+than Langevin, ensemble at each temperature. This is
+the typical ensemble of choice for WDM transport cal-
+culations [29, 30]. The time-dependent free energy and
+total pressure along with their time-averages and stan-
+dard deviations are given in Supplementary Information,
+Table S3, Figs. S5, S6 [43]. We compute the velocity
+autocorrelation function (VACF) and self-diffusion coef-
+ficient (D) [29, 30, 54] of carbon at (ρ, T) = (3.52 g/cc, 5
+eV) with deterministic KS-, mixed and stochastic DFT,
+see Fig.
+4(b).
+For MD simulations, finite simulation
+time leads to its own statistical error, in addition to the
+statistical error due to stochastic calculations in sDFT
+and mDFT. Estimation of this statistical error depends
+on the approach to VACF averaging.
+We see the av-
+erage mDFT diffusion coefficients falls between 1 and 2
+times the statistical error estimate, which is within the
+range of reasonable estimates, see Supplementary Infor-
+mation for details. The pure sDFT diffusion coefficient
+falls slightly outside this range, but is still within 10% of
+the deterministic case. Figure 4(c) shows a comparison
+of temperature-scaled VACF and D for several T, with
+the Nψ/Nχ for mDFT specified in the key. The relation-
+ship between D and T over a temperature range at any
+given density was previously investigated for high-Z ma-
+terials that exhibit multiple ionization states [55]. It was
+argued that, over a large temperature and density range,
+the mutually compensating effects of increased ionization
+and thermal energy result in a constant coupling parame-
+ter Γ, giving rise to a so-called Γ−plateau which, in turn,
+affects quantities such as self-diffusion and viscosity. We
+see that for 1 to 5 eV the change in temperature dom-
+
+(a)
+64
+KS-DFT
+80
+40
+mDFT
+80
+32
+96
+(eV/A)
+60
+32
+96
+16
+128
+^ 40
+16
+128
+128
+16
+V
+1368
+20
+4
++0
+0
+10
+20
+30
+40
+50
+Temperature (eV)
+(b)
+4.0
+Nw= 768, D = (6.16±0.07)
+3.5
+Nu/Nx=128/8,D=(5.97±0.06)
+(1E-3 A2/fs2)
+3.0
+Nu/Nx= 0/128,D = (5.59±0.07)
+2.5
+2.0
+VACF
+1.5
+1.0
+0.5
+0.0
+X
+0
+20
+40
+60
+80
+Time (fs)
+(c)
+0.8
+T=1eV,136/4,D/T=(1.04±0.01)
+2 /eV-fs2)
+0.7
+T=5eV.128/8.D/T=(1.19±0.01)
+T=10eV.128/64,D/T=(1.00±0.02)
+0.6
+T=20eV.96/64,D/T=(0.86±0.02)
+A2/
+0.5
+T=30eV,96/32,D/T= (0.80±0.01)
+(1E-3
+T=40eV,80/32,D/T=(0.76±0.01)
+0.4
+T=50eV.64/40,D/T=(0.73±0.01)
+VACF/T
+0.3
+0.2
+0.1
+0.0
++
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Time (fs)6
+inates correlation, leading to an increase in D/T, while
+for greater than 5 eV the ionization effects become sig-
+nificant leading to a decrease in D/T.
+We have presented the first implementation of the
+mDFT and sDFT methods within the plane-wave PAW
+formalism for DFT. The PAW formalism provides a sig-
+nificant acceleration of stochastic DFT methods due to
+both smaller grids and decreased eigenspectrum range.
+Additionally it opens the door to efficient, all-electron ac-
+curacy, calculations of matter in extreme conditions, as
+is possible in ambient conditions [37]. We have demon-
+strated the efficacy of this approach in the simulation of
+transport properties in isochorically heated warm dense
+carbon up to 50 eV, observing the crossover from kinet-
+ically to Coulomb-dominated correlation effects. Future
+work will include additional transport studies, and ap-
+plication of the PAW method to time-dependent mDFT
+and optical response via the Kubo-Greenwood approach.
+ACKNOWLEDGMENTS
+This work was supported by the U.S. Department of
+Energy through the Los Alamos National Laboratory
+(LANL). Research presented in this article was supported
+by the Laboratory Directed Research and Development
+program of LANL, under project number 20210233ER,
+and Science Campaign 4. We acknowledge the support of
+the Center for Nonlinear Studies (CNLS). This research
+used computing resources provided by the LANL Insti-
+tutional Computing and Advanced Scientific Computing
+programs. Los Alamos National Laboratory is operated
+by Triad National Security, LLC, for the National Nu-
+clear Security Administration of U.S. Department of En-
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+

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+Counterexample Guided Abstraction Refinement with Non-Refined Abstractions
+for Multi-Agent Path Finding
+Pavel Surynek
+Faculty of Information Technology, Czech Technical University in Prague
+Th´akurova 9, 160 00 Praha 6, Czechia
+pavel.surynek@fit.cvut.cz
+Abstract
+Counterexample
+guided
+abstraction
+refinement
+(CEGAR) represents a powerful symbolic tech-
+nique for various tasks such as model checking and
+reachability analysis. Recently, CEGAR combined
+with Boolean satisfiability (SAT) has been applied
+for multi-agent path finding (MAPF), a problem
+where the task is to navigate agents from their start
+positions to given individual goal positions so that
+the agents do not collide with each other.
+The recent CEGAR approach used the initial ab-
+straction of the MAPF problem where collisions
+between agents were omitted and were eliminated
+in subsequent abstraction refinements. We propose
+in this work a novel CEGAR-style solver for MAPF
+based on SAT in which some abstractions are de-
+liberately left non-refined. This adds the necessity
+to post-process the answers obtained from the un-
+derlying SAT solver as these answers slightly dif-
+fer from the correct MAPF solutions. Non-refining
+however yields order-of-magnitude smaller SAT
+encodings than those of the previous approach and
+speeds up the overall solving process making the
+SAT-based solver for MAPF competitive again in
+relevant benchmarks.
+Keywords:
+multi-agent pathfinding (MAPF), counterex-
+ample example guided abstraction refinement (CEGAR),
+Boolean satisfiability (SAT)
+1
+Introduction
+Multi-agent path finding (MAPF) [Silver, 2005; Ryan, 2008;
+Surynek, 2009; Wang and Botea, 2011; Sharon et al., 2013] is
+a task of finding non-conflicting paths for k ∈ N agents A =
+{a1, a2, ..., ak} that move in an undirected graph G = (V, E)
+across its edges such that each agent reaches its goal vertex
+from the given start vertex via its path. Starting configura-
+tion of agents is defined by a simple assignment s : A → V
+and the goal configuration is defined by a simple assignment
+g : A → V . A conflict between agents is usually defined as
+simultaneous occupancy of the same vertex by two or more
+agents or as a traversal of an edge by agents in opposite direc-
+tions. Although MAPF started as purely theoretical studies of
+graph pebbling and puzzle solving [Kornhauser et al., 1984;
+Ratner and Warmuth, 1986; Ratner and Warmuth, 1990], it
+has grown into artificial intelligence mainstream topic with a
+significant impact on many fields including warehouse logis-
+tics [Li et al., 2020b].
+Many problems in robotics [Chud´y et al., 2020; Wen et
+al., 2022; Gharbi et al., 2009], urban traffic optimization
+[Atzmon et al., 2019; Ho et al., 2019], FPGA circuit de-
+sign [Nery et al., 2017], and computer games [Sigurdson et
+al., 2018] can be regarded from the perspective of MAPF
+as listed by various surveys including [Felner et al., 2017;
+Ma et al., 2017].
+Particular aspect that is motivated by practice and makes
+MAPF challenging is the need to find the optimal solutions
+with respect to some cumulative cost [Yu and LaValle, 2013].
+Commonly used cumulative costs in MAPF are makespan
+and sum-of-costs[Stern, 2019]. The makespan corresponds
+to the length of the longest agent’s path. The sum-of-costs
+is the sum of costs of individual paths which corresponds to
+the sum of unit costs of actions, the wait actions including.
+An example of MAPF problem and its sum-of-costs optimal
+solution is shown in Figure 1.
+ 
+path(r1) = 
+[v1, v4, v7, v8, v9] 
+Cost = 4 
+ 
+path(r2) = 
+[v3, v6, v5, v4, v7] 
+Cost = 4 
+ 
+SoC = 8 
+s(a1) = v1 
+a1 
+a2 
+v2 
+s(a1) = v3 
+v4 
+v5 
+v6 
+g(a1) = v7 v8 
+g(a1) = v9 
+a1 
+a2 
+v1 
+v4 
+v7 
+v2 
+v5 
+v8 
+v3 
+v6 
+v9 
+Figure 1: Multi-agent path finding (MAPF) with agents a1 and a2.
+We address MAPF from the perspective of compilation
+techniques that represent a major alternative to search-based
+solvers [Silver, 2005; Wagner and Choset, 2011; Sharon et
+al., 2013; Sharon et al., 2015] for MAPF. Compilation-based
+solvers reduce the input MAPF instance to an instance in a
+different well established formalism for which an efficient
+solver exists. Such formalisms are for example constraint
+programming (CSP) [Dechter, 2003; Ryan, 2010], Boolean
+satisfiability (SAT) [Biere et al., 2021; Surynek, 2012], or
+mixed integer linear programming (MILP) [Rader, 2010;
+Lam et al., 2019].
+The basic compilation scheme for sum-of-costs opti-
+arXiv:2301.08687v1  [cs.AI]  20 Jan 2023
+
+mal MAPF solving has been introduced by the MDD-SAT
+[Surynek et al., 2016] solver that uses so called complete
+models to compile MAPF instances into SAT (see Figure 2).
+The target Boolean formula of the complete model is sat-
+isfiable if and only if the input MAPF has a solution of a
+specified sum-of-costs. The complete model as introduced in
+MDD-SAT consists of three group of constraints:
+• Agent propagation constraints - these constraints en-
+sure that if an agent appears in vertex v at time step t
+then the agent appears in some neighbor of v (including
+v) at time step t + 1. The side effect of these constraints
+is that the agent never disappears. Cost calculation and
+bounding is done together with agent propagation.
+• Path consistency constraints - these constrains ensure
+that agents move along proper paths, that is, agents do
+not multiply and do not appear spontaneously.
+• Conflict elimination constraints - to ensure that agents
+do not conflict with each other according to the MAPF
+rules (vertex and edge conflict).
+ 
+ 
+ 
+Agent 
+Propagation 
+Constraints 
+Solving 
+Answer 
+Interpretation 
+Solution 
+Path 
+Consistency 
+Constraints 
+Conflict 
+Elimination 
+Constraints 
+Complete Model Encoding 
+Figure 2: Schematic diagram of the basic MAPF compilation with
+complete model.
+A significant improvement over complete models in prob-
+lem compilation for MAPF is the introduction of laziness via
+incomplete models in the CSP-based LazyCBS [Gange et al.,
+2019], SAT-based SMT-CBS [Surynek, 2019], and MILP-
+based BCP [Lam et al., 2019]. These solvers use incomplete
+models of MAPF where the conflict elimination constraints
+are omitted for which equivalent solvability no longer holds,
+but only the implication: if the MAPF instance is solvable
+then the instance in the target formalism is solvable too.
+The discrepancy between the original formulation of
+MAPF and its compiled variant in the target formalism is
+eliminated by abstraction refinements similarly as it is done in
+the counterexample guided abstraction refinement (CEGAR)
+[Clarke et al., 2000] approach for model checking (see Fig-
+ure 4). Specifically in MAPF the counterexamples are repre-
+sented by conflicts between agents and their refinements cor-
+respond to elimination of these conflicts.
+1.1
+Contribution
+We further generalize the CEGAR approach for MAPF by
+further omitting the path consistency constraints. Hence only
+agent propagation constraints remain in the initial abstrac-
+tion. In addition to this, in the abstraction refinement we do
+not try to refine with respect to all the omitted constraints
+(path consistency and conflict elimination). Instead we only
+generate counterexamples for conflicts and refine the conflicts
+while path consistency remains non-refined - the abstraction
+refinement in our case is inherently incomplete. To mitigate
+the impact of non-refined path consistency constraints we add
+a new step in the CEGAR compilation architecture in which
+we post-process the answer from the SAT solver.
+The omitted path consistency constraints lead to finding di-
+rected acyclic sub-graphs (DAGs) that connects agents’ initial
+and goal positions instead of proper paths. The desired path
+for agents need to be extracted from the sub-graphs in the new
+post-processing step.
+Our contribution can be understood in a broader perspec-
+tive as a generalization how problems are compiled.
+In
+the classic approaches such as SATPlan [Kautz and Selman,
+1992] the problem solution is read from the assignment of
+decision variables directly. In our approach the assignment
+of decision variables does not represent the problem solution
+directly, but the solution needs to be extracted from the as-
+signment by non-trivial process, though this process should
+be fast (polynomial time) to keep the problem compilation
+viable.
+2
+Background
+One of the first uses of problem compilation can be seen in
+the SATPlan algorithm [Kautz and Selman, 1992; Kautz and
+Selman, 1999] for classical planning [Ghallab et al., 2004].
+A Boolean formula that is satisfiable if and only if a plan
+of specified length exists is constructed and checked for sat-
+isfiability by the SAT solver. To guarantee finding a plan of
+the minimum length, the SATPlan planner iteratively consults
+formulae encoding the existence of a plan of length l0, l0 +1,
+l0 + 2,... where l0 is a lower bound for the plan length, until
+the first satisfiable formula is found.
+Similar approach has been used in SAT-based solvers for
+MAPF such as MDD-SAT [Surynek et al., 2016] where the
+SAT solver is consulted about the existence of a solution
+for the given MAPF instance of specified sum-of-cost SoC.
+To ensure finding sum-of-costs optimal solution for solvable
+MAPF, MDD-SAT checks using the SAT solver existence of
+solutions for SoC 0, SoC 0 + 1, SoC 0 + 2, ... until the first
+positive answer which due to monotonicity of existence of
+solutions w.r.t. bounded sum-of-costs is guaranteed to corre-
+spond to the optimal sum-of-cost.
+2.1
+The MAPF Encoding as SAT
+The complete model for the sum-of-costs optimal MAPF
+as SAT built-in the MDD-SAT solver uses Boolean deci-
+sion variables inspired by direct encoding [Walsh, 2000]
+and multi-valued decision diagrams [Andersen et al., 2007].
+We briefly summarize the complete model as introduced by
+MDD-SAT.
+The decision variables need to represent all possible paths
+of agents such that their sum-of-costs is at most given value
+SoC > 0. Since it is possible for an agent to visit a single
+vertex multiple times, a so-called time expansion [Surynek,
+2017] of the underlying graph G is constructed for each agent
+denoted TEGi.
+TEGi is defined for a given maximum length of indi-
+vidual agent’s path T
+∈ N consisting of T + 1 copies
+of vertices of G called layers indexed by 0, 1, ..., T, where
+
+{(v1, t), (v2, t), ..., (vn, t)} are nodes of the t-th layer of
+TEGi.
+The layers correspond to individual time-steps
+and are interconnected by directed edges that model pos-
+sible transitions of agents, that is there is a directed edge
+[(vj, t), (vj′, t + 1)] whenever there is an edge {vj, vj′} ∈ E.
+Directed edges [(vj, t), (vj′, t + 1)] are added to model wait
+actions of agents.
+The following proposition establishes the correspondence
+between the actual paths traveled by agents in G 1 and di-
+rected paths in TEGs.
+Proposition 1. Any path of length at most T of agent ai going
+from s(ai) to g(ai) is represented by a directed path in TEGi
+connecting (s(ai), 0) and (g(ai), T).
+Having the proposition, we can speak about a representa-
+tion of agent’s path (trajectory) in TEG.
+Since not all nodes (vj, t) of TEGi are actually reachable
+considering the maximum agent’s path length T and its start
+s(ai) and goal g(ai) because either vj is farther than t steps
+from s(ai) or farther than T − t steps from g(ai), such nodes
+be can pruned from TEGi without compromising the repre-
+sentation of all relevant paths. TEDi after pruning unreach-
+able nodes is called a multi-valued decision diagram for agent
+ai and is denoted MDDi (nodes and edges of MDDi are de-
+noted MDDi.V and MDDi.E respectively). Example MDD
+is shown in Figure 3. The complete model for MAPF in the
+MDD-SAT solver is derived from MDDs.
+The decision variable denoted X t
+i,vj is introduced for each
+node (vj, t) ∈ MDDi and expresses that agent ai appears in
+vertex vj at time step t 2.
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+s(a1)=v1 
+a1 
+v2 
+v3 
+v4 
+v5 
+v6 
+v7 
+v8 
+v9=g(a1) 
+(v1,0) 
+(v2,1) 
+(v4,1) 
+(v3,2) 
+(v5,2) 
+(v7,2) 
+(v6,3) 
+(v8,3) 
+(v9,4) 
+MDD1 for T=4 
+Figure 3: Multi-valued decision diagram for the path length T = 4.
+The three groups of constraints expressed on top of the
+X t
+i,vj variables are as follows:
+Agent Propagation Constraints
+X t
+i,vj →
+�
+j′ | [(vj,t);(vj′,t+1)]∈MDDi.E
+X t+1
+i,vj′
+(1)
+This constraint is introduced for each ai, vj, and t and
+specifies that agent must proceed to the next level in MDD.
+1The proper terminology for agents’ paths should be trajectories
+or just sequences of vertices as a vertex can visited multiple times
+by the agent which does not correspond to the standard graph termi-
+nology where a path is a simple sequence of vertices.
+2Auxiliary variables Et
+i,vj,vj′ to express that agent ai moves
+from vj to vj′ between time-steps t and t + 1 can be used as well,
+but we will base our description only on the X t
+i,vj variables as the
+Et
+i,vj,vj′ variables are direct consequence of of the X t
+i,vj variables:
+Et
+i,vj,vj′ ↔ X t
+i,vj ∧ X t+1
+i,vj′ .
+In addition, to this we account among these constraints also
+calculation and bounding of the cost. For each time-step t in
+MDD an auxiliary variable is introduced which is TRUE if
+and only if the agent performed an action. The cost bound-
+ing over auxiliary variables is carried out through cardinal-
+ity constraints encoded as Boolean circuits into the formula
+[Bailleux and Boufkhad, 2003; Silva and Lynce, 2007].
+X 0
+i,s(ai) ∧
+�
+j | vj̸=s(ai)
+¬X 0
+i,vj
+(2)
+X T
+i,g(ai) ∧
+�
+j | vj̸=g(ai)
+¬X T
+i,vj
+(3)
+These equations are introduced for each agent ai and en-
+sure that agent ai starts in vertex s(ai) at time-step 0 and
+finished in its goal vertex g(ai) at time-step T.
+Path Consistency Constraints
+�
+j | (vj,t)∈MDDi.V
+X t
+i,vj = 1
+(4)
+This constraint is introduced for each ai and t and specifies
+that agent can appear in exactly one vertex at a time.
+Conflict Elimination Constraints
+�
+i | (vj,t)∈MDDi.V
+X t
+i,vj ≤ 1
+(5)
+This constraint is introduced for each vj and t and specifies
+that at most one agent can reside in vertex vj at time t. The
+constraint eliminates vertex conflicts, edge conflicts can be
+eliminated analogously.
+As some of the constraint are defined by pseudo-Boolean
+expression, proper translation to CNF is needed which most
+notably concerns the at-most-one constraints [Chen, 2010;
+Nguyen and Mai, 2015]. The combination of pair-wise en-
+coding and sequential counter are used in the MDD-SAT and
+SMT-CBS solvers.
+The formula collecting the above constraints is being built
+for the specific sum-of-costs SoC and is denoted F(SoC).
+The completeness of the model encoded by F(SoC) can be
+summarized as follows [Surynek et al., 2016]:
+Proposition 2. The input MAPF has a solution of the given
+sum-of-costs SoC ⇔ F(SoC) is satisfiable.
+The core of the proof of the proposition is a correspon-
+dence of satisfying assignments of F(SoC) and directed
+paths in MDDs which in turn due to Proposition 1 estab-
+lishes a correspondence between agents’ paths and satisfying
+assignments of F(SoC).
+2.2
+CEGAR for MAPF
+The next step in compilation-based MAPF solving is rep-
+resented by the integration of ideas from counterexample
+guided abstraction and refinement (CEGAR) [Clarke et al.,
+2000; Clarke, 2003]. The two representative solvers SMT-
+CBS [Surynek, 2019] based on SAT and Lazy-CBS [Gange
+et al., 2019] based on CSP resolve conflicts between agents
+as done in the CEGAR approach (though the authors of these
+MAPF solvers do not explicitly mention CEGAR).
+
+ 
+Solving 
+Answer 
+Interpretation 
+Abstraction Refinement 
+Counterexample 
+Generation 
+no counterexamples 
+(no conflicts) 
+counterexamples 
+(conflicts) exist 
+Solution 
+Agent 
+Propagation 
+Constraints 
+Path 
+Consistency 
+Constraints 
+Conflicts 
+Elimination 
+Constraints 
+Initial Abstraction 
+Figure 4: Schematic diagram of counterexample guided abstrac-
+tion refinement (CEGAR) for MAPF. Conflicts between agents are
+treated as counterexamples and eliminated in the abstraction refine-
+ment loop.
+The general CEGAR approach for compilation-based
+problem solving starts with a so called initial abstraction of
+the problem instance being solved in some target formalism
+such as SAT [Biere et al., 2021] or CSP [Dechter, 2003]. The
+initial abstraction do not model the input instance in the full
+details. However still the initial abstraction is passed to the
+solver for the target formalism despite the solver is not pro-
+vided all the details needed to solve it. Then the solver will
+come with some answer and since it could not take some de-
+tails into account during the solving phase, the answer must
+be checked, usually against full details of how the problem
+instance is defined. Two cases need to be distinguished at this
+stage. If the provided answer matches the instance definition
+then it is returned and the solving process finishes. Otherwise
+the CEGAR solving process generates counterexample that
+is determined by the mismatch between the provided answer
+and the requirements the expected answer should satisfy. This
+mismatch is usually represented by the violation of some con-
+straints that were not expressed in the abstraction. Then the
+solving process continues with a so called abstraction refine-
+ment in which the abstraction is augmented to eliminate the
+counterexample and the solving process continues with the
+next iteration of (now refined) abstraction solving.
+To keep the problem solving in the CEGAR approach
+sound, the abstractions must fulfill certain basic requirements
+such as if the problem instance is solvable then any of its ab-
+stractions should be solvable as well (the opposite does not
+hold: the solvable abstraction does not necessarily imply that
+the input instance is solvable).
+The CEGAR approach for MAPF as implemented in SMT-
+CBS and Lazy-CBS is illustrated in Figure 4.
+The initial
+abstraction in both algorithms models existence of paths for
+individual agents but omits the requirement to avoid con-
+flicts between agents. The abstraction refinement hence must
+check the answers of the target solver against the definition of
+conflicts in MAPF. If a conflict is found then the abstraction
+is refined so that the conflict is eliminated.
+The abstraction refinement for a conflict, say between
+agents ai and aj in vertex v at time-step t, is done in the
+case of CSP-based Lazy-CBS by adding a fresh finite domain
+variable pv,t ∈ A that indicates what agent occupies vertex
+v at time step t. The need to assign the variable pv,t a single
+value eliminates the conflict.
+The SAT-based SMT-CBS algorithm refines the conflict by
+adding a new constraint over the existing Boolean variables
+that forbids the occupation of vertex v at time-step t by agents
+ai and aj simultaneously: ¬X t
+i,v ∨ ¬X t
+j,v.
+The edge conflicts and potentially different kinds of con-
+flicts in MAPF and its variants [Andreychuk et al., 2019;
+Bonnet et al., 2018] can be treated analogously.
+The abstraction at any point in SMT-CBS and Lazy-CBS
+forms a so called incomplete model for MAPF. Unlike the
+complete model from MDD-SAT, the equivalent-solvability
+of the model in the target formalism and the input MAPF in-
+stance does not hold for the incomplete MAPF model. Let
+us express this property formally for the SAT-based formula-
+tion. Let F′(SoC) be the formula obtained at some stage in
+CEGAR loop of SMT-CBS used for answering the question
+whether there is a solution of the input MAPF instance of
+the given sum-of-costs SoC. Then the following proposition
+holds [Surynek, 2019]:
+Proposition 3. The input MAPF has a solution of the given
+sum-of-costs SoC ⇒ F′(SoC) is satisfiable.
+The incompleteness property also represents the require-
+ment that keeps the CEGAR approach sound for MAPF as it
+says that F′(SoC) is an abstraction for MAPF.
+3
+Non-Refined Abstractions in MAPF
+We are going further in the CEGAR architecture of the MAPF
+solver. In addition to conflict elimination constrains we also
+omit path consistency constraints in the initial abstraction.
+Moreover we never make any refinement with respect to the
+omitted path consistency constraints - the corresponding ab-
+straction remains non-refined.
+We call the new algorithm based on the non-refined ab-
+stractions Non-Refined SAT or NRF-SAT in short.
+The
+pseudo-code of NRF-SAT is shown as Algorithm 1.
+The high-level function of the algorithm NRF-SAT-
+MAPF() is analogous to SATPlan or MDD-SAT main loop
+where search for the optimal sum-of-costs is done by trying
+to answer questions whether there is a solution of MAPF of
+a specified sum-of-costs SoC (lines 5-6). To answer these
+questions, the algorithm uses low level function NRF-SAT-
+Bounded() that implements the CEGAR architecture with
+non-refined abstractions as shown in Figure 5.
+As the satisfying assignment of formula F′′ representing
+the incomplete model in NRF-SAT does not correspond to
+paths for agents (line 15) further post-processing of the SAT
+solver’s answer is needed (line 16). As we will see later,
+the answer obtained directly from the SAT solver can be in-
+terpreted as special directed acyclic graph (DAG) that con-
+nects agent’s start vertex with its goal vertex. Hence the post-
+processing consists in extraction of proper agent’s path from
+the DAG.
+The rest of the low-level loop (lines 17-22) represents ab-
+straction refinements with respect to conflicts between agents.
+Let us note that conflicts being discovered are collected and
+reused in the next iteration of the high-level loop.
+The NRF-SAT algorithm is sound and optimal, more pre-
+cisely it returns a sum-of-costs optimal solution for a solvable
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Solving 
+Answer 
+Interpretation 
+Counterexample 
+Generation 
+Answer 
+Post-processing 
+Incomplete Abstraction Refinement 
+no counterexamples 
+(no conflicts) 
+counterexamples 
+(conflicts) exist 
+Solution 
+Agent 
+Propagation 
+Constraints 
+Conflict 
+Elimination 
+Constraints 
+Initial Abstraction 
+Figure 5: Schematic diagram of CEGAR problem solver for MAPF
+with non-refined abstractions.
+input MAPF instance. The following series of propositions
+shows the claim.
+Definition 1. DAGi is a directed acyclic sub-graph of
+MDDi such that (g(ai), T) ∈ DAGi.V and there exists a
+directed path from any (vj, t) ∈ DAGi.V to (g(ai), T).
+Proposition 4. The interpretation of satisfying assignment
+of F′′(SoC) at any stage of the NRF-SAT algorithm corre-
+sponds to DAGi such that (s(ai), 0) ∈ DAGi.V .
+Proof. Assume that Boolean decision variables of X t
+i,vj are
+set to TRUE to reflect the choice of vertices in MDDi by a
+given DAGi. Then the agent propagation constraints (1) and
+(3) are satisfied since they directly correspond to existence of
+paths towards (g(ai), T) from any node selected by DAGi
+which is satisfied by the definition of DAGi. Conversely,
+any assignment of Boolean decision variables that satisfies
+constraints (1) and (3) corresponds to DAGi since any setting
+of X t
+i,vj to TRUE must be propagated via (1) and (3) towards
+X T
+i,g(ai). In addition to this, the constraint (2) ensures that
+(s(ai), 0) ∈ DAGi.V .
+The immediate corollary of Proposition 4 and the definition
+of DAGi is as follows:
+Corollary 1.
+There exists a directed path connecting
+(s(ai), 0) and (g(ai), T) in DAGi for each agent ai obtained
+from satisfying assignment of F′′(SoC).
+This directed path will be extracted from the SAT solver
+answer during the answer post-processing step as illustrated
+in Figure 6.
+Additional constraints that bound the cost and those that
+eliminate conflicts included during abstraction refinements
+restrict the set of DAGs that can correspond to satisfying as-
+signments of F′′(SoC).
+The important property of DAGi that directly follows from
+its definition is that a path for agent ai of length T going from
+its start vertex s(ai) to its goal g(ai) can be represented as a
+DAGi (we only add time indices to vertices visited by the
+agent along the path to obtain the DAGi). Hence F′′(SoC)
+is an incomplete model (an abstraction) for MAPF:
+Proposition 5. The input MAPF has a solution of the given
+sum-of-costs SoC ⇒ F′′(SoC) is satisfiable.
+The above propositions establish soundness of the NRF-
+SAT algorithm. In other words, if the algorithm terminates
+Algorithm 1: NRF-SAT: MAPF solving via CEGAR
+with non-refined abstractions.
+1 NRF-SAT-MAPF(M = (G, A, s, g))
+2
+SoC ← lower-Bound(M)
+3
+conflicts ← ∅
+4
+while TRUE do
+5
+(paths, conflicts) ←
+6
+NRF-SAT-Bounded(M, SoC,conflicts)
+7
+if paths ̸= UNSAT then
+8
+return paths
+9
+SoC ← SoC + 1
+10 NRF-SAT-Bounded(M,SoC,conflicts)
+11
+F ′′ ← build-Initial-Abstraction(M,SoC,conflicts)
+12
+while TRUE do
+13
+assignment ← consult-SAT-Solver(F ′′)
+14
+if assignment ̸= UNSAT then
+15
+DAGs ← interpret(M,assignment)
+16
+paths ← extract-Paths(M,DAGs)
+17
+conflicts′ ← validate(M,paths)
+18
+if conflicts′ = �� then
+19
+return (paths, conflicts)
+20
+for each c ∈ conflicts′ do
+21
+F ′′ ← F ′′∪ eliminate-Conflict(c)
+22
+conflicts ← conflicts ∪ conflicts′
+23
+return (UNSAT,conflicts)
+and returns an answer then it is a valid MAPF solution. How-
+ever the termination needs a separate investigation.
+Proposition 6. The abstraction refinement loop for a speci-
+fied SoC is executed by the NRF-SAT algorithm finitely many
+times.
+Proof. Each iteration of the abstraction refinement loop cor-
+responds to a counterexample, a conflict between a pair of
+agents. This conflict appears between two paths extracted
+from a pair of DAGs say DAGi and DAGj. These two spe-
+cific DAGs cannot be interpreted from the satisfying assign-
+ment of F′′(SoC) again in any of the next iterations of the
+abstraction refinement loop since the conflict elimination con-
+straint forbids them to appear simultaneously, either DAGi or
+DAGj must be different. Since there is finitely many pairs of
+DAGs DAGi and DAGj the algorithm either forbids them all
+during the refinements or terminates earlier.
+We are now ready to state the main theoretical property of
+the NRF-SAT algorithm.
+Theorem 1. The NRF-SAT algorithm returns sum-of-costs
+optimal solution for a solvable input MAPF instance.
+Proof. For a solvable MAPF instance, the NRF-SAT finds
+the first sum-of-costs SoC for which the abstraction refine-
+ment loop finishes with a solution since all previous loops are
+guaranteed to terminate due to Proposition 6. Due to mono-
+tonicity of solvability of bounded MAPF with respect to the
+sum-of-costs, this solution is optimal.
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(v2,1) 
+(v3,2) 
+(v1,0) 
+(v6,3) 
+(v5,2) 
+(v9,4) 
+(v4,1) 
+(v7,2) 
+(v8,3) 
+SAT solver answer, DAGi 
+(v2,1) 
+(v3,2) 
+(v1,0) 
+(v6,3) 
+(v5,2) 
+(v9,4) 
+(v4,1) 
+(v7,2) 
+(v8,3) 
+after post-processing 
+Figure 6: Post-processing step in which path is extracted from DAG
+answered by the SAT solver.
+4
+Experimental Evaluation
+We performed an experimental evaluation of NRF-SAT on
+a number of MAPF benchmarks from movingai.com
+[Sturtevant, 2012].
+We compared NRF-SAT against SAT-
+based SMT-CBS [Surynek, 2019] and CSP-based LazyCBS
+[Gange et al., 2019] which are the solvers using similar
+compilation-based approach to MAPF.
+4.1
+Benchmarks and Setup
+We implemented NRF-SAT in C++ via reusing the code of
+the original implementation of SMT-CBS. Both SAT-based
+MAPF solvers are built on top the Glucose 3 SAT solver [Au-
+demard and Simon, 2009; Audemard and Simon, 2018] still
+ranking among top performing SAT solvers according to rel-
+evant competitions [Froleyks et al., 2021]. The extraction of
+agent’s path from DAG is implemented as a simple breadth-
+first search.
+The important feature of the Glucose SAT solver is that it
+provides an interface for adding clauses incrementally that is
+employed during abstraction refinements. After refining the
+formula being answered by the SAT solver with new clauses,
+the solving process does not need to start from scratch. In-
+stead the learned state of the solver is utilized in its run after
+the formula refinement which usually speeds up the process.
+As of LazyCBS, we used its original implementation in
+C++. LazyCBS is built on top of the Geas CSP solver that
+supports lazy clause generation [Stuckey, 2010], a feature
+used by LazyCBS to eliminate MAPF conflicts lazily.
+To obtain instances of various difficulties we varied the
+number of agents from 1 to K, where K is the maximum
+number of agents for which at least one solver is able to
+solve some instance in the given time limit of 300 seconds
+(5 minutes). K varied from approximately 80 agents to 120
+agents depending on the benchmark map. For each number of
+agents, we generated 25 instances according to random sce-
+narios provided on movingai.com (for each benchmark
+map we generated 25 × K instances, i.e. approximately 2500
+MAPF instances per map).
+All experiments were run on a system consisting of Xeon
+2.8 GHz cores, 32 GB RAM per solver instance, running
+Ubuntu Linux 18 3.
+4.2
+The Effect of Non-Refining
+We investigated the effect of non-refining w.r.t. the path con-
+sistency constraints in SAT-based solvers. The comparison of
+3To
+provide
+reproducibility
+of
+presented
+results
+the
+complete
+source
+code
+of
+NRF-SAT
+is
+available
+on
+https://github.com/surynek/boOX.
+the SMT-CBS and NRF-SAT solvers in terms of the number
+of clauses being generated along the entire solving process is
+shown in Figure 8 (this comparison is not relevant for Lazy-
+CBS, hence it is not included in the test).
+The table shows the median number of clauses being gen-
+erated by SMT-CBS for specific map and selected number of
+agents and the number of clauses generated by NRF-SAT for
+the same instance. In this test, small to medium sized maps
+have been used:
+empty-16-16, random-32-32-10,
+and room-64-64-16.
+We can observe that NRF-SAT generates significantly
+fewer clauses than SMT-CBS. This trend is even more pro-
+nounced as the number of agents increases. Order of mag-
+nitude fewer clauses are generated by NRF-SAT for 60 and
+more agents on the presented benchmarks.
+The explanation of this result that path consistency con-
+straints encompass many at-most-one constraints that yields
+many clauses in most of its SAT representations [Nguyen and
+Mai, 2015].
+We also report the total number of abstraction refinements
+for the same set of instances as reported for the number of
+clauses in Figure 8. Surprisingly non-refining often leads to
+a significant reduction of the number of abstraction refine-
+ments. Although this not a rule as sometimes increase in the
+number of abstractions in contrast to SMT-CBS can be ob-
+served in NRF-SAT, the reduction prevails.
+One reason of this difference is that the choice of final
+paths is done by the SAT solver in SMT-CBS while in NRF-
+SAT the final choice is made by the path-processing proce-
+dure that extracts paths from DAGs in a fixed order which
+seems to be more suitable for abstraction refinements.
+In addition to this, we tested the impact of leaving the ab-
+straction non-refined on the overall performance of the SAT-
+based MAPF solver. Runtime results comparing SMT-CBS
+and NRF-SAT on the same set of maps: empty-16-16,
+random-32-32-10, and room-64-64-16 is shown in
+Figure 7.
+The runtime results are presented using cactus plots often
+used to present the results of SAT competitions [Balyo et al.,
+2017], that is runtimes for all instances are sorted so the x-th
+result along the horizontal axis represents the runtime for the
+x-th fastest solved instance by the given MAPF solver. The
+lower plot for the solver means better performance.
+Instances
+with
+up
+to
+approximately
+70
+agents
+for
+empty-16-16, 110 agents for random-32-32-10, and
+60 agents for room-64-64-16 were solved by the solvers
+in the given time limit.
+SMT-CBS solved 1564, 2120,
+and 1152 and NRF-SAT solved 1633, 2266, and 1203
+in total for empty-16-16, random-32-32-10, and
+room-64-64-16 respectively, that is, significantly more
+instances for NRF-SAT. It need to be taken into account that
+the instances that were additionally solved by the NRF-SAT
+solver rank among the difficult ones.
+We can observe in Figure 7 that NRF-SAT dominates in
+harder instances while in easier instances SMT-CBS is some-
+times better (most prominently on empty-16-16).
+The explanation for the better performance of NRF-SAT
+is twofold. First, it generates significantly smaller formulae
+across entire abstraction refinement process hence the pro-
+
+0,01
+0,1
+1
+10
+100
+1000
+0
+200
+400
+600
+800 1000 1200 1400
+Runtime (seconds)
+Instance
+Runtime | empty-16-16
+NRF-SAT
+SMT-CBS
+0,01
+0,1
+1
+10
+100
+1000
+0
+250
+500
+750
+1000 1250 1500 1750 2000
+Runtime (seconds)
+Instance
+Runtime| random-32-32-10
+NRF-SAT
+SMT-CBS
+0,01
+0,1
+1
+10
+100
+1000
+0
+200
+400
+600
+800
+1000
+Runtime (seconds)
+Instance
+Runtime | room-64-64-16
+NRF-SAT
+SMT-CBS
+Figure 7: Runtime comparison of two SAT-based MAPF solvers NRF-SAT and SMT-CBS. Cactus plots of runtimes for the solvers are shown
+(lower plot means better performance).
+Refinements|clauses 
+SMT-CBS  
+NRF-SAT 
+empty-16-16 
+random-32-32-10 
+room-64-64-16 
+Number of agents 
+10 
+2 
+2 
+   1.397 
+       468 
+1 
+2 
+6.018 
+1.459 
+2 
+2 
+6.380 
+1.905 
+20 
+2 
+2 
+2.571 
+919 
+10 
+6 
+44.690 
+8.560 
+3 
+2 
+14.348 
+4.445 
+30 
+22 
+17 
+46.412 
+8.239 
+18 
+14 
+63.672 
+12.564 
+8 
+7 
+273.997 
+40.489 
+40 
+23 
+22 
+271.103 
+36.146 
+9 
+10 
+81.869 
+16.327 
+8 
+8 
+349.156 
+52.137 
+50 
+24 
+22 
+510.689 
+65.999 
+17 
+15 
+1.373.944 
+162.590 
+14 
+14 
+3.644.645 
+384.442 
+60 
+44 
+32 
+4.042.490 
+491.733 
+42 
+32 
+14.094.029 
+1.498.631 
+54 
+45 
+17.694.257 
+1.697.465 
+ 
+Figure 8: The total number of clauses and refinements of SAT-based
+solvers SMT-CBS and NRF-SAT.
+cessing time itself is shorter. Second, the resulting formula
+with omitted path consistency constraints is easier to solve by
+the SAT solver which coupled with the fact the total number
+of abstraction refinements tends to be smaller in NRF-SAT
+leads to overall better performance.
+4.3
+Competitive Comparison
+The competitive comparison of NRF-SAT and LazyCBS in
+terms of runtime is shown in Figure 9.
+This comparison
+is focused on benchmarks that were identified by previous
+studies as those where compilation-based MAPF solvers per-
+form well [Kaduri et al., 2021]. These benchmarks include
+those on mazes, city maps, and game maps. The representa-
+tives we selected for presentation are: maze-32-32-4 (a
+maze map), ost003d (a game map), Berlin 1 256 (a
+city map). Additional experiments we made on other maps
+from these categories yield similar results.
+Again, the runtime results are presented using cactus
+plots. The summary of the results is that LazyCBS solved
+527, 1193, and 1599 while NRF-SAT solved 582, 1335,
+and 2321 in total for maze-32-32-4, ost003d, and
+Berlin 1 256 respectively, that is NRF-SAT solver signif-
+icantly more instances where again these extra instances rank
+among difficult ones.
+Close look at the results reveals that the general trend is
+that LazyCBS has slightly sharper increase in runtimes as in-
+stances are getting harder. This results in reaching the time-
+out by LazyCBS sooner while NRF-SAT can still solve the
+instances within the time limit. The advantage of NRF-SAT
+tends to be more significant as the size of the map grows
+which is surprising for the SAT-based MAPF solvers that are
+notorious to struggle on large maps.
+The explanation for the better performance of NRF-SAT
+on the presented benchmarks is that non-refining in the CE-
+GAR architecture is especially helpful when dealing with
+large maps where it leads to significantly smaller formulae
+than in previous SAT-based solvers. Moreover this combined
+with the rest of the CEGAR architecture leads to a competi-
+tive solver.
+There are many other benchmarks where LazyCBS per-
+forms significantly better than NRF-SAT. Hence we do not
+claim that NRF-SAT is state-of-the-art solver for MAPF.
+However, as we report, there are several important domains
+where NRF-SAT achieves competitive performance which
+shows the importance of non-refined abstractions. Moreover,
+it is needed to take into account that NRF-SAT is in fact a
+vanilla solver for MAPF with no specific MAPF techniques
+such as symmetry breaking [Li et al., 2020a] or rectangle rea-
+soning [Li et al., 2019] being used. Hence there is a potential
+that the SAT-based MAPF solvers can return among top per-
+forming MAPF solvers at least in certain domains and the
+CEGAR architecture with non-refined abstractions can con-
+tribute to this.
+5
+Conclusion
+We proposed a novel solver called NRF-SAT for MAPF based
+on the CEGAR architecture and Boolean satisfiability that
+uses non-refined abstraction during the solving process.
+Unlike previous uses of CEGAR in MAPF we not only
+eliminate conflicts between agents via abstraction refine-
+ments but we also further strengthen the initial abstraction.
+Particularly our new solver NRF-SAT omits large group of
+constraints in the initial abstraction and never makes refine-
+ments with respect to them which as we show can be miti-
+gated by a fast post-processing step.
+From a broader perspective, we not only apply the CEGAR
+architecture with non-refined abstractions for compilation-
+based MAPF solving but we also generalize the architecture
+itself. This generalization consists in finding a solution of a
+
+0,001
+0,01
+0,1
+1
+10
+100
+1000
+0
+100
+200
+300
+400
+Runtime (seconds)
+Instance
+Runtime | maze-32-32-4
+NRF-SAT
+LazyCBS
+0,01
+0,1
+1
+10
+100
+1000
+0
+200
+400
+600
+800 1000 1200 1400
+Runtime (seconds)
+Instance
+Runtime | ost003d
+NRF-SAT
+LazyCBS
+0,01
+0,1
+1
+10
+100
+1000
+0
+250
+500
+750 1000 1250 1500 1750 2000 2250
+Runtime (seconds)
+Instance
+Runtime | Berlin_1_256
+NRF-SAT
+LazyCBS
+Figure 9: Runtime comparison between NRF-SAT and LazyCBS. Cactus plots of runtimes for the solvers are shown (lower plot is better).
+different, more general task, than is the original one using the
+target formalism, rather than finding a solution of the original
+task using the target formalism directly.
+Acknowledgments
+This research has been supported by GA ˇCR - the Czech Sci-
+ence Foundation, grant registration number 22-31346S.
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+

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+arXiv:2301.01300v1  [astro-ph.HE]  3 Jan 2023
+MNRAS 000, 1–4 (2022)
+Preprint 5 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Tidal capture of stars by supermassive black holes: implications for
+periodic nuclear transients and quasi-periodic eruptions
+M. Cufari1★, C. J. Nixon2,3, and Eric R. Coughlin1
+1 Department of Physics, Syracuse University, Syracuse, NY 13244, USA
+2 School of Physics and Astronomy, University of Leicester, Leicester LE1 7RH, UK
+3 School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+Stars that plunge into the center of a galaxy are tidally perturbed by a supermassive black hole (SMBH), with closer encounters
+resulting in larger perturbations. Exciting these tides comes at the expense of the star’s orbital energy, which leads to the naive
+conclusion that a smaller pericenter (i.e., a closer encounter between the star and SMBH) always yields a more tightly bound
+star to the SMBH. However, once the pericenter distance is small enough that the star is partially disrupted, morphological
+asymmetries in the mass lost by the star can yield an increase in the orbital energy of the surviving core, resulting in its ejection
+– not capture – by the SMBH. Using smoothed-particle hydrodynamics simulations, we show that the combination of these two
+effects – tidal excitation and asymmetric mass loss – result in a maximum amount of energy lost through tides of ∼ 2.5% of
+the binding energy of the star, which is significantly smaller than the theoretical maximum of the total stellar binding energy.
+This result implies that stars that are repeatedly partially disrupted by SMBHs many (≳ 10) times on short-period orbits (≲ few
+years), as has been invoked to explain the periodic nuclear transient ASASSN-14ko and quasi-periodic eruptions, must be bound
+to the SMBH through a mechanism other than tidal capture, such as a dynamical exchange (i.e., Hills capture).
+Key words: hydrodynamics — black holes — galaxies: nuclei
+1 INTRODUCTION
+Binary stars may form when two field (i.e., individual) stars pass
+close to one another on nearly parabolic trajectories. As the stars
+pass, tidal oscillatory modes are excited in the stars at the expense
+of the stars’ orbital energy. If the encounter is sufficiently close, the
+tides dissipate enough orbital energy to bind the stars to one another
+(Fabian et al. 1975; Press & Teukolsky 1977).
+A star may also be scattered into the region of parameter space,
+known as the “loss-cone,” that brings the star’s distance of clos-
+est approach very near a supermassive black hole in the nucleus of
+a galaxy (SMBH; Frank & Rees 1976; Lightman & Shapiro 1977;
+Cohn & Kulsrud 1978). Reasoning analogously to the binary for-
+mation scenario, one is tempted to conclude that stellar orbits with
+distances of closest approach nearer the black hole are more strongly
+tidally perturbed by, and thus more tightly bound to, the SMBH.
+However, there is a fundamental upper limit as to how efficient this
+process can be, because tides cannot transfer more energy into oscil-
+lations than the binding energy of the star itself. Additionally, as the
+pericenter distance of the star continues to decrease the perturbative
+tidal limit breaks down, and the star starts to lose a fraction of its outer
+envelope (known as a partial Tidal Disruption Event; TDE). Recent
+simulations have demonstrated that the asymmetric ejection of tidal
+debris in a partial TDE results in the unbinding of the surviving stel-
+lar core from the SMBH (Manukian et al. 2013; Gafton et al. 2015).
+Therefore, the maximum binding energy a star can achieve through
+★ E-mail:mcufari@syr.edu
+tidal interactions with an SMBH could be substantially smaller than
+the theoretical limit of the star’s own binding energy. This maximum
+binding energy generated through tidal dissipation sets a fundamen-
+tal timescale over which one would expect repeating partial TDEs –
+a star that is bound to a SMBH that is partially destroyed on each
+pericenter passage (e.g., Zalamea et al. 2010; Campana et al. 2015;
+Miniutti et al. 2019; Payne et al. 2021) – to recur if the bound star is
+supplied through tidal dissipation.
+In this paper we present the results of smoothed particle hydrody-
+namics (SPH) simulations of partial TDEs from which we obtain the
+maximum binding energy. In Section 2.1 we describe the setup of the
+simulations, and in Section 2.2 we present the results. In Section 3
+we discuss the implications of our findings in the context of periodic
+nuclear transients (PNTs) and quasi-periodic eruptions (QPEs). We
+summarize and conclude in Section 4.
+2 SIMULATIONS
+2.1 Methodology
+We performed simulations of partial TDEs with the SPH code phan-
+tom (Price et al. 2018). The star was modeled as a polytrope with
+a 훾 = 5/3, adiabatic equation of state, with mass 푀★ = 1푀⊙ and
+radius 푅★ = 1푅⊙. In each simulation the star was injected from a
+distance of 12 푟푡, where 푟t = 푅★ (푀•/푀★)1/3, from the SMBH of
+mass 푀• = 106푀⊙ with the center of mass on a parabolic trajectory.
+The impact parameter 훽 ≡ 푟t/푟p, where 푟p is the pericenter distance
+© 2022 The Authors
+
+2
+Cufari, Nixon, & Coughlin
+of the star, was varied between 훽 = 0.4 and 훽 = 0.8 in steps of
+Δ훽 = 0.02. Additional details of the simulation setup can be found in
+Coughlin & Nixon (2015), and the algorithms for, e.g., self-gravity
+can be found in Price et al. (2018).
+We performed simulations with 1.25 × 105, 106 particles and
+8 × 106 particles, and found very little deviation in the outcome.
+All results presented here are from the 8 × 106 particle runs. For
+the purposes of determining the properties of the surviving star,
+we calculate averages using only those particles that have a density
+greaterthan1%ofthemaximumglobal densitywithin thesimulation,
+휌max; we define this subset of particles as the core. For example, the
+location of the core is calculated as the the average position of the
+subset of particles that satisfy this criterion, and the distance from the
+SMBH, 푟c, is determined from the Euclidean norm of the position.
+The analogous statement applies to the core velocity and speed, 푣c.
+Over the range of 훽 we simulated this criterion excludes the tidally
+stripped tails, and reducing the fraction to 0.1% does not impact
+the results because the tidal tails have a substantially lower density
+than ∼ 1% of 휌max. Similarly, increasing the fraction to 10% has a
+negligible effect on the results. However, values > 10% can lead to
+significant noise in the results because in this case there are too few
+particles that contribute to the average.
+We run each simulation until the core energy has settled to a
+constant value that we measure for our results. For disruptions with
+훽 < 0.6, this time is ≲ 1 day after pericenter passage. Simulations
+with 훽 > 0.6 are run up to ∼ 5 days post-pericenter to ensure that the
+energy has converged to a constant value. Running our simulations
+to later times has no affect on our results.
+2.2 Results
+The left panel of Figure 1 shows the specific orbital energy of the
+core, calculated according to
+휖c = 1
+2 푣2
+c − 퐺푀•
+푟c
+.
+(1)
+For 0.4 ≲ 훽 ≲ 0.62, the orbital energy is negative, implying that the
+surviving core is bound to the SMBH. The global minimum energy
+is ∼ 2 − 3% of the binding energy of the star, i.e.,
+휖c,min ≃ −0.025퐺푀★
+푅★
+.
+(2)
+The fact that the star becomes bound to the SMBH following its tidal
+interaction is in agreement with the classical calculations of tidal
+dissipation by Fabian et al. (1975); Press & Teukolsky (1977).
+Conversely, all disruptions with 훽 ≳ 0.62 result in a core with a net
+positive energy following the interaction with the SMBH, indicating
+that the star is on an unbound trajectory. This result agrees with those
+of Manukian et al. (2013); Gafton et al. (2015), who found that the
+“kick” to the star could result in a velocity that is ∼ the escape speed
+of the star for 훽 very close to the value at which complete disruption
+occurs (훽 ≃ 0.9 for a 5/3 polytrope; Guillochon & Ramirez-Ruiz
+2013; Mainetti et al. 2017). Therefore, if a star reaches a pericenter
+distance with 0.62 ≲ 훽 ≲ 0.9, the star is not tidally captured but is
+instead tidally ejected, never to return to pericenter.
+With the energy-semimajor axis and energy-period relationships
+for a Keplerian orbit, we find that the semimajor axis of the captured
+star (for 훽 ≲ 0.62, i.e., for stars that are not ejected) is
+푎c = −퐺푀•
+2휖c
+= 1
+휂c
+푅★
+2
+� 푀•
+푀★
+�
+,
+(3)
+while the orbital period is
+푇c =
+2휋퐺푀•
+(−2휖c)3/2 =
+1
+휂3/2
+c
+� 푅★
+2
+�3/2
+2휋
+√퐺푀★
+� 푀•
+푀★
+�
+.
+(4)
+Here we defined the specific orbital energy of the captured star as 휖c =
+−휂c퐺푀★/푅★, where 휂c ≲ 0.025 (from Figure 1). Using 푀•/푀★ =
+106 in Equation (3), the minimum apocenter distance of the star (with
+휂c = 0.025) is ∼ 1 pc for 푅★ = 1푅⊙.
+The right panel of Figure 1 shows the orbital period of the captured
+star as a function of 훽 from the simulations and using Equation
+(4). We see that the minimum orbital period of the captured star is
+∼ 3 × 104 yr. However, because of the strong dependence on 휂c in
+Equation (4), the period can be much larger for only small changes
+in 훽.
+3 DISCUSSION
+Some current models for PNTs and QPEs suggest that they can be
+produced by stars on bound orbits about SMBHs with periods from
+hours to ∼ 푓 푒푤 years, with the star being partially disrupted and
+feeding an accretion flare every pericenter passage (Miniutti et al.
+2019; King 2020; Payne et al. 2021; King 2022; Wevers et al. 2022).
+PNTs have been suggested to have system parameters that are com-
+parable to 푀• = 107푀⊙, 푅★ = 1푅⊙, 푀★ = 1푀⊙, which gives, from
+Equation (4) with 휂c = 0.025,
+푇PNT ≃ 3 × 105 yr,
+(5)
+which is obviously too long to explain the observed periods ≲ 1
+year (Payne et al. 2021; Wevers et al. 2022). Even if one could inject
+the theoretical maximum energy into the star, and thus set 휂c = 1
+in Equation (4), we would obtain 푇PNT ≃ 103 yr. As argued by
+Cufari et al. (2022), even in this overly optimistic scenario, PNTs
+need an additional mechanism to bind the star sufficiently tightly to
+the SMBH to produce their observed periods in situ.
+On the other hand, it is possible that a PNT could initially be
+generated with an orbital period given by Equation (5), but the period
+then decays through gravitational-wave emission (over millions of
+years) to produce a period as short as ∼ 1 year by the time that we
+observe it1. However, a problem with this interpretation is that even
+for the maximum value of 휂c = 0.025, Equation (3) for the semimajor
+axis of the orbit yields an apocenter distance of ∼ 2푎c ∼ 10 pc for
+푀•/푀★ = 107. Thus, we would expect the star to interact with, and
+be once again perturbed by, the nuclear star cluster, and it seems very
+unlikely that the star will repeatedly return to the same pericenter
+over multiple passages.
+Additionally, while we have assumed that the star is on a parabolic
+orbit, it could have a velocity at infinity that is 푣∞ ≃ 휎, with 휎 the
+galactic velocity dispersion (Miller et al. 2005). From Figure 1, the
+maximum amount of energy able to be dissipated through tides is
+휖c,max ≃ 0.025퐺푀★
+푅★
+≃ 1
+2
+� 푀★
+푀⊙
+� � 푅★
+푅⊙
+�−1 �
+100 km s−1�2
+.
+(6)
+For a 107푀⊙ SMBH, the velocity dispersion from the 푀 −휎 relation
+is 휎 ∼ 110 km s−1(Marsden et al. 2020). Thus, tides may not actually
+be capable of dissipating the true (positive) energy of the orbit, and
+1 As pointed out in Payne et al. (2021), the rate of change of the orbital period
+due to gravitational waves is actually too small to explain the observed value
+for ASASSN-14ko, but it should be possible for gravitational waves to shrink
+the orbit in general and for other systems.
+MNRAS 000, 1–4 (2022)
+
+On tidal capture for PNTs and QPEs
+3
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+-0.05
+0.00
+0.05
+0.10
+β
+ϵc [GM⊙/R⊙]
+0.40
+0.45
+0.50
+0.55
+0.60
+5×104
+1×105
+5×105
+1×106
+5×106
+1×107
+β
+Tc [yrs]
+Figure 1. Left: The energy imparted to the star as a function of the impact parameter 훽. Stars with 휖c < 0 are bound to the black hole. There is a small range of
+훽 for which mass loss occurs on first pericenter passage (i.e., 훽 > 0.5) and for which 휖c < 0, the necessary condition for a repeating partial disruption to occur.
+Right: the period of the orbit of the star for encounters that yield a captured star as a function of the impact parameter 훽. The minimum period corresponds
+minimum orbital energy (i.e. the maximum binding energy), and for these parameters is > 104 yrs.
+hence binding the star through this mechanism may be impossible in
+the first place.
+QPEs – with periods of the order hours – are typically modeled
+with a ∼ 105푀⊙ SMBH partially disrupting a white dwarf star. For
+such a system, Equation (4) with 푀★ = 0.6푀⊙, 푅★ = 0.011푅⊙
+(Nauenberg 1972), 푀• = 4 × 105푀⊙, and 휂c = 0.025 gives
+푇QPE ≃ 28 yr,
+(7)
+which shows that it is not possible to bind a star to a SMBH
+through tidal dissipation and immediately reproduce the observed
+orbital periods of QPEs. As for PNTs, it is nonetheless possible that
+gravitational-wave emission shrinks the orbit2 to ∼ hours before we
+detect them. Because the binding energy of a white dwarf is substan-
+tially larger than that of a main sequence star, the minimum apocenter
+distance is correspondingly smaller (from Equation 3), and the max-
+imum imparted energy through tides is substantially larger than the
+energy at infinity. Thus, without further investigation that is outside
+the scope of the present work, we cannot conclusively state whether
+or not an additional mechanism is required for producing the ob-
+served orbital periods in QPEs (under the paradigm that they are
+powered by repeatedly partially disrupted white dwarfs).
+On the other hand, the tidal breakup of a binary star system (i.e.,
+Hills capture; Hills 1975) can bind the star with a substantially shorter
+period (compared to just tidal dissipation) if the binary is sufficiently
+tight. As demonstrated by Cufari et al. (2022), this is a plausible
+explanation for the origin of the ∼ 114 day period of ASASSN-
+14ko. An additional argument against the gravitational-wave inspiral
+scenario for explaining ASASSN-14ko is that the mass lost by the
+star must be ∼ 1% of the mass of the star to power the emission
+(Cufari et al. 2022), and hence it cannot have survived many (≳ 100
+s) interactions prior to the ∼ 10 that have been observed since the
+initial detection. Similarly, if the white dwarf binary separation is not
+much larger than the radius of the white dwarf itself, then Equation
+(5) from Cufari et al. (2022) with 푎★ = 0.04푅⊙, 푀• = 4 × 105푀⊙,
+and 푀★ = 0.6푀⊙ yields an orbital period of ∼ 8.3 hours for the
+period of the bound star following a Hills-capture event. As also
+suggested in Cufari et al. (2022), it therefore seems plausible that the
+2 The orbital period may also be reduced through the interaction with a
+pre-existing AGN disc (Syer et al. 1991; Cufari et al. 2022; Lu & Quataert
+2022).
+periods of QPEs can be generated with a dynamical exchange process
+without the need for additional dissipation through other means.
+4 SUMMARY AND CONCLUSIONS
+We presented SPH simulations of the interaction between a star and
+an SMBH to determine the maximum degree by which a star can
+be bound to an SMBH through tidal dissipation. Two competing
+mechanisms prevent the star from becoming arbitrarily bound to the
+SMBH. As the distance of closest approach between the star and
+SMBH shrinks, energy from the star’s orbit is expended in exciting
+dynamical tides in the star. However, once the star reaches a dis-
+tance of closest approach comparable to its tidal radius, the star is
+kicked to positive energies as a result of asymmetry in the tidal tails
+that are liberated from the star. The competition between these two
+physical effects results in a minimum possible orbital energy of the
+star following the tidal encounter. For the parameters we simulated
+here, the location of this minimum is at 훽 ∼ 0.55 (pericenter distance
+of 푟t/0.55 with 푟t the usual tidal radius) and the binding energy of
+the orbit is ∼ 2.5% of the star’s binding energy; see Equation (2)
+specifically. This minimum energy is significantly smaller than the
+theoretical maximum, being the entirety of the stellar binding energy.
+In order to produce a repeating partial tidal disruption via a tidal
+dissipation mechanism, the energy kick imparted to the star must be
+negative, otherwise the star will be ejected on a hyperbolic trajec-
+tory. Hence, it would appear that there is a relatively small region of
+parameter space within which the star is only partially destroyed, not
+ejected, and survives for many (≳ 10) pericenter passages, specif-
+ically 0.4 ≲ 훽 ≲ 0.5 for the type of star considered here. Initially
+non-rotating stars in this range of 훽 will be rotating at a nontrivial
+fraction of breakup following the initial interaction, which will move
+the effective tidal radius out (Golightly et al. 2019), but provided that
+the pericenter (effectively unaltered because of the small ratio of the
+maximal angular momentum of the star to the angular momentum of
+the orbit itself) is still within this tidal radius, the star may transfer a
+small amount of mass during each additional pericenter passage. In
+this manner, a star may undergo many cycles of partial disruptions
+before being destroyed or ejected.
+In our simulations we modeled the star as a 5/3 polytrope, which
+is most applicable to low-mass main sequence stars and and low-
+mass white dwarfs. By number, most stars are thought to fall into
+MNRAS 000, 1–4 (2022)
+
+4
+Cufari, Nixon, & Coughlin
+this regime. However, more massive (radiative) stars are consider-
+ably more centrally concentrated than predicted by a 5/3-polytropic
+model. Here we have shown that the minimum energy for the captured
+star – modeled as a 5/3 polytrope – occurs at 훽 ≈ 0.55. This result will
+depend somewhat on the type of star being considered. For example,
+Figure 3 of Manukian et al. (2013) shows that it occurs for 훽 < 1
+when the star is modeled as a 4/3-polytrope and that there is very
+little dependence of the result on the black hole mass. Faber et al.
+(2005) considered the tidal capture of a planet by a star in which the
+mass ratio was 푞 = 0.001, and found a minimum binding energy of
+∼ 14% of the binding energy of the planet at 훽 = 10/19 ≃ 0.523 (see
+their Table 1). Kremer et al. (2022) also recently considered black
+hole-star systems with mass ratios closer to unity, and found a similar
+effect to the one described here if the mass ratio was 0.02 or 0.05
+if the star was modeled as a 훾 = 5/3 polytrope, but that the star
+was able to go from bound to completely disrupted – without being
+ejected – as the mass ratio increased beyond 0.05 and the star was
+modeled with the Eddington standard model (see their Figure 1). We
+defer an analysis of the minimum orbital energy – and the 훽 at which
+the minimum energy occurs – as a function of the mass ratio and the
+type of star to future work.
+In the context of extreme mass-ratio inspirals, Zalamea et al.
+(2010) demonstrated that a white dwarf (or other compact object)
+completes thousands of large-eccentricity orbits before reaching the
+direct capture radius of the SMBH. However, their model accounts
+only for the orbital decay due to gravitational wave emission and
+omits tidal dissipation and mass loss asymmetry effects. Likewise,
+our simulations omit the effects of orbital decay due to gravitational
+wave emission. The pericenter distance of our simulated disruptions
+is > 50 푟g, so orbital decay due to general relativistic effects (at least
+on the first pericenter passage) is negligible. For more compact stars
+with smaller tidal radii nearer the event horizon, orbital decay due to
+general relativistic effects will be more significant over fewer orbital
+periods (though the change in the pericenter will still be extremely
+small, as all of the dissipation occurs near pericenter for these high-
+eccentricity systems; thus the tidal interaction itself may be relatively
+unaltered, aside from the stronger tidal field of the SMBH due to rel-
+ativistic gravity). Future work on partial TDEs nearer the horizon
+of the SMBH should incorporate the change in orbital energy due
+to tidal interaction and mass loss asymmetry alongside gravitational
+wave emission.
+Finally, here we focused on orbits that produce partial TDEs in
+the traditional sense, i.e., the tidal force is not sufficiently strong
+to destroy the star completely. However, Nixon & Coughlin (2022)
+found that at very high 훽 (in their case 훽 = 16), the compression
+experienced by the star near pericenter could revive self-gravity to
+the point that a core reformed with a binding energy (to the SMBH)
+that was much larger than the value predicted by Equation (2) (though
+we caution that while the mass contained in the core was converged,
+the orbital period – and thus the binding energy – was resolution-
+dependent in their simulations). While encounters with high-훽 are
+rare3, it may be possible for tidal capture in this considerably more
+exotic scenario to produce shorter-period orbits than through the
+traditional means.
+3 For example, Equation 16 of Coughlin & Nixon (2022) shows that the
+fraction of TDEs with 훽 > 10 for a 106푀⊙ Schwarzschild SMBH – including
+general relativistic effects – is 0.0046; note that this is a factor of ∼ 4 smaller
+than the value derived by ignoring general relativistic effects, given by their
+Equation 17.
+DATA AVAILABILITY STATEMENT
+Code to reproduce the results in this paper is available upon reason-
+able request to the corresponding author.
+ACKNOWLEDGEMENTS
+M.C. acknowledges support from the Syracuse Office of Undergrad-
+uate Research and Creative Engagement (SOURCE). CJN acknowl-
+edges support from the Science and Technology Facilities Coun-
+cil (grant no. ST/W000857/1), and the Leverhulme Trust (grant
+no. RPG-2021-380). E.R.C. acknowledges support from the Na-
+tional Science Foundation through grant no. AST-2006684 and the
+Oakridge Associated Universities through a Ralph E. Powe Junior
+Faculty Enhancement Award. This work was performed using the
+DiRAC Data Intensive service at Leicester, operated by the Uni-
+versity of Leicester IT Services, which forms part of the STFC
+DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded
+by BEIS capital funding via STFC capital grants ST/K000373/1 and
+ST/R002363/1 and STFC DiRAC Operations grant ST/R001014/1.
+DiRAC is part of the National e-Infrastructure.
+REFERENCES
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+MNRAS 000, 1–4 (2022)
+

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='01300v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='HE]  3 Jan 2023  MNRAS 000, 1–4 (2022)  Preprint 5 January 2023  Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='0  Tidal capture of stars by supermassive black holes: implications for  periodic nuclear transients and quasi-periodic eruptions  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Cufari1★, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Nixon2,3, and Eric R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Coughlin1  1 Department of Physics, Syracuse University, Syracuse, NY 13244, USA  2 School of Physics and Astronomy, University of Leicester, Leicester LE1 7RH, UK  3 School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK  Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' in original form ZZZ  ABSTRACT  Stars that plunge into the center of a galaxy are tidally perturbed by a supermassive black hole (SMBH), with closer encounters  resulting in larger perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Exciting these tides comes at the expense of the star’s orbital energy, which leads to the naive  conclusion that a smaller pericenter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=', a closer encounter between the star and SMBH) always yields a more tightly bound  star to the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' However, once the pericenter distance is small enough that the star is partially disrupted, morphological  asymmetries in the mass lost by the star can yield an increase in the orbital energy of the surviving core, resulting in its ejection  – not capture – by the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Using smoothed-particle hydrodynamics simulations, we show that the combination of these two  effects – tidal excitation and asymmetric mass loss – result in a maximum amount of energy lost through tides of ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='5% of  the binding energy of the star, which is significantly smaller than the theoretical maximum of the total stellar binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  This result implies that stars that are repeatedly partially disrupted by SMBHs many (≳ 10) times on short-period orbits (≲ few  years), as has been invoked to explain the periodic nuclear transient ASASSN-14ko and quasi-periodic eruptions, must be bound  to the SMBH through a mechanism other than tidal capture, such as a dynamical exchange (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=', Hills capture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  Key words: hydrodynamics — black holes — galaxies: nuclei  1 INTRODUCTION  Binary stars may form when two field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=', individual) stars pass  close to one another on nearly parabolic trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' As the stars  pass, tidal oscillatory modes are excited in the stars at the expense  of the stars’ orbital energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' If the encounter is sufficiently close, the  tides dissipate enough orbital energy to bind the stars to one another  (Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Press & Teukolsky 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  A star may also be scattered into the region of parameter space,  known as the “loss-cone,” that brings the star’s distance of clos-  est approach very near a supermassive black hole in the nucleus of  a galaxy (SMBH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Frank & Rees 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Lightman & Shapiro 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  Cohn & Kulsrud 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Reasoning analogously to the binary for-  mation scenario, one is tempted to conclude that stellar orbits with  distances of closest approach nearer the black hole are more strongly  tidally perturbed by, and thus more tightly bound to, the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  However, there is a fundamental upper limit as to how efficient this  process can be, because tides cannot transfer more energy into oscil-  lations than the binding energy of the star itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Additionally, as the  pericenter distance of the star continues to decrease the perturbative  tidal limit breaks down, and the star starts to lose a fraction of its outer  envelope (known as a partial Tidal Disruption Event;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' TDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Recent  simulations have demonstrated that the asymmetric ejection of tidal  debris in a partial TDE results in the unbinding of the surviving stel-  lar core from the SMBH (Manukian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Gafton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  Therefore, the maximum binding energy a star can achieve through  ★ E-mail:mcufari@syr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='edu  tidal interactions with an SMBH could be substantially smaller than  the theoretical limit of the star’s own binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' This maximum  binding energy generated through tidal dissipation sets a fundamen-  tal timescale over which one would expect repeating partial TDEs –  a star that is bound to a SMBH that is partially destroyed on each  pericenter passage (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=', Zalamea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Campana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Payne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2021) – to recur if the bound star is  supplied through tidal dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  In this paper we present the results of smoothed particle hydrody-  namics (SPH) simulations of partial TDEs from which we obtain the  maximum binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='1 we describe the setup of the  simulations, and in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='2 we present the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' In Section 3  we discuss the implications of our findings in the context of periodic  nuclear transients (PNTs) and quasi-periodic eruptions (QPEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' We  summarize and conclude in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  2 SIMULATIONS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='1 Methodology  We performed simulations of partial TDEs with the SPH code phan-  tom (Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' The star was modeled as a polytrope with  a 훾 = 5/3, adiabatic equation of state, with mass 푀★ = 1푀⊙ and  radius 푅★ = 1푅⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' In each simulation the star was injected from a  distance of 12 푟푡, where 푟t = 푅★ (푀•/푀★)1/3, from the SMBH of  mass 푀• = 106푀⊙ with the center of mass on a parabolic trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  The impact parameter 훽 ≡ 푟t/푟p, where 푟p is the pericenter distance  © 2022 The Authors  2  Cufari, Nixon, & Coughlin  of the star, was varied between 훽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='4 and 훽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='8 in steps of  Δ훽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Additional details of the simulation setup can be found in  Coughlin & Nixon (2015), and the algorithms for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=', self-gravity  can be found in Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  We performed simulations with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='25 × 105, 106 particles and  8 × 106 particles, and found very little deviation in the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  All results presented here are from the 8 × 106 particle runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' For  the purposes of determining the properties of the surviving star,  we calculate averages using only those particles that have a density  greaterthan1%ofthemaximumglobal densitywithin thesimulation,  휌max;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' we define this subset of particles as the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' For example, the  location of the core is calculated as the the average position of the  subset of particles that satisfy this criterion, and the distance from the  SMBH, 푟c, is determined from the Euclidean norm of the position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  The analogous statement applies to the core velocity and speed, 푣c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  Over the range of 훽 we simulated this criterion excludes the tidally  stripped tails, and reducing the fraction to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='1% does not impact  the results because the tidal tails have a substantially lower density  than ∼ 1% of 휌max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Similarly, increasing the fraction to 10% has a  negligible effect on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' However, values > 10% can lead to  significant noise in the results because in this case there are too few  particles that contribute to the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  We run each simulation until the core energy has settled to a  constant value that we measure for our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' For disruptions with  훽 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='6, this time is ≲ 1 day after pericenter passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Simulations  with 훽 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='6 are run up to ∼ 5 days post-pericenter to ensure that the  energy has converged to a constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Running our simulations  to later times has no affect on our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='2 Results  The left panel of Figure 1 shows the specific orbital energy of the  core, calculated according to  휖c = 1  2 푣2  c − 퐺푀•  푟c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  (1)  For 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='4 ≲ 훽 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='62, the orbital energy is negative, implying that the  surviving core is bound to the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' The global minimum energy  is ∼ 2 − 3% of the binding energy of the star, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=',  휖c,min ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='025퐺푀★  푅★  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  (2)  The fact that the star becomes bound to the SMBH following its tidal  interaction is in agreement with the classical calculations of tidal  dissipation by Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' (1975);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Press & Teukolsky (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  Conversely, all disruptions with 훽 ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='62 result in a core with a net  positive energy following the interaction with the SMBH, indicating  that the star is on an unbound trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' This result agrees with those  of Manukian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Gafton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' (2015), who found that the  “kick” to the star could result in a velocity that is ∼ the escape speed  of the star for 훽 very close to the value at which complete disruption  occurs (훽 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='9 for a 5/3 polytrope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Guillochon & Ramirez-Ruiz  2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Mainetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Therefore, if a star reaches a pericenter  distance with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='62 ≲ 훽 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='9, the star is not tidally captured but is  instead tidally ejected, never to return to pericenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  With the energy-semimajor axis and energy-period relationships  for a Keplerian orbit, we find that the semimajor axis of the captured  star (for 훽 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='62, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=', for stars that are not ejected) is  푎c = −퐺푀•  2휖c  = 1  휂c  푅★  2  � 푀•  푀★  �  ,  (3)  while the orbital period is  푇c =  2휋퐺푀•  (−2휖c)3/2 =  1  휂3/2  c  � 푅★  2  �3/2  2휋  √퐺푀★  � 푀•  푀★  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  (4)  Here we defined the specific orbital energy of the captured star as 휖c =  −휂c퐺푀★/푅★, where 휂c ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='025 (from Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Using 푀•/푀★ =  106 in Equation (3), the minimum apocenter distance of the star (with  휂c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='025) is ∼ 1 pc for 푅★ = 1푅⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  The right panel of Figure 1 shows the orbital period of the captured  star as a function of 훽 from the simulations and using Equation  (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' We see that the minimum orbital period of the captured star is  ∼ 3 × 104 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' However, because of the strong dependence on 휂c in  Equation (4), the period can be much larger for only small changes  in 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  3 DISCUSSION  Some current models for PNTs and QPEs suggest that they can be  produced by stars on bound orbits about SMBHs with periods from  hours to ∼ 푓 푒푤 years, with the star being partially disrupted and  feeding an accretion flare every pericenter passage (Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' King 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Payne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' King 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Wevers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  PNTs have been suggested to have system parameters that are com-  parable to 푀• = 107푀⊙, 푅★ = 1푅⊙, 푀★ = 1푀⊙, which gives, from  Equation (4) with 휂c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='025,  푇PNT ≃ 3 × 105 yr,  (5)  which is obviously too long to explain the observed periods ≲ 1  year (Payne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Wevers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Even if one could inject  the theoretical maximum energy into the star, and thus set 휂c = 1  in Equation (4), we would obtain 푇PNT ≃ 103 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' As argued by  Cufari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' (2022), even in this overly optimistic scenario, PNTs  need an additional mechanism to bind the star sufficiently tightly to  the SMBH to produce their observed periods in situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  On the other hand, it is possible that a PNT could initially be  generated with an orbital period given by Equation (5), but the period  then decays through gravitational-wave emission (over millions of  years) to produce a period as short as ∼ 1 year by the time that we  observe it1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' However, a problem with this interpretation is that even  for the maximum value of 휂c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='025, Equation (3) for the semimajor  axis of the orbit yields an apocenter distance of ∼ 2푎c ∼ 10 pc for  푀•/푀★ = 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Thus, we would expect the star to interact with, and  be once again perturbed by, the nuclear star cluster, and it seems very  unlikely that the star will repeatedly return to the same pericenter  over multiple passages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  Additionally, while we have assumed that the star is on a parabolic  orbit, it could have a velocity at infinity that is 푣∞ ≃ 휎, with 휎 the  galactic velocity dispersion (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' From Figure 1, the  maximum amount of energy able to be dissipated through tides is  휖c,max ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='025퐺푀★  푅★  ≃ 1  2  � 푀★  푀⊙  � � 푅★  푅⊙  �−1 �  100 km s−1�2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  (6)  For a 107푀⊙ SMBH, the velocity dispersion from the 푀 −휎 relation  is 휎 ∼ 110 km s−1(Marsden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Thus, tides may not actually  be capable of dissipating the true (positive) energy of the orbit, and  1 As pointed out in Payne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' (2021), the rate of change of the orbital period  due to gravitational waves is actually too small to explain the observed value  for ASASSN-14ko, but it should be possible for gravitational waves to shrink  the orbit in general and for other systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  MNRAS 000, 1–4 (2022)  On tidal capture for PNTs and QPEs  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='10  β  ϵc [GM⊙/R⊙]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='60  5×104  1×105  5×105  1×106  5×106  1×107  β  Tc [yrs]  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Left: The energy imparted to the star as a function of the impact parameter 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Stars with 휖c < 0 are bound to the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' There is a small range of  훽 for which mass loss occurs on first pericenter passage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=', 훽 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='5) and for which 휖c < 0, the necessary condition for a repeating partial disruption to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  Right: the period of the orbit of the star for encounters that yield a captured star as a function of the impact parameter 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' The minimum period corresponds  minimum orbital energy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' the maximum binding energy), and for these parameters is > 104 yrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  hence binding the star through this mechanism may be impossible in  the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  QPEs – with periods of the order hours – are typically modeled  with a ∼ 105푀⊙ SMBH partially disrupting a white dwarf star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' For  such a system, Equation (4) with 푀★ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='6푀⊙, 푅★ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='011푅⊙  (Nauenberg 1972), 푀• = 4 × 105푀⊙, and 휂c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='025 gives  푇QPE ≃ 28 yr,  (7)  which shows that it is not possible to bind a star to a SMBH  through tidal dissipation and immediately reproduce the observed  orbital periods of QPEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' As for PNTs, it is nonetheless possible that  gravitational-wave emission shrinks the orbit2 to ∼ hours before we  detect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Because the binding energy of a white dwarf is substan-  tially larger than that of a main sequence star, the minimum apocenter  distance is correspondingly smaller (from Equation 3), and the max-  imum imparted energy through tides is substantially larger than the  energy at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Thus, without further investigation that is outside  the scope of the present work, we cannot conclusively state whether  or not an additional mechanism is required for producing the ob-  served orbital periods in QPEs (under the paradigm that they are  powered by repeatedly partially disrupted white dwarfs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  On the other hand, the tidal breakup of a binary star system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=',  Hills capture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Hills 1975) can bind the star with a substantially shorter  period (compared to just tidal dissipation) if the binary is sufficiently  tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' As demonstrated by Cufari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' (2022), this is a plausible  explanation for the origin of the ∼ 114 day period of ASASSN-  14ko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' An additional argument against the gravitational-wave inspiral  scenario for explaining ASASSN-14ko is that the mass lost by the  star must be ∼ 1% of the mass of the star to power the emission  (Cufari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2022), and hence it cannot have survived many (≳ 100  s) interactions prior to the ∼ 10 that have been observed since the  initial detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Similarly, if the white dwarf binary separation is not  much larger than the radius of the white dwarf itself, then Equation  (5) from Cufari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' (2022) with 푎★ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='04푅⊙, 푀• = 4 × 105푀⊙,  and 푀★ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='6푀⊙ yields an orbital period of ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='3 hours for the  period of the bound star following a Hills-capture event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' As also  suggested in Cufari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' (2022), it therefore seems plausible that the  2 The orbital period may also be reduced through the interaction with a  pre-existing AGN disc (Syer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Cufari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Lu & Quataert  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  periods of QPEs can be generated with a dynamical exchange process  without the need for additional dissipation through other means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  4 SUMMARY AND CONCLUSIONS  We presented SPH simulations of the interaction between a star and  an SMBH to determine the maximum degree by which a star can  be bound to an SMBH through tidal dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Two competing  mechanisms prevent the star from becoming arbitrarily bound to the  SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' As the distance of closest approach between the star and  SMBH shrinks, energy from the star’s orbit is expended in exciting  dynamical tides in the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' However, once the star reaches a dis-  tance of closest approach comparable to its tidal radius, the star is  kicked to positive energies as a result of asymmetry in the tidal tails  that are liberated from the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' The competition between these two  physical effects results in a minimum possible orbital energy of the  star following the tidal encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' For the parameters we simulated  here, the location of this minimum is at 훽 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='55 (pericenter distance  of 푟t/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='55 with 푟t the usual tidal radius) and the binding energy of  the orbit is ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='5% of the star’s binding energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' see Equation (2)  specifically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' This minimum energy is significantly smaller than the  theoretical maximum, being the entirety of the stellar binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  In order to produce a repeating partial tidal disruption via a tidal  dissipation mechanism, the energy kick imparted to the star must be  negative, otherwise the star will be ejected on a hyperbolic trajec-  tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Hence, it would appear that there is a relatively small region of  parameter space within which the star is only partially destroyed, not  ejected, and survives for many (≳ 10) pericenter passages, specif-  ically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='4 ≲ 훽 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='5 for the type of star considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Initially  non-rotating stars in this range of 훽 will be rotating at a nontrivial  fraction of breakup following the initial interaction, which will move  the effective tidal radius out (Golightly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' 2019), but provided that  the pericenter (effectively unaltered because of the small ratio of the  maximal angular momentum of the star to the angular momentum of  the orbit itself) is still within this tidal radius, the star may transfer a  small amount of mass during each additional pericenter passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' In  this manner, a star may undergo many cycles of partial disruptions  before being destroyed or ejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  In our simulations we modeled the star as a 5/3 polytrope, which  is most applicable to low-mass main sequence stars and and low-  mass white dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' By number, most stars are thought to fall into  MNRAS 000, 1–4 (2022)  4  Cufari, Nixon, & Coughlin  this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' However, more massive (radiative) stars are consider-  ably more centrally concentrated than predicted by a 5/3-polytropic  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Here we have shown that the minimum energy for the captured  star – modeled as a 5/3 polytrope – occurs at 훽 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' This result will  depend somewhat on the type of star being considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' For example,  Figure 3 of Manukian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' (2013) shows that it occurs for 훽 < 1  when the star is modeled as a 4/3-polytrope and that there is very  little dependence of the result on the black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Faber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  (2005) considered the tidal capture of a planet by a star in which the  mass ratio was 푞 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='001, and found a minimum binding energy of  ∼ 14% of the binding energy of the planet at 훽 = 10/19 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='523 (see  their Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Kremer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' (2022) also recently considered black  hole-star systems with mass ratios closer to unity, and found a similar  effect to the one described here if the mass ratio was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='02 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='05  if the star was modeled as a 훾 = 5/3 polytrope, but that the star  was able to go from bound to completely disrupted – without being  ejected – as the mass ratio increased beyond 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='05 and the star was  modeled with the Eddington standard model (see their Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' We  defer an analysis of the minimum orbital energy – and the 훽 at which  the minimum energy occurs – as a function of the mass ratio and the  type of star to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  In the context of extreme mass-ratio inspirals, Zalamea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  (2010) demonstrated that a white dwarf (or other compact object)  completes thousands of large-eccentricity orbits before reaching the  direct capture radius of the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' However, their model accounts  only for the orbital decay due to gravitational wave emission and  omits tidal dissipation and mass loss asymmetry effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Likewise,  our simulations omit the effects of orbital decay due to gravitational  wave emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' The pericenter distance of our simulated disruptions  is > 50 푟g, so orbital decay due to general relativistic effects (at least  on the first pericenter passage) is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' For more compact stars  with smaller tidal radii nearer the event horizon, orbital decay due to  general relativistic effects will be more significant over fewer orbital  periods (though the change in the pericenter will still be extremely  small, as all of the dissipation occurs near pericenter for these high-  eccentricity systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' thus the tidal interaction itself may be relatively  unaltered, aside from the stronger tidal field of the SMBH due to rel-  ativistic gravity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' Future work on partial TDEs nearer the horizon  of the SMBH should incorporate the change in orbital energy due  to tidal interaction and mass loss asymmetry alongside gravitational  wave emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  Finally, here we focused on orbits that produce partial TDEs in  the traditional sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=', the tidal force is not sufficiently strong  to destroy the star completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' However, Nixon & Coughlin (2022)  found that at very high 훽 (in their case 훽 = 16), the compression  experienced by the star near pericenter could revive self-gravity to  the point that a core reformed with a binding energy (to the SMBH)  that was much larger than the value predicted by Equation (2) (though  we caution that while the mass contained in the core was converged,  the orbital period – and thus the binding energy – was resolution-  dependent in their simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' While encounters with high-훽 are  rare3, it may be possible for tidal capture in this considerably more  exotic scenario to produce shorter-period orbits than through the  traditional means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  3 For example, Equation 16 of Coughlin & Nixon (2022) shows that the  fraction of TDEs with 훽 > 10 for a 106푀⊙ Schwarzschild SMBH – including  general relativistic effects – is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='0046;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' note that this is a factor of ∼ 4 smaller  than the value derived by ignoring general relativistic effects, given by their  Equation 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  DATA AVAILABILITY STATEMENT  Code to reproduce the results in this paper is available upon reason-  able request to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
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+page_content=' Powe Junior  Faculty Enhancement Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
+page_content=' This work was performed using the  DiRAC Data Intensive service at Leicester, operated by the Uni-  versity of Leicester IT Services, which forms part of the STFC  DiRAC HPC Facility (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
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+page_content=' The equipment was funded  by BEIS capital funding via STFC capital grants ST/K000373/1 and  ST/R002363/1 and STFC DiRAC Operations grant ST/R001014/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfWfwG/content/2301.01300v1.pdf'}
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+Quantile Autoregression-Based Non-causality Testing ∗
+Weifeng Jin†
+January 10, 2023
+click here for the latest version
+Abstract
+Non-causal processes have been drawing attention recently in Macroeconomics and
+Finance for their ability to display nonlinear behaviors such as asymmetric dynamics,
+clustering volatility, and local explosiveness. In this paper, we investigate the statis-
+tical properties of empirical conditional quantiles of non-causal processes. Specifically,
+we show that the quantile autoregression (QAR) estimates for non-causal processes do
+not remain constant across different quantiles in contrast to their causal counterparts.
+Furthermore, we demonstrate that non-causal autoregressive processes admit nonlin-
+ear representations for conditional quantiles given past observations. Exploiting these
+properties, we propose three novel testing strategies of non-causality for non-Gaussian
+processes within the QAR framework. The tests are constructed either by verifying the
+constancy of the slope coefficients or by applying a misspecification test of the linear
+QAR model over different quantiles of the process.
+Some numerical experiments are
+included to examine the finite sample performance of the testing strategies, where we
+compare different specification tests for dynamic quantiles with the Kolmogorov-Smirnov
+constancy test. The new methodology is applied to some time series from financial mar-
+kets to investigate the presence of speculative bubbles. The extension of the approach
+based on the specification tests to AR processes driven by innovations with heteroskedas-
+ticity is studied through simulations. The performance of QAR estimates of non-causal
+processes at extreme quantiles is also explored.
+Keywords: non-causality, quantile autoregression, specification test, KS constancy
+test, non-linearity.
+JEL Classification: C01, C12, C22.
+1
+Introduction
+A stationary autoregressive moving-average (ARMA) process is defined as causal concerning
+the specified innovation sequence if all roots of the autoregressive polynomials are outside
+∗I would like to express my gratitude to my supervisor, Carlos Velasco, who guided me throughout this
+research project. I also want to thank Juan Carlos Escanciano, Miguel Delgado, Jesus Gonzalo, Juan Jose
+Dolado, Nazarii Salish, and Alain Hecq for their constructive comments, and also seminar and workshop
+participants at Universidad Carlos III de Madrid, Time Series Workshop in Zaragoza and CEBA seminar
+2022.
+†Weifeng Jin: Universidad Carlos III de Madrid. jweifeng@eco.uc3m.es
+1
+arXiv:2301.02937v1  [econ.EM]  7 Jan 2023
+
+of the unit circle, so it can be represented by an infinite sum of past innovations. However,
+non-causal autoregressive (AR) processes, due to their ability to display various non-linear
+dynamics, have been drawing increasing attention in the econometrics literature during the
+last two decades. The same concept has been fully exploited as a non-minimum phase in
+the stochastic system with applications to natural sciences. Unlike the causal AR process in
+the classical time series context, mixed causal and non-causal AR processes do not impose
+presumptions on the location of the lag polynomial roots except for the exclusion of the unit
+root, which allows these stationary processes to be dependent on past and future innovations
+at the same time. Breidt et al. (2001) showed that non-causal AR processes can capture styl-
+ized facts like clustering volatility in financial data, which usually is associated with GARCH
+models. The same argument is made in the paper by Lanne et al. (2013), where they derived
+a closed-form expression for the correlation of squared values in levels in the ARMA(1, 1)
+case. Fries and Zakoian (2019); Gouri´eroux and Zako¨ıan (2017) and Cavaliere et al. (2020)
+proposed to model speculative bubbles with non-causal AR or mixed causal and non-causal
+AR processes generated by heavy-tailed innovations because they can display local explosive
+behavior. Regarding forecasting, Lanne et al. (2012) and Hecq and Voisin (2021) argued that
+there is an accuracy gain in forecasting performance after introducing non-causality into the
+modeling procedure. Moreover, Lanne and Luoto (2013) pointed out that non-causal AR
+processes can be an alternative to non-invertible processes for forward-looking behavior, see
+Alessi et al. (2011) for a comprehensive survey of empirical applications of non-invertible
+(non-fundamental) processes to Macroeconomics and Finance.
+The emerging applications of non-causal AR processes promote an interest in testing non-
+causality in practice, given that autocorrelation functions fail to discriminate non-causal
+processes from their causal counterparts. Needless to say, the non-causality check can be
+naturally achieved through testing classical linear hypotheses under robust estimation tech-
+niques applicable to possibly non-causal processes. These estimation methods have been
+developed by Breid et al. (1991) and Lii and Rosenblatt (1992, 1996) through non-Gaussian
+maximum likelihood schemes or minimum distance estimation exploiting information con-
+tained in higher order moments/cumulants or characteristic/cumulative distribution func-
+tions of residuals, see Velasco and Lobato (2018), Velasco (2022), and Jin (2021). With this
+approach, the test procedure is confined to the assumptions required for the corresponding
+estimates, which can be somehow stringent. Apart from that, a factorization of the coef-
+ficients is necessary for disentangling the roots, which becomes rather complicated as the
+order of the AR process increases. Therefore, a testing strategy before the estimation, which
+can potentially work as a model selection, needs investigation. Besides, the testing can serve
+as a detection tool for the existence of speculative bubbles in empirical time series for the
+ability of non-causal processes with heavy tails to exhibit local explosive behavior.
+However, except for the robust estimation techniques introduced before, little has been done
+on the theory of testing non-causality in AR processes.
+Nevertheless, all the estimation
+techniques above deliver the same message that additional information beyond second-order
+moments is required to identify non-causality.
+Following this message, we propose some
+testing strategies for the non-causality of time series within the quantile autoregressions
+framework (QAR hereafter) (Koenker and Xiao (2006)), which allows us to make use of the
+2
+
+complete distribution to measure dependence. Similarly, Hecq and Sun (2021) applied QAR
+to the target process and entertained the sum of rescaled absolute residuals as an information
+criterion to select between purely causal and non-causal models. The approach of Hecq and
+Sun (2021) obtains the proposed test statistic by running QAR in direct and reserved time,
+respectively. This approach can bring up ambiguity in the model selection when the complex-
+ity of the causality structure escalates as the order of the AR model increases. Thereby, this
+strategy may yield misleading results when the time series is mixed causal and non-causal.
+In this paper, we propose three testing procedures based on the well-developed inference for
+QAR estimates exploiting the non-linear characteristics of non-causal processes. Recall that
+the coefficients in the conditional quantile regression for the location model1 are quantile-
+invariant except for the intercept. When the process is causal, the independence of current
+innovation with past observations contributes to the invariant property of coefficients in the
+QAR model across different quantiles. However, under non-causality the true conditional
+quantile of a non-Gaussian AR process exhibits non-linearity in the past information. In-
+duced from this, the best linear approximation to the true conditional quantile is expected to
+show varying coefficients across distinct quantiles. Using this feature, we introduce our first
+strategy for the objective of testing non-causality by carrying out the constancy test over
+the entire quantile interval. Its easy-to-implement attribute makes it a perfect candidate for
+a preliminary check of non-causality. The other approach to detect non-causality is achieved
+through a specification test of the linear functional form for the conditional quantile. With
+this specification-based approach, no distributional knowledge of the innovations beforehand
+is required, nor does the correct specification of the conditional quantile need to be spelled
+out.
+The testing strategies introduced in this paper fill a gap in the theory of testing non-causality.
+Apart from that, they share an appealing property of retaining power when the AR process
+is higher-order with a mixture of causal and non-causal representation. Like the significant
+advantage of quantile regression over conditional mean regression, our approach is robust to
+outliers, making it suitable for heavy-tailed processes commonly employed in finance. Our
+specification-based approach can also be tentatively extended to an AR process with condi-
+tional heteroskedasticity. The tests achieve considerable power in relatively small samples.
+In addition, some Monte Carlo simulation results suggest that the estimates in linear quan-
+tile models approach the true parameters for non-causal processes as the quantile estimand
+approaches to either extremum (0 or 1), which can be a potential viewpoint to investigate
+the nonlinear features of non-causal processes in the future research.
+The rest of the paper is organized as follows. The second section introduces mixed causal and
+non-causal Autoregressions and some of their statistical properties. Section 3 investigates
+the non-causality testing within the QAR framework. Section 4 discusses the finite sample
+performance of our proposed testing procedures through Monte Carlo Simulations. Section
+5 illustrates the tests using financial data with the possible existence of speculative bubbles.
+Section 6 closes the paper with conclusions and some extensions.
+1The location model is any model of the form Y = µ(X)+σ ·u, where u is independent of X. Throughout
+this paper, we refer to Yt = µ (Yt−1, Yt−2, . . . )+ut, where µ(·) is a linear functional form, and ut is a sequence
+of iid innovations, when it comes to the location model.
+3
+
+2
+Mixed Causal and Non-causal Autoregressions
+In the context of classical time series analysis, it is customary to restrict attention to the
+causal representation of autoregressive processes while modeling stationary univariate time
+series. The reason is mentioned in Brockwell and Davis (2009). Every non-causal autoregres-
+sive process is a stationary solution to a future-independent autoregressive process with explo-
+sive roots. It provides the same second-order structure as its causal counterpart. However,
+to feature higher-order dynamics, a general framework of autoregressive processes named
+mixed causal and non-causal autoregressions (MAR(r, s) hereafter) is proposed where tem-
+poral dependence in both past and future is introduced in the processes, defined by
+φ(L)ψ(L−1)Yt = ut
+(2.1)
+where φ(L) = 1 − φ1L − φ2L2 − · · · − φrLr and ψ(L−1) = 1 − ψ1L−1 − ψ2L−2 − · · · − ψsL−s
+are polynomials with backward and forward operators, respectively. {ut}t∈Z is a sequence of
+independent identically distributed (iid) innovations with zero mean. p = r + s is the total
+order encompassing both causal and non-causal polynomials. An equivalent expression of
+equation (2.1) in moving average can be given by
+Yt = φ(L)−1ψ(L−1)−1ut =
+∞
+�
+j=−∞
+ρjut−j,
+(2.2)
+where Yt may depend on both future and past innovations. The stationarity of Yt is assured
+by the absolute summability of ρj, �∞
+j=−∞ |ρj| < ∞ and E |ut|1+δ < ∞ for δ > 0. The former
+condition is guaranteed once the roots of both polynomials φ(z) and ψ(z) are confined to
+locate outside the unit circle.
+When ψ(z) = 1 for all z, the process is reduced to a purely causal process MAR(r, 0)
+like in conventional studies. While if φ(z) = 1 for all z, the process becomes purely non-
+causal. Below, we attach several examples to illustrate the statistical properties of a general
+autoregressive process MAR(r, s).
+Example 2.1 (Second-order Property). Define a purely non-causal MAR(0, 1) sequence
+(1 − ψL−1)Yt = ut starting from an iid sequence of innovations ut of zero-mean and finite
+variance σ2. The autocovariance function of Yt is provided by
+Cov (Yt+h, Yt) = Cov
+�
+�
+∞
+�
+j=0
+ψjut+h+j,
+∞
+�
+j=0
+ψjut+j
+�
+�
+=
+∞
+�
+j=0
+Cov
+�
+ψjut+h+j, ψj+hut+h+j
+�
+=
+ψh
+1 − ψ2 σ2 for h = 0, 1, 2, . . .
+It is worth noting, after some simple calculation, that {Yt}t∈Z shares the same autocovariance
+function as the MAR(1, 0) sequence { ˜Yt}t∈Z defined by (1 − ψL) ˜Yt = ut, which is its causal
+counterpart. A general conclusion can be drawn for MAR(r, s) through the autocovariance
+4
+
+generating function (ACGF). The ACGF of a stationary autoregression is given by
+G(L) =
+1
+|φ(L)ψ(L−1)|2 σ2 =
+1
+φ(L)φ(L−1)ψ(L−1)ψ(L)σ2
+=
+1
+φ(L)ψ(L)φ(L−1)ψ(L−1)σ2
+=
+1
+|φ(L)ψ(L)|2 σ2,
+from where it is clear that MAR(r, s) takes the identical form of ACGF with MAR(r′, s′) as
+long as the total order r + s = r′ + s′ is satisfied and the roots to all polynomials match.
+This second-order property explains the failure to distinguish non-causal processes from
+causal processes based on the ACF. Moreover, it implies that non-Gaussianity of innovations
+is required for identifying non-causal processes since second-order properties are sufficient to
+characterize a Gaussian probabilistic structure but not to other distributions.
+Example 2.2 (Local Explosiveness). Define a MAR(1, 1) process by
+(1 − φL)(1 − ψL−1)Yt = ut, where ut ∼ Lognormal(0, 2) − exp(2)
+A MAR(r, s) process with heavy-tailed innovations can generate multiple phases of local
+explosiveness, which can be employed to model speculative bubbles. A detailed investigation
+of probabilistic properties of a MAR process driven by α-stable non-Gaussian innovations
+is provided by Fries and Zakoian (2019), where they show the properties of the marginal
+and conditional distributions of a stable MAR(r, s).
+But in a general situation, there is
+no closed-form solution to either the marginal or the conditional distribution of a non-
+causal AR process. Here we illustrate the potential applications of MAR(r, s) processes in
+modeling speculative bubbles with some simulated trajectories. In this example, we plot
+four distinct scenarios by varying the parameters of both causal and non-causal components
+of the MAR(1, 1) process with log-normal distributed innovations2. Generally, with non-
+causality, the processes can mimic bubbles by repetitive phases of upward trends followed
+by a sharp drop; see the lower panel in Figure 2.1. The upper panel in the same figure
+indicates more complicated dynamics can be generated by incorporating more causal/non-
+causal components in the data-generating process regarding the number and magnitude of
+bubbles.
+Example 2.3 (Conditional heteroskedasticity). Here we exemplify the applicability of MAR(r, s)
+in characterizing clustering volatility as an alternative to ARCH or other stochastic volatility
+models with a simple MAR(1, 1) process. This argument is originally made by Breidt et al.
+(2001) in an empirical application to New Zealand/US exchange rate. Lanne et al. (2013)
+elaborate on it by deriving explicitly the expression of the autocorrelation of Y 2
+t in an all-pass
+2A log-normal distribution is a heavy-tailed continuous distribution defined on the positive domain, whose
+density function is characterized by the location parameter µ and scale parameter σ. The mean and variance
+are represented by exp(µ + σ2/2) and �
+exp(σ2) − 1�
+exp �
+2µ + σ2�
+, respectively. In this example, we employ
+centered log-normal distribution to be compatible with our setup of mean zero.
+5
+
+Figure 2.1: Trajectories of MAR(1, 1) processes with different parameters (φ, ψ), T=500: the upper
+panel exhibits two MAR(1, 1) processes with different parameters on the causal/non-causal compo-
+nents; the lower panel exhibits a purely non-causal AR(1) process (left) and a purely causal AR(1)
+process (right) when one of the parameters degenerates to zero.
+model of order 1. However, no formal justification for the higher-order dependence structure
+has been made on general non-causal processes. Nevertheless, we present the following ex-
+ample that a non-causal process can exhibit higher-order dependence.
+Continued with data generating process MAR(1, 1), it can be reexpressed by an AR(2) with
+the
+1−ψL
+1−ψL−1 filter on the innovations.
+(1 − φL)(1 − ψL−1)Yt = ut
+ut ∼ IID(0, σ2)
+⇐⇒(1 − φL)(1 − ψL)Yt = ˜ut
+with ˜ut =
+1 − ψL
+1 − ψL−1 ut =
+∞
+�
+j=−∞
+ρjut+j
+with ρj =
+�
+�
+�
+�
+�
+�
+�
+0
+if j < −1
+−ψ
+if j = −1
+ψj − ψj+2
+if j ≥ 0.
+The
+1−ψL
+1−ψL−1 filter introduces higher-order dependence to an iid innovation sequence3
+3Actually, ˜ut is an all-pass time series model, where all roots of the autoregressive polynomial are reciprocals
+of roots in the moving average polynomial and vice versa.
+6
+
+MAR(1,1) with Φ=0.4,=0.8
+MAR(1,1) with @=0.8,b=0.4
+4000
+600
+3500
+500
+3000
+400
+2500
+2000
+300
+1500
+200
+1000
+100
+500
+-500
+-100
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+MAR(0,1) with Φ=0, =0.8
+MAR(1,0) with Φ=0.8, =0
+250
+600
+500
+200
+400
+150
+300
+100
+200
+50
+100
+50
+-100
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500preserving its uncorrelatedness. By simple algebra, we can obtain the formula of the auto-
+correlation function of ˜u2
+t ,
+Corr
+�
+˜u2
+t , ˜u2
+t+h
+�
+=
+κ4(ut)
+��∞
+j=−∞ ρ2
+jρ2
+j+h
+�
++ 2σ4 ��∞
+j=−∞ ρjρj+h
+�2
+κ4(ut)
+��∞
+j=−∞ ρ4
+j
+�
++ 2σ4
+��∞
+j=−∞ ρ2
+j
+�2
+which, in general, does not vanish4. This property demonstrates the capability of non-causal
+processes to exhibit volatility clustering, which is commonly observed in financial data. The
+higher-order dependence analysis of ˜ut is relegated to Appendix 7.1, indicating the possibility
+of accommodating higher-order nonlinear dynamics with a general MAR(r, s) model.
+As shown in the preceding examples, MAR(r, s) models can display various nonlinear char-
+acteristics with a linear process generation scheme. This deviation from ”linearity” can be
+employed as a crucial feature to detect non-causality in the linear time series. A fundamental
+result is formalized by Rosenblatt (2000) on the best one-step predictor in the mean square
+for a general AR process, where he demonstrates that the conditional expectation must be
+nonlinear in the past if there is non-causality involved in the non-Gaussian AR processes
+with finite variance. This statement provides us with the critical theoretical result where our
+tests for non-causality are grounded, which will be elaborated on in the next section. The
+extension to a general VARMA process framework has been developed by Chen et al. (2017)
+and applied to a test for non-invertibility. Afterward, Fries and Zakoian (2019) consider the
+case when the MAR(1, s) process is driven by symmetric α-stable innovations with infinite
+variance. They surprisingly find that the conditional expectation can be explicitly expressed
+by a linear function of the past information, in contrast to a MAR(r, s) model with finite
+variance. However, no study has yet been done on the distributional characterization of a
+MAR(r, s) process due to no closed-form solution. Following a simulation-based approach,
+we perform a preliminary analysis of the conditional density function of a process given the
+past observations, see Appendix 7.1. In short, in the presence of non-causality, the shape
+of the conditional distribution of the process, say f(Yt|Yt−1 = y), is y-dependent. Still, the
+dependence pattern is hard to characterize since it varies across different distributions.
+3
+QAR-based Non-causality Tests
+3.1
+Benchmark model: MAR(r, s)
+In this section, we formalize the test procedures for non-causality. The null hypothesis of
+interest is Yt being a causal process, i.e.
+H0 : ψ1 = · · · = ψs = 0 in (2.1)
+(3.1)
+against the alternative hypothesis denoted by HA, which is {Yt} is non-causal,
+HA : ψk ̸= 0 for some k ∈ {1, 2, . . . , s} in (2.1).
+(3.2)
+4κ4(X) is the fourth cumulant of the random variable X, defined by κ4(X) = E �
+(X − E(X))4�
+−
+3 E2 �
+E (X − E(X))2�
+.
+7
+
+A primitive strategy is to take it as a joint significance test for the coefficients of leads
+{Yt+j}j=1,2,...,s.
+This approach would call for the identification of models and robust in-
+ference for the estimates under both hypotheses. In this paper, we propose a test for H0
+employing the linearity property of {Yt} based on the QAR, which is easy to implement
+empirically and does not require consistent estimates for general autoregressive progresses.
+Denoting the information set generated by the observations up to period t by It = σ (Yt, Yt−1, ...)
+and the τ−th quantile of Yt conditional on the past by QYt (τ|It−1), the following result on
+the linearity of {Yt} justifies our approach.
+Assumption 1. Let {ut}t∈Z be a non-Gaussian iid sequence with (k + 1)th order moment
+finite and (k + 1)th cumulant nonzero for some k ≥ 2.
+Theorem 3.1. Under Assumption 1, a stationary MAR(r, s) process {Yt} has a non-degenerated
+non-causal component, i.e., s ̸= 0 if and only if there exists QYt (τ|It−1) nonlinear in
+{Yt−j}j≥1 for at least one τ ∈ (0, 1).
+As discussed in Example 2.1, there is no meaning to discuss non-causality in a Gaussian
+structure as there is always an equivalent causal representation of a Gaussian AR process for
+its non-causal counterpart. The finiteness of the moment condition and nonzero cumulants
+are also required in Rosenblatt (2000). Theorem 3.1 is deduced from the nonlinearity of the
+best predictor of Yt in the mean square criterion for non-causal processes. The nonlinearity
+in the conditional mean implies dependence in the conditional distribution beyond the linear
+correlation. Note that the theorem does not point out the quantile(s) where this nonlinear
+relationship occurs, nor the manners in which this nonlinearity is expressed. Another remark
+is that the information set considered in the theorem can be replaced by σ-field generated
+by Yt−1 up to Yt−p due to the Markovian property of MAR(r, s), which avoids the infinite-
+dimensional issue arising from It−1. The total number of the lags and leads of Yt included
+to explain the conditional quantile of MAR(r, s) processes, p, is determined by the partial
+autocorrelation function (PACF) of Yt.
+This theorem prompts us to adopt QAR as a medium to detect non-causality. QAR is a
+comprehensive analysis tool in the time series context, providing robust statistical analyses
+against outliers in the measurement of the response variable, which has proven to be rather
+prevailing in recent decades. Given a MAR(r, s) process of order p defined by (2.1), if the
+non-causal component degenerates to 1, i.e., s = 0, r = p, the conditional quantile of Yt can
+be expressed by
+QYt (τ | It−1) =Qut (τ) + φ1Yt−1 + φ2Yt−2 + · · · + φpYt−p
+=θ0(τ) + θ1(τ)Yt−1 + θ2(τ)Yt−2 + · · · + θp(τ)Yt−p
+=X′
+tθ(τ)
+∀τ ∈ (0, 1),
+(3.3)
+where Qut(τ) denotes the τ-quantile of innovations ut, and φj’s are the coefficients of corre-
+sponding Yt−j in the polynomial expansion of (2.1). Therefore, after imposing pure causality,
+the conditional quantile of Yt can be fully characterized by a linear function of past observa-
+tions, X′
+tθ(τ) with Xt = (1, Yt−1, Yt−2, . . . , Yt−p)
+′ and θ(τ) = (θ0(τ), . . . , θp(τ))
+′. The QAR
+estimates of coefficients θ(τ) in this linear quantile model can be obtained by minimizing
+8
+
+the following problem,
+ˆθ(τ) = argmin
+θ∈Rp+1
+T
+�
+t=1
+ρτ(Yt − X′
+tθ),
+(3.4)
+with the check function ρτ(u) = u (τ − I(u < 0)). The asymptotic properties of linear QAR
+estimates were first established by Koenker and Xiao (2006). A brief review of QAR esti-
+mates (3.4) can be found in Appendix 7.2. However, if Yt has a non-degenerated non-causal
+component, the linear dynamic model (3.3) for its conditional quantile is misspecified for at
+least one τ ∈ (0, 1), following Theorem 3.1.
+Within the linear QAR framework, Hecq and Sun (2021) consider a statistic aggregating
+the information of the residuals over quantiles, which they employ as a model selection cri-
+terion between purely causal AR models and purely non-causal AR models.
+Given that
+the calculation of residuals is done either by running QAR with direct or reversed time,
+this methodology may provide misleading conclusions regarding MAR(r, s) in presence of
+causality and non-causality at the same time. By contrast, our approaches address more
+the correctness of the linear specification of the conditional quantile through the QAR. For
+non-causal autoregressive processes, no closed-form of the nonlinear conditional quantile of
+Yt is required. Consequently, our approach is robust to the general MAR(r, s) setting. Before
+we carry out the tests for non-causality, the following assumptions are imposed in the QAR
+framework.
+Assumption 2. The distribution function of innovations ut, F(u) admits a continuous
+density function f(u) away from zero on the domain U = {u : 0 < F(u) < 1}.
+Assumption 3. Denote the family of conditional distribution {P (Yt < y|Xt = x) , y ∈ R, x ∈
+Rr+s} as Fx(y) and its Lebesgue density as fx(y), that is uniformly bounded on the space of
+y × x ⊆ R × Rr+s and uniformly continuous.
+Corollary 3.1.1. Under Assumptions 1-3 and null hypothesis H0, the coefficients except for
+the intercept of the conditional quantile are constant across different quantiles in (0, 1).
+Corollary 3.1.1 is a consequence of the independence between ut and Yt−j for j =
+1, 2, . . . , p when the MAR(r, s) is purely causal. As shown in the equation (3.3), the slope
+coefficients {θj(τ)}j=1,...,p are constant over τ ∈ (0, 1) and uniquely determined by the ex-
+pansion of autoregressive polynomials of Yt. Instead, the performance of the QAR estimates
+is more complicated in the mixed causal and non-causal autoregressive processes since the
+linear model is under misspecification. Angrist et al. (2006) demonstrated in their paper that
+quantile regression is essentially an approximation to the true conditional quantile function
+in a weighted mean squared criterion, with weights associated with the densities of Yt ranging
+from the linear approximation (3.3) to the true conditional quantile QYt(τ|Yt−1, . . . , Yt−p).
+Their statement presents a rough idea of how well the linear function fits the true conditional
+quantile.
+Constancy Test
+Our first approach for detecting non-causality comes along with the
+constancy test of QAR coefficients over the entire quantile range. As stated in Corollary
+9
+
+3.1.1, if the process is causal, the constancy should hold for all θj(τ) for all j = 1, 2, ..., p and
+τ ∈ (0, 1). Whereas for a non-causal process, the estimated coefficients of Yt−j in the quantile
+regression may vary across different quantiles much likely for the following intuitions: i) non-
+causal processes display highly nonlinear dynamics, one of which is asymmetric dynamics;
+ii) linear quantile model is misspecified; iii) the conditional distribution of Yt varies both
+in the location and shape at different values of past observations. For instance, Figure 3.1
+depicts the dynamic performance of QAR(1) estimates of the slope parameter across the
+quantiles for a non-Gaussian autoregressive process, including causal and non-causal cases in
+different colors. The QAR(1) estimates in the non-causal processes (solid blue lines) exhibit
+a trendy pattern over the quantile domain. However, since a general solution for the true
+conditional quantile function of Yt is infeasible, we do not attempt to provide a detailed
+characterization of the varying coefficient property in the non-causal situation. The test for
+non-causality based on the constancy test can only be used to check the necessary condition
+of AR processes being causal. Nevertheless, the accessibility and straightforwardness of this
+method make it a touchstone for non-causality testing in practice.
+(a) QAR(1) estimates of the slope of Yt−1
+over (0,1): exponential distribution
+(b) QAR(1) estimates of the slope of Yt−1
+over (0,1): chi-square distribution
+Figure 3.1: QAR(1) estimates of a pair of AR(1) processes: one is causal, and the other is non-
+causal. The left part of the figure is applied to the processes generated by exponential innovations,
+and the right part is to the processes generated by chi-square innovations. The true parameter is
+0.6(1/0.6) for the causal(non-causal) case.
+To implement the constancy test of coefficients under the QAR framework, we consider the
+approach developed by Koenker and Xiao (2006), where the hypothesis is formulated in the
+manner analog to the classical linear hypothesis:
+H1
+0 : Rθ(τ) = φ with R =
+�
+0p×1
+... Ip
+�
+for all τ ∈ (0, 1)
+with the unknown τ-invariant parameter vector φ = (φ1, φ2, . . . , φp)′ which needs to be
+estimated5.
+Naturally, the naive test for this hypothesis is constructed on the quantile
+process
+VT (τ) =
+√
+T
+�
+RˆΣ−1
+1 ˆΣ0 ˆΣ−1
+1 R′�−1/2 �
+Rˆθ(τ) − ˆφ
+�
+5For purely causal processes, the constant vector consists of the parameters in the autoregressive polyno-
+mials, i.e., φ1, φ2, . . . , φp in (3.3)
+10
+
+5
+causal
+coefficients
+noncausal
+5
+0
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.05
+coefficients
+causal
+noncausal
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Tand the Kolmogorov-Smirnov (KS) type of test statistic is adopted for the interest of testing
+a compact set of quantiles
+KSVT =
+sup
+τ∈T ⊂(0,1)
+VT (τ) , where T is a compact interval,
+where ˆθ(τ) is the linear QAR estimate, and ˆΣ1, ˆΣ0 are the corresponding estimates of the
+asymptotic variance, see appendix 7.4. ˆφ is a
+√
+T consistent estimator of φ and a simple
+choice is the QAR estimator ˆθ(τ ∗) at any τ ∗ ∈ T .6 A closed interval [ϵ1, 1 − ϵ2] with trivial
+numbers ϵ1, ϵ2 is proposed for T to avoid missing much information from the entire quantile
+interval (0, 1).
+Under the hypothesis of constancy H1
+0,
+VT (τ) =⇒ Bp(τ) − f
+�
+F −1(τ)
+� �
+RΣ−1
+0 R′�−1/2 Z
+where Bp(τ) is a p-dimensional standard Brownian bridge and Z = plimT→∞
+√
+T
+�ˆφ − φ
+�
+is a drift brought up by the estimation of the nuisance parameter φ. To annihilate this
+non-trivial effect, a martingale transformation K was introduced into VT (τ) to retrieve the
+distribution-free merit of the KS test. Denote the derivative of the density function by ˙f
+and define
+g(x) =
+�
+1,
+� ˙f
+�
+F −1(x)
+�
+/f
+�
+F −1(x)
+���′ and
+C(z) =
+� 1
+z
+g(x)g(x)′dx,
+and the martingale transformation on the process VT (τ) is constructed as follows
+˜VT (τ) =KVT (τ)
+=VT (τ) −
+� τ
+0
+�
+g′
+T (x)C−1
+T (x)
+� 1
+x
+g(s)dVT (s)
+�
+dx,
+where gT (x) and CT (x) are uniformly consistent estimators of g(x) and C(x) in the considered
+domain, respectively. The proposed KS-type norm on the transformed process becomes
+KS ˜VT = sup
+τ∈T
+��� ˜VT (τ)
+��� .
+Corollary 3.1.2 (Constancy test for non-causality). Under Assumptions 1-3 and the causal-
+ity hypothesis H0,
+˜VT (τ) ⇒ W p(τ)
+KS ˜VT ⇒ sup
+τ∈T
+∥W p(τ)∥ ,
+where W p(τ) represents a p-dimensional standard Brownian motion.
+The related discussion on the estimation of density and score functions is given in Koenker
+6Another appropriate choice is the estimator from the ordinary least square of φj, j = 1, 2, . . . , p in (2.1).
+11
+
+and Xiao (2002), providing suggestions on the choice of bandwidth in detail. In the command
+KhmaladzeTest implemented in R studio, Hall/Sheather bandwidth (Hall and Sheather
+(1988)) for sparsity estimation is set as default. The critical values are obtained through
+approximating W p(τ) by a Gaussian random walk, and the corresponding values at different
+significance levels can be found in tables in the Appendix of Koenker and Xiao (2002). One
+remark on the quantile interval in the proposition, typically a symmetric interval [ϵ, 1 − ϵ]
+trimmed by a small number ϵ close to 0 is considered for simplicity in practice. Monte Carlo
+experiments have evidenced that an appropriate trimming in the entire quantile interval al-
+leviates the over-rejection of the null hypothesis ascribable to the instability of estimation
+at extremal quantiles without sacrificing the power.
+Specification test-based approach
+Another direction to test non-causality is based on
+Theorem 3.1, where non-causality in the linear processes is translated into the misspecifi-
+cation of conditional quantiles of non/causal processes by linear dynamic quantile models
+(3.3). Equivalently, we aim to test
+E
+�Ψτ
+�Yt − X′
+tθ0
+� |Yt−1, . . . , Yt−p
+� = 0 a.s. for some θ0 ∈ B and ∀τ ∈ Υ ⊂ (0, 1),
+(3.5)
+where Ψτ(·) = I (· ≤ 0)−τ, and B is a family of uniformly bounded functions from Υ ⊂ (0, 1)
+to Θ ⊂ Rp+1.7 Both Υ and Θ are compact sets. Escanciano and Velasco (2010) (hereafter
+EV) characterize this restriction by unconditional moments
+E
+�Ψτ
+�Yt − X′
+tθ0
+� exp
+�ix′Xt
+�� = 0,
+∀x ∈ Rp+1, for some θ0 ∈ B and ∀τ ∈ Υ ⊂ (0, 1),
+(3.6)
+where i = √−1. Following the strategy of the EV test, we consider the statistic based on
+the residual processes indexed by quantiles τ, θ ∈ B and x ∈ Rp+1
+REV
+T
+�
+x, τ; ˆθ
+�
+= T −1/2
+T
+�
+t=1
+Ψτ
+�
+Yt − X′
+tˆθ
+�
+exp
+�ix′Xt
+�
+(3.7)
+with true parameters θ0 replaced by QAR estimates ˆθ from a given sample
+�
+Yt, X
+′
+t
+�
+t=1,2,...,T .
+Xt is composed by a constant and the lags of Yt up to order p. Theoretically, T −1/2REV
+T
+approaches to zero at a certain rate when T goes to infinity for any x ∈ Rp+1 if X′
+tθ0(τ) is the
+correct specification for QYt(τ|It) and ˆθ(τ) is a
+√
+T-consistent estimator of θ0(τ) for τ ∈ Υ.
+Thus, the distance between this statistic (3.7) and zero naturally turns into a measure of
+the deviation of X′
+tθ0(τ) from the true QYt(τ|It). The suggested Cram´er-von Mises (CvM)
+norm on (3.7) is defined by
+�
+τ∈Υ,x∈Rp+1
+���REV
+T
+�
+x, τ; ˆθ
+����
+2 dΦ(x)dW(τ),
+(3.8)
+7In our case, actually only θ0(τ) is required to be a uniformly bounded function from Υ ⊂ (0, 1) to Θ ⊂ R,
+and the rest θj(τ) are mapped to a constant for j = 1, 2, . . . , p. ∀τ ∈ (0, 1).
+12
+
+where Φ(x) and W(τ) are weighting functions defined on Rp+1 and Υ, respectively with pos-
+itive derivative in the corresponding domain. This CvM norm permits us to consider infinite
+model specifications for conditional quantiles at all τ’s of interest. Other possible options of
+norms aggregating the information over the quantiles and x, for instance, Kolmogorov-type,
+are also applicable here. The following proposition presents the asymptotic behavior of the
+EV test applied for testing non-causality.
+Corollary 3.1.3 (EV Test for non-causality). Under H0 and Assumptions 1-3, let E (XtX′
+t)
+be nonsingular in a neighborhood of θ(τ) = θ0(τ) for all τ ∈ Υ,
+REV
+T
+�
+x, τ; ˆθ
+�
+=⇒ ˜REV
+∞ (x, τ) ,
+and
+�
+τ∈Υ,x∈Rp+1
+���REV
+T
+�
+x, τ; ˆθ
+����
+2 dΦ(x)dW(τ) −→d
+�
+τ∈Υ,x∈Rp+1
+��� ˜REV
+∞ (x, τ)
+���
+2 dΦ(x)dW(τ)
+where ˜REV
+∞ = R∞ − ∆R. R∞ is a Gaussian process with mean zero and covariance function
+defined by
+Cov (x1, x2; τ1, τ2) = (τ1 ∧ τ2 − τ1τ2) E
+�exp
+�i(x1 − x2)′X0
+��
+and the drift ∆R is introduced due to the asymptotic effect from estimation error of ˆθ,
+∆R(x, τ) = G′(x, θ0(τ))Q(τ),
+where G(x, θ0(τ)) = E [Xtf (Qut(τ)) exp(ix′Xt)] and Q(·) is Σ−1/2
+0
+Bp+1/f
+�F −1(·)
+� . Bp+1
+is a (p + 1)-dimensional standard Brownian bridge. f and F are the density and cumulative
+distribution functions of the innovation ut, respectively. Σ0 = E (XtX′
+t).
+The result immediately follows from Escanciano and Velasco (2010), where they develop
+the result for a general class of quantile estimates covering various linear and nonlinear
+models of interest with corresponding assumptions. Those conditions are satisfied under
+Assumptions 1-3 in the context of MAR(r, s) processes under the null hypothesis.
+The
+limiting distribution of the test statistic is no longer distribution-free due to the estimation
+of nuisance parameters. Hence, a subsampling method is proposed to approximate the critical
+value. The operation for calculating the residual process (3.7) and the test statistic (3.8) is
+applied to a given subsample (Yt, . . . , Yt+b) of size b, denoted by REV
+b
+(x, τ; ˆθb,t) and Γ
+�
+REV
+b,t
+�
+respectively, and repeated for t = 1, 2, . . . , T − b + 1. The cdf of the limiting distribution of
+the proposed statistic is approximated by the empirical cdf across resamples, i.e.,
+ˆP
+�
+Γ
+�
+REV
+b
+�
+≤ ω
+�
+=
+1
+T − b + 1
+T−b+1
+�
+t=1
+I
+�
+Γ
+�
+REV
+b,t
+�
+≤ ω
+�
+.
+Therefore, the 1 − αth sample quantile, cEV
+T,b (α) defined as
+cEV
+T,b (α) = inf
+�
+ω : ˆP
+�
+Γ
+�
+REV
+b
+�
+≤ ω
+�
+≥ 1 − α
+�
+,
+intuitively serves as the critical value for this test at the α-level of the significance. The
+13
+
+validity of this subsampling approach has been verified by Escanciano and Velasco (2010),
+who suggest an appropriate choice for bandwidth b = ⌊kT 2/5⌋ for the sake of optimal minimax
+accuracy8. Some numerical evidence has demonstrated that a diverse range of values of k,
+like 4, 5, and 6 provide reasonably good performance in finite samples. A centering strategy
+can be adopted for the resampling statistic to achieve better performance power-wise in finite
+samples.
+Alternatively, Escanciano and Goh (2014) (hereafter EG) translate the restriction (3.5) into
+E
+�Ψτ
+�Yt − X′
+tθ0
+� I (Xt ≤ x)
+� = 0
+∀x ∈ Rp+1, for some θ0 ∈ B and for all τ ∈ Υ ⊂ (0, 1).
+(3.9)
+Naturally, a new test statistic can be constructed on the sample analog of moment conditions
+(3.9) with the replacement of θ0 by their QAR estimates ˆθ,
+T −1/2
+T
+�
+t=1
+Ψτ
+�
+Yt − X′
+tˆθ
+�
+I (Xt ≤ x)
+τ ∈ Υ, x ∈ Rp+1.
+(3.10)
+Unlike the approach in the EV test, where the asymptotic behavior of the test statistic is
+derived by incorporating the non-negligible effect from the estimates of nuisance parameters
+into the final limiting distribution, Escanciano and Goh (2014) introduce a variant of weight-
+ing functions I (Xt ≤ x) satisfying the orthogonality condition for the Taylor expansion of
+the statistic (3.10) around the true parameter θ0. With this consideration, the test statistic
+becomes
+REG
+T
+�
+x, τ; ˆθ
+�
+=
+√
+T
+T
+�
+t=1
+�
+�
+�Ψτ
+�
+Yt − X′
+tˆθ
+�
+�
+�I (Xt ≤ x) − ˆD′
+T
+�
+x, ˆθ(τ)
+� �
+T −1
+T
+�
+t=1
+ˆδt,τ ˆδ′
+t,τ
+�−1
+ˆδt,τ
+�
+�
+�
+�
+�
+(3.11)
+with ˆDT
+�
+x, ˆθ(τ)
+�
+= T −1 �T
+t=1 ˆδt,τI (Xt ≤ x) and
+ˆδt,τ = ˆf
+�
+X′
+tˆθ(τ) | Xt
+�
+Xt,
+where ˆf (y | Xt) is a consistent estimator of the conditional density function of Yt given the
+past information, f (y | Xt). One suggested kernel estimator is proposed by Escanciano and
+Goh (2012), defined by
+ˆf
+�
+X′
+tˆθ(τ) | Xt
+�
+=
+1
+MhM
+M
+�
+j=1
+K
+�
+X′
+tˆθ(τ) − X′
+tˆθ(τj)
+hM
+�
+,
+(3.12)
+where {τj}M
+j=1 is a sequence randomly selected from Υ following the uniform distribution
+with M −→ ∞ as well as T −→ ∞. K(·) is a kernel function, and hM denotes a smoothing
+parameter which may depend on the data and the quantiles considered for the estimation.
+Compared to other density estimator candidates, this estimator is computationally less cum-
+bersome but still preserves the same convergence rate as Rosenblatt estimator when the
+following assumption is imposed for the kernel function and the corresponding smoothing
+8⌊z⌋ denotes the largest integer that does not exceed z.
+14
+
+parameter.
+Assumption 4.
+1. For kernel function K(s):
+(a) K(s) is continuously differentiable;
+(b)
+� ∞
+−∞ K(s) = 1;
+(c) K(s) is uniformly bounded;
+(d) K(s) is of second order, i.e.
+� ∞
+−∞ sK(s) = 0,
+� ∞
+−∞ s2K(s)ds ∈ (0, ∞) and
+� ∞
+−∞ K2(s)ds ∈
+(0, ∞).
+2. The convergence rate of smoothing parameter hM to 0 has to satisfy P (aM ≤ hM ≤ bM) →
+1, for some deterministic sequences of positive numbers aM and bM such that bM −→ 0
+and ap+2
+M M/logM → ∞ as T → ∞.
+These regularity conditions for kernel functions apply to commonly used options in prac-
+tice, for instance, the Gaussian kernel.
+Corollary 3.1.4 (EG Test for non-causality). Under H0 and Assumptions 1-4, let the matrix
+E
+�
+δt,τδ′
+t,τ
+�
+be nonsingular in a neighborhood of θ(τ) = θ0(τ) for all τ ∈ Υ,
+REG
+T
+�
+x, τ; ˆθ
+�
+=⇒ REG
+∞ (x, τ) ,
+and
+�
+τ∈Υ,x∈Rp+1
+���REG
+T
+�
+x, τ; ˆθ
+����
+2 dΦ(x)dW(τ) −→d
+�
+τ∈Υ,x∈Rp+1
+���REG
+∞ (x, τ)
+���
+2 dΦ(x)dW(τ),
+where REG
+∞
+is a Gaussian process with mean zero and covariance function characterized by
+(τ1 ∧ τ2 − τ1τ2) E {Πτ1I (Xt ≤ x1) Πτ2I (Xt ≤ x2)} ,
+with the so-called orthogonal projection operator on the weighting function
+ΠτI (Xt ≤ x) ≡ I (Xt ≤ x) − D′ (x, θ0(τ)) E−1 (δt,τδt,τ) δt,τ.
+and δt,τ = f (X′
+tθ0(τ) | Xt) Xt, D (x, θ0(τ)) = E (δt,τI (Xt ≤ x)).
+As stated before, the main advantage of the EG test is the limiting distribution of the
+test statistic being invariant to the estimation effect of ˆθ(τ). Therefore, compared to the
+asymptotic distribution of the EV test, there is no ”drift” term subtracted from a Gaussian
+process. However, this asymptotic distribution still depends on the data-generating process.
+Consequently, we cannot tabulate the critical values for the considered statistic. This is
+coped with the aid of a multiplier bootstrap approach.
+The approximation based on a
+transformation on REG
+T
+�
+x, τ; ˆθ
+�
+is obtained by multiplying by a sequence of iid random
+15
+
+variables {Wt}T
+t=1 with zero mean and unit variance, independent on Xt,
+˜REG
+T,t
+�
+x, τ; ˆθ
+�
+=
+√
+T
+T
+�
+t=1
+�
+�
+�Ψτ
+�
+Yt − X′
+tˆθ
+�
+�
+�I (Xt ≤ x) − ˆD′
+T
+�
+x, ˆθ(τ)
+� �
+T −1
+T
+�
+t=1
+ˆδt,τ ˆδ′
+t,τ
+�−1
+ˆδt,τ
+�
+�
+�
+�
+� Wt.
+(3.13)
+One common choice of {Wt}T
+t=1 is
+�
+�
+�
+P (W = 1 − ω) = ω/
+√
+5
+P (W = ω) = 1 − ω/
+√
+5, with ω =
+�√
+5 + 1
+�
+/2.
+(3.14)
+This transformation has been proven by Escanciano and Goh (2014) to restore the limiting
+distribution of the original statistic. It allows us to use the empirical distribution of any
+continuous functional, including CvM form, Γ
+� ˜REG
+T,t
+�
+, i.e.,
+ˆP
+�
+Γ
+� ˜REG
+T,t
+�
+≤ ω | {Wt}T
+t=1
+�
+= 1
+T
+T
+�
+t=1
+I
+�
+Γ
+� ˜REG
+T,t
+�
+≤ ω
+�
+to consistently estimate the limiting distribution of the original statistic Γ
+�
+REG
+T
+�
+. Likewise,
+the (1 − α)-th empirical quantile of the transformed statistic,
+cEG
+T
+(α) = inf
+�
+ω : ˆP
+�
+Γ
+� ˜REG
+T
+�
+≤ ω | {Wt}T
+t=1
+�
+≥ 1 − α
+�
+,
+will is a consistent estimate of the critical value at α significance level.
+In the presence of non-causality, ψ(L−1) does not vanish from general MAR(r, s) processes.
+Then, by Corollary 3.1.1,
+E
+�Ψτ
+�Yt − X′
+tθ1(·)
+� exp (i · Xt)
+� ̸= 0
+and
+E
+�Ψτ
+�Yt − X′
+tθ1(·)
+� I (Xt ≤ ·)
+� ̸= 0
+in a set with a positive Lebesgue measure on Rp+1×Υ, provided that Φ and W are absolutely
+continuous on Rp+1 × Υ with respect to the Lebesgue measure. Correspondingly, under the
+alternative,
+Γ
+�
+REV
+T
+�
+=
+�
+τ∈Υ,x∈Rp+1
+���REV
+T
+�
+x, τ; ˆθ
+����
+2 dΦ(x)dW(τ) →p ∞
+and
+Γ
+�
+REG
+T
+�
+=
+�
+τ∈Υ,x∈Rp+1
+���REG
+T
+�
+x, τ; ˆθ
+����
+2 dΦ(x)dW(τ) →p ∞,
+so both specification-based tests are consistent.
+16
+
+4
+Monte Carlo Simulations
+In this section, we study the performance of the three proposed tests in finite samples and
+compare them with each other in terms of size and power.
+Constancy Test
+In the first experiment, we focus on the approach based on the constancy
+test. In the simulation, we consider a pair of MAR(1, 0) and MAR(0, 1) models
+�
+�
+�
+(1 − φL) Yt = ut
+�1 − ψL−1� Y ∗
+t = ut,
+(4.1)
+which are generated from 11 non-Gaussian distributed innovations, which cover a majority
+of distributions commonly used in the empirics, ranging from symmetric to asymmetric,
+bounded to unbounded support, with mixed types of tail behavior. The parameters (φ, ψ)
+with values (0.3, 0.6, 0.9) enable us to investigate the sensitivity of the method responding to
+data-generating processes with different persistence. The sample sizes are 200 and 500 with
+500 replications. ϵ = 0.05 is the default choice for the quantile interval [ϵ, 1 − ϵ] ⊂ (0, 1).
+The empirical size and power of rejecting the constancy hypothesis to detect non-causality
+under the QAR framework are summarized in Table 1.
+Regarding the size, the constancy test has an empirical size close to the nominal level in most
+cases but suffers from a severe over-rejection for heavy-tailed distributions. This corresponds
+to the estimation of conditional quantiles of these processes when τ is extremely close to 0
+or 1, which generally calls for a larger sample size to produce less biased and more stable
+estimates. The volatility of QAR estimates of extremal quantiles triggers the over-rejection
+of the constant coefficients under the causality hypothesis. Therefore, as seen in Figures
+4.1c and 4.1d, where innovations follow truncated Cauchy distribution9 and log-normal dis-
+tribution, respectively, the QAR estimates at quantiles close to 0 and 1 turn rather volatile
+compared to the estimates at other levels in (0, 1). To alleviate this issue, we propose to
+check different trimming strategies in the quantile interval for the test. There is an obvious
+trade-off in the selection of truncation of the quantile interval: over-trimmed, the power of
+the test will decrease due to the loss of valuable information; under-trimmed, the volatile
+estimate of extremal quantiles is not excluded, so the distortion in size will remain as be-
+fore. Consequently, we conduct some experiments to analyze the sensitivity of the constancy
+test in response to truncated intervals; see Table 2. The results suggest a trimmed quantile
+interval [0.10, 0.90] would be appropriate for the sample size considered because the power
+remains relatively high while the size is close to the nominal level. Another possible reason
+highlighted by Koenker and Xiao (2002) is that the default smoothing parameter selection
+for estimating the density function of innovations, which comprises the test statistic in the
+KhmaladzeTest command in R studio, produces satisfactory performance for the class of dis-
+tributions considered there, but is not designed for heavy-tailed distributed innovations. A
+more adaptive bandwidth choice of density function estimation of heavy-tailed distributions
+at extreme quantiles needs further investigation.
+9Truncated at a sufficiently large value to ensure the existence of variance.
+17
+
+From the perspective of power, the test achieves a significant leap in power as the sample size
+increases from 200 to 500 in most scenarios. More particularly, the method provides favorable
+performance in the presence of asymmetry in the distributions of innovations with rejection
+rates of 40% ∼ 90% in 200-sized samples and 70% ∼ 90% in 500-sized samples, respec-
+tively. This finding coincides with the idea shared in Velasco and Lobato (2018), where the
+third-order moments contribute most to the identification of non-causal AR processes. This
+phenomenon is exceptionally well-illustrated in the first four cases (Exponential, Gamma,
+Beta, and F distributions) when the distribution of innovations is skewed but has no heavy
+tails. However, in the cases where innovations do not display skewness or heavy-tailedness,
+the test can barely distinguish non-causal processes from their causal counterparts, see Fig-
+ure 4.2. The point of the failure is illustrated in the same figure. The QAR estimates for
+non-causal AR(1) with symmetric innovations appear to be indistinguishable from the ones
+for the causal counterparts, even under misspecification.
+Table 1: Empirical size and power of non-causality test using constancy⋆ test in QAR in various cases
+Distribution
+test
+T=200
+T=500
+ut
+φ(ψ) = 0.3†
+φ(ψ) = 0.6
+φ(ψ) = 0.9
+φ(ψ) = 0.3
+φ(ψ) = 0.6
+φ(ψ) = 0.9
+Exp(1)-1
+size
+4.20%
+4.00%
+4.80%
+6.40 %
+6.80%
+4.80%
+power
+33.80%
+38.80%
+40.20%
+65.40%
+69.60%
+72.40%
+Gamma(1,1)-1
+size
+4.40%
+3.00%
+3.80%
+5.00 %
+5.80%
+4.40%
+power
+34.40%
+37.00%
+42.20%
+64.40%
+68.40%
+70.20%
+Beta(5,1)-5/6
+size
+6.80%
+6.60%
+5.40%
+6.80 %
+9.40%
+6.40%
+power
+15.40%
+27.20%
+39.00%
+24.80%
+65.00%
+77.20%
+F(5,5)-5/3
+size
+5.00%
+6.60%
+3.40%
+8.40 %
+6.00%
+4.20%
+power
+80.00%
+81.80%
+70.00%
+97.20%
+98.20%
+96.20%
+χ2
+5 − 5
+size
+5.00%
+4.40%
+4.00%
+4.00 %
+4.80%
+5.20%
+power
+11.40%
+11.40%
+8.80%
+27.40%
+35.60%
+17.40%
+skewed normal
+size
+5.20%
+6.60%
+7.40%
+9.80 %
+7.40%
+9.60%
+power
+8.00%
+7.80%
+10.60%
+25.40%
+20.00%
+12.80%
+truncated Cauchy
+size
+26.60%
+34.80%
+43.00%
+20.80 %
+19.20%
+33.20%
+power
+70.00%
+96.40%
+92.40%
+80.00%
+99.80%
+91.80%
+log normal
+size
+21.20%
+19.60%
+23.60%
+23.20 %
+26.40%
+26.60%
+power
+98.20%
+99.00%
+99.60%
+100.00%
+100.00%
+100.00%
+t3
+size
+5.40%
+4.00%
+5.00%
+4.00 %
+6.80%
+4.80%
+power
+9.80%
+13.00%
+15.80%
+11.60%
+20.00%
+25.80%
+uniform
+size
+6.20%
+7.60%
+6.00%
+6.00 %
+6.80%
+5.80%
+power
+13.40%
+8.00%
+13.20%
+40.40%
+7.80%
+48.80%
+Laplace
+size
+3.40%
+4.00%
+4.60%
+2.60 %
+3.80%
+4.60%
+power
+8.40%
+5.60%
+17.00%
+10.80%
+8.60%
+29.00%
+⋆: constancy test over the quantile interval [0.05, 0.95].
+†: φ(ψ) = 0.3 means the coefficient in the lag polynomial of the MAR(1, 0) process (purely causal) is 0.3; the
+coefficient in the lead polynomial of the MAR(0, 1) process (purely non-causal) is 0.3 as well, for comparison.
+EV Test
+With the same setting as the one in the Constancy Test, we take the case with
+the coefficient φ(ψ) = 0.6 for MAR(1, 0) (MAR(0, 1)) as the representative to examine the
+performance of EV test in discriminating non-causal processes from their causal alternatives.
+The sample size varies from 100 to 200 and 500. The number of replications is 500. Regarding
+the bandwidth for the subsampling scheme to approximate the critical value, we choose
+b = ⌊4T 2/5⌋.
+As for the weighting functions involved in the expression (3.8), a uniform
+18
+
+(a) χ2
+5 − 5 distribution
+(b) skewed normal distribution
+(c) truncated Cauchy distribution
+(d) log normal distribution
+Figure 4.1: Four cases of QAR(1) on MAR(1, 0) and MAR(0, 1), T=500, φ(ψ) = 0.6 : asymmetric
+distributions
+(a) t3 distribution
+(b) uniform distribution
+(c) Laplace distribution
+Figure 4.2: Three cases of QAR(1) on MAR(1, 0) and MAR(0, 1), T=500, φ(ψ) = 0.6: symmetric
+distributions
+19
+
+5
+coefficients
+causal
+noncausal
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+T5
+causal
+noncausal
+coefficients
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+T5
+coefficients
+0.5
+causal
+noncausal
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+L0
+2
+5.
+causal
+noncausal
+coefficients
+0.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+T1.0
+coefficients
+0.5
+0'0
+causal
+noncausal
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+tcoefficients
+0.6
+causal
+0.4
+noncausal
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+T1.0
+coefficients
+5
+0
+causal
+noncausa
+5
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+tTable 2: Empirical size and power of non-causality test using constancy test with different trimmed quantile interval
+Sample size
+[0.05, 0.95]
+φ(ψ) = 0.3
+φ(ψ) = 0.6
+φ(ψ) = 0.9
+φ(ψ) = −0.4
+φ(ψ) = −0.6
+φ(ψ) = −0.8
+T=100
+size
+2.40%
+2.20%
+2.60%
+2.80%
+2.00%
+2.40%
+power
+22.40%
+24.60%
+22.00%
+4.80%
+32.80%
+26.60%
+T=200
+size
+3.60%
+3.60%
+4.00%
+5.00%
+2.80%
+4.40%
+power
+40.80%
+38.60%
+41.80%
+8.40%
+48.00%
+44.00%
+T=500
+size
+3.40%
+6.20%
+4.80%
+6.60 %
+4.80%
+3.40%
+power
+67.20%
+67.00%
+72.40%
+14.20%
+66.00%
+72.20%
+Sample size
+[0.10, 0.90]
+φ(ψ) = 0.3
+φ(ψ) = 0.6
+φ(ψ) = 0.9
+φ(ψ) = −0.4
+φ(ψ) = −0.6
+φ(ψ) = −0.8
+T=100
+size
+3.40%
+3.20%
+3.20%
+3.80%
+2.40%
+3.40%
+power
+28.00%
+25.40%
+22.80%
+6.00%
+32.80%
+28.20%
+T=200
+size
+3.20%
+2.80%
+5.00%
+3.00%
+4.20%
+4.00%
+power
+37.40%
+43.40%
+43.20%
+5.80%
+49.40%
+48.80%
+T=500
+size
+4.80%
+5.40%
+6.00%
+4.60 %
+3.80%
+5.00%
+power
+71.20%
+67.80%
+69.20%
+14.60%
+65.40%
+72.80%
+Sample size
+[0.15, 0.85]
+φ(ψ) = 0.3
+φ(ψ) = 0.6
+φ(ψ) = 0.9
+φ(ψ) = −0.4
+φ(ψ) = −0.6
+φ(ψ) = −0.8
+T=100
+size
+4.40%
+2.60%
+3.60%
+5.00%
+3.00%
+3.00%
+power
+27.80%
+28.20%
+21.80%
+8.80%
+37.60%
+30.20%
+T=200
+size
+5.60%
+3.80%
+3.60%
+6.60%
+3.40%
+4.40%
+power
+39.00%
+40.00%
+42.40%
+9.60%
+51.60%
+46.60%
+T=500
+size
+5.40%
+4.40%
+5.00%
+4.20 %
+4.80%
+5.40%
+power
+64.80%
+65.80%
+69.00%
+21.80%
+68.00%
+71.60%
+The innovations follow exponential distribution.
+distribution is applied to W(τ) defined on the evenly discretized quantile interval Υ =
+[0.01, 0.99] and 2-dimensional standard normal distribution to Ψ(x) for the sake of simplicity
+in the calculation. The results of the empirical size and power are displayed in Table 3.
+Concerning size, the results present some fluctuation around the nominal level. The test
+tends to under-reject the correct null hypothesis in the light-tailed scenarios while over-rejects
+in the heavy-tailed scenarios. This comes from the subjective choice in the subsampling size,
+whose performance can be sensitive to the quantile interval included in the test and the data-
+generating process. Nevertheless, the distortion in size is not so significant, and we believe
+this can be eased by choosing different subsampling schemes. As for power, the test achieves
+reasonably good performance generally. From the simulation results, we observe that the
+power of detecting non-causality is close to 100% in most cases when the sample size is 500,
+which is as expected. It is worth noting that the convergence rate of power towards 100%
+varies across different distributions. The numerical evidence indicates that the further the
+innovations depart from Gaussianity, the faster the convergence rate is. In such log-normal
+and uniform distributions, a reasonably high power, like 86% or 92%, has been obtained in
+relatively small samples. While others (chi-square) may need larger samples to reach the
+same level of power.
+EG Test
+The setup of DGP keeps unchanged, like in the EV test. The sample size varies
+from 50 to 100 and 200.
+The weighting functions Ψ(x) and W(τ) in the CvM form of
+REG
+T
+�
+x, τ; ˆθ
+�
+are chosen to be the empirical distribution of Xt and uniform distribution
+over the grid of quantiles from Υ = [0.01, 0.99] considered in the estimation. The critical
+20
+
+Table 3: Empirical size and power of non-causality test using EV test
+in QAR in various cases
+Distribution
+parameter:
+φ(ψ) = 0.6
+ut
+T:
+100
+200
+500
+Gaussian
+size
+2.20%
+3.40%
+3.40%
+power
+0.40%
+0.60%
+2.60%
+exponential
+size
+4.40%
+1.80%
+2.00%
+power
+29.20%
+43.20%
+97.20%
+Gamma
+size
+1.80%
+3.00%
+2.80%
+power
+10.60%
+41.60%
+97.60%
+Beta
+size
+3.20%
+3.20%
+3.40%
+power
+26.40%
+67.80%
+99.00%
+F
+size
+3.60%
+5.80%
+3.40%
+power
+24.00%
+67.60%
+99.00%
+χ2
+5 − 5
+size
+4.80%
+4.40%
+3.60%
+power
+13.40%
+22.20%
+68.40%
+log normal
+size
+5.40%
+4.60%
+6.80%
+power
+54.00%
+92.40%
+100.00%
+t3
+size
+6.20%
+7.00%
+6.40%
+power
+13.80%
+42.40%
+93.40%
+Uniform
+size
+5.00%
+6.00%
+3.40%
+power
+44.00%
+86.80%
+100.00%
+Laplace
+size
+4.40%
+4.80%
+4.40%
+power
+14.00%
+47.20%
+98.20%
+EV test: the choice of size for subsampling is ⌊4T 2/5⌋.
+value is obtained through the multiplier bootstrap introduced in the methodology section.
+Implementing the multiplier bootstrap avoids computing the estimates for each subsample.
+The results are summarized in Table 4. Regarding the empirical size performance, this ap-
+proach delivers stable rejection frequencies under the null hypothesis associated with their
+nominal level in all cases considered in the simulation exercise. This is anticipated in accor-
+dance with the argument in the previous section that there is no subjective choice involved
+in the approximation of the critical value. In terms of power, the EG test has an increasing
+trend as the sample is enlarged. Similar to the EV test, the EG approach outperforms in the
+presence of skewness and excess kurtosis (regardless of negative or positive), with a rejection
+probability over 70% in relatively small samples (200). For the cases where the performance
+is less satisfactory, such as t3 and Laplace distributions, power still increases when the sample
+size becomes larger.
+An overall comparison of the three methods is exhibited in Table 5. Size-wise, the EG test
+has an appealing attribute of undistorted size in general scenarios compared to the other two
+approaches. On the contrary, the constancy test suffers from over-rejection in heavy-tailed
+cases, and the EV test delivers less accuracy than the EG test in some cases. Power-wise,
+the EG test is the most robust one as it produces relatively good results in all situations but
+is extraordinarily competent for asymmetric distributions. By contrast, the EV test achieves
+the highest power among the three in the symmetric cases. In comparison, the constancy
+test can be a powerful tool in detecting non-causality in processes with heavy tails. However,
+given that the consistency of the constancy test for non-causality is not guaranteed and the
+size is distorted, it may give misleading conclusions in empirical analysis. Thus, the con-
+21
+
+Table 4: Empirical size and power of non-causality test using EG test
+in QAR in various cases
+Distribution
+parameter:
+φ(ψ) = 0.6
+ut
+T:
+50
+100
+200
+Gaussian
+size
+6.00%
+5.00%
+4.00%
+power
+4.80%
+5.60%
+5.80%
+exponential
+size
+5.20%
+4.60%
+6.00%
+power
+24.80%
+49.20%
+76.40%
+Gamma
+size
+5.00%
+5.60%
+5.00%
+power
+23.80%
+45.80%
+75.80%
+Beta
+size
+6.40%
+5.80%
+6.00%
+power
+24.20%
+42.20%
+75.80%
+F
+size
+4.40%
+5.00%
+5.40%
+power
+22.60%
+37.40%
+67.00%
+χ2
+5 − 5
+size
+6.60%
+5.40%
+4.60%
+power
+12.80%
+22.60%
+41.60%
+log normal
+size
+5.60%
+5.40%
+4.00%
+power
+32.00%
+60.60%
+85.80%
+t3
+size
+4.40%
+6.60%
+7.00%
+power
+6.80%
+14.80%
+36.20%
+Uniform
+size
+4.80%
+6.60%
+6.60%
+power
+10.20%
+25.40%
+64.80%
+Laplace
+size
+5.60%
+5.00%
+6.80%
+power
+8.20%
+14.40%
+41.60%
+EG test: critical value obtained from multiplier bootstrap.
+stancy test can only be considered a preliminary test to check whether the process is likely
+non-causal, followed by the implementation of the EV or EG tests, which serve as the formal
+tests for non-causality. In practice, a combination of the constancy test and EV (EG) test
+is suggested.
+5
+Empirical Applications
+In this section, we apply our non-causality tests to six financial series studied in Fries and
+Zakoian (2019): cotton price, soybean price, sugar price, coffee price, Hang Seng Index (HSI),
+and Shiller Price/Earning ratio (Shiller PE), where single or multiple spikes and asymmetric
+dynamics are exhibited.
+Fries and Zakoian (2019) found numerical evidence in favor of
+MAR(r, s) models in fitting time series with local explosiveness phases compared to purely
+causal AR models. The frequency of data is quarterly for the Shiller PE series and monthly
+for the rest10. The trajectories of these series are displayed in Figure 5.1.
+Before proceeding to our testing strategies, we must ensure that the series is stationary. The
+augmented Dickey-Fuller tests indicate no unit roots in the cotton, soybean, sugar, and coffee
+price. Neither the evidence of unit root is found in the series of HSI after a linear trend is
+subtracted. The stationarity of the Shiller PE series is secured after the first difference in the
+levels. The sample partial autocorrelation function for each series is computed, see Figure
+5.2, to determine the order of the lag orders: cotton: 2; soybean: 2; sugar: 4; coffee: 3; HSI:
+10The access of the replication data can be found in Fries and Zakoian (2019)
+22
+
+Table 5: Comparison of QAR-based non-causality tests
+Distribution
+test type:
+constancy test
+EV test
+EG test
+ut
+T:
+100
+200
+100
+200
+100
+200
+Gaussian
+size
+2.80%
+5.40%
+2.20%
+3.40 %
+5.00%
+4.00%
+power
+4.20%
+3.40%
+0.40%
+0.60%
+5.60%
+5.80%
+exponential
+size
+3.20%
+4.00%
+4.40%
+1.80%
+4.60%
+6.00%
+power
+25.40%
+38.80%
+29.20%
+43.20%
+49.20%
+76.40%
+Gamma
+size
+3.80%
+3.00%
+1.80%
+3.00%
+5.60%
+5.00%
+power
+26.60%
+37.00%
+10.60%
+41.60%
+45.80%
+75.80%
+Beta
+size
+4.20%
+6.60%
+3.20%
+3.20%
+5.80%
+6.00%
+power
+18.60%
+27.20%
+26.40%
+67.80%
+42.20%
+75.80%
+F
+size
+6.80%
+6.60%
+3.60%
+5.80%
+5.00%
+5.40%
+power
+62.40%
+81.80%
+24.00%
+67.60%
+34.70%
+67.00%
+χ2
+5
+size
+5.80%
+4.40%
+4.80%
+4.40%
+5.40%
+4.60%
+power
+9.60%
+11.40%
+13.40%
+22.20%
+22.60%
+41.60%
+log normal
+size
+20.20%
+19.60%
+5.40%
+4.60%
+5.40%
+4.00%
+power
+78.80%
+99.60%
+54.00%
+92.40%
+60.60%
+85.80%
+t3
+size
+3.40%
+4.00%
+6.20%
+7.00%
+6.60%
+7.00%
+power
+10.20%
+13.00%
+13.80%
+42.40%
+14.80%
+36.20%
+Uniform
+size
+6.80%
+7.60%
+5.00%
+6.00%
+6.60%
+6.60%
+power
+7.40%
+8.00%
+44.00%
+86.80%
+25.40%
+64.80%
+Laplace
+size
+5.80%
+4.00%
+4.40%
+4.80%
+5.00%
+6.80%
+power
+5.20%
+5.60%
+14.00%
+47.20%
+14.40%
+41.60%
+Constancy test: with trimmed quantile interval [0.05, 0.95]
+1; Shiller PE: 7.
+Three non-causality testing procedures: the constancy test, the EV test, and the EG test,
+are applied to each series, respectively. The corresponding results are reported in Table 6. For
+the constancy test, we consider two trimmed quantile intervals: [0.05, 0.95] and [0.10, 0.90]
+to mitigate the instability effect on the power from the subjective choice in the quantiles.
+Both in the cases of cotton and sugar series, a significant fluctuation in the coefficients is
+observed. Yet no strong evidence against linear quantile specification is found based on the
+EV or EG test. The strong rejection in the constancy test for these series may result from
+the over-rejection issue in heavy-tailed scenarios, which makes the results from the EV or
+EG tests more reliable. This does not deviate much from the results obtained by Fries and
+Zakoian (2019), where they found one non-causal root (0.94) in the cotton series and a root
+(0.92) in the sugar series. As seen in the numerical experiments, a non-causal AR model with
+coefficients near the unity makes it harder to distinguish from its causal counterpart, as well
+as generates less distinct dynamics than its causal counterpart. For three series (soybean,
+coffee, and Shiller PE), the tests show strong evidence favoring non-causal processes at 5%
+or even 1% level from all three testing strategies. For HSI, the test shows mild evidence of
+non-causality at the significance level of 10% based on the EG test. This result is compatible
+with the conclusion drawn in Fries and Zakoian (2019), where mixed models with nontrivial
+non-causal components11 are selected.
+11The coefficients of the non-causal components in the MAR(r, s) models are not closed to the unity.
+23
+
+Figure 5.1: Financial series trajectories
+24
+
+cottonprice,monthly,08/1972-07/2017
+soybeanprice,monthly,01/1973-05/2006
+2
+12
+1.5
+10
+8
+0.5
+0
+4
+0
+100
+200
+300
+400
+500
+600
+0
+100
+200
+300
+400
+500
+sugarprice,monthly,11/1962-08/2018
+coffee price,monthly,04/1976-05/2018
+0.6
+3
+0.4
+0.2
+0
+0
+0
+100
+200
+300
+400
+500
+600
+700
+0
+100
+200
+300
+400
+500
+600
+x1HangSengIndex,monthly,11/1986-03/2017
+Shiller Price/Earning ratio,quarterly,Q1/1881-Q2/2017
+50
+3
+40
+2
+30
+20
+10
+0
+0
+100
+200
+300
+400
+0
+100
+200
+300
+400
+500
+600Figure 5.2: Sample partial autocorrelation functions of six financial series
+25
+
+Sample PACF of cotton
+Sample PACF of soybean
+ Sample Partial Autocorrelation 
+Sample Partial Autocorrelation
+0.5
+0.5
+0.5
+-0.5
+0
+5
+10
+15
+20
+0
+5
+10
+15
+20
+Lag
+Lag
+Sample PACF of sugar
+SamplePACFof coffee
+ Sample Partial Autocorrelation
+Sample Partial Autocorrelation
+0.5 
+0.5 
+0
+-0.5
+-0.5
+0
+5
+10
+15
+20
+0
+5
+10
+15
+20
+Lag
+Lag
+Sample PACF of detrended HSI
+Sample PACF of first difference of Shiller P/E
+Autocorrelation
+Sample Partial Autocorrelation
+0.8
+0.6
+0.5
+ Partial
+0.4
+Sample
+0.2
+0
+-0.5
+0
+5
+10
+15
+20
+0
+5
+10
+15
+20
+Lag
+LagTable 6: Non-causality tests for six financial series in Fries and Zakoian (2019)
+Financial series
+test type:
+constancy test
+EV test
+EG test
+total AR order(r + s)
+[0.05, 0.95]†
+[0.10, 0.90]
+Cotton
+2
+statistic
+6.897∗∗
+4.738∗∗
+0.012
+0.025
+critical value12
+(3.393)
+(3.287)
+(0.046)
+(0.032)
+Soybean
+2
+statistic
+4.703∗∗
+4.445∗∗
+0.209∗∗
+0.027∗∗
+critical value
+(3.393)
+(3.287)
+(0.158)
+(0.021)
+Sugar
+4
+statistic
+306.043∗∗
+144.025∗∗
+0.009
+0.065
+critical value
+(5.560)
+(5.430)
+(0.102)
+(0.100)
+coffee
+3
+statistic
+8.457∗∗
+6.992∗∗
+0.149∗∗
+0.044∗∗
+critical value
+(4.523)
+(4.383)
+(0.089)
+(0.025)
+Hang Seng Index
+1
+statistic
+0.017
+0.012
+0.168
+0.078∗
+critical value
+(2.140)
+(2.102)
+(0.176)
+(0.083)
+Shiller’s P/E ratio
+7
+statistic
+15.105∗∗
+9.027∗∗
+0.197∗∗
+0.135∗∗
+critical value
+(8.578)
+(8.368)
+(0.189)
+(0.066)
+† the trimmed quantile interval considered in the constancy test.
+∗∗ stands for significance at level 5% and ∗ stands for significance at 10%.
+the critical value at 10% significance level for the EG test in the case of Hang Seng Index is 0.061.
+6
+Extensions and conclusion
+6.1
+Some Extensions
+So far, the preceding discussion has been confined to MAR(r, s) driven by iid innovations.
+Within this framework, the only possible source of non-linearity in MAR(r, s) is non-causality,
+which contributes to the consistency of the test in the aforementioned methods. However,
+stylized nonlinear dynamics like conditional heteroskedasticity or asymmetric dynamics are
+prevalently observed in the financial and macroeconomic data, which renders it more demand-
+ing to detect non-causality in a time series process. A more robust methodology applicable
+to AR models accommodating non-linear features needs investigation. The critical point is
+how to disentangle non-linearity induced by non-causality from the other alternatives. If
+these non-linear features can be captured by a parametric model, one plausible solution is to
+incorporate these non-linear terms into the baseline model (2.1). Below we list some possible
+extensions where this strategy is employed.
+Asymmetric Dynamics
+This can be solved by allowing varying coefficients in the MAR(r, s)
+model in the spirit of the random coefficient model, defined by
+˜φ(L) ˜ψ(L−1)Yt = ut
+(6.1)
+where ˜φ(L) = 1 − φ1(Ut)L − · · · − φr(Ut)Lr and ˜ψ = 1 − ψ1(Ut)L−1 − · · · − ψs(Ut)L−s, Ut
+is an iid sequence of random variables following standard uniform distribution, and ut is an
+iid innovation sequence satisfying Assumptions 1. Denote
+Ωc =
+�
+φ1(Ut)
+. . .
+φr−1(Ut)
+φr(Ut)
+Ir−1
+0(r−1)
+�
+and
+Ωnc =
+�
+ψ1(Ut)
+. . .
+ψs−1(Ut)
+ψs(Ut)
+Is−1
+0(s−1)
+�
+.
+26
+
+Similar to the p-th order autoregressive process, which is designed to accommodate asym-
+metric dynamics in Koenker and Xiao (2006) for linear QAR model, we need to assume
+E (Ωc ⊗ Ωc) and E (Ωnc ⊗ Ωnc) have eigenvalues with moduli less than one. This equation
+(6.1) is able to mimic asymmetric dynamics since φj, ψj’s are functions [0, 1] → R. The
+definition of non-causality in this context will be adapted to that ˜ψ(L−1) does not decline
+to constant. In the causal situation, this model (6.1) works like a random coefficient model
+with lags. The linearity of the conditional mean is restored, and the linear quantile dynamic
+model with varying coefficients over different quantiles remains the correct specification for
+conditional quantiles of Yt. On the other hand, when the process is non-causal, it is con-
+ceivable that linearity will not hold anymore. Therefore, the methodology relying on the
+specification tests is applicable here.
+Volatility Clustering
+Concerning volatility clustering, which is routinely modeled by the
+quadratic ARCH/GARCH model in squared residuals. Another popular choice is to replace
+the squared value with the absolute value suggested by Taylor (2008) and make the model a
+linear ARCH.
+φ(L)ψ(L−1)Yt = vt
+where vt = σtut
+σt = γ0 + γ1|vt−1| + · · · + γq|vt−q|
+(6.2)
+where φ(L) and ψ(L−1) are defined following (2.1), and ut is a sequence of iid innovations.
+The linear ARCH model is able to capture the correlation in the variance and meanwhile
+preserves a relatively simple linear specification compared to other alternatives like GARCH.
+Under the H0 where ψ(L−1) degenerates to 1, the linearity of conditional quantile specifica-
+tion still holds after adding {|vt−j|}q
+j=1 into the regression equation.
+QYt (τ|Yt−1, Yt−2, . . . ) = φ1Yt−1 + · · · + φrYt−r + Qut(τ) (γ0 + γ1|vt−1| + · · · + γq|vt−q|)
+= Qut(τ)γ0
+�
+��
+�
+˜γ0(τ)
++ Qut(τ)γ1
+�
+��
+�
+˜γ1(τ)
+|vt−1| + · · · + Qut(τ)γq
+�
+��
+�
+˜γq(τ)
+|vt−q| + φ1Yt−1 + · · · + φrYt−r
+= ˜γ0(τ) + ˜γ1(τ)|vt−1| + · · · + ˜γq(τ)|vt−q| + φ1Yt−1 + · · · + φrYt−r,
+(6.3)
+where |vt−j| can be recovered by Yt−j − φ1Yt−j−1 − · · · − φrYt−j−r. Under non-causality,
+the explicit expression of the conditional quantile of Yt remains unclear. Nevertheless, it
+cannot be characterized by encompassing linear combinations of residuals in the model.
+Consequently, the linear dynamic quantile model would not be the correct specification
+conceivably.
+Overall, these two possible extensions to cases with nonlinear dynamics are tentative since
+the statistical properties of conditional quantiles of Yt defined by (6.1) and (6.2) require
+further investigation, which opens a couple of lines for future research. Some simulation
+trials in Appendix 7.4 have shown the validity of the proposed strategies.
+27
+
+6.2
+Perspective from Extreme Quantiles
+One intriguing observation from the simulation is that QAR estimates at extreme quantiles
+might be informative for identifying the true models even though linear quantile specification
+is incorrect for the conditional quantile of non-causal processes. As depicted in Figure 4.1,
+for MAR(0, 1) processes with coefficient 0.6 driven from asymmetric innovations,
+�
+1 − 0.6L−1�
+Yt = ut ⇔ Yt = (0.6)−1Yt−1 − (0.6)−1ut−1.
+The estimated slope of Yt−1 approaches (0.6)−1 when the quantile gets close to 0 or 1.
+Somehow it indicates that the linear correlation at extreme quantiles between Yt and Yt−1
+can help to discriminate causality and non-causality. A similar idea has been adopted for
+model selection based on the extreme clustering of residuals by Fries and Zakoian (2019), but
+the method is restricted to α−stable distributions. Rich data is required for further analysis
+to get a less biased estimator for conditional quantiles close to 0 or 1. This opens a possible
+avenue for future research in line with identifying causal and non-causal processes using tail
+information of processes.
+6.3
+Conclusion
+This paper introduces three novel testing strategies for non-causality in linear time series
+within the quantile regression framework. The tests exploit the non-linearity of autoregressive
+processes with non-causality and achieve the objective of detecting non-causality based on the
+well-developed inference under the QAR framework. The constancy test shares the simplicity
+of implementation but lacks consistency since the behavior of linear quantile autoregression
+for non-causal processes is not clear yet.
+This issue from the constancy test is tackled
+by testing the specification of linear conditional quantile models to detect non-causality.
+Specification-based non-causality testing procedures like EV and EG tests yield stable size
+at a nominal level and fairly satisfactory power. On the one hand, the EV test outperforms
+the EG test in platykurtic and leptokurtic situations or symmetric distributions. On the
+other hand, the EG test is less computationally cumbersome and shows better performance
+when the process is skewed. However, no method is placed in a dominating situation. Thus,
+a combined testing procedure with the constancy test as a preliminary check complemented
+with either EV or EG test is suggested for practitioners.
+Some possible extensions to accommodate different dependence in model innovations, which
+might bring obstacles in detecting non-causality, are proposed at the end of the paper. Some
+simulation results in QAR estimate at extreme quantiles indicate the possibility of identifying
+non-causal processes by employing information from the tails of processes.
+28
+
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+
+7
+Appendices
+7.1
+Some Properties of Non-causal Autoregressive Processes
+Higher-order Dependence of All-pass Time Series Models
+Following the same setup
+in Example 2.3,
+˜ut =
+1 − ψL
+1 − ψL−1 ut =
+∞
+�
+j=−∞
+ρjut+j.
+The skewness of ˜ut is
+E
+�
+˜u3
+t
+�
+=
+∞
+�
+j=−∞
+ρ3
+j E
+�
+u3
+t
+�
+=
+�
+1 − 3ψ2(ψ + 1)
+ψ2 + ψ + 1
+�
+E
+�
+u3
+t
+�
+, |ψ| < 1,
+where −1 <
+�
+1 − 3ψ2(ψ+1)
+ψ2+ψ+1
+�
+< 1.
+From the above expression, it is easy to tell that the
+all-pass filter preserves the symmetry of the innovations if the original ones are not skewed
+but might alter the direction of skewness if ut is asymmetric by varying values of ψ. Apart
+from the correlation in the squared value of ˜ut that has been explicitly shown in Example
+2.3, here we derive the closed-form solution for the dependence at order 3. It suffices to show
+Cov
+�
+˜u3
+t , ˜u3
+t+h
+�
+is nonzero for h ̸= 0.
+E
+�
+˜u3
+t ˜u3
+t+h
+�
+= E
+�
+�
+∞
+�
+j=−∞
+∞
+�
+i=−∞
+∞
+�
+m=−∞
+∞
+�
+n=−∞
+∞
+�
+l=−∞
+∞
+�
+k=−∞
+ρjρiρmρnρlρkut+jut+iut+mut+h+nut+h+kut+h+l
+�
+�
+=
+�
+�
+∞
+�
+j=−∞
+ρ3
+jρ3
+j+h
+�
+�
+�
+E(u6
+t ) − 15 E(u4
+t ) E(u2
+t ) − 10 E2(u3
+t ) − 15 E3(u2
+t )
+�
++ 3
+�
+�
+∞
+�
+j=−∞
+�
+ρj+hρ3
+j + ρ3
+j+hρj
+�
+�
+� E(u4
+t ) E(u2
+t )
++
+�
+�
+�
+�
+�
+∞
+�
+j=−∞
+ρ3
+j
+�
+�
+2
++ 9
+�
+�
+∞
+�
+j=−∞
+ρ2
+j+hρj
+�
+�
+�
+�
+∞
+�
+j=−∞
+ρj+hρ2
+j
+�
+�
+�
+�
+� E2(u3
+t )
+after using �∞
+j=−∞ ρjρj+h = 0 (coincides with no correlation property of all-pass time series
+process) and �∞
+j=−∞ ρ2
+j = 1 (variance preserving property),
+Cov
+�
+˜u3
+t , ˜u3
+t+h
+�
+= E
+�
+˜u3
+t ˜u3
+t+h
+�
+− E
+�
+˜u3
+t
+�
+E
+�
+˜u3
+t+h
+�
+generally is not zero. For instance, h = 1, after simplification
+Cov
+�
+˜u3
+t , ˜u3
+t+1
+�
+= α1 E(u6
+t ) +
+�
+α2 E(u4
+t ) + α4 E2(u2
+t )
+�
+E(u2
+t ) + α3 E2(u3
+t )
+with
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+α1 = −3ψ5(1−ψ2)3
+ψ4+ψ2+1
+α2 = −3ψ3(1 − ψ2)3 + 45ψ5(1−ψ2)3
+ψ4+ψ2+1
+α3 = 30ψ5(1−ψ2)3
+ψ4+ψ2+1
+− 9(1−ψ2)3(2ψ+1)ψ2
+(ψ2+ψ+1)2
+α4 = 45ψ5(1−ψ2)3
+ψ4+ψ2+1
+31
+
+where zeros are not attained at the same value of ψ ∈ (−1, 1) \ {0}.
+Conditional Density Function of Non-causal Autoregressions
+In this section, we
+try to study the properties of the conditional density function of the response variable in the
+presence of non-causality in the autoregressive process through simulations. Consider a pair
+of MAR(1, 0) and MAR(0, 1) processes generated from iid innovations following the same
+distribution with the density function fu(·),
+�
+�
+�
+Yt = 0.6Yt−1 + ut
+˜Yt = 0.6−1 ˜Yt−1 + ut = 0.6 ˜Yt+1 − 0.6ut+1.
+(7.1)
+One is purely causal, and the other is purely non-causal with a coefficient of 0.6. We start
+by analyzing f (Yt ≤ y|Yt−1 = x) in the causal case.
+f (Yt = y|Yt−1 = x) = dP (Yt ≤ y|Yt−1 = x)
+dy
+= dP (0.6Yt−1 + ut ≤ y|Yt−1 = x)
+dy
+= dP (ut ≤ y − 0.6x|Yt−1 = x)
+dy
+= fu (y − 0.6x) = fu (y − 0.6Yt−1) ,
+(7.2)
+It is not difficult to conclude from equation 7.2 that the conditional density of Yt given Yt−1
+is shifting horizontally as the value of Yt−1 varies. Despite the change in the location of the
+density function, the rest remains the same across different values of Yt−1.
+Similarly, we derive the conditional density function for the non-causal case,
+f
+� ˜Yt = y| ˜Yt−1 = x
+�
+=
+f
+� ˜Yt−1 = x| ˜Yt = y
+�
+f
+� ˜Yt = y
+�
+f( ˜Yt−1 = x)
+by Bayes rule
+=
+f
+�
+0.6 ˜Yt − 0.6ut = x | ˜Yt = y
+�
+f
+� ˜Yt = y
+�
+f (Yt−1 = x)
+by definition of ˜Yt−1
+=
+fu
+�y − 0.6−1x
+� f
+� ˜Yt = y
+�
+f (Yt−1 = x)
+by the independence of ut and ˜Yt
+=
+fu
+�y − 0.6−1x
+� f
+� ˜Yt = y
+�
+� ∞
+−∞ f
+� ˜Yt−1 = x| ˜Yt = s
+�
+f
+� ˜Yt = s
+�
+ds
+law of total probability
+=
+fu
+�y − 0.6−1x
+� f
+� ˜Yt = y
+�
+� ∞
+−∞ fu (s − 0.6−1x) f
+� ˜Yt = s
+�
+ds
+.
+(7.3)
+There is no general closed-form solution to this expression. Note that no x plays a role in
+f(Yt = y), and the denominator is x−dependent but highly nonlinear due to the integration.
+32
+
+If we assign additive property13 to the marginal distribution of Yt. This nonlinearity can be
+shown more clearly. Say ut follows an exponential distribution with rate λ, then the equation
+7.3 has the explicit form
+f (Yt = y|Yt−1 = x)
+=
+�� ∞
+−∞ e−isy �∞
+j=0
+λ
+λ−is(0.6)j ds
+�
+λe−(x−0.6y)
+�� ∞
+−∞ e−isx �∞
+j=0
+λ
+λ−is(0.6)j ds
+�
+I (x − 0.6y ≥ 0) .
+This suggests that the shape (functional form) of the density function would differ, corre-
+sponding to the choice of x. The following simulation experiment demonstrates this argu-
+ment. In this simulation, we generate two AR(1) processes (7.1) by exponentially distributed
+(a) conditional density of Yt in the causal
+case
+(b) conditional density of ˜Yt in the non-
+causal case
+Figure 7.1: conditional density of Yt given different x
+innovations with rate 1. The sample size is 500. Estimated conditional density functions
+f (y|Yt−1 = x) are plotted in Figure 7.1, given five choices of x: 10%, 30%, 50%, 70%, and
+90% percentiles of Yt−1 sample. The density function is estimated by akj command in R
+Studio, which is a univariate adaptive kernel estimation used by Portnoy and Koenker (1989).
+The left panel displays the result for the causal case. As shown in (7.2), density functions
+in different colors (values of x) share the same shape but the location. Whereas in the right
+panel, where the estimated density is plotted, five estimated conditional density functions
+present distinct modes, skewness, and kurtosis.
+7.2
+Asymptotic Properties of QAR Estimates
+QAR is first proposed by Koenker and Xiao (2006) to study the conditional quantile functions
+of the following pth-order AR process,
+Yt = θ0(Ut) + θ1(Ut)Yt−1 + · · · + θp(Ut)Yt−p,
+(7.4)
+where Ut is an iid sequence distributed as a standard uniform. This expression can be re-
+garded as an AR(p) process allowing coefficients of lags to be random but somewhat depen-
+13Additive property states the sum of independent variables from the same distribution would follow the
+distribution from the same family. The common distributions which share this property are α-stable distri-
+bution, exponential distribution, geometric distribution, etc.
+33
+
+3
+0.10
+Density
+0.30
+N
+0.50
+0.70
+0.90
+0000
+-2
+0
+2
+4
+6
+present3
+Density
+2
+0.10
+0.30
+0.50
+0.70
+0.90
+OD
+-3
+-2
+-1
+0
+presentdent on each other. By the property of monotone transformation, our target, the conditional
+quantile at each τ ∈ (0, 1), can be written as
+QYt (τ|Yt−1, . . . , Yt−p) = θ0(τ) + θ1(τ)Yt−1 + · · · + θp(τ)Yt−p,
+τ ∈ (0, 1).
+(7.5)
+The estimates of θ(τ) = (θ0(τ), θ1(τ), . . . , θp(τ))′ are obtained by minimizing the following
+objective function,
+ˆθ(τ) = argmin
+θ∈Rp+1
+T
+�
+t=1
+ρτ(Yt − X′
+tθ),
+(7.6)
+where X′
+t = (1, Yt−1, . . . , Yt−p) and check function ρτ(u) = u (τ − I(u < 0)). A vectorized
+form of (7.4) is introduced to facilitate the asymptotic analysis of the estimates ˆθ(τ),
+Y t = AtY t−1 + V t,
+with
+At =
+�
+θ1(Ut)
+θ2(Ut)
+. . .
+θp(Ut)
+Ip−1
+0(p−1)×1
+�
+and V t =
+�
+ϵt
+0(p−1)×1
+�
+where ϵt = θ0(Ut) − E (θ0(Ut)) and Y t = (Yt, Yt−1, . . . , Yt−p+1)′. The study of asymptotic
+properties is based on the following conditions
+1. {ϵt} are iid innovations with mean 0 and finite variance σ2 < ∞. The distribution
+function of ϵt, F, admits a continuous density f(ϵ) away from zero on E = {ϵ : 0 <
+F(ϵ) < 1}.
+2. The eigenvalues of E (At ⊗ At) have moduli within unity.
+3. The conditional distribution function P(Yt < ·|Yt−1, Yt−2, . . . ) denoted by Ft−1(·) has
+a density function ft−1(·) uniformly integrable on E.
+Under these three assumptions,
+Σ−1/2√
+T
+�ˆθ(τ) − θ(τ)
+�
+→d Bp+1(τ)
+where Bk(τ) is a k-dimensional Brownian bridge. By definition it can be written as N(0, τ(1−
+τ)Ik) for any given τ. Σ is a matrix characterized by density and distribution function of ϵt.
+Let Σ0 = E (XtX′
+t) and Σ1 = E
+�
+ft−1
+�
+F −1
+t−1(τ)
+�
+XtX′
+t
+�
+. Then Σ is defined as Σ−1
+1 Σ0Σ−1
+1 .
+In the special case where the data generating process is a conventional causal AR model
+with fixed coefficients, the conditional density would be independent of Xt. We will have
+Σ1 = f(F −1(τ)) E (XtX′
+t). Further we can simplify the Σ to f−2(F −1(τ)) E−1 (XtX′
+t).
+7.3
+Proof to Theorem 3.1
+Non-causality ⇒ Nonlinear conditional quantile
+We prove this statement by con-
+tradiction. Consider a stationary MAR(r, s) with innovations satisfying Assumption 1 and
+s > 0.
+Assume all conditional quantiles of Yt are linear in (Yt−1, Yt−2, . . . , Yt−p), where
+34
+
+p = r + s. That is,
+QYt (τ|Yt−1, Yt−2, . . . , Yt−p) = θ0(τ) + θ1(τ)Yt−1 + · · · + θp(τ)Yt−p for any given τ ∈ (0, 1).
+By aggregating QYt (τ|Yt−1, Yt−2, . . . , Yt−p) over the entire quantile range, the linearity is
+maintained for the aggregation. Therefore, we have
+� 1
+0
+QYt (τ|Yt−1, Yt−2, . . . , Yt−p) dτ =
+� 1
+0
+θ0(τ)dτ +
+� 1
+0
+θ1(τ)dτYt−1 + · · · +
+� 1
+0
+θp(τ)dτYt−p
+(7.7)
+Equivalently, we can yield
+E (Yt|Yt−1, Yt−2, . . . , Yt−p) = θ0 + θ1Yt−1 + · · · + θpYt−p,
+which contradicts the statement in Corollary 5.2.3 by Rosenblatt (2000) on the nonlinearity
+of conditional expectation in past information with the presence of non-causality. Thus, the
+presumed statement is not valid. That is to say, there exists a conditional quantile of Yt
+which is nonlinear in the past information for at least one τ ∈ (0, 1) if s > 0.
+Nonlinear conditional quantile ⇒ Non-causality
+This can be demonstrated equiva-
+lently by its contrapositive statement: an AR process Yt being causal implies its conditional
+quantile QYt (τ | It−1) is linear for all τ ∈ (0, 1).
+If a MAR(r, s) is purely causal, i.e., s=0 and r=p. Then we have
+Yt = φ1Yt−1 + φ2Yt−2 + · · · + φrYt−r + ut,
+where ut is independent of past observations.
+The conditional quantile of the response
+variable can be directly expressed as a linear combination of {Yt−j}j=1,...,r,
+QYt (τ|Yt−1, . . . , Yt−r) = Qut (τ|Yt−1, . . . , Yt−r) + φ1Yt−1 + φ2Yt−2 + · · · + φrYt−r for ∀τ ∈ (0, 1)
+= Qut(τ)
+� �� �
+θ0(τ)
++ φ1
+����
+θ1(τ)
+Yt−1 + φ2
+����
+θ2(τ)
+Yt−2 + · · · + φr
+����
+θr(τ)
+Yt−r.
+7.4
+Simulations of Extensions
+In this section, we present some simulation results of non-causality testing strategies extended
+to the autoregressive processes with heteroskedasticity. In this stage, we assume the form of
+heteroskedasticity is known.
+Consider a pair of MAR(1, 0)-ARCH(1) and MAR(0, 1)-ARCH(1) processes defined by
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Yt = 0.7Yt−1 + vt
+Y ∗
+t = 0.7−1Y ∗
+t−1 + vt = 0.7Y ∗
+t+1 − 0.7vt+1
+vt = σtut
+σt = 0.2 + 0.8 |vt−1| where ut ∼ IID(0, 1).
+(7.8)
+35
+
+As explained in the extension section, we want to detect non-causality by checking whether
+the coefficients (except the intercept) in the following linear dynamic quantile model are
+τ-invariant
+QYt (τ|Yt−1, vt−1) = θ0(τ) + θ1(τ)Yt−1 + θ2(τ)|vt−1|,
+τ ∈ Υ ⊂ (0, 1)
+(7.9)
+provided that vt−1 is recovered with 100% accuracy.
+Another approach is to check whether the linear model (7.8) is the correct specification for the
+conditional quantile of Yt and Y ∗
+t . The innovation ut varies from exponential to t student and
+Laplace distributions. The sample size in this trial is 100, 200, 500, and 1000. The trimmed
+quantile interval for the constancy test is [0.05, 0.95] and Υ = [0.01, 0.99] for the specification-
+based test (here in this experiment, we only apply the EV test to see its performance). The
+empirical size and power of non-causality tests for cases with heteroskedasticity are displayed
+in Table 7. It is clear that both methods have fairly good performance, even in relatively small
+samples. Regarding the empirical size, both approaches have a slight distortion compared to
+the nominal level. In the case of the EV test, a different bandwidth can be applied to adjust
+the empirical size to the expected level. Concerning the empirical power, the specification-
+based approach dominates the constancy test in most cases. Except in the case of asymmetric
+distribution, the constancy test outperforms the EV test in small samples ( T=100 and 200
+). It is conceivable that when we replace vt by its estimate ˆvt, the asymptotic effect of the
+estimation needs to be taken into account when we construct test statistics. This is beyond
+the scope of this paper.
+Table 7: Empirical size and power of non-causality tests for AR-ARCH models with known heteroskedasticity
+Distribution
+test type
+T=100
+T=200
+T=500
+T=1000
+ut
+size
+power
+size
+power
+size
+power
+size
+power
+Exponential
+constancy test
+3.60%
+39.00%
+4.40%
+54.60%
+6.00%
+75.80%
+7.00%
+90.80%
+EV test
+5.20%
+16.80%
+4.40%
+34.80%
+4.60%
+82.80%
+5.00%
+97.80%
+t student
+constancy test
+5.20%
+24.60%
+6.60%
+39.40%
+6.00%
+63.60%
+7.80%
+78.00%
+EV test
+8.20%
+51.00%
+7.20%
+83.20%
+8.20%
+99.80%
+7.20%
+100.00%
+Laplace
+constancy test
+4.40%
+14.40%
+3.80%
+25.00%
+7.60%
+38.80%
+4.40%
+55.00%
+EV test
+7.00%
+45.40%
+6.40%
+82.60%
+8.60%
+99.20%
+7.20%
+10.00%
+The bandwidth for approximating the critical value using subsampling for the EV test is 7.
+36
+

T9E3T4oBgHgl3EQfEQll/content/tmp_files/2301.04294v1.pdf.txt

@@ -0,0 +1,4073 @@
+Prepared for submission to JHEP
+Thrust distribution in Higgs decays up to the fifth
+logarithmic order
+Wan-Li Ju,a Yongqi Xu,b Li Lin Yang,c Bin Zhoud
+aINFN, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy
+bSchool of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University,
+Beijing 100871, China
+cZhejiang Institute of Modern Physics, School of Physics, Zhejiang University, Hangzhou 310027,
+China
+dINPAC, Shanghai Key Laboratory for Particle Physics and Cosmology, School of Physics and
+Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
+E-mail: Wanli.Ju@mi.infi.it, xuyongqi@pku.edu.cn,
+yanglilin@zju.edu.cn, zb0429@sjtu.edu.cn
+Abstract: In this work, we extend the resummation for the thrust distribution in Higgs
+decays up to the fifth logarithmic order. We show that one needs the accurate values of
+the three-loop soft functions for reliable predictions in the back-to-back region. This is
+especially true in the gluon channel, where the soft function exhibits poor perturbative
+convergence.
+arXiv:2301.04294v1  [hep-ph]  11 Jan 2023
+
+Contents
+1
+Introduction
+2
+2
+Theoretical framework
+2
+3
+Numeric results
+7
+3.1
+Choice of parameters and estimation of uncertainties
+7
+3.2
+The resummed thrust distributions in the gluon channel
+7
+3.3
+The resummed thrust distributions in the quark-antiquark channel
+10
+4
+Summary and outlook
+10
+A Choice of scales in the momentum space
+11
+B Fixed order ingredients
+14
+C Anomalous dimensions
+20
+– 1 –
+
+1
+Introduction
+The hadronic decays of the Higgs boson provide a unique windows to study the Yukawa
+couplings of the lighter quarks such as the charm quark and the strange quark. These
+rare decays might be enhanced by new physics effects beyond the standard model (see,
+e.g., [1, 2]), and can be probed at the Large Hadron Collider (LHC) and the future Higgs
+factories [3–7]. However, the hadronic decays of the Higgs boson can also proceed through
+the H → gg partonic channel. While the gluon channel is also useful, it is desirable to
+distinguish it from the H → q¯q channel to gain maximal information about the Yukawa
+couplings. To this end, it is important to study various differential distributions in these
+two channels.
+A classical differential distribution for hadronic final states is the event shape variable
+“thrust” [8]. It was extensively studied in the process e+e− → hadrons. In the context of
+H → hadrons, the next-to-leading order (NLO) and approximate next-to-next-to-leading
+order (NNLO) predictions were calculated in [9]. These fixed-order results suffer from large
+logarithms in the endpoint region, which need to be resummed to all orders in the strong
+coupling αs. The resummation framework is also well-established for e+e− → hadrons
+[10–18]. The applications to the Higgs case were carried out in [19, 20] at the next-to-
+next-to-leading logarithmic (NNLL and NNLL′) accuracies. In this work, we extend the
+resummation accuracy up to the fifth order, and present the results at N3LL′ and N4LL.
+The paper is organized as follows. In Section 2 we briefly review the factorization for-
+mula for the thrust distribution, and give technical details of the resummation framework.
+In Section 3 we provide numeric results for the resummed thrust distributions, with jet and
+soft scales chosen in the Laplace space. The summary and outlook come in Section 4. The
+alternative results with jet and soft scales chosen in the momentum space are presented in
+Appendix A, and we leave some lengthy expressions to the remaining Appendices.
+2
+Theoretical framework
+We consider the process H → hadrons induced by the following effective Lagrangian
+Leff = αs(µ)Ct(mt, µ)
+12πv
+Og +
+�
+q
+yq(µ)
+√
+2 Oq
+≡ αs(µ)Ct(mt, µ)
+12πv
+HGµν,aGa
+µν +
+�
+q
+yq(µ)
+√
+2 H ¯ψqψq ,
+(2.1)
+where v is the Higgs vacuum expectation value; H represents the physical Higgs boson
+after electroweak symmetry breaking; Ga
+µν is the field strength tensor of the gluon field;
+ψq represent the light quark fields. We will ignore the masses of the light quarks, but keep
+the Yukawa couplings yq non-vanishing. The strong coupling αs, the Yukawa coupling yq
+and the Wilson coefficient Ct of the effective operator are renormalized in the MS scheme
+at the scale µ.
+The thrust variable T is defined as
+T ≡ max
+⃗n
+�
+i |⃗n · ⃗pi|
+�
+i |⃗pi|
+,
+(2.2)
+– 2 –
+
+where ⃗pi denote the 3-momenta of final state particles. The unit vector ⃗n that maximize the
+above ratio is called the thrust axis. For convenience we introduce the variable τ ≡ 1 − T.
+In this work we are concerned with the limit T → 1 or τ → 0. Physically this corresponds
+to two back-to-back jets in the final state. In this limit the differential decay rate can be
+factorized into the product (convolution) of a hard function, a soft function and two jet
+functions [9–12, 21–24]:
+dΓq
+dτ = Γq
+B(µ) |Cq
+S(mH, µ)|2
+�
+dp2
+n dp2
+¯n dk δ
+�
+τ − p2
+n + p2
+¯n
+m2
+H
+−
+k
+mH
+�
+× Jq
+n(p2
+n, µ) Jq
+¯n(p2
+¯n, µ) Sq(k, µ) ,
+dΓg
+dτ = Γg
+B(µ) |Ct(mt, µ)|2 |Cg
+S(mH, µ)|2
+�
+dp2
+n dp2
+¯n dk δ
+�
+τ − p2
+n + p2
+¯n
+m2
+H
+−
+k
+mH
+�
+× Jg
+n(p2
+n, µ) Jg
+¯n(p2
+¯n, µ) Sg(k, µ) ,
+(2.3)
+where the superscript q or g labels the partonic subprocesses H → q¯q or H → gg, with
+Γq
+B and Γg
+B being the corresponding total decay rates at the Born level. Their explicit
+expressions are
+Γq
+B = y2
+q(µ) mH CA
+16π
+,
+Γg
+B = α2
+s(µ) m3
+H
+72 π3 v2 .
+(2.4)
+In the factorization formula, Ci
+S (with i = q, g) are hard Wilson coefficients arising when
+matching the full theory of QCD to the soft-collinear effective theory (SCET) [25–30]; Si
+are soft functions defined as the vacuum expectation values of soft Wilson-loop operators;
+Ji
+n and Ji
+¯n are jet functions along the two light-like directions nµ = (1,⃗n) and ¯nµ = (1, −⃗n),
+where ⃗n is the thrust axis.
+The various ingredients satisfy renormalization group (RG) equations
+d
+d ln µyq(µ) = γy(αs(µ)) yq(µ) ,
+d
+d ln µCt(mt, µ) = γt(αs(µ)) Ct(µ2) ,
+d
+d ln µCi
+S(mH, µ) =
+�
+Γi
+cusp(αs(µ)) ln −m2
+H − iϵ
+µ2
++ γi
+H(αs(µ))
+�
+Ci
+S(mH, µ) ,
+d
+d ln µJi(p2, µ) =
+�
+−2Γi
+cusp(αs(µ)) ln p2
+µ2 − 2γi
+J(αs(µ))
+�
+Ji(p2, µ)
++ 2Γi
+cusp(αs(µ))
+� p2
+0
+dq2 Ji(p2, µ) − Ji(q2, µ)
+p2 − q2
+,
+d
+d ln µSi(k, µ) =
+�
+4Γi
+cusp(αs(µ)) ln k
+µ − 2γi
+S(αs(µ))
+�
+Si(k, µ)
+− 4Γi
+cusp(αs(µ))
+� k
+0
+dq Si(k, µ) − Si(q, µ)
+k − q
+.
+(2.5)
+Note that the evolution equations for the jet and soft functions involve convolutions. It is
+useful to introduce the Laplace transformed functions
+˜ji(LJ, µ) =
+� ∞
+0
+dp2 exp
+�
+−Np2
+m2
+H
+�
+Ji(p2, µ) ,
+– 3 –
+
+˜si(LS, µ) =
+� ∞
+0
+dk exp
+�
+− Nk
+mH
+�
+Si(k, µ) ,
+(2.6)
+where
+LJ = ln m2
+H
+µ2 ¯N ,
+LS = ln mH
+µ ¯N ,
+(2.7)
+with ¯N ≡ NeγE and γE being the Euler’s constant. The Laplace-space jet and soft functions
+satisfy local RG equations
+d
+d ln µ
+˜ji(LJ, µ) =
+�
+−2Γi
+cusp(αs(µ))LJ − 2γi
+J(αs(µ))
+� ˜ji(LJ, µ) ,
+d
+d ln µ ˜si(LS, µ) =
+�
+4Γi
+cusp(αs(µ))LS − 2γi
+S(αs(µ))
+�
+˜si(LS, µ) .
+(2.8)
+Under the Laplace transform, the differential decay rates are expressed as
+� ∞
+0
+dτ e−τN dΓq
+dτ = Γq
+B(µ) |Cq
+S(mH, µ)|2 �˜jq(LJ, µ)
+�2 ˜sq(LS, µ) ,
+� ∞
+0
+dτ e−τN dΓg
+dτ = Γg
+B(µ) |Ct(mt, µ)|2 |Cg
+S(mH, µ)|2 �˜jg(LJ, µ)
+�2 ˜sg(LS, µ) .
+(2.9)
+For small τ, the dominant contribution arises from the region of large N. In this case the
+large logarithms LJ and LS appear with increasing powers at each order in the perturbative
+expansions of the jet and soft functions. We will resum these logarithms to all orders in
+αs using RG evolution.
+In the RG equations, Γi
+cusp are the cusp anomalous dimensions, which are known up
+to the four-loop accuracy [31–34], and their fifth order contributions were estimated in
+[35]. The beta function governing the running strong coupling is known up to the five-loop
+order in [36–41]. The anomalous dimension γy for the Yukawa coupling is the same as
+that for the quark masses in the MS scheme, and is known up to the fifth order in [42–46].
+The non-cusp anomalous dimension γt governing scale evolution of the Wilson coefficient
+Ct is also known on the fifth level [47–52]. The up-to four-loop results for the remaining
+anomalous dimensions can be found for γi
+H in [53–62], for γi
+S in [63–66], and for γi
+J in
+[66–73]. The Wilson coefficient Ct is known up to the four-loop order [49–51, 74–77] and
+so is the hard sector [55, 57–62, 78] in the factorization of Eq. (2.3). As for the fixed-order
+expansions of the jet functions and the soft functions, the analytic results are known up
+to the three-loop order [11, 12, 67–73, 79, 80], with the exception of the scale-independent
+terms of the three-loop soft function. We collect all these ingredients in the Appendices B
+and C. They allow us to perform the resummation of large logarithms to the N3LL′ order
+and approximately to the N4LL order. For the counting of logarithmic orders, we refer to
+Table 1.
+To resum the large logarithms, we choose appropriate scales for each of the functions
+in the factorization formula, and use the RG equations to evolve them to a common scale.
+The choice of scales can be done either in the Laplace N-space or in the momentum τ-
+space. In the following, we present results with scale choices in the Laplace space, while
+– 4 –
+
+Logarithmic accuracy
+Γcusp, β
+γt,y,H,j,s
+Ct, CS, ˜j, ˜s
+NNLL′
+3-loop
+2-loop
+2-loop
+N3LL
+4-loop
+3-loop
+2-loop
+N3LL′
+4-loop
+3-loop
+3-loop
+N4LL
+5-loop
+4-loop
+3-loop
+Table 1. Definitions of the logarithmic orders.
+those in the momentum space will be discussed in Appendix A. We choose the scales for
+the Ct, CS, ˜j and ˜s functions to be
+µt = etmt ,
+µh = ehmH ,
+µj = ej
+mH
+√ ¯N
+,
+µs = es
+mH
+¯N ,
+(2.10)
+where by default we take et = eh = ej = es = 1, and we vary them up and down by a factor
+of two to estimate the associated uncertainties. The resummed differential decay rates in
+the Laplace space can be written as
+�Γq(N) = Γq
+B(µh) U q(µh, µj, µs) |Cq
+S(mH, µh)|2 �˜jq(LJ, µj)
+�2 ˜sq(LS, µs)
+� mH
+µs ¯N
+�ηq
+,
+�Γg(N) = Γg
+B(µh) U g(µt, µh, µj, µs) |Ct(mt, µt)|2 |Cg
+S(mH, µh)|2
+×
+�˜jg(LJ, µj)
+�2 ˜sg(LS, µs)
+� mH
+µs ¯N
+�ηg
+,
+(2.11)
+where the evolution functions are given by
+U q(µh, µj, µs) = exp
+�
+4Sq(µh, µj) + 4Sq(µs, µj) − 2Aq
+cusp(µh, µj) ln m2
+H
+µ2
+h
+− 2Aq
+S(µh, µs) − 4Aq
+J(µh, µj)
+�
+,
+U g(µt, µh, µj, µs) = exp
+�
+2At(µh, µt) + 4Sg(µh, µj) + 4Sg(µs, µj) − 2Ag
+cusp(µh, µj) ln m2
+H
+µ2
+h
+− 2Ag
+S(µh, µs) − 4Ag
+J(µh, µj)
+�
+,
+(2.12)
+in which the functions Sq,g and Aq,g
+i
+are defined by [81]
+Sq,g(ν, µ) = −
+� αs(µ)
+αs(ν)
+dαs
+Γq,g
+cusp(αs)
+β(αs)
+� αs
+αs(ν)
+d˜αs
+β(˜αs) ,
+Aq,g
+i (ν, µ) = −
+� αs(µ)
+αs(ν)
+dαs
+γq,g
+i
+(αs)
+β(αs) ,
+(2.13)
+for i = cusp, t, S, J, and ηq,g = 4Aq,g
+cusp(µj, µs). The momentum-space differential decay
+rates can then be obtained through an inverse Laplace transform
+dΓq,g
+dτ
+=
+1
+2πi
+� +i∞
+−i∞
+dN eNτ �Γq,g(N) .
+(2.14)
+– 5 –
+
+The integration contour should, in principle, be chosen such that all singularities of the
+integrand are situated to the left side. However, the resummed integrand develops a Landau
+pole at large N due to the scale choices µj ∼ mH/
+√ ¯N and µs ∼ mH/ ¯N, which signals
+the breakdown of perturbation theory in that region. Correspondingly, the inverse Laplace
+transform of the perturbatively resummed integrand suffers from an ambiguity of non-
+perturbative origin. We adopt the so-called Minimal Prescription [82], in which the contour
+lies to the right of all physical singularities but to the left of the Landau pole.
+With the generic framework, we still need to specify a few details in the evaluation of
+the resummed differential decay rates at a given logarithmic accuracy. The strong coupling
+αs at a given scale µ is evaluated according to
+αs(µ) = αs(ν)
+X
+�
+1 − αs(ν)
+4πX
+β1 ln(X)
+β0
++
+�αs(ν)
+4πX
+�2 �β2
+1
+β2
+0
+�
+ln2(X) − ln(X) − 1 + X
+�
++ β2
+β0
+(1 − X)
+�
++
+�αs(ν)
+4πX
+�3 �β3
+1
+β3
+0
+�
+− X2
+2 + X − ln3(X) + 5 ln2(X)
+2
++ 2(1 − X) ln(X) − 1
+2
+�
++ β3
+2β0
+(1 − X2) + β1β2
+β2
+0
+�
+2X ln(X) − 3 ln(X)
+− X(1 − X)
+��
++
+�αs(ν)
+4πX
+�4 �
+− β4(X3 − 1)
+3β0
++ β3β1
+6β2
+0
+�
+(X − 1)(4X2 + X + 1)
++ 6(X2 − 2) ln(X)
+�
++ β2
+2(X − 1)2(X + 5)
+3β2
+0
++ β2β2
+1
+β3
+0
+�
+− (X + 3)(X − 1)2
++ ln(X)(−2X2 + 5X − 3) − 3(X − 2) ln2(X)
+�
++ β4
+1
+6β4
+0
+�
+(X − 1)2(2X + 7)
++ 6 ln4(X) − 26 ln3(X) + 9(2X − 1) ln2(X) + 6(X − 4)(X − 1) ln(X)
+���
+,
+(2.15)
+where the initial scale is chosen at the Z boson mass, ν = mZ, and
+X = 1 + αs(ν)
+2π β0 ln µ
+mZ
+.
+(2.16)
+The coefficients of the beta function are defined through
+dαs
+d ln µ = −2αs
+∞
+�
+n=0
+�αs
+4π
+�n+1
+βn .
+(2.17)
+The Yukawa coupling yq at a given scale is evaluated with
+yq(µ) = yq(mH) exp
+�
+Aq
+y(µ, mH)
+�
+.
+(2.18)
+The evolution factors U q,g and the factors involving ηq,g are expanded on the expo-
+nent up to a given logarithmic accuracy defined in Table 1. The expansion is done by
+counting the large logarithms ln(ν/µ) as O(1/αs). The fixed-order factors (|Cq
+S|2 �˜jq�2 ˜sq
+and |Ct|2|Cg
+S|2 �˜jg�2 ˜sg) are also expanded up to a given loop order. We are now ready
+to perform numeric evaluations of the resummed differential decay rates. The results are
+presented in the next Section.
+– 6 –
+
+3
+Numeric results
+3.1
+Choice of parameters and estimation of uncertainties
+In this section we are devoted to the numeric results. Throughout this paper, we choose
+αs(mZ) = 0.1181, mH = 125.1 GeV and mt = 172.9 GeV [83]. The scales are chosen as in
+Eq. (2.10). Note that this choice is conventional in the small-τ region considered in this
+work. On the other hand, if one wants to match the resummed distributions to the fixed-
+order ones, it is necessary to deal with the intermediate regime between the resummation
+dominated small-τ region and the fixed-order dominated large-τ region.
+We leave this
+subtlety to future investigations. We estimate the perturbative uncertainties by varying
+each of et, eh, ej and es up and down by a factor of two, while keeping the others at
+their defaults. The resulting variations of the differential decay rates are then added in
+quadrature.
+The scale-independent constant terms of the three-loop soft functions are not known
+yet. The term in the quark channel was extracted in [71] through a numeric fit to the
+fixed-order thrust distribution, which has a large uncertainty. In this work we set
+cS
+3q = −19988 ± 5000 ,
+(3.1)
+and estimate the corresponding variation of the resummed distribution.
+For the gluon
+channel we apply the Casimir scaling and set
+cS
+3g = −45433 ± 11250 .
+(3.2)
+The five-loop cusp anomalous dimensions are also unknown, with only a rough estimation
+available [35]. However, we have checked that they only have a rather mild effect on the
+resummed thrust distributions.
+3.2
+The resummed thrust distributions in the gluon channel
+We now show the resummed thrust distributions in the gluon channel at various logarithmic
+accuracies. The baseline for comparison is the NNLL′ result, that is the state-of-the-art
+accuracy in the literature (see Refs. [19, 20], although they adopted scale choices in the
+momentum space).
+In Fig. 1, we show the comparison between NNLL′ and N3LL, and that between NNLL′
+and N3LL′, for the τ range [0.01, 0.15]. This covers the small-τ and intermediate-τ regions,
+but cuts out the large-τ region where fixed-order matching would be important. One can
+see that the N3LL result has a slightly reduced scale uncertainty compared to the NNLL′
+one. The reduction is most significant in the small-τ region, where resummation effects are
+expected to be important. The N3LL′ result further reduces the scale uncertainty in the
+intermediate τ region, with the three-loop hard, jet and soft functions included. However,
+we observe an unusual increase of scale uncertainty in the small τ region, as is clear from the
+right plot of Fig. 1. It can be seen that the two bands even do not overlap below τ ∼ 0.03.
+This fact can be traced to the unusually large constant term cS
+3g of the three-loop soft
+– 7 –
+
+0.00
+0.03
+0.06
+0.09
+0.12
+0.15
+0
+3
+6
+9
+12
+15
+1
+B
+d
+d
+H
+gg, Laplace
+NNLL′
+N3LL
+0.00
+0.03
+0.06
+0.09
+0.12
+0.15
+0
+3
+6
+9
+12
+15
+1
+B
+d
+d
+H
+gg, Laplace
+NNLL′
+N3LL′
+Figure 1. The resummed thrust distributions in the gluon channel. Left: NNLL′ vs. N3LL; Right:
+NNLL′ vs. N3LL′.
+0.00
+0.03
+0.06
+0.09
+0.12
+0.15
+0
+3
+6
+9
+12
+15
+1
+B
+d
+d
+H
+gg, Laplace
+N3LL′, default
+N3LL′, cS
+3g
+N3LL′, cS
+3g
+0.00
+0.03
+0.06
+0.09
+0.12
+0.15
+0
+3
+6
+9
+12
+15
+1
+B
+d
+d
+H
+gg, Laplace
+NNLL′
+N3LL′, cS
+3g
+Figure 2. The effects of cS
+3g on the N3LL′ results in the gluon channel. Left: N3LL′ results with
+3 values of cS
+3g; Right: NNLL′ vs. N3LL′ where cS
+3g is taken to its “upper” value (with a smaller
+absolute value).
+function. It is instructive to show the soft function at its default scale µs = mH/ ¯N, where
+LS = 0, for cS
+3g = −45433:
+˜sg(0, µs) = 1 − 2.356 αs(µs) + 1.617 α2
+s(µs) − 22.90 α3
+s(µs) + · · · .
+(3.3)
+For small τ, one expects that the dominant contributions in the Laplace space come from
+the region where τN ∼ 1. This means that, below τ ∼ 0.03, µs is typically only about a
+few GeVs, where αs ∼ 0.2 is not so small. Therefore, the gluon soft function has a rather
+poor perturbative convergence if we take the fitted central value of cS
+3g. It is highly desired
+to calculate the exact value of cS
+3g to settle down this issue: either its absolute value is in
+fact smaller, and the N3LL′ result is already sufficient; or it is indeed that large, then one
+needs to have even higher order corrections for reliable predictions. Efforts towards this
+goal are being actively pursued in the literature [84, 85].
+To demonstrate the effects of different values of cS
+3g, we show in the left plot of Fig. 2
+– 8 –
+
+H → gg, N3LL′
+µt
+µh
+µj
+µs
+cS
+3g
+max
+12.128
+12.120
+12.120
+13.077
+12.219
+min
+12.106
+11.967
+12.063
+12.044
+12.020
+Table 2. Variations of the N3LL′ differential decay rate at τ = 0.05 induced by changing the scales
+and cS
+3g. The central value is 12.120.
+0.00
+0.03
+0.06
+0.09
+0.12
+0.15
+0
+3
+6
+9
+12
+15
+1
+B
+d
+d
+H
+gg, Laplace
+N3LL′
+N4LL
+0.00
+0.03
+0.06
+0.09
+0.12
+0.15
+0
+3
+6
+9
+12
+15
+1
+B
+d
+d
+H
+gg, Laplace
+N4LL, default
+N4LL, c(3)
+S
+N4LL, c(3)
+S
+Figure 3. The resummed thrust distributions in the gluon channel. Left: N3LL′ vs N4LL; Right:
+N4LL results with 3 values of cS
+3g.
+H → gg, N4LL
+µt
+µh
+µj
+µs
+cS
+3g
+Γg(4)
+cusp
+max
+12.089
+12.084
+12.084
+12.980
+12.183
+12.085
+min
+12.073
+11.936
+12.025
+12.022
+11.985
+12.083
+Table 3. Variations of the N4LL differential decay rate at τ = 0.05 induced by changing the scales,
+cS
+3g and the five-loop cusp anomalous dimension Γg(4)
+cusp. The central value is 12.084.
+the resummed distributions for three values of cS
+3g: the default value −45433, the “lower”
+value −56683, and the “upper” value −34183. Note that since the fitted value of cS
+3g is
+negative, the “upper” value has a smaller absolute value, and leads to a better perturbative
+convergence.
+Indeed, as can be seen from the plot, the result with the “upper” value
+exhibits a smaller scale uncertainty, especially in the small-τ region. It is also evident from
+the right plot of Fig. 2, that the N3LL′ band is better overlapped with the NNLL′ one with
+cS
+3g taking the “upper” value. Finally for reference, we list in Table 2 the variations of the
+N3LL′ differential decay rate at τ = 0.05 induced by changing the values of various scales
+as well as cS
+3g. It is clear that the main source of the scale uncertainty comes from the soft
+scale, as expected. It can also be seen that varying cS
+3g has a larger effect than varying
+µt, µh or µj. All these emphasize again that we need a better understanding of the soft
+function at and beyond three loops.
+We now add another layer of resummation on top of N3LL′, and present the results at
+N4LL. The results are shown in Fig. 3, with explicit numbers at τ = 0.05 given in Table 3.
+We find that the additional order of resummation has a mild effect on the distribution,
+– 9 –
+
+0.01
+0.03
+0.05
+0.07
+0.09
+0.11
+0.13
+0.15
+0
+5
+10
+15
+20
+25
+30
+35
+1
+B
+d
+d
+H
+qq, Laplace
+NNLL′
+N3LL′
+0.01
+0.03
+0.05
+0.07
+0.09
+0.11
+0.13
+0.15
+0
+5
+10
+15
+20
+25
+30
+35
+1
+B
+d
+d
+H
+qq, Laplace
+N3LL′, default
+N3LL′, c(3)
+S
+N3LL′, c(3)
+S
+Figure 4. The resummed thrust distributions in the q¯q channel. Left: NNLL′ vs. N3LL′; Right:
+N3LL′ results with 3 values of cS
+3q.
+that is only clearly visible in the peak region. It is also evident that the five-loop cusp
+anomalous dimension does not have important impacts.
+3.3
+The resummed thrust distributions in the quark-antiquark channel
+We now briefly discuss the results in the quark-antiquark channel.
+In the left plot of
+Fig. 4, we compare the NNLL′ result against the N3LL′ one, where the three-loop constant
+term cS
+3q of the soft function is chosen at the default value. We again observe that the
+uncertainty band of N3LL′ is broader than the NNLL′ one, especially at the lower end of
+the distribution. In the right plot of Fig. 4, we show the N3LL′ distributions for 3 values of
+cS
+3q: the default value −19988, the “upper” value −14988 and the “lower” value −24988. As
+expected, the band becomes narrower for the “upper” value, where the absolute value of cS
+3q
+is smaller, and the soft function has a perturbative convergence. Overall, the uncertainties
+of the resummed thrust distributions in the q¯q channel are significantly smaller than those
+in the gluon channel. This can be partly explained by the smaller color factor CF compared
+to CA.
+4
+Summary and outlook
+In this work, we extend the resummation for the thrust distribution in Higgs decays up
+to the fifth logarithmic order. A main conclusion that can be drawn from our results is
+that one needs the accurate values of the three-loop soft functions for reliable predictions
+in the small-τ region. This is especially true in the gluon channel, where the perturbative
+convergence of the soft function seems to be rather bad with a large three-loop constant
+term.
+Once the three-loop soft functions become available, the ingredients collected in this
+work will allow for faithful numeric predictions at the N3LL′ and N4LL accuracies. Depend-
+ing on the size of the three-loop constant, it is possible that one even needs the four-loop
+gluon soft function to reduce the scale uncertainties and obtain reliable predictions in the
+small-τ region.
+– 10 –
+
+Acknowledgments
+This work was supported in part by the National Natural Science Foundation of China
+under Grant No. 11975030 and 12147103, and the Fundamental Research Funds for the
+Central Universities.
+A
+Choice of scales in the momentum space
+In the main text, we have chosen the jet and soft scales in the Laplace space, and performed
+the inverse Laplace transform numerically. A different approach is to set the jet and soft
+scales independent of the Laplace variable N. In this case, the inverse Laplace transform
+(2.14) can be carried out analytically [12, 81, 86]. For simplicity we only discuss the gluon
+channel in this Appendix. The result can be written as
+dΓg
+dτ = Γg
+B(µh) U g(µt, µh, µj, µs) |Ct(mt, µt)|2 |Cg
+S(mH, µh)|2
+×
+�
+˜jg
+�
+ln µsmH
+µ2
+j
++ ∂ηg, µj
+��2
+˜sg(∂ηg, µs)
+�
+1
+τ Γ(ηg)
+� τmH
+µseγE
+�ηg�
+.
+(A.1)
+The common practice is then to choose µs and µj as a function of τ, such that µj ∼ √τ mH
+and µs ∼ τmH in the small-τ region. In this work we don’t care about matching with the
+fixed-order results in the large-τ region (otherwise one needs to introduce “profile scales”
+as in [19, 20]). Therefore we can adopt the simplest choices
+µt = etmt ,
+µh = ehmH ,
+µj = ej
+√τ mH ,
+µs = esτmH ,
+(A.2)
+where the parameters et, eh, ej and es are set to 1 by default, and are varied up and down
+by a factor of two to estimate the uncertainties. The numeric results with the above scale
+choices are shown in Fig. 5. We observe very large scale uncertainties in the small-τ region,
+much larger than those seen in Fig. 1. As it turns out, these large uncertainties originate
+from the jet and soft scales, i.e., ej and es.
+The prescription in Eq. (A.2) actually has a subtle problem related exactly to the jet
+and soft scales. The formula (A.1) is based on the solutions to the RG equations (2.5).
+Taking the gluon jet function as an example, the RG equation is
+d
+d ln µJg(p2, µ) =
+�
+−2Γg
+cusp(αs(µ)) ln p2
+µ2 − 2γg
+J(αs(µ))
+�
+Jg(p2, µ)
++ 2Γg
+cusp(αs(µ))
+� p2
+0
+dq2 Jg(p2, µ) − Jg(q2, µ)
+p2 − q2
+.
+(A.3)
+And the solution reads [81, 86]
+Jg(p2, µ) = exp
+�
+−4Sg(µj, µ) + 2Ag
+J(µj, µ)
+� ˜jg(∂η, µj)
+�
+1
+p2
+�
+p2
+µ2
+j
+�η�
+∗
+e−γEη
+Γ(η) ,
+(A.4)
+– 11 –
+
+0.01
+0.03
+0.05
+0.07
+0.09
+0.11
+0.13
+0.15
+0
+3
+6
+9
+12
+15
+1
+B
+d
+d
+H
+gg, Momentum, Old
+NNLL′
+N3LL′
+Figure 5. The resummed thrust distributions at NNLL′ and N3LL′ in the gluon channel with the
+scale choices (A.2) at the level of differential decay rates.
+where η = 2Ag
+cusp(µj, µ), and the star-distribution is defined by
+� Q2
+0
+dp2
+�
+1
+p2
+�
+p2
+µ2
+j
+�η�
+∗
+f(p2) =
+� Q2
+0
+dp2 f(p2) − f(0)
+p2
+�
+p2
+µ2
+j
+�η
++ f(0)
+η
+�Q2
+µ2
+�η
+.
+(A.5)
+The solution is of course formally independent of µj. However, at any finite order there is
+a residue dependence. Assuming that µj is independent of p2, the above solution indeed
+satisfies the RG equation (A.3), where µj is the same in both Jg(p2, µ) and Jg(q2, µ).
+However, the scale choice in Eq. (A.2) actually makes µj correlated with p2 ∼ τm2
+H. That
+is, µ2
+j ∼ p2 in Jg(p2, µ), and µ2
+j ∼ q2 in Jg(q2, µ). This immediately renders the convolution
+in Eq. (A.3) ill-defined due to the singularity as q2 → 0.
+One may of course ignore the problem with Eq. (A.3) and insist on using Eq. (A.4)
+with µj ∼ p2 as the resummed jet function. As long as one does not take p2 → 0 (where
+non-perturbative physics enters anyway), this does not pose any difficulty at face value.
+However, let us take a closer look. We set µj = ej
+�
+p2 in Eq. (A.4), and truncate the
+solution at NLL′ accuracy (that means two-loop Γg
+cusp, one-loop γg
+J and one-loop ˜jg). We
+then expand the resummed jet function in terms of αs(µ). This gives
+Jg,NLL′ �
+p2, µ; µj = ej
+�
+p2
+�
+= αs(µ)
+4π
+1
+p2
+�
+12 ln p2
+µ2 − 23
+3
+�
++
+�αs(µ)
+4π
+�2 1
+p2
+×
+�
+576 ln3(ej) + 736 ln2(ej) +
+�9364
+9
+− 144π2
+�
+ln(ej) + 72 ln3
+� p2
+µ2
+�
++ · · ·
+�
++ O(α3
+s) ,
+(A.6)
+where the ellipsis denotes further ej-independent terms. We can see that the ej-dependence
+cancels at order α1
+s, as it should at the NLL′ accuracy. However, there exist rather high
+powers of ln(ej) at order α2
+s (and beyond). While ln(ej) is counted as a “small” logarithm,
+these high powers lead to large uncertainties of the resummed jet function when ej is
+varied between 1/2 and 2. The resummed soft function Sg(k, µ) with µs = esk exhibits
+the similar behavior. These explain the large uncertainty bands observed in Fig. 5. In
+– 12 –
+
+practice, it is often argued that ej and es are not independent and should be correlated.
+This can result in partial cancellations between the lnn(ej) and lnn(es) terms, and lead to
+smaller uncertainty estimations.
+The above behavior does not occur with the scale choices in Eq. (2.10). We can do the
+same exercise with µj = ejmH/
+√ ¯N in the Laplace space. The resummed Laplace-space
+jet function is given by
+˜jg(LJ, µ) = exp
+�
+−4Sg(µj, µ) + 2Ag
+J(µj, µ)
+�
+�
+m2
+H
+¯N µ2
+j
+�η
+˜jg(LJ, µj) .
+(A.7)
+Again truncating at the NLL′ accuracy and expanding in terms of αs(µ), we can perform
+the inverse Laplace transform analytically to arrive at
+Jg,NLL′ �
+p2, µ; µj = ejmH/
+�
+¯N
+�
+= αs(µ)
+4π
+1
+p2
+�
+12 ln p2
+µ2 − 23
+3
+�
++
+�αs(µ)
+4π
+�2 1
+p2
+�
+72 ln3
+� p2
+µ2
+�
++ · · ·
+�
++ O(α3
+s) .
+(A.8)
+Evidently, all the lnn(ej) terms drop out at order α2
+s. This explains the smaller uncertainty
+bands in Fig. 1 compared to Fig. 5.
+There is an alternative way of choosing the jet and soft scales in the momentum space
+[18, 87, 88]. We define the accumulative decay rate as
+ˆΓg(τ) ≡
+� τ
+0
+dΓg
+dτ ′ dτ ′ .
+(A.9)
+In the above integrals, µj and µs are set to the same expressions as in Eq. (A.2), where τ
+is the upper bound of the integration. The jet and soft scales are hence independent of the
+integration variable τ ′. Before resummation, the factorization formula for the accumulative
+decay rates can be obtained by integrating Eq. (2.3) over τ. The result reads
+ˆΓg(τ) = Γg
+B(µ) |Ct(mt, µ)|2 |Cg
+S(mH, µ)|2
+�
+dp2
+n dp2
+¯n dk ˆJg
+n(p2
+n, µ) ˆJg
+¯n(p2
+¯n, µ) Sg(k, µ) ,
+(A.10)
+where the integration domain is determined by the constraints
+τ ≥ p2
+n + p2
+¯n
+m2
+H
++
+k
+mH
+,
+p2
+n, p2
+¯n, k ≥ 0 .
+(A.11)
+Now, since the jet scale µj is a function of τ and is independent of p2, the problem with
+the convolution in Eq. (A.3) is absent here.
+After resummation, the accumulative decay rate is
+ˆΓg(τ) = Γg
+B(µh) U g(µt, µh, µj, µs) |Ct(mt, µt)|2 |Cg
+S(mH, µh)|2
+×
+�
+˜jg
+�
+ln µsmH
+µ2
+j
++ ∂ηg, µj
+��2
+˜sg(∂ηg, µs)
+�
+1
+Γ(1 + ηg)
+� τmH
+µseγE
+�ηg�
+.
+(A.12)
+– 13 –
+
+0.01
+0.03
+0.05
+0.07
+0.09
+0.11
+0.13
+0.15
+0
+3
+6
+9
+12
+15
+1
+B
+d
+d
+H
+gg, Momentum, New
+NNLL′
+N3LL′
+Figure 6. The resummed thrust distributions at NNLL′ and N3LL′ in the gluon channel with the
+scale choices (A.2) at the level of accumulative decay rates.
+We set the jet and soft scales at the accumulative level following Eq. (A.2), i.e., µj =
+ej
+√τmH and µs = esτmH. We then take the derivative with respect to τ to obtain the
+resummed differential decay rate:
+dΓg
+dτ = d
+dτ
+ˆΓg(τ) .
+(A.13)
+The numeric results with this new prescription are plotted in Fig. 6.
+To compare the “new” way of scale setting at the accumulative level with the “old”
+way at the differential level, we can study the resummed integrated jet function
+ˆJg(p2, µ) =
+� p2
+0
+dq2 Jg(q2, µ)
+= exp
+�
+−4Sg(µj, µ) + 2Ag
+J(µj, µ)
+� ˜jg(∂η, µj)
+�
+p2
+µ2
+j
+�η
+e−γEη
+Γ(1 + η) ,
+(A.14)
+with µj = ej
+�
+p2. This is part of the resummed accumulative decay rate (A.12). We still
+truncate at the NLL′ accuracy and expand in terms of αs(µ). We then take the derivative
+with respect to p2 and arrive at
+∂
+∂p2 ˆJg,NLL′(p2, µ; µj = ej
+�
+p2) = αs(µ)
+4π
+1
+p2
+�
+12 ln p2
+µ2 − 23
+3
+�
++
+�αs(µ)
+4π
+�2 1
+p2
+�
+72 ln3
+� p2
+µ2
+�
++ · · ·
+�
++ O(α3
+s) .
+(A.15)
+We see that the order α2
+s term is indeed independent of ej.
+B
+Fixed order ingredients
+In this Appendix we list the fixed-order ingredients appearing in the resummation formula
+(2.11).
+– 14 –
+
+The Wilson coefficient Ct is known up to the four-loop order [47–52]. For our purpose,
+we need its expression up to three loops, which is given by
+Ct(mt, µ) = 1 + αs
+4π 11 +
+�αs
+4π
+�2 �
+Lt
+�
+19 + 16
+3 nf
+�
++ 2777
+18
+− 67
+6 nf
+�
++
+�αs
+4π
+�3 �
+L2
+t
+�
+209 + 46nf − 32
+9 n2
+f
+�
++ Lt
+�4834
+9
++ 2912
+27 nf + 77
+27n2
+f
+�
+− 2761331
+648
++ 897943ζ3
+144
++
+�58723
+324
+− 110779ζ3
+216
+�
+nf − 6865
+486 n2
+f
+�
+, (B.1)
+where Lt = ln(µ2/m2
+t ), and we have explicitly set the number of colors Nc = 3 to shorten
+the expression.
+The hard Wilson coefficients Cq,g
+S
+are expanded as
+Cq,g
+S (mH, µ) =
+�
+n=0
+�αs
+4π
+�n
+Cq,g(n)
+S
+,
+(B.2)
+where the gluon channel coefficients up to three-loop order are given by [59]
+Cg(0)
+S
+= 1 ,
+Cg(1)
+S
+=
+�π2
+6 − L2
+�
+CA ,
+Cg(2)
+S
+= nfCA
+�
+−2L3
+9
++ 10L2
+9
++
+�52
+27 + 2π2
+9
+�
+L − 46ζ3
+9
+− 5π2
+18 − 916
+81
+�
++ C2
+A
+�L4
+2 + 11L3
+9
++
+�π2
+6 − 67
+9
+�
+L2 + L
+�
+−2ζ3 − 11π2
+9
++ 80
+27
+�
+− 143ζ3
+9
++π4
+72 + 67π2
+36
++ 5105
+162
+�
++ nfCF
+�
+2L + 8ζ3 − 67
+6
+�
+,
+Cg(3)
+S
+= nfCACF
+�
+−8L3
+3
++ L2(13 − 16ζ3) + L
+�
+−376ζ3
+9
++ 4π4
+45 + π2 + 3833
+54
+�
++32π2ζ3
+9
++ 14564ζ3
+81
++ 608ζ5
+9
+− 34π2
+27
+− 16π4
+405 − 341219
+972
+�
++ n2
+fCA
+�
+−2L4
+27 + 40L3
+81
++
+�116
+81 + 4π2
+27
+�
+L2 + L
+�
+−128ζ3
+27
+− 40π2
+81
+− 14057
+729
+�
++4576ζ3
+243
++ π4
+243 + 2π2
+27 + 611401
+13122
+�
++ nfC2
+A
+�2L5
+9
+− 8L4
+27 +
+�
+−734
+81 − π2
+9
+�
+L3
++L2
+�118ζ3
+9
++ 377
+27 − 103π2
+54
+�
++ L
+�28ζ3
+9
++ 1910π2
+243
+− 4π4
+15 + 133036
+729
+�
++ 428ζ5
+9
+−41π2ζ3
+27
+− 460ζ3
+81
++ 73π4
+1620 − 14189π2
+4374
+− 3765007
+6561
+�
++ C3
+A
+�
+−L6
+6 − 11L5
+9
++
+�281
+54 − π2
+4
+�
+L4 + L3
+�
+2ζ3 + 11π2
+18
++ 1540
+81
+�
++ L2
+�143ζ3
+9
++ 685π2
+108
+− 6740
+81
+− 73π4
+360
+�
++L
+�17π2ζ3
+9
++ 2048ζ3
+27
++ 16ζ5 + 44π4
+45
+− 6710π2
+243
+− 373975
+1458
+�
++ 2222ζ5
+9
+– 15 –
+
+−104ζ2
+3
+9
+− 605π2ζ3
+54
+− 152716ζ3
+243
++ 105617π2
+4374
+− 1939π4
+9720
++ 29639273
+26244
+− 24389π6
+408240
+�
++ n2
+fCF
+�4L2
+3
++ L
+�32ζ3
+3
+− 52
+3
+�
+− 112ζ3
+3
+− 10π2
+27
++ 4481
+81
+− 4π4
+405
+�
++ nfC2
+F
+�
+−2L + 296ζ3
+3
+− 160ζ5 + 304
+9
+�
+,
+(B.3)
+and the quark channel coefficients are given by [60]
+Cq(0)
+S
+= 1 ,
+Cq(1)
+S
+= 1
+6
+�
+−6L2 + π2 − 12
+�
+CF ,
+Cq(2)
+S
+= C2
+F
+�L4
+2 +
+�
+2 − π2
+6
+�
+L2 + 6(4L − 5)ζ3 − 2π2L + 7π2
+3
+− 83π4
+360 + 6
+�
++ CF
+�11L3CA
+9
++ 1
+3π2L2CA − 67L2CA
+9
+− 26Lζ3CA + 11
+9 π2LCA + 242LCA
+27
++
+151ζ3CA
+9
++ 11π4CA
+45
+− 467CA
+81
+− 103π2CA
+108
+− 10L3
+9
++ 50L2
+9
+− 10π2L
+9
+− 280L
+27
++ 10ζ3
+9
++25π2
+54
++ 1000
+81
+�
+,
+Cq(3)
+S
+= C3
+F
+�
+−L6
+6 + π2L4
+12
+− L4 − 24L3ζ3 + 2π2L3 + 30L2ζ3 + 83π4L2
+360
+− 7π2L2
+3
+− 6L2
+−240Lζ5 − 4
+3π2Lζ3 + 20Lζ3 + 19π4L
+15
++ 7π2L − 50L + 16ζ2
+3 + 89π2ζ3
+3
+−654ζ3 + 424ζ5 + 37729π6
+136080 + 575
+3
+− 353π2
+18
+− 77π4
+36
+�
++ C2
+F CA
+�
+−11L5
+9
+− π2L4
+3
++67L4
+9
++ 26L3ζ3 − 308L3
+27
+− 55π2L3
+54
+− 943L2ζ3
+9
++ 689π2L2
+108
++ 1673L2
+81
+− 17π4L2
+90
+−5
+3π2Lζ3 + 1660Lζ3
+3
++ 120Lζ5 + π4L
+6
++ 614L
+27
+− 3506π2L
+81
++ 296ζ2
+3
+3
+− 1676ζ5
+9
+−4820ζ3
+27
+− 3049π2ζ3
+54
++ 31819π2
+486
+− 9335
+81
+− 893π4
+9720 − 3169π6
+17010
+�
++ C2
+F
+�10L5
+9
+− 50L4
+9
++25π2L3
+27
++ 250L3
+27
++ 350L2ζ3
+9
+− 335π2L2
+54
++ 3625L2
+162
+− 4160Lζ3
+9
++ 7π4L
+9
++ 2200π2L
+81
+−7075L
+54
++ 59980ζ3
+81
+− 2080ζ5
+9
+− 95π2ζ3
+27
+− 305π4
+972
+− 30655π2
+972
++ 179375
+972
+�
++ CF C2
+A
+�
+−121L4
+54
+− 22π2L3
+27
++ 1780L3
+81
++ 88L2ζ3 + 13π2L2
+27
+− 11π4L2
+45
+− 11939L2
+162
++44
+9 π2Lζ3 + 136Lζ5 − 13900Lζ3
+27
++ 4822π2L
+243
+− 47π4L
+54
++ 10289L
+1458
++ 163π2ζ3
+9
++107648ζ3
+243
++ 106ζ5
+9
+− 1136ζ2
+3
+9
++ 10093π4
+4860
+− 769π6
+5103 − 264515π2
+8748
++ 5964431
+26244
+�
++ CACF
+�110L4
+27
++ 20π2L3
+27
+− 2890L3
+81
+− 40L2ζ3 + 40π2L2
+9
++ 8635L2
+81
++ 3620Lζ3
+9
+– 16 –
+
++11π4L
+27
+− 8180π2L
+243
+− 37495L
+729
++ 10π2ζ3
+9
+− 20ζ5
+3
+− 14300ζ3
+27
++ 166295π2
+4374
+−119π4
+243
+− 2609875
+13122
+�
++ CF
+�
+−50L4
+27
++ 1000L3
+81
+− 100π2L2
+27
+− 2500L2
+81
++ 400Lζ3
+27
++1000π2L
+81
++ 23200L
+729
+− 5000ζ3
+243
+− 235π4
+243
+− 2650π2
+243
++ 51800
+6561
+�
+,
+(B.4)
+where L = ln
+�
+(−m2
+H − iϵ)/µ2�
+. Although not used in this paper, the fourth-order coeffi-
+cients can already be extracted from the form factors calculated in Ref. [61] and Ref. [62].
+The jet functions are expanded as
+˜jq,g(LJ, µ) =
+�
+n=0
+�αs
+4π
+�n ˜jq,g(n)(LJ) .
+(B.5)
+The expansion coefficients for the quark jet function are [12, 71]
+˜jq(1)(LJ) = CF
+�
+2L2
+J − 3LJ − 2π2
+3
++ 7
+�
+,
+˜jq(2)(LJ) = CF nf
+�4
+9L3
+J − 29
+9 L2
+J +
+�247
+27 − 2π2
+9
+�
+LJ + 13π2
+18
+− 4057
+324
+�
++ CF CA
+�
+−22
+9 L3
+J +
+�367
+18 − 2π2
+3
+�
+L2
+J +
+�
+40ζ3 + 11π2
+9
+− 3155
+54
+�
+LJ − 18ζ3 − 37π4
+180
+−155π2
+36
++ 53129
+648
+�
++ C2
+F
+�
+2L4
+J − 6L3
+J +
+�37
+2 − 4π2
+3
+�
+L2
+J +
+�
+4π2 − 24ζ3 − 45
+2
+�
+LJ
+−6ζ3 + 61π4
+90
+− 97π2
+12
++ 205
+8
+�
+,
+˜jq(3)(LJ) = CF n2
+f
+�
+4
+27L4
+J − 116
+81 L3
+J +
+�470
+81 − 4π2
+27
+�
+L2
+J +
+�58π2
+81
+− 8714
+729 − 64
+27ζ3
+�
+LJ
+�
++ CF CAnf
+�
+− 44
+27L4
+J +
+�1552
+81
+− 8π2
+27
+�
+L3
+J +
+�28π2
+9
+− 7531
+81
++ 8ζ3
+�
+L2
+J +
+�32π4
+135
+− 1976ζ3
+27
+− 2632π2
+243
++ 160906
+729
+�
+LJ
+�
++ CF C2
+A
+�
+121
+27 L4
+J +
+�44π2
+27
+− 4649
+81
+�
+L3
+J +
+�22π4
+45
+− 132ζ3 − 389π2
+27
++ 50689
+162
+�
+L2
+J +
+�18179π2
+486
+− 53π4
+135 − 599375
+729
+− 232ζ5 − 88π2ζ3
+9
++ 6688ζ3
+9
+�
+LJ
+�
++ C2
+F nf
+�
+8
+9L5
+J − 70
+9 L4
+J +
+�875
+27 − 20π2
+27
+�
+L3
+J +
+�151π2
+27
+− 15775
+162
+�
+L2
+J
++
+�32ζ3
+9
++ 4π4
+27 − 2833π2
+162
++ 7325
+36
+�
+LJ
+�
++ C2
+F CA
+�
+− 44
+9 L5
+J +
+�433
+9
+− 4π2
+3
+�
+L4
+J
++
+�164π2
+27
+− 10537
+54
++ 80ζ3
+�
+L3
+J +
+�
+− 68ζ3 + π4
+30 − 2045π2
+54
++ 157943
+324
+�
+L2
+J
++
+�290ζ3
+3
+− 120ζ5 − 88π2ζ3
+3
+− 923π4
+540
++ 35075π2
+324
+− 151405
+216
+�
+LJ
+�
++ C3
+F
+�
+4
+3L6
+J − 6L5
+J
+– 17 –
+
++
+�
+23 − 4π2
+3
+�
+L4
+J +
+�
+8π2 − 99
+2 − 48ζ3
+�
+L3
+J +
+�
+60ζ3 + 61π4
+45
+− 151π2
+6
++ 349
+4
+�
+L2
+J
++
+�
+240ζ5 + 64π2ζ3
+3
+− 218ζ3 − 149π4
+30
++ 145π2
+4
+− 815
+8
+�
+LJ
+�
++ cJ
+3q ,
+(B.6)
+The scale-independent constant term cJ
+3q is given by [71]
+cJ
+3q = 25.06777873C3
+F + 32.81169125CAC2
+F − 0.7795843561C2
+ACF − 31.65196210CACF nfTF
+− 61.78995095C2
+F nfTF + 28.49157341CF n2
+fT 2
+F .
+(B.7)
+The expansion coefficients for the gluon jet function are [70, 72]
+˜jg(1)(LJ) = CA
+�
+2L2
+J − 11
+3 LJ + 67
+9 − 2π2
+3
+�
++ nf
+�2
+3LJ − 10
+9
+�
+,
+˜jg(2)(LJ) = n2
+f
+�4
+9L2
+J − 40
+27LJ − 2π2
+27 + 100
+81
+�
++ CF nf
+�
+2LJ + 8ζ3 − 55
+6
+�
++ CAnf
+�
+16
+9 L3
+J
+− 28
+3 L2
+J +
+�224
+9
+− 10π2
+9
+�
+LJ − 8ζ3
+3 + 67π2
+27
+− 760
+27
+�
++ C2
+A
+�
+2L4
+J − 88
+9 L3
+J +
+�389
+9
+− 2π2
+�
+L2
+J
++
+�55π2
+9
++ 16ζ3 − 2570
+27
+�
+LJ − 88ζ3
+3
++ 17π4
+36
+− 362π2
+27
++ 20215
+162
+�
+,
+˜jg(3)(LJ) = n3
+f
+�
+8
+27L3
+J − 40
+27L2
+J +
+�200
+81 − 4π2
+27
+�
+LJ
+�
++ CF n2
+f
+�
+10
+3 L2
+J +
+�
+16ζ3 − 24
+�
+LJ
+�
+− C2
+F nfLJ + CAn2
+f
+�
+4
+3L4
+J − 292
+27 L3
+J +
+�3326
+81
+− 4π2
+3
+�
+L2
+J +
+�508π2
+81
+− 116509
+1458
+− 256ζ3
+27
+�
+LJ
+�
++ CACF nf
+�
+16
+3 L3
+J +
+�
+32ζ3 − 55
+�
+L2
+J +
+�
+− 8π4
+45 − 10π2
+3
++ 5599
+27
+− 1096ζ3
+9
+�
+LJ
+�
++ C2
+Anf
+�
+20
+9 L5
+J − 64
+3 L4
+J −
+�88π2
+27
+− 3106
+27
+�
+L3
+J +
+�586π2
+27
+− 8ζ3
+3 − 10067
+27
+�
+L2
+J
++
+�449π4
+270
+− 16831π2
+243
++ 1052135
+1458
+− 1280ζ3
+27
+�
+LJ
+�
++ C3
+A
+�
+4
+3L6
+J − 110
+9 L5
+J +
+�
+85 − 8π2
+3
+�
+L4
+J
++
+�484π2
+27
+− 9623
+27
++ 32ζ3
+�
+L3
+J +
+�169π4
+90
+− 484ζ3
+3
+− 2362π2
+27
++ 85924
+81
+�
+L2
+J
++
+�
+− 4411π4
+540
++ 52678π2
+243
+− 1448021
+729
+− 112ζ5 − 160π2ζ3
+9
++ 6316ζ3
+9
+�
+LJ
+�
++ cJ
+3g . (B.8)
+From Ref. [72], we have cJ
+3g = 647.7843434644 for nf = 5.
+The soft functions are expanded as
+˜sq,g(LS, µ) =
+�
+n=0
+�αs
+4π
+�n
+˜sq,g(n)(LS) .
+(B.9)
+– 18 –
+
+The expansion coefficients for the quark soft function are [12, 80]
+˜sq(1)(LS) = CF
+�
+−8L2
+S − π2�
+,
+˜sq(2)(LS) = CF nf
+�
+− 32
+9 L3
+S + 80
+9 L2
+S −
+�8π2
+9
++ 224
+27
+�
+LS − 52ζ3
+9
++ 77π2
+27
++ 40
+81
+�
++ CF CA
+�
+176
+9 L3
+S +
+�8π2
+3
+− 536
+9
+�
+L2
+S +
+�44π2
+9
+− 56ζ3 + 1616
+27
+�
+LS + 286ζ3
+9
++ 14π4
+15
+− 871π2
+54
+− 2140
+81
+�
++ C2
+F
+�
+32L4
+S + 8π2L2
+S + π4
+2
+�
+,
+˜sq(3)(LS) = CF n2
+f
+�
+− 64
+27L4
+S + 640
+81 L3
+S −
+�32π2
+27
++ 800
+81
+�
+L2
+S +
+�64π2
+9
+− 3200
+729 − 64ζ3
+9
+�
+LS
+�
++ CF CAnf
+�
+704
+27 L4
+S +
+�64π2
+27
+− 9248
+81
+�
+L3
+S +
+�64π2
+9
++ 16408
+81
+�
+L2
+S +
+�6032ζ3
+27
++ 64π4
+45
+− 19408π2
+243
+− 80324
+729
+�
+LS
+�
++ CF C2
+A
+�
+− 1936
+27 L4
+S −
+�352π2
+27
+− 28480
+81
+�
+L3
+S +
+�104π2
+27
+− 88π4
+45
+− 62012
+81
++ 352ζ3
+�
+L2
+S +
+�50344π2
+243
+− 88π4
+9
++ 556042
+729
++ 384ζ5 + 176π2ζ3
+9
+− 36272ζ3
+27
+�
+LS
+�
++ C2
+F nf
+�
+256
+9 L5
+S − 640
+9 L4
+S +
+�32π2
+3
++ 1504
+27
+�
+L3
+S +
+�5620
+81
+− 856π2
+27
+− 160ζ3
+9
+�
+L2
+S +
+�608ζ3
+9
++ 56π4
+45
++ 152π2
+27
+− 3422
+27
+�
+LS
+�
++ C2
+F CA
+�
+− 1408
+9
+L5
+S
++
+�
+4288
+9
+− 64π2
+3
+�
+L4
+S +
+�
+448ζ3 − 176π2
+3
+− 12928
+27
+�
+L3
+S +
+�5092π2
+27
+− 2288ζ3
+9
+− 152π4
+15
++ 17120
+81
+�
+L2
+S +
+�
+56π2ζ3 − 44π4
+9
+− 1616π2
+27
+�
+LS
+�
++ C3
+F
+�
+− 256
+3 L6
+S − 32π2L4
+S − 4π4L2
+S
+�
++ cS
+3q .
+(B.10)
+The three-loop scale-independent term cS
+3q is not precisely known at the moment. Its calcu-
+lation is under active investigation [84, 89, 90]. In Ref. [71], this term was extracted through
+a numeric fit to the fixed-order thrust distribution. The value (with large uncertainties)
+reads
+cS
+3q = −19988 ± 1440(stat.) ± 4000(syst.) .
+(B.11)
+The results for the gluon soft function can be obtained from the quark one employing
+the non-Abelian exponential theorem [91–93]. Up to three loops, they are related by the
+Casimir scaling
+ln [˜sg(LS, µ)] = CA
+CF
+ln [˜sq(LS, µ)] .
+(B.12)
+Hence the expansion coefficients for the gluon soft function are
+˜sg(1)(LS) = CA
+�
+−8L2
+S − π2�
+,
+– 19 –
+
+˜sg(2)(LS) = CAnf
+�
+− 32
+9 L3
+S + 80
+9 L2
+S −
+�8π2
+9
++ 224
+27
+�
+LS + 77π2
+27
++ 40
+81 − 52ζ3
+9
+�
++ C2
+A
+�
+32L4
+S + 176
+9 L3
+S +
+�32π2
+3
+− 536
+9
+�
+L2
+S +
+�44π2
+9
++ 1616
+27
+− 56ζ3
+�
+LS + 286ζ3
+9
++ 43π4
+30
+− 871π2
+54
+− 2140
+81
+�
+,
+˜sg(3)(LS) = CAn2
+f
+�
+− 64
+27L4
+S + 640
+81 L3
+S −
+�32π2
+27
++ 800
+81
+�
+L2
+S +
+�64π2
+9
+− 3200
+729 − 64ζ3
+9
+�
+LS
+�
++ CF CAnf
+�
+− 32
+3 L3
+S +
+�220
+3
+− 64ζ3
+�
+L2
+S +
+�608ζ3
+9
++ 16π4
+45
+− 8π2
+3
+− 3422
+27
+�
+LS
+�
++ C2
+Anf
+�
+256
+9 L5
+S − 1216
+27 L4
+S +
+�352π2
+27
+− 3872
+81
+�
+L3
+S +
+�416ζ3
+9
+− 664π2
+27
++ 16088
+81
+�
+L2
+S
++
+�6032ζ3
+27
++ 104π4
+45
+− 17392π2
+243
+− 80324
+729
+�
+LS
+�
++ C3
+A
+�
+− 256
+3 L6
+S − 1408
+9
+L5
+S
++
+�10928
+27
+− 160π2
+3
+�
+L4
+S +
+�
+448ζ3 − 1936π2
+27
+− 10304
+81
+�
+L3
+S +
+�880ζ3
+9
+− 724π4
+45
++ 1732π2
+9
+− 4988
+9
+�
+L2
+S +
+�680π2ζ3
+9
+− 36272ζ3
+27
++ 384ζ5 − 44π4
+3
++ 35800π2
+243
++ 556042
+729
+�
+LS
+�
++ cS
+3g ,
+(B.13)
+where
+cS
+3g = CACF nf
+�
+−52
+9 π2ζ3 + 77π4
+27
++ 40π2
+81
+�
++ C2
+Anf
+�52π2ζ3
+9
+− 77π4
+27
+− 40π2
+81
+�
++ C3
+A
+�
+−286
+9 π2ζ3 − 11π6
+10
++ 871π4
+54
++ 2140π2
+81
+�
++ C2
+ACF
+�286π2ζ3
+9
++ 14π6
+15
+− 871π4
+54
+− 2140π2
+81
+�
++ CAC2
+F
+1
+6π6 + CA
+CF
+cS
+3q .
+(B.14)
+C
+Anomalous dimensions
+In this Appendix we list the expressions of the various anomalous dimensions appearing in
+the resummation formula.
+For the cusp anomalous dimensions, we write
+Γq
+cusp(αs) = CF
+�
+γcusp(αs) + δγq
+cusp(αs)
+�
+,
+Γg
+cusp(αs) = CA
+�
+γcusp(αs) + δγg
+cusp(αs)
+�
+.
+(C.1)
+Up to three loops the cusp anomalous dimensions satisfy Casimir scaling, so that δγq
+cusp(αs)
+and δγg
+cusp(αs) only start at α4
+s. We define the expansion as
+γcusp(αs) =
+�
+n=0
+γ(n)
+cusp
+�αs
+4π
+�n+1
+,
+δγq,g
+cusp(αs) =
+�
+n=3
+δγq,g(n)
+cusp
+�αs
+4π
+�n+1
+(C.2)
+– 20 –
+
+The expansion coefficients are [31, 33–35]
+γ(0)
+cusp = 4 ,
+γ(1)
+cusp = −1
+34π2CA + 268CA
+9
+− 80nfTF
+9
+,
+γ(2)
+cusp = −
+16n2
+f
+27
+− 208nfζ(3)
+3
++ 80π2nf
+9
+− 1276nf
+9
++ 264ζ3 + 44π4
+5
+− 536π2
+3
++ 1470nf ,
+γ(3)
+cusp = n3
+f
+�64ζ3
+27 − 32
+81
+�
++ n2
+f
+�4160ζ3
+27
+− 304π2
+81
+− 104π4
+135
++ 17875
+243
+�
++ nf
+�416π2ζ3
+3
+−153920ζ3
+27
++ 19504ζ5
+9
++ 12800π2
+27
+− 616π4
+45
+− 344345
+81
+�
+− 432ζ2
+3 − 528π2ζ3
++ 20944ζ3 − 10824ζ5 + 2706π4
+5
++ 84278
+3
+− 44200π2
+9
+− 2504π6
+105
+,
+δγq(3)
+cusp = nf
+�
+−80ζ3
+9
+− 400ζ5
+9
++ 40π2
+9
+�
+− 720ζ2
+3 + 80ζ3 + 2200ζ5 − 40π2 − 124π6
+63
+,
+δγg(3)
+cusp = nf
+�
+−80ζ3
+3
+− 400ζ5
+3
++ 40π2
+3
+�
+− 2160ζ2
+3 + 240ζ3 + 6600ζ5 − 120π2 − 124π6
+21
+.
+(C.3)
+As far as we know, there is no complete results for the five-loop cusp anomalous dimensions.
+In this paper, we make use of the approximate results estimated in [35],
+γ(4)
+cusp + δγq(4)
+cusp = 50000 ± 40000 ,
+γ(4)
+cusp + δγg(4)
+cusp = 30000 ± 60000 .
+(C.4)
+The anomalous dimension for the quark Yukawa coupling reads
+γy = −
+�
+n=0
+�αs
+4π
+�n+1
+γ(n)
+y
+,
+(C.5)
+where [44, 45]
+γ(0)
+y
+= 6CF ,
+γ(1)
+y
+= 97CACF
+3
+− 10CF nf
+3
++ 3C2
+F ,
+γ(2)
+y
+= −48ζ3CACF nf − 556
+27 CACF nf − 129
+2 CAC2
+F + 11413
+54
+C2
+ACF + 48ζ3C2
+F nf
+− 46C2
+F nf − 70
+27CF n2
+f + 129C3
+F ,
+γ(3)
+y
+= n3
+f
+�128ζ3
+27
+− 664
+243
+�
++ n2
+f
+�1600ζ3
+9
+− 32π4
+27
++ 10484
+243
+�
++ nf
+�
+−68384ζ3
+9
++36800ζ5
+9
++ 176π4
+9
+− 183446
+27
+�
++ 271360ζ3
+27
+− 17600ζ5 + 4603055
+81
+.
+(C.6)
+The anomalous dimension for the Wilson coefficient Ct is
+γt(αs) =
+�
+n=0
+�αs
+4π
+�n+1
+γ(n)
+t
+,
+(C.7)
+– 21 –
+
+where [47–51]
+γ(0)
+t
+= 0 ,
+γ(1)
+t
+= 40
+3 CAnfTF − 1
+368C2
+A + 8CF nfTF ,
+γ(2)
+t
+= − 1
+27650n2
+f + 10066nf
+9
+− 5714 ,
+γ(2)
+t
+= −
+2186n3
+f
+243
++ n2
+f
+�
+−12944ζ3
+27
+− 50065
+27
+�
++ nf
+�13016ζ3
+9
++ 1078361
+27
+�
+− 21384ζ3 − 149753 .
+(C.8)
+The anomalous dimension for the jet functions are
+γq,g
+j (αs) =
+�
+n=0
+�αs
+4π
+�n+1
+γq,g(n)
+j
+,
+(C.9)
+where [66, 81]
+γq(0)
+j
+= −3CF ,
+γq(1)
+j
+= 8π2nf
+27
++ 484nf
+81
++ 352ζ3
+3
+− 4π2
+3
+− 3610
+27 ,
+γq(2)
+j
+= −256
+81 ζ3n2
+f − 14272ζ3nf
+81
+− 80
+243π2n2
+f +
+13828n2
+f
+2187
++ 1172π4nf
+1215
++ 1592π2nf
+243
++ 100984nf
+729
+− 25696ζ5
+9
+− 9632π2ζ3
+81
++ 153136ζ(3)
+27
+− 6818π4
+405
++ 7588π2
+81
+− 470183
+243
+,
+γq(3)
+j
+= 4483.56 ,
+(C.10)
+and [66, 70]
+γg(0)
+j
+= −β0 ,
+γg(1)
+j
+= −2
+9π2nfCA + 184nfCA
+27
++ 16ζ3C2
+A + 11
+9 π2C2
+A − 1096C2
+A
+27
++ 2nfCF ,
+γg(2)
+j
+= −304
+9 nfζ3CACF − 8
+45π4nfCACF − 2
+3π2nfCACF + 4145
+54 nfCACF − 112
+27 n2
+fζ3CA
++
+1811n2
+fCA
+1458
++ 20
+81π2n2
+fCA − 8
+27nfζ3C2
+A + 77
+135π4nfC2
+A − 1306
+243 π2nfC2
+A
++ 42557nfC2
+A
+1458
+− 64
+9 π2ζ3C3
+A + 260ζ3C3
+A − 112ζ5C3
+A + 6217
+243 π2C3
+A − 583
+270π4C3
+A
+− 331153C3
+A
+1458
+−
+11n2
+fCF
+9
+− nfC2
+F ,
+γg(3)
+j
+= 26138.7 .
+(C.11)
+The hard anomalous dimensions are
+γq,g
+H (αs) =
+�
+n=0
+�αs
+4π
+�n+1
+γq,g(n)
+H
+,
+(C.12)
+– 22 –
+
+where the coefficients up to three loops can be obtained from Eq. (B.3) and Eq. (B.4). For
+the gluon case, they read [53–57, 59, 62]
+γg(0)
+H
+= 0 ,
+γg(1)
+H
+= −2π2nf
+3
+− 152nf
+9
++ 36ζ3 + 11π2 − 160
+3 ,
+γg(2)
+H
+= −
+112n2
+fζ3
+9
++
+20π2n2
+f
+27
++
+7714n2
+f
+243
++ 920nfζ3
+9
++ 214π4nf
+45
+− 1270π2nf
+27
+− 76256nf
+81
+− 120π2ζ3 + 2196ζ3 − 864ζ5 + 6109π2
+9
+− 319π4
+5
++ 37045
+27
+,
+γg(3)
+H
+= n3
+f
+�400ζ3
+81
++ 8π2
+81 − 64π4
+405 + 38426
+2187
+�
++ n2
+f
+�368π2ζ3
+9
+− 49342ζ3
+81
++8000ζ5
+9
++ 19675π2
+486
+− 2474π4
+405
++ 3605645
+1944
+�
++ nf
+�32ζ2
+3
+3
+− 504π2ζ3
++528362ζ3
+27
+− 90140ζ5
+9
++ 52153π4
+135
+− 262193π2
+162
+− 7927313
+216
+− 54917π6
+2835
+�
++ 21384ζ2
+3 + 2004π4ζ3
+5
+− 576π2ζ3 + 119536ζ3
+3
++ 1152π2ζ5
+− 62796ζ5 − 22734ζ7 + 18051π6
+70
++ 1041691π2
+54
+− 123029π4
+30
++ 5481844
+81
+.
+(C.13)
+For the quark case, they are given as [53, 60, 61, 94]
+γq(0)
+H
+= 0 ,
+γq(1)
+H
+= 8π2nf
+9
++ 160nf
+81
++ 368ζ3
+3
+− 68π2
+9
+− 352
+27 ,
+γq(2)
+H
+= −
+64n2
+fζ3
+81
+−
+80π2n2
+f
+81
++
+11776n2
+f
+2187
+− 23584nfζ3
+81
++ 136π4nf
+1215
++ 9128π2nf
+243
+− 47192nf
+729
++ 164144ζ3
+27
+− 30656ζ5
+9
+− 9760π2ζ3
+81
+− 10124π2
+81
++ 213772
+243
+− 4132π4
+405
+,
+γq(3)
+s
+= n3
+f
+�
+−2240ζ3
+729
+− 32π2
+243 − 128π4
+3645 + 95744
+19683
+�
++ n2
+f
+�
+− 1
+2432816π2ζ3
++18848ζ3
+729
++ 25984ζ5
+81
++ 6856π4
+10935 − 130486π2
+2187
++ 126461
+4374
+�
++ nf
+�
+−110848ζ2
+3
+27
++115504π2ζ3
+243
+− 3066064ζ3
+243
++ 59488ζ5
+9
++ 29264π4
+729
++ 755786π2
+729
+− 1641457
+486
+−975668π6
+229635
+�
++ 175712ζ2
+3
+9
++ 992696π4ζ3
+3645
++ 1365152ζ3
+9
++ 243872π2ζ5
+81
++ 2888332ζ7
+27
+− 81584π2ζ3
+9
+− 2158832ζ5
+9
++ 2058794π6
+25515
+− 362842π2
+243
++ 19161124
+729
+− 1095218π4
+1215
+.
+(C.14)
+– 23 –
+
+Due to the RG invariance of physical observables, the soft anomalous dimensions satisfy
+the consistency relations
+γq
+s = γq
+H + γy − 2γq
+j ,
+γg
+s = γg
+H + γt + β
+αs
+− 2γg
+j .
+(C.15)
+We expand them in αs
+γq,g
+s (αs) =
+�
+n=0
+�αs
+4π
+�n+1
+γq,g(n)
+s
+,
+(C.16)
+where
+γq(0)
+s
+= 0 ,
+γq(1)
+s
+= 8π2nf
+27
+− 448nf
+81
+− 112ζ3 − 44π2
+9
++ 3232
+27 ,
+γq(2)
+s
+=
+448ζ3n2
+f
+81
++ 13600ζ3nf
+81
+− 80
+243π2n2
+f −
+8320n2
+f
+2187
+− 736π4nf
+405
++ 5944π2nf
+243
+− 129496nf
+729
++ 2304ζ5 + 352π2ζ3
+3
+− 5264ζ3 + 352π4
+15
+− 25300π2
+81
++ 547124
+243
+,
+γq(3)
+s
+= −5350.4 ,
+(C.17)
+and
+γg(0)
+s
+= 0 ,
+γg(1)
+s
+= 2π2nf
+3
+− 112nf
+9
+− 252ζ3 − 11π2 + 808
+3 ,
+γg(2)
+s
+=
+112n2
+fζ3
+9
+−
+20π2n2
+f
+27
+−
+2080n2
+f
+243
++ 3400nfζ3
+9
++ 1486π2nf
+27
+− 184π4nf
+45
+− 32374nf
+81
++ 264π2ζ3 − 11844ζ3 + 5184ζ5 + 264π4
+5
+− 6325π2
+9
++ 136781
+27
+,
+γg(3)
+s
+= −14715.4 .
+(C.18)
+Up to three loops, the quark and gluon soft anomalous dimensions satisfy the Casimir
+scaling γg
+s/CA = γq
+s/CF . However, starting at four loops, due to the emergence of new
+Casimir operators, the relation must be generalized as appropriate [66, 95].
+– 24 –
+
+Finally, the beta function coefficients are [40, 41]
+β0 = 11CA
+3
+− 4nfTF
+3
+,
+β1 = 34C2
+A
+3
+− 20CAnfTF
+3
+− 4CF nfTF ,
+β2 =
+325n2
+f
+54
+− 5033nf
+18
++ 2857
+2
+,
+β3 =
+1093n3
+f
+729
++ n2
+f
+�6472ζ3
+81
++ 50065
+162
+�
++ nf
+�
+−6508ζ3
+27
+− 1078361
+162
+�
++ 3564ζ3
++ 149753
+6
+,
+β4 = n4
+f
+�1205
+2916 − 152ζ3
+81
+�
++ n3
+f
+�
+−48722ζ3
+243
++ 460ζ5
+9
++ 809π4
+1215 − 630559
+5832
+�
++ n2
+f
+�698531ζ3
+81
+− 381760ζ5
+81
+− 5263π4
+405
++ 25960913
+1944
+�
++ nf
+�
+−4811164ζ3
+81
++1358995ζ5
+27
++ 6787π4
+108
+− 336460813
+1944
+�
++ 621885ζ3
+2
+− 288090ζ5 + 8157455
+16
+− 9801π4
+20
+.
+(C.19)
+– 25 –
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+FORTIFYING THE YOMDIN-GROMOV ALGEBRAIC LEMMA
+DMITRY NOVIKOV, BENNY ZACK
+Abstract. We provide sharp cylindrical parametrizations of cylindrical cell de-
+compositions by maps with bounded Cr norm in the
+#o-minimal setting, thus
+generalizing and strengthening the Yomdin-Gromov Algebraic Lemma.
+We introduce forts, geometrical objects encoding the combinatorial structure of
+cylindrical cell decompositions in o-minimal geometry. Cylindical decompositions,
+refinements of such decompositions, and cylindrical parametrizations of such de-
+composition become morphisms in the category of forts. We formulate and prove
+the above results in the language of forts.
+Contents
+1.
+Introduction
+2
+1.1.
+The Yomdin Gromov Algebraic Lemma
+2
+1.2.
+Cells and Cylindrical Decompositions
+4
+1.3.
+Sharp cylindrical decomposition
+5
+1.4.
+Cylindrical Parametrizations and the Main Result
+6
+1.5.
+Applications of the Main Result
+8
+1.6.
+Structure of this paper
+9
+2.
+Preliminary results for Sharp Structures
+9
+2.1.
+Definition of sharply o-minimal structures
+9
+2.2.
+Sharpness of arithmetic operations
+10
+2.3.
+Sharpness of Refinement
+11
+2.4.
+Sharpness of Cr locus
+11
+2.5.
+Sharp definable choice
+11
+3.
+Forts
+12
+3.1.
+Forts and Morphisms
+12
+3.2.
+Combinatorial Equivalence
+15
+3.3.
+Inverse image fort and proof of Proposition 3.18
+17
+3.4.
+Pulling back along extensions
+18
+3.5.
+The Tower construction
+18
+3.6.
+Smoothing
+19
+Date: January 13, 2023.
+1
+arXiv:2301.04953v1  [math.LO]  12 Jan 2023
+
+2
+DMITRY NOVIKOV, BENNY ZACK
+3.7.
+The main result reformulated
+20
+3.8.
+Final Remarks
+21
+4.
+The step F1,k
+22
+5.
+The step Sℓ−1,k + Fℓ−1,k → Sℓ,k
+25
+6.
+The step S≤ℓ,k + F≤ℓ−1,k → Fℓ,k
+26
+6.1.
+Reduction to the case where fC,j,λ is Cr and fC,j,λ(x1, ·) is an r-function
+for all x1.
+28
+6.2.
+Induction on the first unbounded derivative
+29
+References
+30
+1. Introduction
+1.1. The Yomdin Gromov Algebraic Lemma.
+Denote I = (0, 1). For m ≥ n
+we denote by πm
+n : Im → In the projection on the first n coordinates. Often we will
+omit m from the notation.
+Additionally, the symbol Oa(1) denotes a specific universally fixed function a �→
+Ca where Ca > 0, and the symbol polya(b) denotes a polynomial pa with positive
+coefficients evaluated at b, where a �→ pa is a universally fixed map.
+Definition 1.1. Let U ⊂ Rℓ be a domain. For a Cr function f : U → I, we denote
+||f|| to be its supremum norm, and define
+||f||r := max
+|α|≤r
+||f (α)||
+α!
+.
+For a Cr map f : U → Rm, we define ||f||r := max
+1≤i≤m||fi||r where fi are the coordinate
+functions of f. If a Cr function (resp. map) satisfies ||f||r ≤ 1 we shall call it an
+r-function (resp. r-map).
+The Yomdin-Gromov algebraic lemma and its generalizations to the o-minimal
+setting have remarkable applications in the fields of dynamics and diophantine ge-
+ometry. We now state the original semialgebraic version. Fix r ∈ N.
+Theorem 1.2 ([8], Section 3.3). Let X ⊂ Iℓ be a µ-dimensional semialgebraic set,
+and let β the sum of the degrees of the equations and the inequalities that define X.
+Then there exists a constant C = C(β, r, ℓ) and r-maps f1, . . . , fC : Iµ → X, such
+that X = ∪ifi(Iµ).
+Remark 1.3. G.Binyamini and D.Novikov [2] prove that C can be bounded from
+above by polyℓ(β) · rµ, and moreover that fi may be taken to be semialgebraic of
+complexity polyℓ(β, r).
+
+FORTIFYING THE YOMDIN-GROMOV ALGEBRAIC LEMMA
+3
+Pila and Wilkie [9] generalized Theorem 1.2 to the o-minimal setting in order
+to prove their celebrated Pila-Wilkie counting theorem about rational points on
+definable sets. We will now state their version. In the general o-minimal setting
+the notion of complexity is not available, and will be replaced by uniformity over
+families. For an introduction on o-minimal sturtures, see [10]. Fix an o-minimal
+structure, and an r ∈ N.
+Theorem 1.4 ([9], Corollary 5.2). Let {Xλ ⊂ Iℓ}λ∈Ik be a family of definable sets
+such that dim Xλ ≤ µ for all λ. Then there exists a constant C = C(X, r) such that
+for each λ there are definable maps f1,λ, . . . , fC,λ : Iµ → Xλ whose images cover Xλ.
+For a more detailed exposition on this lemma, its applications and limitations, see
+[2, 3]. In [3], Binyamini and Novikov strengthen Theorem 1.4 by showing that the
+maps fi,λ can be chosen to be cellular, see Definition 1.7 below.
+Definition 1.5. A basic cell C ⊂ Rℓ of length ℓ is a product of ℓ points {0} and
+intervals I.
+Remark 1.6. While a basic cell C is not generally a domain, we will often implic-
+itly identify it with the basic cell obtained by omitting the {0} factors from C. In
+particular there is a natural meaning for a map f : C → R to be Cr.
+Definition 1.7. Let X, Y ⊂ Rℓ. A map f = (f1, . . . , fℓ) : X → Y is called precel-
+lular if
+(1) It is triangular, that is, the coordinate function fi depends only on x1, . . . , xi
+for every i = 1, . . . , ℓ.
+(2) For every 1 ≤ i ≤ ℓ and every fixed x1, . . . , xi−1, the function fi(x1, . . . , xi−1, ·)
+is a strictly increasing function.
+A cellular map is a continuous precellular map.
+Remark 1.8. If f : X → Y is cellular, then for any 1 ≤ j ≤ ℓ the map f1...i :=
+(f1, . . . , fi) : πi(X) → πi(Y ) is cellular as well.
+Also, for any 1 ≤ i ≤ n, if
+(x1, . . . , xi) ∈ πi(X) the map f(x1, . . . , xi, ·) : π−1
+i (x1, . . . , xi) → π−1
+i (f1...i(x1, . . . , xi))
+is cellular.
+We now state the main result of [3]. The notation Sℓ and Fℓ below come from
+“set” and “function” respectively.
+Theorem 1.9 ([3], Theorem 19). Let ℓ, r ∈ N, then:
+Sℓ: For every definable set X ⊂ Iℓ there exists a finite collection {Cα} of basic cells of
+length ℓ and a collection of cellular r-maps {φα : Cα → X} such that X = ∪αφα(Cα).
+Fℓ: For every pair (X, F) of a definable set X ⊂ Iℓ and a definable map F : X → Iq
+(for any q ∈ N) there exists a finite collection {Cα} of basic cells of length ℓ and a
+
+4
+DMITRY NOVIKOV, BENNY ZACK
+collection of cellular r-maps {φα : Cα → X} such that X = ∪αφα(Cα), and for each
+α the map (φα)∗F is an r-map.
+Remark 1.10. This formulation is automatically uniform over families, due to Re-
+mark 1.8. Also, note that Fℓ follows from Sℓ+1.
+Though the proof of [3, Theorem 19] is less technically involved than previous
+proofs, it does not provide polynomial bounds in q in the semi-algebraic or Pfaffian
+cases. Originally one of the main purposes of this paper was to augment the proof
+of Theorem 1.9 to obtain polynomial bounds in q in these cases. However, due to
+the recent development of sharply o-minimal structures (#o-minimal for short, see
+[6, 4, 5]), it is natural to generalize this proof to work in the setting of #o-minimal
+structures.
+Recall that in #o-minimal structures every definable set is associated with two
+natural numbers, “format” F and “degree” D, generalizing ambient dimension and
+complexity respectively from semialgebraic geometry. For the complete definition,
+see Definition 2.2. For a detailed introduction, see [5].
+Our main result (see Theorems 1.18,1.19 below) in particular implies the following
+version of Theorem 1.9 for #o-minimal structures, with polynomial bounds in q.
+Theorem 1.11. Let Σ = (S, Ω) be a sharply o-minimal structure.
+#Sℓ: Let X ⊂ Iℓ have format F and degree D. Then there exists a collection {Cα}
+of polyF(D) basic cells and cellular r-maps {φα : Cα → X} such that X = ∪αφα(Cα).
+#Fℓ: Let a definable set X ⊂ Iℓ have format F and degree D, and a definable map
+F : X → Iq such that for every j = 1, . . . , q the coordinate functions Fj of F have
+format F and degree D. Then there exists a collection {Cα} of polyF(D, q) basic
+cells of length ℓ and a collection of cellular r-maps {φα : Cα → X} of format OF(1)
+and degree polyF(D) such that X = ∪αφα(Cα), and for each α the map (φα)∗F is
+an r-map.
+The statement #Sℓ in the (restricted) subPfaffian case is due to Binyamini, Jones,
+Schmidt and Thomas [1], and a rewording of their proof works in the general #o-
+minimal case. The statement #Fℓ however is stronger and it does not follow from
+#Sℓ+1. In fact, our main result is notably stronger than Theorem 1.11. We prove
+the existence of a cellular r-parametrization with strict control on the combinatorial
+and geometric structure of the parametrizing maps. See Theorems 1.18,1.19 below.
+We start with reviewing the notion of cylindrical decomposition, and introducing the
+notion of cylindrical parametrization.
+1.2. Cells and Cylindrical Decompositions.
+Fix an o-minimal expansion of R.
+We review the notions of cells as they are presented in [10].
+
+FORTIFYING THE YOMDIN-GROMOV ALGEBRAIC LEMMA
+5
+Figure 1. An example of a cellular decomposition of I2 which is
+not a cylindrical decomposition, and a cylindrical decomposition of I2
+refining (see Definition 1.17) it.
+Definition 1.12 (Cells). A cell of length 1 is a subset C ⊂ R1 which is either a point
+or an open interval, and its type is defined to be either (0) or (1) respectively.
+Let C ⊂ Rℓ be a cell of length ℓ with type τ ∈ {0, 1}ℓ, and let f, g : C → R>0 be
+continuous definable functions with f < g everywhere. A cell �C of length ℓ + 1 is one
+of the following sets:
+• �C = C ⊙ (f, g) := {(x, y) : x ∈ C, f(x) < y < g(x)}, in which case the type
+of �C is (τ, 1) ∈ {0, 1}ℓ+1.
+• �C = C ⊙ f := {(x, y) :
+x ∈ C ; y = f(x)}, in which case the type of �C is
+(τ, 0) ∈ {0, 1}ℓ+1.
+A cell C is compatible with a set X if either C ⊂ X or C ∩ X = ∅. A collection
+{C1, . . . , Cn} is a cellular decomposition of X if the cells Ci are pairwise disjoint and
+∪iCi = X.
+We will use the following stronger notion, see Figure 1.
+Definition 1.13 (Cylindrical decompositions). Let X ⊂ I be a definable set. A
+cylindrical decomposition of X is a cellular decomposition of X.
+Let X ⊂ Iℓ be
+definable. A cellular decomposition Φ of X is called a cylindrical decomposition of
+X if the collection πℓ−1(Φ) := {πℓ−1(C)| C ∈ Φ} is a cylindrical decomposition of
+πℓ−1(X).
+1.3. Sharp cylindrical decomposition.
+Let us recall the definition of sharp
+cellular decomposition from [5].
+
+6
+DMITRY NOVIKOV, BENNY ZACK
+Definition 1.14. Let S be an o-minimal structure and Ω be an FD-filtration on
+S. We say that (S, Ω) has sharp cylindrical decomposition (or
+#CD for short) if
+whenever X1, . . . , Xs ⊂ Iℓ are definable sets of format F and degree D, there exists
+a cylindrical decomposition of Iℓ into polyF(D, s) cells of format OF(1) and degree
+polyF(D) compatible with X1, . . . , Xs.
+It is not generally known if every #o-minimal structure has #CD. However, it was
+shown in [5] that for quantitative applications one can always reduce to the case of a
+#o-minimal structure with #CD, see [5, Remark 1.3, Theorem 1.9] for more details.
+We will therefore assume that our #o-minimal structure has #CD, and this will be
+sufficient for quantitative applications.
+For example, while our main results are not known to be true as stated for general
+#o-minimal structures, they are sufficient to prove a sharp version of the Pila-Wilkie
+counting theorem for every #o-minimal structure, see section 1.5 for more details.
+1.4. Cylindrical Parametrizations and the Main Result.
+Our goal is to
+strengthen Theorem 1.9 in such a way that the maps φα have additional combinatorial
+properties. More precisely, we prove that they can be chosen to form a Cylindrical
+parametrization, see Definition 1.15 below.
+Definition 1.15. Let Φ = {Cα}α∈A be a cylindrical decomposition of X ⊂ Iℓ. A
+cylindrical parametrization of Φ is a collection of surjective cellular maps {φα : Cα →
+Cα}α∈A, where Cα is a basic cell of the same type as Cα, such that the following holds:
+For any two cells Cα, Cγ, and for every 1 ≤ k ≤ ℓ − 1, if πk(Cα) = πk(Cγ) then
+φα
+k ◦ πk = φγ
+k ◦ πk.
+Example 1.16. A cylindrical parametrization of the cylindrical decomposition from
+Figure 1 can be viewed as a surjective, precellular map φ : F → I2, where F is
+the set in figure 2 below. Notice that F has a natural decomposition into cells that
+are integer translations of basic cells, and φ is continuous when restricted to any
+such cell. The sets F, I2 are examples of forts, and φ is an example of a morphism
+between forts, see Definition 3.9.
+Definition 1.17. Given two cylindrical decompositions Φ = {Cα}α∈A and Φ′ =
+{Cβ}β∈B of a set X, we say that Φ′ is a refinement of Φ if there exists a map s :
+B → A such that Cβ ⊂ Cs(β) for all β ∈ B.
+A cylindrical parametrization will be called an r-parametrization if all its maps
+are r-maps. We now state our main results.
+Theorem 1.18. Fix ℓ, r ∈ N.
+S∗
+ℓ : Let Φ be a cylindrical decomposition of Iℓ. Then there exists a refinement Φ′ of
+Φ that admits a cylindrical r-parametrization.
+
+FORTIFYING THE YOMDIN-GROMOV ALGEBRAIC LEMMA
+7
+Figure 2. The fort (0, 1)×(0, 1) �{1}×(0, 2) �(1, 2)×(0, 3) �{2}×
+(0, 3) �(2, 3) × (0, 4) �{3} × (0, 5) �(3, 4) × (0, 1).
+F ∗
+ℓ : Let Φ = {Cα}α∈A be a cylindrical decomposition of Iℓ, together with a col-
+lection of definable maps {fCα,j : Cα → I}α∈A ,j∈J.Then there exists a refinement
+Φ′ = {C′
+γ}γ∈A′ of Φ that admits a cylindrical r-parametrization {φγ : Cγ → C′
+γ}γ∈A′
+such that if C′
+γ ⊂ Cα, then (φγ)∗fCα,j is an r-function for every j.
+This theorem can be rather easily deduced from Theorem 1.9. However, its sharp
+counterpart is much more meaningful.
+Theorem 1.19. Fix ℓ, r ∈ N.
+S∗
+ℓ : Let Φ be a cylindrical decomposition of Iℓ of size N into cells of format F and
+degree D. Then there exists a refinement Φ′ of Φ of size polyF,r(D, N), such that Φ′
+admits a cylindrical r-parametrization whose maps have format OF,r(1) and degree
+polyF,r(D).
+F ∗
+ℓ : Let Φ = {Cα}α∈A be a cylindrical decomposition of Iℓ of size N into cells
+of format F and degree D, together with a collection of definable maps {fCα,j :
+Cα → I}α∈A, j∈J such that each fCα,j has format F and degree D. Then there ex-
+ists a refinement Φ′ = {C′
+γ}γ∈A′ of Φ of size polyF,r(D, N, |J|), and a cylindrical r-
+parametrization {φγ : Cγ → C′
+γ} of Φ′, such that the maps φγ have format OF,r(1) and
+degree polyF,r(D), where (φγ)∗fCα,j is an r-function for every j whenever C′
+γ ⊂ Cα.
+
+8
+DMITRY NOVIKOV, BENNY ZACK
+Remark 1.20. (On the index set J) A priopri, J may depend on α. However we
+can, and always will, add constant functions to the collections so that the index set
+J can be chosen to be independent of the cells C.
+Our main results generalize Theorem 1.9 (and Theorem 1.11) in the following
+way. Let X1, . . . , Xs ⊂ Iℓ be definable. By o-minimality, there exists a cylindrical
+decomposition Φ compatible with Xi for 1 ≤ i ≤ s. By Theorem 1.18, we may assume
+that Φ admits a cylindrical r-parametrization {φα}α∈I. Fix i, and let {Cβ}β∈J be
+the cells of Φ contained in Xi. Then the collection {φβ}β∈J (which is a cylindrical
+r-parametrization of Xi) satisfies the requirements of Theorem 1.9 for Xi. The proof
+of Theorem 1.11 is analogous. Indeed, if the sets Xi are defined in a sharply o-
+minimal structure with #CD, we obtain sharp cylindrical parametrization using #CD
+and Theorem 1.19 instead of ordinary cylindrical decomposition and Theorem 1.18.
+Remark 1.21. Binyamini and Novikov have shown that in Theorem 1.2, the com-
+plexity of the parametrizing maps can be taken to be polynomial in r. This is essen-
+tially due to the analytic nature of the semialgebraic structure. In the general sharp
+case, we are unable obtain polynomial bounds in r.
+However, in [6], Binyamini,
+Novikov and B.Zack prove an approximation version of the Yomdin-Gromov Lemma
+with polynomial dependence in r under the assumption of sharp derivatives (see [6]
+or section 3 in [5]). While we conjecture that with sharp derivatives the dependence
+on r in Theorems 1.18,1.19 can be taken to by polynomial, we do not see a clear way
+to utilize sharp derivatives in this direction.
+Remark 1.22 (Effectivity). If one begins with an effective #o-minimal structure in
+the sense of [5, Remark 1.24], then all of our results are effective in the same sense.
+1.5. Applications of the Main Result.
+Binyamini, Jones, Schmidt and Thomas
+[1] prove a sharp version of the Pila-Wilkie counting theorem (see [9]) for the re-
+stricted subPfaffian structure using the results of [7] and a version of #Sℓ from The-
+orem 1.11 for this structure. The same argument will yield a sharp version of the
+Pila-Wilkie counting theorem for any #o-minimal structure with sharp cylindrical
+decomposition, and thus for any
+#o-minimal structure, see Theorem 1.25 below.
+We note that under the assumption of sharp derivatives, a polylog version of the
+Pila-Wilkie Theorem generalizing the Wilkie conjecture, holds, see [6, Definition 8,
+Theorem 1].
+Definition 1.23. Let A ⊂ Rℓ. We denote Aalg to be the union of all connected 1
+dimensional semialgebraic subsets of A, and Atran = A\Aalg.
+Definition 1.24. For a rational number x = p
+q where p, q are coprime integers, we
+denote H(x) := max{|p|, |q|}, H(x) is called the height of x. For a vector x ∈ Qℓ we
+denote H(x) to be the maximum among the height of x′s coordinates.
+
+FORTIFYING THE YOMDIN-GROMOV ALGEBRAIC LEMMA
+9
+Theorem 1.25. Let (S, Ω) be a sharply o-minimal structure.
+Let A ⊂ Rℓ be a
+definable set of format F and degree D. Then for every ϵ > 0 we have
+(1)
+|{x ∈ Atran ∩ Qℓ : H(x) ≤ H}| ≤ polyF,ϵ(D)Hϵ.
+1.6. Structure of this paper.
+In [3], standard o-minimal technique is used to
+conclude a family version for Fℓ from Fℓ. This technique is not sharp. For example,
+the dimension of the space of all semialgebraic sets with complexity ≤ β is poly-
+nomial in β, and so we cannot “interact” with it in the framework of #o-minimal
+structures. Instead, we prove a family version for every single step, keeping track of
+the formats and degrees of the total spaces involved.
+In section 2, we review sharp versions of some standard o-minimal constructions.
+In section 3, we formally introduce forts and study their elementary properties and
+constructions. We reformulate S∗
+ℓ , F ∗
+ℓ in terms of forts, and denote them by Sℓ,k, Fℓ,k,
+where k is the amount of parameters in the family.
+We prove Sℓ,k, Fℓ,k by induction. We first prove F1,k for every k ∈ Z≥0. We then
+show the steps S≤ℓ,k + F≤ℓ,k → Sℓ+1,k and Sℓ+1,k + F≤ℓ,k+1 → Fℓ+1,k. Sections 4,5,6
+are dedicated to F1,k and the two induction steps respectively.
+Finally, we remark that our proof for Theorem 1.19 works completely verbatim for
+Theorem 1.18, one just needs to ignore the FD-filtration and carry through the same
+constructions. Therefore we will always work with sharply o-minimal structures.
+2. Preliminary results for Sharp Structures
+2.1. Definition of sharply o-minimal structures.
+We follow the definition for
+#o-minimal structures as it appears in [5].
+Definition 2.1 (FD-filtrations). Let S be an o-minimal expansion of the real field.
+An FD-filtration on S is a filtration Ω = {ΩF,D}F,D∈N on the collection of definable
+sets by two natural numbers, such that the following holds.
+(1) Every definable set is in ΩF,D for some F, D ∈ N,
+(2) For every F, D ∈ N we have
+(2)
+ΩF,D ⊂ ΩF+1,D ∩ ΩF,D+1.
+We say that a definable set has format F and degree D if X ∈ ΩF,D. We say that a
+definable map has format F and degree D if its graph has format F and degree D.
+
+10
+DMITRY NOVIKOV, BENNY ZACK
+Definition 2.2 (Sharp o-minimality). Let S be an o-minimal expansion of the real
+field, and let Ω be an FD-filtration on S. The pair (S, Σ) is called a sharply o-minimal
+structure (#o-minimal for short) if the following holds.
+(1) For every F ∈ N there exists a polynomial PF of one variable with positive
+coefficients such that if X ⊂ R has format F and degree D then it has at
+most PF(D) components.
+(2) If X ⊂ Rℓ has format F and degree D then the sets R × X, X × R, Rℓ \ X
+and πℓ−1(X) have format F + 1 and degree D. Moreover, F ≥ ℓ neccasarily
+holds.
+(3) If Xi ⊂ Rℓ are definable sets of format Fi and degree Di for i = 1, . . . , k
+then the union ∪Xi has format F and degree D, and the intersection ∩Xi
+has format F + 1 and degree D, where F := max
+i Fi and D := �
+i Di.
+(4) If P ∈ R[x1, . . . , xℓ] has degree d, then the zero set {P = 0} ⊂ Rℓ has format
+ℓ and degree d.
+Fix a sharply o-minimal structure. We will often work with families, so we intro-
+duce the following definition to make notation easier.
+Definition 2.3. Let Fλ = {fλ : X → Y }λ∈Λ be a family of definable maps. We say
+that the family Fλ is an (F, D)-family if the total space FΛ = {(λ, x, y) ∈ Λ×X ×Y :
+y = fλ(x)} has format F and degree D. Similarly, if Xλ is a family of subsets of Rℓ,
+we say it is an (F, D)-family if the total space XΛ = {(λ, x) : λ ∈ Λ, x ∈ Xλ} has
+format F and degree D.
+2.2. Sharpness of arithmetic operations.
+The following basic lemmas about
+sharpness of arithmetic operations were not strictly treated in [5, 6], even though
+they were used there implicitly. We prove them here, for they serve as an additional
+illustration of the mechanism of the #o-minimal axioms.
+Lemma 2.4. If Xλ ⊂ Ra, Yλ ⊂ Rb are definable (F, D)-families then the fiber
+product Xλ × Yλ is an (OF(1), polyF(D)) family.
+Proof. The non-family version immediately from the axioms and from the equality
+X × Y =
+�
+X × Rb�
+∩ (Ra × Y ). Therefore the set
+(3)
+XΛ × YΛ = {(λ, x, µ, y) : λ, µ ∈ Λ, x ∈ Xλ, y ∈ Yµ}
+has format OF(1) and degree polyF(D). It is left to notice that the total space of
+the family Xλ × Yλ is given by a projection of (XΛ × YΛ) ∩ {λ = µ}.
+□
+Similarly, one can prove the following Lemma.
+Lemma 2.5. Let fλ, gλ : X → Y and hλ : Y → Z be definable (F, D)-families.
+Then the families fλ ± gλ, hλ ◦ fλ are (OF(1), polyF(D))-families. If Y ⊂ R, then so
+is fλgλ, and if in addition gλ ̸= 0, then so is fλ/gλ.
+
+FORTIFYING THE YOMDIN-GROMOV ALGEBRAIC LEMMA
+11
+2.3. Sharpness of Refinement.
+We will often use the following Lemmas implic-
+itly.
+Lemma 2.6.
+(1) Let Φ be a cylindrical decomposition of Iℓ, whose cells are of format F and
+degree D, and for every cell C ∈ Φ let ΦC be a cylindrical decomposition of
+C whose cells are of format F and degree D. Then there exists a cylindrical
+decomposition Ψ of Iℓ of size polyF(D, �
+C∈Φ |ΦC|) whose cells have format
+OF(1) and degree polyF(D) that refines all the decompositions ΦC.
+More
+precisely, every cell of Ψ is contained in a cell of ΦC for some C ∈ Φ.
+(2) Let Φ1, . . . , Φs be cylindrical decompositions of Iℓ whose cells have format F
+and degree D. Then there exists a cylindrical decomposition Ψ of Iℓ of size
+polyF(D, �s
+i=1 |Φi|) whose cells have format OF(1) and degree polyF(D) that
+refines Φ1, . . . , Φs.
+Proof. For the first item, use #CD on all the collection of all cells in ∪CΦC. For the
+second item, use #CD on the collection of all cells in ∪iΦi.
+□
+2.4. Sharpness of Cr locus.
+We need the following slightly stronger formulation
+of [5, Proposition 3.1]. The proof remains the same however.
+Proposition 2.7. Let f1, . . . , fs : Iℓ → I be definable functions of format F and
+degree D. Then f1, . . . , fs are Cr outside a definable set V of codimension ≥ 1 of
+format OF,r(1) and degree polyF,r(D, s).
+Proof. See the proof of [5, Proposition 3.1].
+□
+Remark 2.8. The same proof shows that if f is Cr, then for any α ∈ Nℓ with |α| ≤ r
+the partial derivative
+∂
+∂xαf has format OF,|α|(1) and degree polyF,|α|(D). We will use
+this fact without referring to this remark.
+2.5. Sharp definable choice.
+Proposition 2.9. Let {Xy ⊂ Rℓ}y∈Y be an (F, D)-family of non empty definable
+sets. Then there exists a definable map g : Y → Rℓ of format OF(1) and degree
+polyF(D) such that g(y) ∈ Xy for all y ∈ Y .
+Proof. See [5, Section 4].
+□
+Remark 2.10. The following is an equivalent formulation: Let f : X → Y be a
+surjective definable function of format F and degree D. Then there exists a definable
+section g : Y → X of f that has format OF(1) and degree polyF(D).
+
+12
+DMITRY NOVIKOV, BENNY ZACK
+3. Forts
+3.1. Forts and Morphisms.
+Let qm
+i
+: Im → Im−i be the projection on the last
+m − i coordinates. m will often be omitted.
+For a set X ⊂ Rℓ and x ∈ πi(X) where 1 ≤ i ≤ ℓ, we denote Xx := qi(πi|−1
+X (x)).
+For a point x ∈ Rℓ and an integer 1 ≤ i ≤ ℓ, we denote x1...i := (x1, . . . , xi).
+Definition 3.1. An integer cell of length 1 is either a point {k} or an interval
+(k, k + 1) where k ∈ Z. An integer cell of length ℓ is a product of ℓ integer cells of
+length 1.
+Remark 3.2. From now on the notation C is reserved for integer cells, and not
+general cells as in Definition 1.12.
+There are two useful inductive definitions of forts.
+Definition 3.3 defines forts
+via their structure along the x1 axis, while Definition 3.5 is similar to the inductive
+definition of cells in o-minimal geometry. We prove in Proposition 3.8 below that
+these two definitions are equivalent.
+Definition 3.3. A fort F of length 1 is an interval (0, n) for a positive integer n. It
+is naturally partitioned into integer cells, and we define C(F) to be the set of these
+integer cells. We denote by Forts(1) the set of forts of length 1.
+Let F 1 be a fort of length 1, and let s : C(F 1) → Forts(ℓ − 1) be any function.
+A fort of length ℓ is the set
+(4)
+F 1 ⋉ s :=
+�
+C∈C(F 1)
+C × s(C).
+The fort F 1 ⋉ s is naturally partitioned into integer cells C × C′ where C ∈ C(F 1)
+and C′ ∈ C(s(C)). We denote by C(F 1 ⋉ s) the set of these cells. Finally, denote by
+Forts(ℓ) the set of forts of length ℓ.
+Remark 3.4. We will often write F with an upper index to indicate its length. i.e.
+if not stated otherwise F ℓ means that F is a fort of length ℓ.
+Definition 3.5. Let F ℓ−1 be a fort, and ϕ : F ℓ−1 → N a function constant on each
+cell C ∈ C(F ℓ−1). Then the set
+(5)
+F ℓ−1 ⊙ ϕ := {(x1, . . . , xℓ) : (x1, . . . , xℓ−1) ∈ F ℓ−1, 0 < xℓ < ϕ(x1, . . . , xℓ−1)}
+is a fort of length ℓ.
+Example 3.6. Let F ℓ be a fort. Then for any 1 ≤ i < ℓ, the set πi(F ℓ) is a fort.
+We will also say that F ℓ extends πi(F ℓ).
+
+FORTIFYING THE YOMDIN-GROMOV ALGEBRAIC LEMMA
+13
+Example 3.7. Let F ℓ be a fort. Then for any 1 ≤ i ≤ ℓ and every (x1, . . . , xi) ∈
+πi(F ℓ), the set F ℓ
+(x1,...,xi) is a fort.
+Proposition 3.8. Definition 3.5 is equivalent to Definition 3.3.
+Proof. First note that it is straightforward to verify Example 3.6 with both defini-
+tions. We argue by induction on ℓ, the cases ℓ = 1, 2 being clear. Let F ℓ be a fort
+in the sense of Definition 3.3, and write F ℓ = F 1 ⋉ s. For every cell C ∈ C(F 1),
+denote by s′(C) := πℓ−2(s(C)). By induction, there exists a function ϕC : s′(C) → N,
+constant on the cells of s′(C), such that s′(C)⊙ϕC = s(C). We extend ϕC to C ×s′(C)
+by making it not depend on x1 ∈ C, and keep the same notation ϕC.
+Next, note that F ℓ−1 := πℓ−1(F ℓ) = F 1 ⋉ s′, so ϕ := �
+C∈C(F 1) ϕC is a function
+F ℓ−1 → N which is constant on the cells of F ℓ−1. It remains to show that F ℓ =
+F ℓ−1 ⊙ ϕ.
+Fix a cell in F ℓ, it is of the form C′ = C × C′′ where C ∈ C(F 1) and C′′ ∈ C(s(C)).
+Since s′(C) ⊙ ϕC = s(C), the value of ϕC on πℓ−2(C′′) is not smaller than sup qℓ−2(C′′).
+By the definition of ϕ, this means that this cell is also a cell of F ℓ−1 ⊙ ϕ. The same
+argument shows that every cell of F ℓ−1 ⊙ ϕ is a cell of F ℓ.
+The proof that Definition 3.5 implies Definition 3.3 is completely analogous.
+□
+We next equip Forts(ℓ) with morphisms, making it a category.
+Definition 3.9. Let F1, F2 ∈ Forts(ℓ). A definable map φ : F1 → F2 is called a
+morphism if:
+(1) φ is surjective
+(2) φ is precellular.
+(3) For every C1 ∈ C(F1) there exists C2 ∈ C(F2) such that φ(C1) ⊂ φ(C2), and
+moreover the restriction Φ|C1 is continuous.
+Example 3.10. Let F be a fort. Then the set dF = {d · x, x ∈ F} for d ∈ N is
+also a fort, and the map dF → F given by x �→ d−1x is a morphism called linear
+subdivision of order d.
+Example 3.11. Let φ : F ℓ
+1 → F ℓ
+2 be a morphism.
+Then for any 1 ≤ i < ℓ,
+φ1...i := (φ1, . . . , φi) is a morphism πi(F ℓ
+1) → πi(F ℓ
+2).
+Example 3.12. Let φ : F ℓ → G ℓ be a morphism. Then for any 1 ≤ i ≤ ℓ, and
+every (x1, . . . , xi) ∈ πi(F ℓ), the map φ(x1,...,xi) := φ(x1, . . . , xi, ·, . . . , ·) is a morphism
+F ℓ
+(x1,...,xi) → G ℓ
+φ1...i(x1,...,xi).
+The following is an important example of morphisms. They are constructed using
+the canonical coordinate-wise affine parmaterization of o-minimal cells.
+
+14
+DMITRY NOVIKOV, BENNY ZACK
+Definition 3.13. Let X ⊂ Rm, Y ⊂ Rn be definable sets. A map f : X → Y is
+called affine if f is a restriction of an affine map Rm → Rn, i.e if f = Ax + b for
+A ∈ Mn×m, b ∈ Rn.
+Definition 3.14. Let C be an integer cell of length ℓ. A cellular map φ : C → Rℓ is
+called natural if for every 1 ≤ i ≤ ℓ and every (x1, . . . , xi−1) ∈ πi−1(C), the function
+φi(x1, . . . , xi−1, ·) is affine.
+The following Lemma is straightforward.
+Lemma 3.15. Let C be a cell (not necessarily integer) of length ℓ, and let C be any
+integer cell with the same type as C. Then there exists a unique surjective natural
+cellular map AC,C : C → C. If C is definable in a sharply o-minimal structure and
+has format F and degree D, then AC,C has format OF(1) and degree polyF(D).
+Definition 3.16. A morphism φ : F ℓ
+1 �� F ℓ
+2 is called natural if for every C1 ∈
+C(F ℓ
+1), the restriction φ|C1 is natural.
+Given a cylindrical decomposition Φ of Iℓ, there exists a unique pair (F ℓ, φ) where
+φ : F ℓ → Iℓ is a natural morphism such that for every C ∈ C(F ℓ), the restriction
+φ|C is a homeomorphism between C and a cell of Φ. The fort F ℓ will be called the
+type of Φ. We will now decribe a series of bijections illustrating the equivalence of
+morphisms and cylindrical parametrizations.
+(1) Cylindrical decompositions of Iℓ of type F ℓ are in 1 − 1 correspondence with
+natural morphisms F ℓ → Iℓ.
+(2) Cylindrical parametrizations of cylindrical decompositions of Iℓ of type F ℓ
+are in 1 − 1 correspondence with morphisms F ℓ → Iℓ.
+(3) Given a cylindrical decomposition Φ which corresponds to a natural morphism
+φ : F ℓ → Iℓ, cylindrical parametrizations of refinements Φ′ of Φ of type G ℓ
+are in 1 − 1 correspondence with commutative triangles of morphisms of the
+kind
+F ℓ
+Iℓ
+G ℓ
+φ
+.
+In particular, this means that the ordinary cylindrical decomposition theorem from
+o-minimality can be reformulated in terms of forts and natural morphisms. We will
+use the following somewhat stronger formulation.
+Definition 3.17. Let F be a fort, and let {XC,j}C∈C(F), j∈J be a collection of de-
+finable subsets of F such that XC,j ⊂ C. We say that a morphism φ : �
+F → F is
+compatible with {XC,j}C∈C(F), j∈J if φ( �C) is compatible with every XC,j.
+
+FORTIFYING THE YOMDIN-GROMOV ALGEBRAIC LEMMA
+15
+Proposition 3.18. . Let F be a fort, and let {XC,j}C∈C(F), j∈J be a collection of
+definable subsets of F such that XC,j ⊂ C. Then there exists a natural morphism
+φ : �
+F → F compatible with {XC,j}C∈C(F), j∈J.
+If in addition the structure is #o-minimal, and the sets XC,j have format F and
+degree D, then �
+F can be taken to have size polyF(D, |C(F)|, |J|) and the morphism
+φ can be chosen so that for every C ∈ C(F) the restriction φ|C has format OF(1)
+and degree polyF(D).
+We postpone the proof of Proposition 3.18 until we develop the theory of forts
+further.
+3.2. Combinatorial Equivalence.
+Here is a general useful lemma about mor-
+phisms.
+Lemma 3.19. Let φ : F ℓ
+1 → F ℓ
+2 be a morphism.
+Then φ is one-to-one, and
+moreover for every 1 ≤ i ≤ ℓ and every (x1, . . . , xi−1) ∈ πi−1(F ℓ
+1) the function
+φi(x1, . . . , xi−1, ·) is continuous.
+Proof. We prove it by induction on ℓ. For ℓ = 1, since φ is a strictly monotone
+increasing bijection between two intervals, it is clearly one-to-one and continuous.
+Suppose the claim is true for ℓ − 1. Fix (x1, . . . , xℓ−1) ∈ πℓ−1(F ℓ
+1).The function
+φℓ(x1, . . . , xℓ−1, ·) is continuous by Example 3.12 and the ℓ = 1 case. If φ(x) = φ(x′),
+we know that πℓ−1(x) = πℓ−1(x′) by induction, and then x = x′ follows from the fact
+that φℓ(x1, . . . , xℓ−1, ·) is strictly monotone.
+□
+Suppose one has a definable family {φλ : F → G }λ∈I of morphisms from the fort
+F to the fort G . Then the map I × F → I × G defined by (λ, x) �→ (λ, φλ(x)) is an
+onto precellular map which is not necessarily a morphism. The first obstruction is
+the continuity in λ of the restrictions φλ|C for all cells C ∈ C(F). To formulate the
+second obstruction, we introduce the following definition.
+Definition 3.20. Let φ, ψ : F → G be morphisms. We say that φ, ψ are combina-
+torially equivalent if for every C ∈ C(F), the images φ(C), ψ(C) are contained in the
+same cell of G .
+Example 3.21. Let F ∈ Forts(ℓ), then all morphisms F → Iℓ are combinatorially
+equivalent.
+Example 3.22. Consider the following two natural morphisms φ, ψ : (0, 3) → (0, 2).
+(6)
+φ =
+�
+x
+2, x ∈ (0, 2],
+x − 1, x ∈ (2, 3)
+ψ =
+�
+x, x ∈ (0, 1],
+x+1
+2 , x ∈ (1, 3)
+
+16
+DMITRY NOVIKOV, BENNY ZACK
+They are not combinatorially equivalent, since φ(1) = 1
+2 and so φ maps the cell {1}
+into the cell (0, 1), but ψ(1) = 1, and so maps the cell {1} into itself.
+Proposition 3.23. Let {φλ : F → G }λ∈I be a definable family of combinatorially
+equivalent morphisms. Suppose further that for every C ∈ C(F), the map (λ, x) �→
+φλ(x) is continuous on I × C.
+Then the map ψ : I × F → I × G defined by
+(λ, x) → (λ, φλ(x)) is a morphism.
+Proof. Clearly, ψ is onto and precellular. By the second assumpstion, we also know
+that ψ is continuous on the cells of I × F. Now let C ∈ C(I × F) be a cell. Then
+there exists C′ ∈ C(F) such that C = I × C′. Since the φλ are combinatorially
+equivalent, there exists C′′ ∈ C(G ) such that φλ(C′) ⊂ C′′ for all λ. Thus, ψ maps
+I × C′ into I × C′′, as needed.
+□
+The following is an important structural proposition on forts and morphisms.
+Proposition 3.24. Let φ : F ℓ → G ℓ be a morphism, and let C ∈ C(πi(F ℓ)) for
+some 1 ≤ i ≤ ℓ.
+(1) For every x, x′ ∈ C we have F ℓ
+x = F ℓ
+x′. Thus the notation F ℓ(C) := F ℓ
+x is
+justified.
+(2) The map C(F ℓ(C)) → C(F ℓ) defined by C′ �→ C × C′ is a bijection between
+C(F ℓ(C)) and the cells �C′ of F ℓ satisfying πi(�C′) = C.
+(3) For every x, x′ ∈ C the morphisms φx, φx′ are combinatorially equivalent.
+Proof. We prove the first item by ascending induction on i.
+1. For i = 1, it is clear from Definition 3.3. Now suppose the claim is true for i − 1.
+Let (xi+1, . . . , xℓ) ∈ F ℓ
+x. Since x1...i−1, x
+′
+1...i−1 are in the same cell of πi−1(F ℓ), by
+induction we have F ℓ
+x1...i−1 = F ℓ
+x′
+1...i−1, thus (xi, xi+1, . . . , xℓ) ∈ F ℓ
+x′
+1...i−1, or in other
+words (xi+1, . . . , xℓ) ∈
+�
+F ℓ
+x′
+1...i−1
+�
+xi.
+We claim that xi, x′
+i are in the same cell of
+π1
+�
+F ℓ
+x′
+1...i−1
+�
+. Indeed, since x, x′ are in the same cell of πi(F ℓ), we have that xi, x′
+i
+are bounded between the same two consecutive integers, or are both equal to an in-
+teger. This precisely means that they are in the same cell of π1(F ℓ
+x′
+1...i−1). So, again
+by the induction hypothesis, we have
+�
+F ℓ
+x′
+1...i−1
+�
+xi =
+�
+F ℓ
+x′
+1...i−1
+�
+x′
+i
+. We conclude that
+(xi+1, . . . , xℓ) ∈
+�
+F ℓ
+x′
+1...i−1
+�
+x′
+i
+= F ℓ
+x′
+1...i. So F ℓ
+x ⊂ F ℓ
+x′, and by symmetry we even have
+equality.
+2. Since the cells of F ℓ(C) are disjoint, obviously this map is one to one.
+Let
+C′ ∈ C(F ℓ(C)), clearly, C × C′ is not disjoint from F ℓ, and since it is an integer
+cell, it then follows that C × C′ ∈ C(F ℓ), and obviously πi(C × C′) = C. Finally, let
+
+FORTIFYING THE YOMDIN-GROMOV ALGEBRAIC LEMMA
+17
+�C′ ∈ C(F ℓ) such that πi(�C′) = C. Then �C′ = C × C′ for some integer cell C′. Again,
+clearly C′ is not disjoint from F ℓ(C), and thus C′ ∈ C(F ℓ(C)), and so the map is
+surjective.
+3. Let C′ ∈ C(F ℓ(C)), and let �
+C′′ ∈ C(G ℓ) be the cell that contains φ(C × C′).
+By the second item �
+C′′ corresponds to a cell C′′ ∈ C(G ℓ
+φ1...i(x1...i)). We claim that
+φx(C′), φx′(C′) ⊂ C′′.
+Indeed, let y ∈ C′.
+Then (x, y), (x′, y) ∈ C × C′, and so
+φ(x, y), φ(x′, y) ∈ �
+C′′. Again by the second item, this exactly means that φx(y), φx′(y) ∈
+C′′.
+□
+3.3. Inverse image fort and proof of Proposition 3.18.
+Lemma 3.25. Let F, G ∈ Forts(ℓ), and let φ : F → G be a morphism. Let G ′ ⊂ G
+be a fort. Then F ′ := φ−1(G ′) is a fort and φ|F ′ : F ′ → G ′ is a morphism.
+Proof. If F ′ is a fort, then the second part of the lemma is obvious.
+We prove
+that it is a fort by induction on ℓ. The case ℓ = 1 is clear. By the iductive hy-
+pothesis, φ−1
+1...ℓ−1(πℓ−1(G ′)) is a fort, and since φ is precellular, we have πℓ−1(F ′) =
+φ−1
+1...ℓ−1(πℓ−1(G ′)). Let C be one of the cells of φ−1
+1...ℓ−1(πℓ−1(G ′)), and let C′′ be the cell
+of πℓ−1(G ′) that contains φ1...ℓ−1(C). Then G ′(C′′) is naturally a subfort of G (C′′), and
+so by the induction hypothesis for any x ∈ C we have that F ′
+x is a fort. Moreover, ac-
+cording to proposition 3.24, for any x, x′ ∈ C, the morphisms φx, φx′ : F(C) → G (C′′)
+are combinatorially equivalent, which means precisely that F ′
+x = F ′
+x′ for any such
+x, x′. We’ve shown that F ′ is a fort by Definition 3.5.
+□
+Proof of Proposition 3.18. The fort F is contained in a fort D which is a linear
+subdivision of (0, 1)ℓ of order d ≤ |C(F)|.
+By
+#CD there exists a cylindrical
+decomposition of D compatible with the sets XC,j and the cells P ∈ C(D) into
+polyF(D, |C(D)|, |{XC,j}|) = polyF(D, |C(F)|, |J|) cells of format OF(1) and degree
+polyF(D). Indeed, an integer cell of length ℓ always has format OF(1) and degree
+poly(ℓ) = poly(F) which is within our scale since polyF(D, F) = polyF(D) and
+moreover |C(D)| = polyℓ(|C(F)|).
+In other words, there exists a surjective precellular map φ : F ′ → D, where F ′
+is a fort with |C(F ′)| = polyF(D, |C(F)|, |J|), such that for any C′ ∈ C(F ′), φ|C′
+is continuous and has format OF(1) and degree polyF(D). Moreover, since φ(C′) is
+compatible with the cells of D which are disjoint, it follows that there exists a cell
+of D which contains φ(C′). Thus, φ is morphism. Denote �
+F := φ−1(F). It is now
+easy to see that φ| �
+F : �
+F → F satisfies the requirements of the proposition.
+□
+
+18
+DMITRY NOVIKOV, BENNY ZACK
+3.4. Pulling back along extensions.
+The pullback of a fort F ℓ along a morphism
+to the fort πℓ−1(F ℓ) is a useful natural construction.
+Definition 3.26. Let φ : F ℓ−1
+1
+→ F ℓ−1
+2
+be a morphism, and let F ℓ
+2 extend F ℓ−1
+2
+.
+We define the fort φ∗F ℓ. Suppose F ℓ
+2 = F ℓ−1
+2
+⊙ ϕ (as in Definition 3.5), then
+we define φ∗F ℓ
+2 := F ℓ−1
+1
+⊙ (ϕ ◦ φ).
+The morphism φ extends to the morphism
+(φ, Id) : φ∗F ℓ
+2 → F ℓ
+2.
+If one has a morphism φ : F k
+1 → F k
+2 and for 1 ≤ k < ℓ and a fort F ℓ
+2 extend-
+ing F k
+2 , then φ∗F ℓ
+2 ∈ Forts(ℓ) is defined by induction, and φ will again extend to a
+morphism φ∗F ℓ
+2 → F ℓ
+2.
+We will need the following lemma, which we leave as an exercise for the reader.
+Lemma 3.27. Let φ, ψ : F k → G k be combinatorially equivalent, and let G ℓ extend
+G k. Then φ∗G ℓ = ψ∗G ℓ, and moreover the extensions of φ, ψ to φ∗G ℓ are combina-
+torially equivalent.
+3.5. The Tower construction.
+Given a morphism φ : F → G , it will be more
+convenient to keep track of the formats and degrees of the restrictions of φ to the
+integer cells of F, rather than the format and the degree of φ itself. We therefore
+introduce the following useful slight abuse of notation.
+Definition 3.28. We say that the family {φλ : F → G }λ∈Λ of morphisms is an
+(F, D)-family of morphisms if for every C ∈ C(F) the family {φλ|C}λ∈Λ is an (F, D)-
+family as in Definition 2.3.
+Remark 3.29. Let φ be any morphism, then in particular, the format and degree of
+φ as a morphism are different from its format and degree as a map.
+If C is an integer cell, let b(C) be the basic cell with the same type as C. If m ∈ Z,
+denote tm(x1, . . . , xℓ) := (x1 + m, x2, . . . , xℓ).
+The point of Proposition 3.30 below is, that in order to construct a morphism to a
+fort F 1 ⋉s, it is enough to construct for each C ∈ C(F 1) a morphism to b(C)×s(C).
+Proposition 3.30 (The tower construction). Let F 1 ⋉ s be a fort of length ℓ, and
+let C1, . . . , Cn be the cells of F 1, ordered compatibly with their order in R. Suppose
+that for every 1 ≤ i ≤ n one has a morphism φCi : F ℓ
+Ci → b(Ci) × s(Ci) of format F
+and degree D. Let mi := sup π1(F ℓ
+Ci), and m0 = 0. Then
+(7)
+F ℓ :=
+n−1
+�
+i=0
+tm0+···+mi(F ℓ
+Ci+1)
+is a fort, and the map φ : F ℓ → F 1 ⋉ s defined by φ|tm0+···+mi(F ℓ
+Ci+1) := φCi ◦
+t−(m0+···+mi) is a morphism of format OF(1) and degree polyF(D).
+
+FORTIFYING THE YOMDIN-GROMOV ALGEBRAIC LEMMA
+19
+The proof is immediate. Note that the derivatives of φ are equal to the derivatives
+of φCi, which will be crucial for constructing morphisms with bounded derivatives.
+3.6. Smoothing.
+We say that a morphism of forts is a Cr-morphism if its restric-
+tion to every integer cell is Cr. We use Proposition 2.7 in the context of forts in the
+following way.
+Proposition 3.31.
+Smℓ: Let φ : F ℓ → G ℓ be a natural morphism of format F and degree D, and fix
+r ∈ N. Then there exists a fort K ℓ of size polymax{F,r}(D, |C(F|)) and a natural
+Cr-morphism ψ : K ℓ → F ℓ of format OF,r(1) and degree polyF,r(D) such that φ◦ψ
+is a Cr-morphism.
+Fsmℓ: Let F ℓ be a fort and for every cell C ∈ C(F) let FC = {fC,j : C → I}j∈J
+be a collection of definable functions of format F and degree D. Then there ex-
+ists a natural Cr-morphism φ : K ℓ → F ℓ of format OF,r(1) and degree polyF,r(D)
+where |C(K ℓ)| = polymax{F,r}(D, |F|, |J|) such that for every C′ ∈ C(K ℓ) and every
+C ∈ C(F) with φ(C′) ⊂ C, the function φ|∗
+C′fC,j is Cr for every j ∈ J.
+Proof. The proof is by an induction similar in structure to the induction proof of the
+main result. That is, we prove FSm1, FSmℓ−1 → Smℓ, Smℓ → FSmℓ.
+FSm1: By the tower construction we may assume that F 1 = I, and we omit C from
+the notation fC,j. According to Proposition 2.7 each fj is Cr outside polyF,r(D)
+points. Therefore all the fj are Cr outside |J| · polyF,r(D) points. We finish by
+applying Propositon 3.18 to these points, and note that any natural morphism of
+length 1 is automatically Cr.
+FSmℓ−1 → Smℓ: Let C ∈ C(F ℓ) be a cell. Since φ is natural, if type(C) = (. . . , 1)
+then there are fC, gC : πℓ−1(C) → R such that
+(8)
+φ|C(x1, . . . , xℓ) = (1 − xℓ)fC(x1, . . . , xℓ−1) + xℓgC(x1, . . . , xℓ−1).
+If type(C) = (. . . , 0), then φℓ can be identified with a function on C′ = πℓ−1(C). Let
+F ℓ−1 := πℓ−1(F ℓ) and for every C′ ∈ C(F ℓ−1) define
+F 1
+C′ := {fC, gC : C ∈ C(F ℓ), πℓ−1(C) = C′, type(C) = (. . . , 1)},
+F 2
+C′ := {φℓ|C : C ∈ C(F ℓ), πℓ−1(C) = C′, type(C) = (. . . , 0)},
+F 3
+C′ := {φ1|C′, . . . , φℓ−1|C′}.
+(9)
+In other words, F 1
+C′ and F 2
+C′ correspond to the ℓ′th coordinate of φ, while F 3
+C′ cor-
+responds to the first ℓ − 1 coordinates. Now, apply FSmℓ−1 to the fort F ℓ−1 and
+FC′ = F 1
+C′ ∪ F 2
+C′ ∪ F 3
+C′. We obtain a natural Cr-morphism ψ′ : K ℓ−1 → F ℓ−1. Since
+
+20
+DMITRY NOVIKOV, BENNY ZACK
+F 3
+C′ ⊂ FC, the morphism φ1...ℓ−1 ◦ ψ′ : K ℓ−1 → πℓ−1(G ℓ) is a Cr-morphism. Now
+let K ℓ := (ψ′)∗F ℓ, so that ψ′ extends to a natural Cr-morphism ψ : K ℓ → F ℓ.
+Finally, due to F 1
+C′, F 2
+C′ ⊂ FC′, we see that φ ◦ ψ is Cr, as needed. It is easy to track
+the formats and degrees of this construction.
+Smℓ → FSmℓ : Let k be the smallest integer such that there exists a cell C with at
+least k zeros in its type and a j ∈ J such that fC,j is not Cr on C. Due to Proposition
+2.7, for every C ∈ C(F) there exists a definable set VC ⊂ C of format OF,r(1) and
+degree polymax{F,r}(D, |J|) with dim(C) − dim(VC) ≥ 1 such that outside it, fC,j is
+Cr for every j ∈ J. By Proposition 3.18 there is a natural morphism ψ : G ℓ → F ℓ
+compatible with the collection {VC}C∈C(F ℓ). By Smℓ we may assume that ψ is a
+Cr-morphism. Thus we have increased k by 1, we repeat this procedure until k = ℓ,
+at which point we are done.
+□
+3.7. The main result reformulated.
+A morphism is called an r-morphism if all
+its restrictions to integer cells are r-maps. We are now ready to state the family
+version of our main result in terms of forts and morphisms. We first state the case
+without parameters for simplicity. The reader should keep Definition 3.28 in mind.
+Theorem 3.32.
+Sℓ,0: Let φ : F ℓ → G ℓ be a natural morphism of format F and degree D. Then
+there exists a fort K ℓ with |C(K ℓ)| = polyF,r(D, |C(F ℓ)|) and an r-morphism
+ψ : K ℓ → F ℓ of format OF(1) and degree polyF(D, r) such that the composition
+φ ◦ ψ is an r-morphism.
+Fℓ,0: Let F ℓ be a fort and for every C ∈ C(F ℓ) let FC = {fC,j : C → I}j∈J
+be a collection of definable functions such that each fC,j has format F and degree
+D. Then there exists a fort K ℓ with |C(K )| = polymax{F,r}(D, |C(F ℓ)|) and an
+r-morphism φ : K ℓ → F ℓ of format OF,r(1) and degree polyF,r(D), such that for
+every C ∈ C(F ℓ), j ∈ J and every C′ ∈ C(K ) with φ(C′) ⊂ C, the function (φ|C′)∗fC,j
+is an r-function.
+Sℓ,0 implies S∗
+ℓ of Theorem 1.19 in the following way. Let Φ be a cylindrical de-
+composition of Iℓ of size N into cells of format F and degree D. Then Φ corresponds
+to a natural morphism φ : F ℓ → Iℓ such that |C(F ℓ)| = N and the restrictions φ|C
+are of format OF(1) and degree polyF(D) for all C ∈ C(F ℓ). Let ψ : K ℓ → F ℓ
+be the morphism whose existence is guaranteed by Sℓ,0 for φ. Then the composition
+φ ◦ ψ : K ℓ → Iℓ corresponds to cylindrical r-parametrization of a refinment Φ′ of
+Φ that has size |C(K )| = polymax{F,r}(D, |C(F ℓ)|) = polyF,r(D, N). Moreover, for
+every C′ ∈ K , the restriction (φ◦ψ)|C′ is a composition of two maps that have format
+OF(1), and have degrees polyF(D), polyF,r(D). Therefore by Lemma 2.5, (φ ◦ ψ)|C′
+
+FORTIFYING THE YOMDIN-GROMOV ALGEBRAIC LEMMA
+21
+has format OF(1) and degree polyF,r(D), as needed. Fℓ,0 implies F ∗
+ℓ in a similar way.
+We now state the family version of our main result.
+Theorem 3.33.
+Sℓ,k: Let {φλ : F ℓ → G ℓ}λ∈Ik be an (F, D)-family of natural morphisms. Then there
+exists
+• A natural morphism ϕ : K k → Ik of format OF,r(1), degree polyF,r(D) where
+|C(K k)| = polymax{F,r}(D, |C(F ℓ)|),
+• For every P ∈ C(K ) a fort KP with |C(KP)| = polymax{F,r}(D, |C(F ℓ)|)
+and an (OF,r(1), polyF,r(D))-family of combinatorially equivalent r-morphisms
+{ψλ : KP → F ℓ}λ∈ϕ(P),
+such that for every λ ∈ ϕ(P) the composition φλ ◦ ψλ is an r-morphism.
+Fℓ,k: Let F ℓ be a fort and for every C ∈ C(F ℓ) let FC,λ = {fC,j,λ : C → I}λ∈Ik,j∈J be a
+collection of |J| families such that for every fixed j ∈ J the family {fC,j,λ : C → I}λ∈Ik
+is an (F, D)-family of definable functions. Then there exists
+• A natural morphism ϕ : K k → Ik of format OF,r(1), degree polyF,r(D) where
+|C(K k)| = polymax{F,r}(D, |C(F ℓ)|, |J|),
+• For every P ∈ C(K ) a fort KP with |C(KP)| = polyF(D, |C(F ℓ)|, |J|) and
+an (OF,r(1), polyF,r(D))-family of combinatorially equivalent r-morphisms
+{φλ : KP → F ℓ}λ∈ϕ(P),
+such that for every λ ∈ ϕ(P) and every fC,j,λ ∈ FC,λ where φλ(C′) ⊂ C, the pullback
+�
+φλ|C′�∗ fC,j,λ is an r-function.
+3.8. Final Remarks.
+Remark 3.34. It is easy to notice that composing a morphism φ with linear sub-
+division of order d decreases the partial derivatives of φ by a factor of at least d−1.
+Similarly if φ : F → G is a morphism and f is a function on some cell of G , then
+the d-subdivision of F will also decrease the partial derivatives of φ∗f by a factor of
+at least d. Linear subdivision of order d increases the size of a fort of length ℓ by a
+factor of dℓ, and quite often we shall use linear subdivision with d = polyF,r(D) or
+d = OF,r(1), we shall use this reduction freely. For example, the composition of two
+r-morphisms is an r-morphism up to linear subdivision of order Oℓ,r(1) = OF,r(1).
+Remark 3.35. Let F ℓ be a fort and for every C ∈ C(F ℓ) let FC,λ = {fC,j,λ : C →
+I}{λ∈Ik,j∈J} be a collection of |J| families such that for every fixed j ∈ J the family
+{fC,j,λ : C → I}λ∈Ik is an (F, D)-family of definable functions. Suppose we wish to
+find
+
+22
+DMITRY NOVIKOV, BENNY ZACK
+• A natural morphism ϕ : K k → Ik of format OF,r(1), degree polyF,r(D) where
+|C(K k)| = polymax{F,r}(D, |C(F ℓ)|, |J|),
+• For every P ∈ C(K ) a fort KP with |C(KP)| = polyF(D, |C(F ℓ)|, |J|) and
+an (OF,r(1), polyF,r(D))-family of combinatorially equivalent r-morphisms
+{φλ : KP → F ℓ}λ∈ϕ(P),
+such that for every λ ∈ ϕ(P) and every fC,j,λ ∈ FC,λ where φλ(C′) ⊂ C, the pull-
+back
+�
+φλ|C′�∗ fC,j,λ has a property P. Suppose moreover that this can be done if the
+functions fC,j,λ have the property Q. Then it is sufficient to find
+• A natural morphism ϕ′ : K ′k → Ik of format OF,r(1), degree polyF,r(D)
+where |C(K k)| = polymax{F,r}(D, |C(F ℓ)|, |J|),
+• For every P ∈ C(K ′k) a fort KP
+′ with |C(K ′
+P)| = polyF(D, |C(F ℓ)|, |J|)
+and an (OF,r(1), polyF,r(D))-family of combinatorially equivalent r-morphisms
+{φ′λ : K ′
+P → F ℓ}λ∈ϕ(P),
+such that for every λ ∈ ϕ(P) and every fC,j,λ ∈ FC,λ where φλ(C′) ⊂ C, the pullback
+�
+φ′λ|C′�∗ fC,j,λ has the property Q. Indeed, we first decompose Ik into cells P on which
+the functions fC,j,λ have the property Q, and then we decompose P into cells on which
+the functions fC,j,λ have the property P. The upper bounds on format and degree and
+the size of the forts above follow from Lemma 2.6 and linear subdivision.
+In particular, we will prove Fℓ,k step by step, at each step reducing to the case
+where the functions fC,j,λ have an additional property.
+4. The step F1,k
+We will need a series of lemmas.
+Lemma 4.1. Let {f1,λ : I → I}λ∈Ik, . . . , {fs,λ : I → I}λ∈Ik be (F, D)-families, and
+let r ∈ N. Then there exists
+• A natural morphism ϕ : K k → Ik of format OF,r(1), degree polyF,r(D, s)
+where |C(K k)| = polyF,r(D, s),
+• For every P ∈ C(K k) a fort K 1
+P with |C(K 1
+P )| = polyF,r(D, s) and an
+(OF,r(1), polyF,r(D))-family of combinatorially equivalent natural morphisms
+{φλ : K 1
+P → I}λ∈ϕ(P),
+such that for every λ ∈ ϕ(P) and every C ∈ C(KP) the pullbacks (φλ|C)∗fi,λ are
+Cr-functions.
+Proof. By Proposition 3.31 there exists a natural morphism ϕ′ : K k+1 → Ik+1 such
+that for every P′ ∈ C(K k+1) and every 1 ≤ j ≤ s, the pullbacks (φ′|P′)∗fj,λ(x) are
+Cr as functions of (λ, x). Denote K k = πk(K k+1), ϕ = ϕ′
+1...k and for every cell
+P ∈ C(K k) define K 1
+P := K k+1(P). Then for every λ ∈ ϕ(P) the morphism ϕ′
+
+FORTIFYING THE YOMDIN-GROMOV ALGEBRAIC LEMMA
+23
+induces a morphism φλ := ϕ′
+λ : K 1
+P → I, and these morphisms are combinatorially
+equivalent by Proposition 3.24, this finishes the proof.
+□
+The following lemma is a family version of the original argument by Gromov in
+[8].
+Lemma 4.2. Suppose r ≥ 2, and let {f1,λ : I → I}λ∈Ik, . . . , {fs,λ : I → I}λ∈Ik be
+(F, D)-families of (r − 1)-functions. Then there exists
+• A natural morphism ϕ : K k → Ik of format OF,r(1),degree polyF,r(D) where
+|C(K k)| = polymax{F,r}(1)(D, s),
+• For every P ∈ C(K k) a fort K 1
+P with |C(K 1
+P )| = polyF,r(D, s), and an
+(OF,r(1), polyF,r(D))-family {φλ : KP → I}λ∈ϕ(P) of combinatorially equiva-
+lent r-morphisms,
+such that for every C ∈ C(KP) and every λ ∈ ϕ(P) the pullbacks (φλ|C)∗fi,λ are
+r-functions.
+Proof. Note that the φλ obtained in Lemma 4.1 are natural, and this means that up
+to a linear division of order d = Or(1), by Lemma 4.1, the tower construction and
+Remark 3.35, we can assume that the functions fi,λ are Cr+1-functions.
+Let the coordinates of Ik+1 be (λ, x) and let ϕ′ : K k+1 → Ik+1 be a natural mor-
+phism compatible with the sets {f (r+1)
+j,λ
+= 0} and {f (r)
+j,λ = 0} for every 1 ≤ j ≤ s. De-
+note K k = πk(K k+1), ϕ = ϕ′
+1...k and for every P ∈ C(K k) define K 1
+P := K k+1(P).
+For every λ ∈ ϕ(P) the morphism ϕ′ induces a morphism φλ := ϕ′
+λ : K 1
+P → I, and
+these morphisms are combinatorially equivalent by Proposition 3.24. Since φλ are
+natural, up to a linear subdivision of order d = Or(1), the tower construction and
+Remark 3.35, we have reduced to the case where the family {f (r)
+j,λ}λ∈Ik is either a
+family of monotone increasing functions or a family of monotone decreasing func-
+tions (or a family of constant functions, in which case we are done) of constant sign
+for every 1 ≤ j ≤ s. Moreover, since an r-parametrization of f is the same as an
+r-parametrization of −f, we may additionally assume that f (r)
+i,λ (x) ≥ 0 for all x and
+all λ.
+Define φ : I → I by x �→ 3x2 − 2x3. It is an r-morphism up to linear subdivision
+of order 3. If f (r)
+i,λ is monotone decreasing, then
+(10)
+2
+x ≥ f (r−1)
+i,λ
+(x) − f (r−1)
+i,λ
+(0)
+x
+= f (r)
+i,λ (cx) ≥ f (r)
+i,λ (x).
+
+24
+DMITRY NOVIKOV, BENNY ZACK
+If f (r)
+i,λ is monotone increasing, then
+(11)
+2
+1 − x ≥ f (r−1)
+i,λ
+(1) − f (r−1)
+i,λ
+(x)
+x
+= f (r)
+i,λ (cx) ≥ f (r)
+i,λ (x).
+We have
+(12)
+(fi,λ(φ(x)))(r) = Or(1) + Or
+�
+(φ′)rf (r)
+i,λ (φ(x))
+�
+Note that φ′ = 6x(1−x). If f (r)
+i,λ is monotone decreasing then it is bounded near 1 by
+equation 10, and so (φ′)rf (r)
+i,λ (φ(x)) = Or(1) near 1. Near 0 we have (φ′)rf (r)
+i,λ (φ(x)) =
+Or(xr−2) by equation 10 and since r ≥ 2 we get overall that (fi,λ(φ(x)))(r) = Or(1).
+If f (r)
+i,λ is monotone increasing then it is bounded near 0 by equation 11, and so
+(φ′)rf (r)
+i,λ (φ(x)) = Or(1) near 0. Near 1 we have (φ′)rf (r)
+i,λ (φ(x)) = Or((1 − x)r−2) by
+equation 11, and again since r ≥ 2 we have (φ′)rf (r)
+i,λ (φ(x)) = Or(1). Thus we are
+done by linear subdivision of order Or(1).
+□
+Proof of F1,k. The idea of the proof is as follows. By Lemma 4.2 above, it is suffi-
+cient to reduce to the case where the functions fj,λ are 1-functions. Suppose we have
+definable Cr functions f1, . . . , fs : I → I, and suppose that |f ′
+1| < · · · < |f ′
+s|, and
+that either |f ′
+s| ≥ 1 or |f ′
+s| ≤ 1 globally on I. If |f ′
+s| ≤ 1, we are done. If |f ′
+s| ≥ 1, we
+can reparametrize I by f −1
+s , and then we will be done. We will now reduce to one of
+these two cases.
+By the tower construction we may assume that F 1 = I. By Lemma 4.1 we may
+further assume that the functions fi,λ(x) are in C2. Now define
+gij(λ, x) := |f ′
+i,λ(x)| − |f ′
+j,λ(x)|,
+hi(λ, x) := |f ′
+i,λ(x)| − 1.
+(13)
+Let the coordinates of Ik+1 be (λ, x) and let ϕ′ : K k+1 → Ik+1 be a natural mor-
+phism compatible with the sets {gij = 0}, {hi = 0} and {f ′
+j,λ(x) = 0} for every
+1 ≤ i, j ≤ s. Denote ϕ = ϕ′
+1...k, and for every P ∈ C(K k) define K 1
+P := K k+1(P).
+For every λ ∈ ϕ(P) define Φλ
+P = {ϕλ(C) : C ∈ C(K 1
+P )}, a cylindrical decomposition
+of I of size polyF(D, s).
+We aim to construct a family of r-morphisms {φλ : K 1
+P → I}λ∈ϕ(P) which will be
+a family of cylindrical r-parametrizations of Φλ
+P, such that for every λ ∈ ϕ(P), every
+1 ≤ j ≤ s and every C ∈ C(K 1
+P ) the pullback (φλ|C)∗fj,λ is an 1-function. To do
+this, for every interval Lλ ∈ Φλ we define a monotone increasing surjective r-map
+
+FORTIFYING THE YOMDIN-GROMOV ALGEBRAIC LEMMA
+25
+φLλ : I → Lλ such that for every 1 ≤ j ≤ s the pullback (φLλ)∗fj,λ|Lλ is an 1-function.
+Fix P ∈ C(K k), a λ ∈ ϕ(P) and an interval Lλ ∈ Φλ
+P. Since ϕ′ is compatible
+with f ′
+j,λ, on Lλ some of the functions fj,λ are constant (and their indices j are con-
+stant as λ ranges over ϕ(P)) and those that aren’t constant are strictly monotone.
+Constant functions are automatically r-functions, so we may assume that none of
+the fj,λ are constant. Since ϕ′ is compatible with gij, we can relabel the indices j
+so that |f ′
+1,λ(x)| < · · · < |f ′
+s,λ(x)| for every λ ∈ ϕ(P) and every x ∈ Lλ. Since ϕ′ is
+compatible with hj, we have that either |f ′
+s,λ(x)| ≤ 1 for all λ ∈ ϕ(P), x ∈ Lλ, or
+|f ′
+s,λ(x)| > 1 for all λ ∈ ϕ(P), x ∈ Lλ.
+If |f ′
+s,λ(x)| ≤ 1, we simply parametrize the interval Lλ by an affine map φLλ : I →
+Lλ. Otherwise, suppose without loss of generality that fs,λ is monotone increasing.
+We parametrize Lλ by the map φLλ = f −1
+s,λ ◦ Aλ : I → Lλ, where Aλ : I → fs,λ(Lλ)
+is an affine map. We claim that |φ′
+Lλ| ≤ 1. Indeed,
+(14)
+|φ′
+Lλ(x)| = length(fs,λ(Lλ))
+|f ′
+s,λ(φLλ(x))|
+≤ length(I)
+1
+= 1.
+Finally, we also claim that (φLλ)∗fj,λ is a 1-function for every 1 ≤ j ≤ s. Indeed,
+(15)
+|((φLλ)∗fj,λ)′(x)| = length(fs,λ(Lλ)) · |f ′
+j,λ(φLλ(x))|
+|f ′
+s,λ(φLλ(x))| ≤ 1
+where the last inequality follows as |f ′
+j,λ| < |f ′
+s,λ|. Thus we may assume that the
+functions fj,λ are 1-functions. We are now in position to use Lemma 4.2 r − 1 times,
+and we are done.
+□
+5. The step Sℓ−1,k + Fℓ−1,k → Sℓ,k
+Let {φλ : F ℓ → G ℓ}λ∈Ik be an (F, D)-family of natural morphisms. Fix a cell
+C ∈ C(πℓ−1(F ℓ)) and a cell C′ ∈ C(F ℓ) with πℓ−1(C′) = C. Since φλ is natural, if
+type(C′) = (. . . , 1) then there are definable functions fC′,λ, gC′,λ on C such that
+(16)
+φλ
+ℓ |C′ = (1 − xℓ)fC′,λ(x1, . . . , xℓ−1) + xℓgC′,λ(x1, . . . , xℓ−1),
+and if type(C′) = (. . . , 0) then φλ
+ℓ can be identified with a function on C, and we
+maintain the notation φλ
+ℓ . Define
+F 1
+C,λ := {fC′,λ, gC′,λ : C′ ∈ C(F ℓ), πℓ−1(C′) = C, type(C′) = (·, 1)},
+F 2
+C,λ := {φλ
+ℓ |C′ : C′ ∈ C(F ℓ), πℓ−1(C′) = C, type(C′) = (·, 0)},
+F 3
+C,λ := {φλ
+1|C, . . . , φλ
+ℓ−1|C},
+FC,λ := F 1
+C,λ ∪ F 2
+C,λ ∪ F 3
+C,λ.
+(17)
+
+26
+DMITRY NOVIKOV, BENNY ZACK
+Lemma 2.5 implies that for every C′ ∈ C(F ℓ) with πℓ−1(C′) = C and type(C′) = (·, 1),
+the families {fC′,λ}λ∈Ik and {gC′,λ}λ∈Ik are (OF(1), polyF(D))-families.
+Now use
+Fℓ−1,k on the fort F ℓ−1 := πℓ−1(F ℓ) and the family of functions FC,λ.
+We obtain a natural morphism ϕ : K k → Ik, where K k is a fort of size |C(K k)| =
+polymax{F,r}(D, |C(F ℓ)|), for each cell P ∈ C(K k) a fort K ′
+P of size polymax{F,r}(D, |C(F ℓ)|)
+and an (OF,r(1), polyF,r(D))-family of combinatorially equivalent r-morphisms {ψ′λ :
+K ′
+P → F ℓ−1}λ∈ϕ(P).
+Since F 3
+C,λ ⊂ FC,λ we see that φλ
+1...ℓ−1 ◦ ψ′λ is an r-morphism.
+Define KP :=
+(ψ′λ)∗F ℓ. Since the ψ′λ are combinatorially equivalent, KP indeed does not depend
+on λ. ψ′λ extends to an r-morphism ψλ : KP → F ℓ, and since F 1
+C,λ, F 2
+C,λ ⊂ FC,λ we
+see that φλ ◦ ψλ is an r-morphism as needed.
+6. The step S≤ℓ,k + F≤ℓ−1,k → Fℓ,k
+We will need the following lemmas.
+Lemma 6.1. Let f : Iℓ → I be a definable function of format F and degree D, and
+suppose that for all x1 ∈ I the function f(x1, ·) is L-Lipshitz. Then outside polyF(D)
+hyperplanes of the form {x1 = const} the function f is continuous.
+Proof. Consider the set Var := {x : limy→x|f(y) − f(x)| > 0}.
+Obviously, f is
+continuous on the set Iℓ\Var. We claim that for every t ∈ I, the intersection Var ∩
+{x1 = t} is open in the hyperplane {x1 = t}. Indeed, suppose that for some x ∈
+{x1 = t}
+(18)
+ϵ = limy→x|f(y) − f(x)| > 0.
+If x′ ∈ {x1 = t} and |x − x′| <
+ϵ
+4L then
+|f(y) − f(x′)| ≥ |f(y − x′ + x) − f(x)| − |f(x′) − f(x)| − |f(y − x′ + x) − f(y)| ≥
+≥ |f(y − x′ + x) − f(x)| − ϵ
+2,
+(19)
+since y − x′ + x, y also have the same x1 coordinate. Considering the lim of both
+sides of the above equation as y → x′, we conclude that limy→x′|f(y)−f(x′)| ≥ ϵ
+2 > 0.
+As f is definable, there is a cylindrical decomposition Φ of Iℓ where |Φ| = polyF(D)
+such that f is continuous on the cells of Φ. Clearly, Var can only intersect the cells
+of Φ that are of dimension ≤ ℓ − 1. Let X ∈ Φ such that X ∩ Var ̸= ∅. We claim
+that X necessarily has type (0, . . . ). Indeed, assume that type(X) = (1, . . . ). Then
+
+FORTIFYING THE YOMDIN-GROMOV ALGEBRAIC LEMMA
+27
+for any t ∈ I the intersection X ∩ {x1 = t} has dimension ≤ ℓ − 2. Let t ∈ π1(X).
+On the one hand, Var ∩ {x1 = t} has dimension ℓ − 1 by the above. On the other
+hand, Var ∩ {x1 = t} lies in the finite union of sets Xi ∩ {x1 = t}, where Xi ∈ Φ are
+the cells intersecting Var of type (1, . . . ) (as projections on x1 of cells of Φ of type
+(0, . . . ) and cells of Φ of type (1, . . . ) do not intersect). Thus Var ∩ {x1 = t} is of
+dimension ≤ ℓ − 2, and this is a contradiction.
+In conclusion, Var can only intersect cells of type (0, . . . ) and is therefore contained
+in polyF(D) hyperplanes {x1 = const}, as needed.
+□
+The following is a family version of Lemma 6.1.
+Lemma 6.2. Let {fλ : Iℓ → I}λ∈Ik be an (F, D) family, and suppose that for every
+x1 ∈ I, λ ∈ Ik the function fλ(x1, ·) is L-Lipshitz.Then there exists:
+• A natural morphism ϕ : K k → Ik of format OF(1), degree polyF(D) where
+|C(K k)| = polyF(D),
+• for every P ∈ C(K k) a fort K 1
+P of size polyF(D) and an (OF(1), polyF(D))-
+family {φ′λ : K 1
+P → I}λ∈ϕ(P) of natural morphisms,
+such that if φλ is the extension of φ′λ to (φ′λ)∗Iℓ, then for every C ∈ C((φ′λ)∗Iℓ), the
+pullback (φλ|C)∗fλ is continuous.
+Proof. Let
+d(λ, x1, x′
+1) := sup
+y∈Iℓ−1 |f(x1, y) − f(x′
+1, y)|,
+Σλ := {x1 : limx′
+1→x1d(λ, x′
+1, x1) > 0}.
+(20)
+By Lemma 6.1, for each fixed λ ∈ Ik, the function fλ is continuous outside the hy-
+perplanes {x1 = c} where c ∈ Σλ. We use Proposition 3.18 on the fort Ik+1 (with
+coordinates (λ, x1)) and the set Σ := {(λ, x1) : x1 ∈ Σλ} to obtain a natural mor-
+phism ϕ′ : K k+1 → Ik+1 compatible with Σ.
+Define K k := πk(K k+1), ϕ := ϕ′
+1...k, and for every C ∈ C(K k) let K 1
+P :=
+K k+1(P), and φ′λ := ϕ′
+λ. It is easy to see that this construction satisfies the require-
+ments of the lemma.
+□
+The following is a version of Lemma 12 from [3] for sharp structures, which is a
+key lemma in this subsection.
+Lemma 6.3. Let f : Iℓ → I be a definable C1 function of format F and degree D,
+and suppose that for any 2 ≤ i ≤ ℓ one has | ∂
+∂xif| ≤ 1. Then the function
+∂
+∂x1f(x1, ·)
+is bounded for all but polyF(D) values of x1.
+
+28
+DMITRY NOVIKOV, BENNY ZACK
+Proof. Consider the set
+(21)
+B :=
+�
+x1 : ∀M > 0 ∃(x2, . . . , xℓ) ∈ Iℓ−1
+����
+∂
+∂x1
+f(x1, x2, . . . , xℓ)
+���� > M
+�
+By the axioms of #o-minimal structures and Remark 2.8 the set B has format OF(1)
+and degree polyF(D). By Lemma 12 from [3], the set B is finite. Therefore it consists
+of polyF(D) points, as needed.
+□
+The following is a family version of Lemma 6.3.
+Lemma 6.4. Let {fλ : Iℓ}λ∈Ik be an (F, D)-family of definable C1 functions. Sup-
+pose that for every 2 ≤ i ≤ s, λ ∈ Ik one has | ∂
+∂xif| ≤ 1. Then there exists
+• A natural morphism ϕ : K k → Ik of format OF(1), degree polyF(D) where
+|C(K k)| = polyF(D),
+• for every P ∈ C(K k) a fort K 1
+P of size polyF(D) and an (OF(1), polyF(D))-
+family {φ′λ : K 1
+P → I}λ∈ϕ(P) of natural morphisms
+such that if φλ is the extension of φ′λ to (φ′λ)∗Iℓ, then for every C ∈ C((φ′λ)∗Iℓ) and
+every fixed x1 ∈ I the derivative
+∂
+∂x1
+�
+(φλ|C)∗fλ(x1, ·)
+�
+is bounded.
+Proof. This lemma follows from Lemma 6.3 in the same way as Lemma 6.2 follows
+from Lemma 6.1.
+□
+6.1. Reduction to the case where fC,j,λ is Cr and fC,j,λ(x1, ·) is an r-function
+for all x1.
+Let F ℓ be a fort and for every C ∈ C(F) let FC,λ = {fC,j,λ : C →
+I}λ∈Ik,j∈J be a collection of |J| families such that for every fixed j ∈ J the family
+{fC,j,λ : C → I}λ∈Ik is an (F, D)-family of definable functions. By the tower con-
+struction and the induction hypothesis we may assume that F ℓ = I × F ℓ−1. Let
+C ∈ C(F ℓ−1) and define FC,(λ,x1) = {fI×C,j,λ(x1, ·) : fI×C,j,λ ∈ FI×C,λ}. That is, we
+consider each of the target families of functions {fI×C,j,λ} as a family of functions on
+cells of F ℓ−1 with parameters λ, x1.
+We use Fℓ−1,k+1 on F ℓ−1 and FC,(λ,x1). We obtain a natural morphism ϕ′ : K k+1 →
+Ik+1 (where the coordinates of Ik+1 are (λ, x1)) and for each P′ ∈ C(K k+1) a family
+of combinatorially equivalent r-morphisms φ(λ,x1) : KP′ → F ℓ−1 such that for every
+(λ, x1) ∈ ϕ′(P′) and every j ∈ J the function (φ(λ,x1)|C′)∗fI×C,j,λ(x1, ·) is an r-function
+whenever C′ ∈ C(KP′) and φ(λ,x1)(C′) ⊂ C.
+Define K k := πk(K k+1), and fix a cell P ∈ C(K k). We in particular have a
+combinatorially equivalent family {ϕλ : K k(P) → I}λ∈ϕ′
+1...k(P) given by ϕλ := φ′
+λ.
+By Proposition 3.24 we can identify C(K k(P)) with the cells of K k+1 lying above
+
+FORTIFYING THE YOMDIN-GROMOV ALGEBRAIC LEMMA
+29
+P. Define s : C(K k(P)) → Forts(ℓ − 1) by s(P′) := KP′ and KP := K k(P) ⋉ s.
+Finally, define φλ : KP → F ℓ by φλ(x1, . . . , xℓ) := (ϕλ(x1), φ(λ,x1)(x2, . . . , xℓ)).
+We would like to use Proposition 3.23 to assert that the φλ are morphisms, for as
+x1 ranges over a cell of K k(P), the morphisms φλ(x1, ·) are combinatorially equiva-
+lent. However, we do not know that φ(λ,x1) are continuous as functions of x1, . . . , xℓ.
+Still, we do know that for every x1 the morphism φλ(x1, ·) is an r-morphism, and
+in particular 1-Lipshitz. Therefore, by Lemma 6.2, we may assume that φ(λ,x1) is
+continuous as a function of x1, . . . , xℓ. And so, by Proposition 3.23 the map φλ is a
+morphism for every λ ∈ ϕ′
+1...k(P) . We have thus reduced to the case where for every
+λ ∈ Ik, x1 ∈ π1(F ℓ), j ∈ J and C ∈ C(F ℓ) the function fC,j,λ(x1, ·) is an r-function.
+The last goal of this subsection is to further reduce to the case where the functions
+fC,j,λ are Cr. Due to Proposition 3.31 we can find a natural morphism ϕ : K k → Ik
+and for each cell P ∈ C(K k) a family of combinatorially equivalent Cr morphisms
+{φλ : KP → F ℓ}λ∈ϕ(P) such that (φλ|C′)∗fC,j,λ is Cr for every j ∈ J, λ ∈ ϕ(P) when-
+ever C′ ∈ C(KP) and φλ(C′) ⊂ C. By Sℓ,k we may assume that φλ are r-morphisms.
+Note that by the chain rule, since the first coordinate of a cellular map depends only
+on x1, we see that ||(φλ|C′)∗fC,j,λ(x1, ·)||r = Or,ℓ(1) and so up to a linear subdivision
+of order d = Oℓ,r(1) the reduction to Cr functions has not spoiled our first reduction
+to r-morphisms for every x1.
+6.2. Induction on the first unbounded derivative.
+Order the indices α ∈ Nℓ
+by lexicographic order. Let α be the first index such that |α| ≤ r and such that
+there exists C ∈ C(F ℓ), j ∈ J and λ ∈ Ik such that ||fC,j,λ||r > 1. By the previous
+subsection α1 > 0. By Lemma 6.4, the tower construction and the induction hypoth-
+esis, we may assume that F ℓ = I × F ℓ−1 and for every x1 ∈ I, and every C, j, λ the
+function f (α)
+C,j,λ(x1, ·) is bounded.
+We shall now reparametrize x1. For every C ∈ C(F ℓ) define
+(22)
+SC,j,λ := {|f (α)
+C,j,λ(x1, . . . , xℓ)| ≥ 1
+2 · sup
+x2...ℓ
+|f (α)
+C,j,λ(x1, ·)|} ⊂ C
+By Proposition 2.9 there exists definable families of curves {γC,j,λ : I → SC,j,λ}λ∈Ik
+such that γC,j,λ
+1
+(x1) = x1. Let
+F 1
+λ := {γC,j,λ : C ∈ C(F ℓ), j ∈ J, λ ∈ Ik},
+F 2
+λ := {f (α)
+C,j,λ ◦ γC,j,λ : C ∈ C(F ℓ), j ∈ J, λ ∈ Ik},
+Fλ := F 1
+λ ∪ F 2
+λ.
+(23)
+
+30
+DMITRY NOVIKOV, BENNY ZACK
+We apply F1,k to the fort I and the functions in Fλ. We obtain a natural morphism
+ϕ : K k → Ik and for every P ∈ C(K k) a family of combinatorially equivalent mor-
+phisms φ′λ : K ′
+P → I such that for every C′ ∈ C(K ′
+P) the pullbacks (φ′λ|C′)∗γC,j,λ and
+(φ′λ|C′)∗(f (α)
+C,j,λ ◦γC,j,λ) are r-functions. Define KP := (φ′λ)∗F ℓ and let φλ : KP → F ℓ
+extend φ′λ.
+Let C′′ ∈ C(KP) such that φλ(C′′) ⊂ C. By the induction on α and the chain
+rule we have that
+�
+(φλ|C′′)∗fC,j,λ
+�(β) = Oℓ,r(1) when β < α.
+When computing
+�
+(φλ|C′′)∗fC,j,λ
+�(α) by the chain rule, again by the induction on α all the terms except
+for (φλ|C′′)α1 · (φλ|C′′)∗f (α)
+C,j,λ are bounded by Oℓ,r(1). Now
+(φλ|C′′)α1 · (φλ|C′′)∗f (α)
+C,j,λ ≤
+�∂φλ|C′′
+∂x1
+�α1
+· 2(φλ
+1|C′′)∗(f (α)
+C,j,λ ◦ γC,j,λ) ≤
+≤ ∂φλ|C′′
+∂x1
+· 2(φλ
+1|C′′)∗(f (α)
+C,j,λ ◦ γC,j,λ).
+(24)
+We proceed to bound the right hand side. Consider (φλ
+1|C′′)∗(f (α−11)
+C,j,λ
+◦ γC,j,λ)′, which
+is bounded by Oℓ,r(1) by the induction hypothesis. By the chain rule it is equal to
+(25)
+∂φλ|C′′
+∂x1
+· (φλ
+1|C′′)∗
+�
+f (α)
+C,j,λ ◦ γC,j,λ +
+ℓ
+�
+i=2
+(γC,j,λ
+j
+)′ · f
+(α−11+1j)
+C,j,λ
+◦ γC,j,λ
+�
+.
+Due to the induction on α and the definition of φλ, all of the terms above except for
+∂φλ|C′′
+∂x1
+· (φλ
+1|C′′)∗(f (α)
+C,j,λ ◦ γC,j,λ) are bounded by Oℓ,r(1). Thus, ∂φλ|C′′
+∂x1
+· 2(φλ
+1|C′′)∗(f (α)
+C,j,λ ◦
+γC,j,λ) is also bounded by Oℓ,r(1), and we are done by linear subdivision of order
+d = Oℓ,r(1). □
+References
+[1] G. Binyamini, G. Jones, H. Schmidt, and M. Thomas. Effective Pila-Wilkie in the restricted
+sub-Pffafian structure. In preperation, 2023.
+[2] G. Binyamini and D. Novikov. Complex Cellular Structures. Annals of Mathematics,
+190(1):145–248, 2019.
+[3] G. Binyamini and D. Novikov. The Yomdin-Gromov Algebraic Lemma revisited. Arnold Math
+J., 7:419–430, 2021.
+[4] G. Binyamini and D. Novikov. Tameness in geometry and arithmetic: beyond o-minimality.
+EMS press, 2022.
+[5] G. Binyamini, D. Novikov, and B. Zak. Sharply o-minimal structures and sharp cellular de-
+composition. Preprint, 2022.
+[6] G. Binyamini, D. Novikov, and B. Zak. Wilkie’s conjecture for Pfaffian structures. Preprint,
+2022.
+
+FORTIFYING THE YOMDIN-GROMOV ALGEBRAIC LEMMA
+31
+[7] G. Binyamini and N. Vorobjov. Effective cylindrical cell decompositions for restricted sub-
+Pfaffian sets. International Mathematics Research Notices, 2020.
+[8] M. Gromov. Entropy, homology and semialgebraic geometry. S´eminaire Bourbaki, 86(145-
+146):225–240, 1987.
+[9] J. Pila and A.J. Wilkie. The rational points of a definable set. Duke Math. J, 133(3):591–616,
+2006.
+[10] Lou van den Dries. Tame Topology and O-minimal Structures. Cambridge University Press,
+1998.
+

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+Conservation Tools: The Next Generation of
+Engineering–Biology Collaborations
+Andrew K. Schulz1,2,+,∗, Cassie Shriver3,+, Suzanne Stathatos4,+,
+Benjamin Seleb3, Emily Weigel3, Young Hui Chang3,
+M. Saad Bhamla5, David Hu1,3, Joseph R. Mendelson III3,6
+Schools of Mechanical Engineering1, Biological Sciences3, and Chemical and Biomolecular Engineering5
+Georgia Institute of Technology, Atlanta, GA 30332, USA
+Max Planck Institute for Intelligent Systems2, Stuttgart, Germany
+School of Computing and Mathematical Sciences4
+California Institute of Technology, Pasadena, CA 91125, USA
+Zoo Atlanta6, Atlanta, GA 30315, USA
+January 4, 2023
++ indicates co-first author
+Corresponding author:
+Andrew Schulz
+Heisenbergstraße 3, Stuttgart, Germany 70569
+(405)780-0542
+andrew.schulz1994@gmail.com
+Keywords:
+Conservation Tech, Human-Centered Design, AI4Good, Tech4Wildlife
+Abstract
+The recent increase in public and academic interest in preserving biodiversity has led to the growth
+of the field of conservation technology. This field involves designing and constructing tools that
+utilize technology to aid in the conservation of wildlife. In this article, we will use case studies
+to demonstrate the importance of designing conservation tools with human-wildlife interaction in
+mind and provide a framework for creating successful tools. These case studies include a range of
+complexities, from simple cat collars to machine learning and game theory methodologies. Our goal
+is to introduce and inform current and future researchers in the field of conservation technology and
+provide references for educating the next generation of conservation technologists. Conservation
+technology not only has the potential to benefit biodiversity but also has broader impacts on fields
+such as sustainability and environmental protection. By using innovative technologies to address
+conservation challenges, we can find more effective and efficient solutions to protect and preserve
+our planet’s resources.
+Background and Motivation
+The term ”conservation technology” was first proposed by Berger-Tal in 20181 to broadly describe
+the use of technology to manage and conserve wildlife. While a commonly referenced example is
+unmanned aerial vehicles (UAVs, also known as drones), there are many other conservation tech-
+nologies, including camera traps, mobile applications, spatial mapping, and environmental DNA.
+Much of the existing technology uses modern hardware and software design processes to improve
+1
+arXiv:2301.01103v1  [q-bio.QM]  3 Jan 2023
+
+upon ongoing conservation efforts and initiate previously under-addressed efforts1. Some of the
+major goals of conservation technology are to iterate more quickly to improve outdated equipment,
+increase accessibility to tools, and use modern technology to address conservation problems in
+entirely new ways. Conservation technology is being developed for animals in both natural envi-
+ronments and captive settings (e.g. foxes in urban settings and elephants in zoos, respectively)2
+and applications for this technology may include conservation challenges and monitoring needs for
+animals, plants, habitats, geological phenomena such as volcanoes, climate, and atmosphere, and
+more.
+Why has there been a revolution in implementing these new and old tools in the conservation
+sector in the last few years? The recent broader recognition of the significant threats to biodiversity
+has driven demand for conservation technology.
+Since 1970, wildlife populations have plunged
+by 69%3.
+With advancing technological revolutions over the past millennium, many scientists,
+engineers, and other conservation stakeholders see conservation tools as valuable methods to address
+serious ongoing conservation challenges.
+Historically, the field of conservation technology has taken an opportunistic approach wherein
+stakeholders invest in developing technology for a specific, non-conservation need that is then
+applied to wildlife management1, a prime example being the development of drones by the military.
+While opportunistic technologies certainly aid in wildlife management efforts, they also tend to be
+expensive, less accessible to the conservation community, and rarely the most idealized solution for
+specific wildlife management issues. Given the alarming rates of biodiversity loss amidst the current
+mass extinction4, there has been an increasing push towards purpose-driven technology designed
+in consultation with members of the conservation community1. Critically, new technology is not
+considered to represent proper conservation technology until its genuine usefulness and success for
+managing and conserving wildlife has been demonstrated.
+Producing purpose-built technology requires a variety of skills that are typically beyond the
+scope of a single person, so establishing successful interdisciplinary collaborations is crucial. To
+properly synthesize these perspectives, conservation technology must establish the necessary bridges
+between the conservation community, technologists, and policymakers1,5. However, these interdis-
+ciplinary collaborations are dependent on effective communication across domains, which can be
+difficult given differences in objectives and goals. While technology development and outcomes are
+often derived from an engineering design mindset, biological conservation is more hypothesis-driven
+and grounded in the scientific method.
+Despite grants, training, and other efforts towards these collaborations, the conservation com-
+munity has encountered many solutions claiming to be universally minded but lacking in necessary
+interdisciplinary knowledge and partners in respective fields. Inadequate communication between
+fields initially led the wildlife community to be distrustful of new technologies because many engi-
+neering solutions were poorly applied to serve the needs of conservationists, especially in natural
+outdoor situations. However, the pressing nature of the sixth mass extinction and climate change
+has made the necessity of interdisciplinary solutions evident and worth pursuing despite communi-
+cation difficulties. And with this expansion of interdisciplinary collaborations comes the realization
+that novel contributions to conservation technology can be designed in a variety of ways.
+The term ”conservation technology” has received much criticism from the conservation com-
+munity for the implication of requiring advanced technologies. This is misleading when plenty of
+modern innovations involve using simple non-hardware/software devices to serve novel conservation
+problems. Conservation technology can be as complex as machine learning used to identify and
+track species or as simple as chili pepper fences used to deter African Elephants from damaging
+farmland6,7. It is for this reason that we will utilize the term conservation tools (CT) instead of
+conservation technology for the remainder of this manuscript. A tool is broadly defined as a device
+2
+
+to carry out a particular function, and we believe using the term ”conservation tools” better encom-
+passes the diversity of the field. Rephrasing also intentionally includes indigenous solutions utilized
+in traditional conservation practices around the globe, which may not be accurately described in
+the scope of conservation technology.
+In this manuscript, we will describe a diverse array of case studies of conservation tools
+that have been implemented globally. We also emphasize how the importance of the project to
+work alongside the communities impacted with the communities as the stakeholder to minimize the
+traditional practices of parachute science while maximizing community partnerships, access, and
+conservation impact. Parachute science, where scientists ”parachute” into places for purposes of
+research or conservation but leave without a trace of co-authorship to community members who
+made the work possible, is an unfortunately common phenomenon8. We describe tools that are
+heavily focused on hardware and heavily focused on software but can still be simple to use, build,
+and adapt to additional conservation needs. This manuscript is meant to be a starting guide to
+introduce the field of creating conservation tools to those without experience. We hope that the
+glossary of terms also allows current practitioners of CT to understand the diversity of this field
+better. This manuscript hopes to allow the reader to understand that biodiversity is essential not
+just in the wild but also in the technologies utilized to help the conservation tools be as effective
+as possible.
+Conservation Tools Vocabulary
+As the conservation technology field has grown, it has adopted many terms from other fields to
+describe conservation tools accurately. Unfortunately, many of these technical terms are domain-
+specific and can alienate stakeholders. We list and define many of the terms commonly used to
+describe conservation tools and provide corresponding publications with more details on these spe-
+cific terms (Table 1). We elaborate on each term in the case studies and introduction. Throughout
+this paper, we present these terms in bold font to indicate where readers can refer to this table for
+additional details.
+Discussion
+We propose to shift the phrase ”conservation technology” to conservation tools because the tradi-
+tional process of creating conservation technology relies on what is known as opportunistic tech-
+nology1. Purpose-built technology is common in the hardware and software industry, considered
+collectively under the term Human-Centered Design (HCD). HCD operates by using a design
+mindset that focuses on the context of the use of the idea. A common example is the difference
+between checkout interfaces in different environments. Purchasing a beverage at a bar versus at
+a supermarket are similar situations, but the context of use for exchanging money for goods is
+completely different. In each scenario, there are a set number of individuals, m, and a set number
+of items each individual has, n. For the bar, there is a very large m with a very small n, but the
+opposite is true of the supermarket. Thus the design of the technology and processes that enable
+the purchasing of goods are in very different contexts of use. We will apply the same logic to
+conservation tools in utilizing a Human–Wildlife-Centered Design (HWCD) approach.
+A HWCD approach for conservation tools requires consideration of not just the human interac-
+tion with the device but also the interaction between humans and wildlife. While ”human-wildlife
+conflict”9 is used to describe interactions in urban, farming, or wild settings that can cause large
+amounts of harm to human interests10, ”Human-Wildlife interaction” is broadly used to describe
+3
+
+both positive and negative interactions. HWCD is not a new concept, as it has been implemented
+for millennia by indigenous peoples that live, interact, and move with the land.
+For designers
+from non-indigenous backgrounds, it is essential to understand that you will never be able to
+achieve true indigenous design unless the primary designers of a technology solution are from the
+native indigenous lands where the solutions will be implemented11. To ignore indigenous or other
+community-derived knowledge is to create a solution with only partial expertise or knowledge of
+the problem; for this reason, we implore readers to understand that the effectiveness of your tool
+relies on the active collaboration of the community, scientists, and engineers. As we go forward
+in this manuscript, it is paramount for authors to understand that the best tools are created by
+indigenous researchers, scientists, and engineers working collaboratively as they are the most knowl-
+edgeable folks in the world about the conservation challenges non-indigenous members outside the
+community have imposed.
+We will now discuss five case studies to introduce the core principles which we highlight in
+Figure 1, which we believe conservation technology solutions should be implemented. The themes
+are listed at the beginning of each section.
+• What – what is its use?
+• How – how is it used?
+• Where – what are the use cases/how it helps
+• Why – future directions/open questions/etc.
+The following case studies cover specific conservation tools that are created, drawing from both
+new and old technologies.
+Each of these solutions utilizes some measure of HWCD., although
+some utilize frugal materials and are simple, while others take advantage of advanced hardware
+and software that have become more accessible in recent years. The themes of these case studies
+relate to what the surveyed community of conservationists believe are the most important tools for
+assisting in advancing conservation from the most recent state of conservation technology report12.
+We proceed with discussing a case study on accessibility in technology.
+Case Study 1: AudioMoth
+Principle: Solutions should be open source, and accessible in cost and function
+Open-source software often describes the ability to access the code and customize and edit the code
+how we see fit. Undergraduate biologists are often taught R, an open-source programming language
+geared toward statistical modeling; comparatively, engineers and computer scientists frequently
+learn Python, an open-source all-purpose programming language. Regardless of the coding language
+used, open-source code can then be appreciated by the collaboration community where the prior
+knowledge only differs by which open-source software is used in their curriculum. Workshops13
+and online forums14 have begun to bridge the gap between the two groups; for example, in the
+CV4Ecology workshop, engineers teach conservation biologists at the graduate level and post-
+doctoral levels specific tools in Python. One example of how open-source software and hardware
+are used for conservation is AudioMoth.
+What is its use? Effective wildlife management decisions require abundant data on the organ-
+isms. Acoustic monitoring has become one of the more ubiquitous methods of recording information
+in field situations where sound is relevant15,16. While early practices required individuals to actively
+note the sounds they heard, passive monitoring devices can be deployed into areas of interest to
+record information for animals located within a certain proximity15–17.
+4
+
+How is it used? The AudioMoth costs ten times less than commercial products, is energy
+efficient, records both human-audible sounds and ultrasonic frequencies, is the size of a credit card,
+and was created by two PhD students with the intention of increasing scientific accessibility(Figure
+2A)18 .
+Furthermore, this device is open-source, meaning the code and operating features are
+made public for distribution and modifications for individual projects19. Immediately popular, it
+has been used to monitor animal populations20, track migrations21, identify poaching activity19,
+detect sounds underwater22, and even discover new species19.
+What is a use case? The AudioMoth exemplifies how designing technology that is open-source
+and accessible can dramatically increase scientific participation, with substantial implications for
+informing future wildlife management policies and practices. The term open source solution can
+mean several different things when looking at a device such as AudioMoth and we will discuss these
+in a set of categories: open-source hardware, open-source software, and open-source code.
+These devices can substantially increase monitoring coverage both in terms of land area and
+recording time15–17. While the multi-functional device has applications throughout the biological
+world it has seen greater use recently with the release of the United Nations’s sustainable develop-
+ment goals (SDGs), specifically investigating Life below Water (SDG 14) and Life on Land (SDG
+15). However, initial productions were far too expensive and complex for mass implementation in
+the scientific community until the creation of the AudioMoth.
+A core feature of open-source hardware is that the project can be built of mechanical parts
+that are all easy to acquire. This can mean a variety of things from easy to purchase or easy
+to create using different advanced manufacturing techniques such as 3D printing or laser cutting.
+specifically 3D printing, sometimes referred to as additive manufacturing, allows for specific parts of
+a hardware model to be built at low expense23. Truly open-sourced hardware will not just tell you
+the techniques utilized but will also include the exact parts, files, models, etc. required for this. A
+new journal publication type is leveraging the future of open-source hardware through publications.
+These journals currently include the Journal of Open Hardware, The Journal of Open Engineering,
+and HardwareX. These journals require all submissions to include complete information for all
+hardware and software included in the device24. It should, however, be mentioned that some of
+these publishers, including Elsevier, are for-profit publishers.
+What is the Potential? Open-sourced hardware does not just mean the mechanical devices
+such as nuts and bolts, but also the electrical devices, such as the circuit board and circuitry diagram
+(Figure 2B). By providing these schematics, users can build the entire AudioMoth system using
+the specifications sheets provided in their publication in HardwareX. There can be large gaps in
+the idea of open source between hardware and software. Many devices allow you full access to the
+sourcing and schematics of the hardware but require you to purchase specific software from the
+organization.
+Two of the commonly applied to be used in this context are front-end interface and back-end
+interface. The front-end interface is what the primary user sees on a screen, or the user-interface
+(UI)25 . The back-end interface is internal equipment that is actually doing much of the coding
+work25. Many of the devices will not feature a customizable front end because it will directly reflect
+a customizable back end known as the application programming interface or API. An API is a pri-
+mary way of allowing open-source code not only to be accessed but also updated and the outputs to
+be changed. This is necessary for researchers because acquiring an API lets the researcher customize
+the data collection, for example extracting location information, or measurements of temperature or
+velocity all of which allow the open-source tool to be customizable for both inputs and outputs. The
+manufacturer of AudioMoth, OpenAcoustics (https://www.openacousticdevices.info/audiomoth),
+provides the researcher not only an API but also all of the operational code, in a user manual for
+each device thus permitting customization of any device aspect (Figure 2C-D). Finally, its ease
+5
+
+of use for the end user is a crucial design component. In designing this device the engineering
+developers have a feedback mechanism for those in the field to help continuously improve the use
+of this. The engineers and scientists developed this tool to be used by indigenous researchers and
+people in which they could place these devices wherever they see fit to start receiving data. Those
+that have occupied indigenous lands are the most knowledgeable about where the placement of
+these devices would be most effective.
+In designing a tool with HWCD in mind, it is vital to think of the context of use. The AudioMoth
+is intended to be used by ecologists attempting to get bio-acoustic data from their open-source
+sensor. Biologists differ significantly from computer scientists and engineering researchers; biology
+is a hypothesis-based field, whereas engineering is design based. The AudioMoth is designed by and
+for biologists to be deployed quickly and repaired easily in various applications and environments.
+AudioMoth utilizes advanced technology in both software and hardware while utilizing context-of-
+use to consider how it will be used.
+Case Study 2: Environmental DNA (eDNA)
+Principle: Solutions should take advantage of increasing hardware technologies
+DNA is a well-established scientific tool for an ever-expanding scope of biological studies and
+beyond26. An enormous challenge in the use of DNA for purposes of conservation is that traditional
+methods of DNA collection require biological samples such as urine, hair, skin, or other tissue27.
+Traditional biological methods have historically required the restraint, capture, or rapid collection
+of fresh DNA samples that can be either logistically infeasible or actively at-odds with observing
+organisms in the wild. Focal organisms in many conservation programs often are extremely rare or
+secretive, and it may not be possible or logistically feasible to get samples from them. Thomsen
+and Willerslev (2015) reviewed the use of eDNA as an emerging tool in conservation28. One of
+the primary challenges they highlighted in conservation is the trade-off between the invasiveness of
+studies and data collection. Applications of eDNA are reducing the needs for invasive studies and
+enabling locating and monitoring of creatures too rare or secretive for traditional survey methods.
+What is its use? Environmental DNA allows for the analysis of diets, geographical ranges,
+population sizes, demographics, and genetics, as well as the assessment of the presence/absence
+of species at sites. These can be quantified using environmental samples such as feces left behind
+and analyzed for different genetic information. The field of eDNA benefits the conservation space
+as being a holistically non-invasive method of DNA extraction, making it very repeatable. The
+techniques for the collection of eDNA still require biological sample collection, but the samples can
+be in much lower quantity and make use of vacuums to process the needed concentrations.
+How is it used? This is a previously developed tool, more recently used by wildlife conserva-
+tionists, which allows using DNA samples found in the environment for understanding endangered
+species in their natural environment utilizing only DNA samples. The novelty in this tool is its
+ability to not only detect DNA information about animals, but it is applicable across a variety of
+environments including land, sea, and in polar ice samples28. As a tool, eDNA is useful in a variety
+of conservation and ecological fields, but this solution has had a significant history of colonial-style
+parachute science29. It is important to note that while solutions and technology like this can be
+leveraged, they must be thought of in the HWCD framework. Working with local and indigenous
+communities as eDNA is not the sole solution and additionally on the ground conservation work is
+necessary for the long-term conservation of wildlife30.
+What is a use case?: One of the first documented uses of eDNA was in 1992 when Amos
+utilized shed skin from cetacean mammals species to inform a population analysis31. Although
+not considered as conservation technology at the time, this was one of the first applications of
+6
+
+non-invasive eDNA for biological conservation and population assessment. Now eDNA is utilized
+to monitor not just populations but to catalog local bio-diversity of fishes32, manage reptile popu-
+lations33, and forest conservation34 using the interface between remote sensing and environmental
+DNA.
+What is the potential? This solution appears to be a universal (or cure-all) tool. Universally
+designed solutions typically will only work for specific use cases. The non-universality in eDNA is
+that it leverages the well-established scientific tool of DNA for new and innovative applications for
+purposes of conservation data and management. Despite its broad range of potential applications,
+eDNA nevertheless is not a universal solution for all situations, especially because the methodology
+is complex in terms of sampling and acquisition of data. As in a wet lab technique it is prone
+to the same human errors as other lab based risks including contamination, biased results and
+interpretations, or even as simple as not having adequate reference databases for identifying DNA
+sequences for all regions or applications.
+These pitfalls do not discount eDNA as an example of utilizing new advances in hardware
+and scientific progress to advance conservation practices. Tools such as this continually are being
+improved and innovated. This field is expanding in the past years with increasing establishments
+of DNA Barcodes that permit identification of species using online DNA databases35.
+Case Study 3: Computer Vision
+Principle: Solutions should take advantage of increasing software technologies
+Machine learning is the science and art of developing computer algorithms to learn automatically
+from data and experience36. Computer vision is a sub-field of machine learning, in which com-
+puters and systems are trained to extract meaningful information (aka ”see”) from images, videos,
+and other inputs. Computer vision lets computers understand visual inputs37.
+What is its use? While humans have been “trained” during their lifetime to identify objects,
+understand their depth, and see their interactions, computer models require thousands of images
+to teach machines to “see” new scenarios. Computer vision has expanded in recent years, too,
+from only being able to work on super-computers to now working on edge devices like cell phones
+and laptops in the wild38–40. With hardware advances and algorithms designed for lower-resource
+devices, computer vision has become less expensive and more accessible to many organizations that
+wish to use it. Users of computer vision applications today include, but are not limited to, (1)
+iPhone users to unlock their phones with their face, (2) drivers of self-driving cars, and (3) traffic
+enforcers who use red-light traffic cameras.
+How is it used? Conservationists use camera traps to capture images of wildlife. Computer
+vision techniques are applied to these camera trap images to help scientists detect, track, classify,
+and re-identify (recognize) individual animals, among other things41,42.
+A typical camera trap
+apparatus is shown in Figure 3A. A camera is placed in a region of interest. It passively collects
+information about what goes through that region. Camera traps collect data over a specified period,
+either writing to an external hard drive or pushing data to a cloud-hosted framework. Camera traps
+often include infrared and/or motions sensors that can identify warm-bodied or moving objects.
+When an animal triggers the sensor, the camera records (writes images to memory), as shown
+in Figure 3B. Afterward, the data is fed into a machine learning model to learn and recognize
+patterns in the data, Figure 3C.
+Traditionally, computer vision has used classical supervised machine learning algorithms (al-
+gorithms that need human-labeled data). These algorithms let the model understand identifying
+characteristics of the animals within the regions of interest (for example, color histograms, texture
+differences, locomotor gait, etc. Figure 3C)6. From those characteristics, the model can learn to
+7
+
+detect and classify wildlife species in the images. Two key use cases of this include classifying ani-
+mal species (classification) and recognizing and identifying individual animals (re-identification
+or re-id). In these tasks, extraction of the foreground of the image is an important pre-processing
+step to focus the model on the animal of interest. For example, the first step in classifying urban
+wildlife is often to crop the image to focus on the animal and not to focus on cars, leaves, trees,
+etc .43. The model could then take these cropped images and classify them as different species
+types (i.e. squirrels, dogs, coyotes, etc.). Alternatively,43 could be used to identify which photos
+to ignore. For example, several urban wildlife monitoring projects use it to crop humans out of
+images and ignore empty images.44.
+What is a use case? Recent advances in hardware have allowed computer vision to expand
+to underwater locations. The Caltech Fish Counting task leverages sonar cameras placed in rivers
+to detect, track, and count salmon as they swim upstream45. The setup of these types of cameras
+within rivers is illustrated in Figure 4. They cannot rely on infrared sensors, so they capture
+images continuously across a specified period. Fisheries managers review the videos and manually
+count the number of salmon. Caltech researchers are working on automating this with computer
+vision45.
+What is the potential? Computer vision has led to a set of technologies that can aid wildlife
+conservation across terrestrial, aquatic, and lab environments. Using computer vision as a tool
+can help solve limitations in manual data analysis by saving time and by limiting external bias.
+Processing large amounts of data quickly allows ecologists to then identify ecological patterns,
+trends, etc. in their scientific space and facilitates quicker lead times on field observations. Their
+science, then, informs ecological actions and goals. The integration of computer vision into wildlife
+conservation is dynamically automating animal ecology and conservation research using data-driven
+models.6
+Case Study 4: Game Theory and Optimization
+Principle: Economics and Artificial Intelligence should be leveraged in conservation
+challenges to optimize decision-making
+Artificial intelligence is actively being used to combat wildlife threats. When designing con-
+servation tools, like sensors, one key challenge is where to place them in an animal’s ecosystem to
+collect relevant data. Researchers are looking into ways to leverage artificial intelligence methods
+to optimize conservation/resource planning and policy-making. One such field in computer science
+that differs from computer vision is the use of game theory for more effective data collection. Game
+theory is a collection of analytical tools that can be used to make optimal choices in inter-actional
+and decision-making problems. The use of game theory for conservation has only recently become
+a field of study.
+What is its use?: In non-mathematical terms, optimization is the study of how to make the
+best or most efficient decision given a certain set of constraints. In probability theory and machine
+learning, the multi-armed bandit problem is one type of optimization problem in which a limited
+set of resources must be split among/between competing choices to maximize expected gain. This
+problem is a sub-class of a broader set of problems called stochastic scheduling problems. In these
+problems, each machine provides a reward randomly from a probability distribution that is not
+known a-priori. The user’s objective is to maximize the sum of the rewards. These techniques
+are commonly used for logistics (routing) coordination and financial portfolio design, though they
+have also been adapted to be used for modeling nefarious actors and optimally countering them.
+In wildlife scenarios, biologists often have to use a small number of tools to collect data in a vast
+environment often hundreds of square kilometers. The use of optimization strategies has recently
+8
+
+begun to help ecologists and biologists pinpoint locations to effectively collect data descriptive of a
+large ecological habitat.
+How is it used? Patrol Planning. Wildlife poaching and trading threaten key species across
+ecosystems. Illegal wildlife trade facilitates the introduction of invasive species, land degradation,
+and biodiversity loss46. Historically, park rangers have recorded where poachers have struck. How-
+ever, in most national parks, there is a limited supply of park rangers. They often are limited
+to driving, walking, or biking around the parks. Several parks have repositories of historical data
+detailing poaching locations identified in the past. This data can be used to predict likely poaching
+threats and locations in the future. Work has been done in the game theory and optimization
+space to leverage machine learning (on the historical data) and optimize multi-modal (i.e. driving
+and walking) patrol planning. Ultimately, parks and wildlife conservation organizations want to
+find the optimal answer to the questions, “How should I organize my patrols?” and “How will
+adversaries respond?”47. This optimization technique provides them with a way to answer those
+questions directly.
+Economic Modeling. Additional researchers, including Keskin and Nuwer, are working toward
+understanding the economics behind these wildlife threats. Poaching functions as an additional
+source of income for individuals in rural communities who may rely primarily on tourism for income.
+If these communities cannot rely on tourism, they may focus on wildlife trafficking, as those species
+are prevalent near them48. A review of wildlife tracking49 focusing on operations and supply chain
+management recognized four challenges that limit preventative measures:
+1. the difficulties of understanding the true scale of illegal wildlife trade (IWT) from available
+data;
+2. the breadth of the issue - trafficked animals are used for food, status symbols, traditional
+medicine, exotic pets, and more (this requires the policy remedy to be multifaceted), and
+sometimes IWT operates in countries with corrupt governments or limited infrastructures for
+law enforcement and monitoring;
+3. IWT groups are geared toward undetectable operations, especially from financial institutions;
+4. IWT is considered less serious than other trafficking, i.e. human, drugs, weapons.
+There are several suggested ways to apply research in supply-chain operations toward combating
+IWT49. These include: bolstering data through satellite data, acoustic monitoring, news scraping,
+and finding online markets; strengthening data detection and prediction through network analysis
+and understanding data bias; modeling the problem as a network interdiction problem to see how
+to disrupt the supply chain network; more effective resource management and reducing corruption.
+By analyzing the complex supply chain and operations behind IWT, Keskin et al. illuminated a
+more clear picture of each location/scenario individually, which allows an informed and targeted
+response to prevent illegal wildlife trafficking50.
+What are some use cases?
+Evidence from parks in Uganda suggests that poachers are
+deterred by ranger patrols, illuminating the increased need for robust, sequential planning47. Com-
+puter science economists have worked on adversarial modeling to demonstrate poachers’ deterrence
+to patrols along with other poacher behavior patterns47. An illustration of poaching patterns with
+increased patrols are shown in Figure 5.
+Researchers working at the Jilin Huangnihe National Nature Reserve in China first used machine
+learning to predict poaching threats and then used an algorithm to optimize a patrol route. When
+rangers were dispatched in December 2019, they successfully found forty-two snares, significantly
+more than they had found in previous months and patrols51. Combining machine learning and
+9
+
+optimization techniques, therefore, has proven to increase the efficiency of patrol planning and can
+be expanded to more conservation management applications as well.
+What is the potential? Applying optimization techniques across conservation-oriented tasks
+will provide insight and better resource usage to historically under-resourced applications and
+programs.
+In addition, these optimization techniques and economically-focused viewpoints can
+prompt organizations and governments to identify and quell issues more efficiently. Programs can
+best utilize the limited resources they have and do so in an efficient data-driven manner. This
+can, in theory, be scaled to any resource-limited situation, too.
+Those with camera traps, for
+example, can study where to best place them to capture the most data-rich images. Those with
+limited AudioMoths, similarly, can study where to place them to ensure optimal and most realistic
+acoustic captures.
+Case Study 5: Cat Collar
+Principle: Solutions can be simple and should not be over-engineered This case study
+serves to provide an example that fails to fit the restrictive title of conservation technology that
+has taken hold and the overly technocentric vision that often results. A significant challenge to be
+considered when designing conservation tools is viability. A device or solution that is functional may
+not be enough. It must also be accessible in terms of parts availability and costs. The intended
+user base of any conservation tool is likely quite small. While the invention of a powerful tool
+may provide substantial functionality and/or opportunity, it is possible that (through traditional
+avenues) consumer demands and means are insufficient to support its development and distribution.
+When a consumer cannot fully utilize or understand a conservation solution, it can fail, such as
+when there is little to no local adoption of the tools developed52. While frugal science is defined as
+reducing the cost of equipment in terms of bio-engineering, its driving principles are uniquely suited
+to conservation developments. Frugal science is subtly different than the Do-It-Yourself (DIY) and
+Free and Open-Source Hardware (FOSH) as it is focused on utilizing the most frugal ingredients to
+build your tool. While not all devices can be designed frugally, many can have dramatically reduced
+costs as much of the conservation space is not purpose-built designs, but instead, like camera traps,
+are designed for hunters but also utilized also by ecologists and conservationists around the world.
+What is its use? The collar is used to help combat nearly 1.3–4.0 billion birds per year killed
+by domestic cats in the US alone53. This makes domestic cats one of the nation’s most significant
+anthropogenic threats to wildlife, yet very little counter-action has been taken. Common fondness
+towards cats makes enforcing restrictions on them difficult and makes enacting equivalent invasive
+species eradication methods extremely unpopular, even for feral cats54.
+Despite how silly the
+Birdsbesafe® collar looks, it has been found that collared cats killed 19 times fewer birds than did
+their uncollared cats55. The collar is far less costly and controversial than alternative measures and
+allows cats to continue prowling freely. While no one would expect it at first glance, this demeaning
+accessory could be the most realistic, effective solution to preventing cat-caused bird extinction.
+How is it used? The Birdsbesafe® collar is a simple woven fabric collar with bright collars
+painted along the fabric. The application of this device is simple as a large percent of the US popu-
+lation has outdoor cats. These collars are around $10 and are made of two-inch wide cotton fabric
+tubes connecting to a quick-release collar. The collars are meant to be worn by both domesticated
+and wild cats as a means to reduce bird deaths. Much like other Felidae, domesticated cats (Felis
+catus) have stealthy prey-stalking behavior. The collar has bright colors that can be seen from
+far away by both birds and small mammals that the cats may prey upon. This solution is simple
+as no technology is required, and the tool is used by hooking onto a collar around the cat’s neck
+so the cat cannot pull them off. Additionally, these collars were designed with cats in mind as it
+10
+
+does not impede on any of the cat’s primary needs, including eating, drinking, sleeping, urinating,
+or defecating. Functionally this tool is a wearable technology that utilizes color to assist in the
+reduction of bird deaths, and it is as simple as hooking it around your pet cat’s collar.
+What is a use case? In contrast to many of the previously discussed case studies, the idea of
+a conservation tool can be as simple as a brightly colored collar. In biomechanics, space footwear
+is designed to reduce the stress on the joints of the foot. This is verified using scientific data and
+techniques, but an essential factor in shoes is not just biomechanical support but how they look on
+your feet. This is the human element of human-centered design and is an important consideration
+when working with domestication species. When working with a domesticated species, like the cat,
+the solution amongst biologists is very simple: do not let your cat outside. However, the human
+element in this solution is that not all folks will follow these recommendations, and therefore other
+techniques need to be used. Less complex tools, like the cat collar, whose simplicity and low cost
+facilitate broader implementation, can make essential steps in conservation. This solution does
+not solve the issue of invasive cats killing birds and other animals but is dramatically reduces the
+impact that cats have on those that purchase these for their cats.
+What is the potential? Financing the development of conservation tools is a significant issue,
+and there may be few solutions that allow financing of crucial pilot tools and prototypes. Support
+from philanthropic organizations or technology organizations working in the technology space, such
+as Google Earth, Bezos Earth Fund, or AI4Good from Microsoft, can allow small startups to fund
+grants for these tool creations. But instead of this support, we encourage the implementation of
+frugal science methodology56.
+Conclusions
+Conservation tools vary but are united in their potential to aid conservation. There
+is no single solution to the many challenges in conservation. Conservation tools are designed to
+be a part of a community’s toolkit to help conserve and protect wildlife, and we discuss the key
+themes that make them successful in Figure 6. As these case studies show, conservation tools
+are not meant to solve all problems, but they can be useful in contexts where previous methods
+are too onerous or costly. Developers of conservation tools must understand that their designs
+need to be user-friendly for conservation practitioners and be viewed as a resource rather than a
+complete solution for addressing biodiversity decline. The most effective solutions are those that are
+realistically implementable and take into consideration the context of human-wildlife interactions
+in the design process. In this paper, we review five case studies of specific conservation tools that
+are advancing wildlife conservation. When examining these tools, it is important to consider the
+context in which they are used and the specific conservation issues they are addressing.
+To develop effective conservation tools, biologists, computer scientists, and engineers must col-
+laborate and apply their expertise. These interdisciplinary teams must also work with community
+members who have a deep understanding of conservation challenges. The wide range of perspectives
+and challenges addressed through these partnerships allow conservation tools to take many forms.
+We highlight five key characteristics of successful conservation tools. Open-source and accessible
+solutions like the AudioMoth offer opportunities for crowd-sourcing and additional improvements,
+as well as the ability to adapt existing frameworks to similar problems. Hardware from other fields
+can be repurposed in innovative ways to benefit conservation, such as using eDNA to reduce the
+invasiveness of data collection techniques. Existing software like computer vision can also be ap-
+plied to the conservation field to streamline and expand data analyses. Successful conservation
+tools are not limited to biology, engineering, and computer science; they can also benefit from
+11
+
+non-traditional fields like math for identifying ideal collection sites. Finally, not all solutions need
+to be high tech to be effective. Simple solutions, like cat collars with bells to protect birds, can
+also be effective conservation tools.
+In this paper, we aim to provide a foundation for future conservation tool creators by reviewing
+case studies of successful tools and highlighting key themes. These case studies demonstrate the
+diverse range of approaches that can be taken in conservation technology, from simple cat collars to
+complex machine learning and game theory methodologies. By drawing on the expertise of inter-
+disciplinary teams that include biologists, computer scientists, engineers, and community members,
+we can develop effective tools that address the unique challenges of each conservation context. As
+we work to conserve and protect wildlife, it is essential to remember that conservation tools are just
+one part of a larger toolkit and should be integrated into traditional and indigenous approaches
+to conservation. Through this review, we hope to inspire the development of innovative solutions
+to address the pressing needs of biodiversity conservation.
+Ultimately, conservation technology
+is essential for addressing the challenges of biodiversity preservation and promoting sustainable
+solutions for human-wildlife interactions.
+Acknowledgements
+Thank you to all of the members of the Georgia Tech Tech4Wildlife Student Organization for their
+support.
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+not
+the
+tool
+by
+itself.
+Journal
+of
+Fish
+Biology,
+98(2):383–386,
+2021.
+eprint:
+https://onlinelibrary.wiley.com/doi/pdf/10.1111/jfb.14177.
+[31] W.
+Amos,
+H.
+Whitehead,
+M.
+J.
+Ferrari,
+D.
+A.
+Glockner-Ferrari,
+R.
+Payne,
+and
+J. Gordon.
+Restrictable Dna from Sloughed Cetacean Skin;
+Its Potential for Use
+in
+Population
+Analysis.
+Marine
+Mammal
+Science,
+8(3):275–283,
+1992.
+eprint:
+https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1748-7692.1992.tb00409.x.
+[32] Mei Shen, Nengwen Xiao, Ziyi Zhao, Ningning Guo, Zunlan Luo, Guang Sun, and Junsheng Li.
+eDNA metabarcoding as a promising conservation tool to monitor fish diversity in Beijing water
+systems compared with ground cages. Scientific Reports, 12(1):11113, June 2022. Number: 1
+Publisher: Nature Publishing Group.
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+
+[33] Bethany Nordstrom, Nicola Mitchell, Margaret Byrne, and Simon Jarman. A review of applica-
+tions of environmental DNA for reptile conservation and management. Ecology and Evolution,
+12(6):e8995, 2022.
+eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/ece3.8995.
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+Conservation Policies with Essential Biodiversity Variables Incorporating Remote Sensing and
+Environmental DNA Technologies. Forests, 13(3):445, March 2022. Number: 3 Publisher:
+Multidisciplinary Digital Publishing Institute.
+[35] Morgan R. Gostel and W. John Kress. The Expanding Role of DNA Barcodes: Indispensable
+Tools for Ecology, Evolution, and Conservation. Diversity, 14(3):213, March 2022. Number:
+3 Publisher: Multidisciplinary Digital Publishing Institute.
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+Challenges and Opportunities, March 2022. arXiv:2203.02544 [cs].
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+The iwildcam 2020 competition dataset.
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+preprint arXiv:2004.10340, 2020.
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+of the European conference on computer vision (ECCV), pages 456–473, 2018.
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+and people. Frontiers in Sustainable Cities, 4, 2022.
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+Van Horn, and Pietro Perona. The caltech fish counting dataset: A benchmark for multiple-
+object tracking and counting. In European Conference on Computer Vision (ECCV), 2022.
+[46] Unep. analysis of the environmental impacts of illegal trade in wildlife., 2017.
+https://
+wedocs.unep.org/handle/20.500.11822/17554.
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+Artificial Intelligence (UAI-21), 2021.
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+MacDonald, Rowan Hilend, Stanley Griffis, and Meredith L. Gore. Quantitative investigation
+of wildlife trafficking supply chains: A review. Omega, page 102780, 2022.
+15
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+[50] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep
+convolutional neural networks. 25, 2012.
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+for anti-poaching. In AAAI Conference on Artificial Intelligence, 2021.
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+servation Agriculture Practices in Ethiopia. International Journal of Environmental Chem-
+istry, 5(1):7, May 2021. Number: 1 Publisher: Science Publishing Group.
+[53] Scott R. Loss, Tom Will, and Peter P. Marra. The impact of free-ranging domestic cats on
+wildlife of the United States. Nature Communications, 4(1):1396, January 2013. Number: 1
+Publisher: Nature Publishing Group.
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+Anne
+T.
+Loyd
+and
+Sonia
+M.
+Hernandez.
+Public
+Perceptions
+of
+Domes-
+tic
+Cats
+and
+Preferences
+for
+Feral
+Cat
+Management
+in
+the
+Southeastern
+United
+States.
+Anthrozo¨os,
+25(3):337–351,
+September 2012.
+Publisher:
+Routledge
+eprint:
+https://doi.org/10.2752/175303712X13403555186299.
+[55] S.K. Willson, I.A. Okunlola, and J.A. Novak. Birds be safe: Can a novel cat collar reduce
+avian mortality by domestic cats (Felis catus)? Global Ecology and Conservation, 3:359–366,
+January 2015.
+[56] Gaurav Byagathvalli, Elio J. Challita, and M. Saad Bhamla.
+Frugal Science Powered by
+Curiosity. Industrial & Engineering Chemistry Research, 60(44):15874–15884, November 2021.
+[57] Botswana Predator Conservation Trust (2022). Panthera pardus csv custom export. https:
+//lila.science/datasets/leopard-id-2022/.
+16
+
+Figure 1: Visual Abstract displaying the Conservation Tool framework discussed in this piece. Silhouettes
+created by Gabriela Palomo-Munoz and Undraw.co.
+17
+
+Affordable
+Accessible
+Simple
+New Hardware
+New Software
+.": Table 1: Terms used by different groups practicing utilizing advanced and new technology to develop con-
+servation tools and references to find more information about each term.
+18
+
+Conservation Technology Terminology
+Term
+Defenition
+Reference
+Conservation Tools
+Devices that are made and developed to be applied to conservation of wildlife
+This Study
+Conservation
+An interdisciplinary field that works to design technology to help prevent the sixth-mass
+Bergertal 2018
+Technology
+extinction
+In-situ conservation
+Conservation of a species at the original place, e.g. in the wild
+Braverman 2014
+Ex-situ conservation
+Conservation of a species in a captive setting, e.g. at a zoo
+Braverman 2014
+Devices that are built for a particular industry, such as camera traps designed for
+Opportunistic
+hunters, but utilized for a different purpose, such as biologits using camera traps for
+Bergertal 2018
+technology
+ecology studies
+Silver-bullet solutions
+A one size fits all solution that can address and solve any issue
+Shaw 2017
+A design thinking that takes the context-of-use of the exact device as the primary
+Human-Centered Design
+Jacobson 2000
+design component
+A design thinking that takes the context-of-use of the exact device as the primary
+Context of Use
+Jacobson 2000
+design component
+A design thinking that is designed by the indigenous populations that are the most
+Indigenous design
+Nawrotski 2010
+familiar with the conservation initiatives
+Colonization of
+The historical implication that is conservation is performed by those that colonized the
+Loss 2011
+conservation
+states where the conservation is performed
+Solutions that are open access and solutions are fully accessible by the public to re-
+open-source solutions
+Lerner 2005
+create, re-design, and re-invent
+Frugal technology
+Technology that is built using frugal science techniques and
+Byagathvalli 2021
+The concept of re-purposing existing materials or products for uses other than that
+Frugal science
+Byagathvalli 2021
+they were originally intended is not new
+Lahoz-Monfort
+Passive solutions
+Technologies for species monitoring and conservation
+2012
+Technologies that are specifically for active solutions in the conservation tech field that
+This Study
+Active Solutions
+directly interact with the species, such as invasive species eradication
+Supervised Learning
+A machine learning subset of problems where the available data has labeled examples
+Stuart 2010.
+Unsupervised Learning
+A machine learning subset of problems that analyzes and clusters unlabeled data
+Schmarje 2021
+Self-supervised
+A machine learning subset in which a model trains itself to learn part of the input from
+Hendrycks 2019
+Learning
+another part of data, often leveraging the underlying structure of the data
+Object Detection
+Lin 2014
+class or set of classes
+Classification
+The computer vision process of predicting a class of one object to an image
+Krizhevsky 2012
+Object Re- Identification
+Takes object detection one step further by matching a given object in a new
+Stewart 2021
+(ReID)
+environment to the same object in a different environment
+The computer vision task of taking a set of initial object detections, creating a unique
+Tracking
+Yilmaz, 2009
+identifier for each detection, and tracking each object over a series of time
+Fine-Tuning
+The computer vision process of taking a model that has been trained on one task and
+This Study
+tuning it to make it perform a different, similar task
+A machine learning method that uses a pre-trained model as the starting point for a
+Transfer- Learning
+model in a new task (i.e. it has already learned how to "see" one set of things, and will
+Zhuang, 2019
+be trained again to get better focus on another set of things)
+Created with DatawrapperFigure 2: A) Audiomoth, B) Open source printed circuit board that audiomoth includes on website, C)
+Open source code for controlling and interpreting data from the audiomoth via Github, D) Online
+and app-based user-interface for audiomoth users. Images were taken from AudiomMoth website
+with permission.
+19
+
+B
+Power input
+LED signal output
+3.3V - 6V (IN)
+Green LED (Out)
+Red LED (Out)
+GND
+GPIO
+AL
+Power output
+BV(OUT)
+Manual bootloader
+Switch input 
+JSB/OF
+MEMSMic
+GND
+DEFAULT
+CUSTOM (C)
+USB/OFF (U)
+DEFAULT (D)
+Switch
+on
+off
+on
+Cc-U
+-D
+...
+AudioMoth Configuration App
+C
+OpenAcousticDevices/
+D
+16:54:1903/09/2021UTC
+Device ID:
+24A46B055DD2F953
+AudioMoth-...
+Firmware description:
+AudioMoth-Firmware-Basic
+1.6.0
+Battery:
+≤3.6V
+Schedule
+Rocording
+Fitering
+Advancod
+AnElectron-basedapplicationcapableof
+00.00
+06:00
+12:00
+18:00
+24:00
+Start recording:
+06:00
+configuring thefunctionalityof theAudioMoth
+End recording:
+00:60
+Add recording period
+recording deviceand settingthe onboard clock.
+Remove selected perid
+Clear all periods.
+03/09/2021
+☆ 22
+83
+03
+95
+03/09/2021
+Contributors
+Stars
+Forks
+Each day this wll produco 180 fios, ach 5280 kB, totating 950 MB
+Issues
+Daily energy consumption will be approximately 38 mAh
+Configure AudioMothFigure 3: Basic camera trap setup. A) A camouflaged camera trap is often placed on a tree or a pole.
+B) Camera trap model. It is equipped with a motion-triggering sensor, a digital camera, and a
+memory card. When an animal passes in the region of interest, the camera captures photos/video
+at a specified frame rate of the animal. C) The figure was made using a dataset from LilaBC57
+and images from Flickr.
+20
+
+A
+B
+9000
+666
+DIGITAL CAMERA
+animal
+PADLOCK READY
+approaches
+Camera Trap
+WEATHERPROOF CASE
+forest
+INFRARED SENSOR
+Detection Cone
+CapturedImage Data
+C
+Training()
+Machine Learning Re-ID Method:
+Ele1()
+Trained
+Ele2()
+Image
+Ele3()
+Classifier
+Model
+Ele99()
+Ele
+Input Data → Feature ID
+Prediction
+Trained
+Image
+Classifier
+ModelFigure 4: Sonar camera arrangement: Here are some diagrams of the camera deployment. On the left, you
+can see the camera shooting out multiple acoustic sonar beams - they are used to pick up fish in
+high resolution. The closer the fish are to the camera, the higher their resolution is. In the top
+right, you can see how these sonar cameras are placed to ”see” all areas where the salmon might
+swim. There are 2 sonar cameras - the one with the red triangle only captures one field of view;
+the one with the three narrow triangles oscillates between capturing three different stratas (20
+min at one, 20 min at the second, 20 min at the third). The bottom right image shows what these
+three strata images look like when combined. The white boxes are the annotated fish swimming
+through the stream. Images have been provided from the Alaska Department of Fish and Game
+and Caltech45
+21
+
+Nearshoretransducet
+River Flow
+0.30
+[w)
+unde
+5
+10
+15
+20
+30
+35
+40
+45
+50
+Source: ADFG
+Stratum 3:
+1151x1497px,6FPS,-1.8°
+Stratum2:784x1922px,9FPS,-0.7°
+Stratum 1
+River Flow
+288 x624px
+9 FPS-3.6°
+3m
+8m
+23m
+35mFigure 5: Utilizing game theory and optimization for conservation practices. A) Data mapping of a con-
+servation issue to determine which states conservation funding is most important. B) Raw map
+of the United States. C) Overlapped image of the clustering depicted in A with the raw map of
+the United States. D) Data interpreted map displaying large arrows in the states where the most
+conservation is needed with smaller arrows (in light green) displaying states where clustering is
+beginning. Images made using DataWrapper.
+22
+
+B
+c
+DFigure 6: Process that goes into designing and utilizing technology to develop new conservation tools.
+Silhouettes created by Gabriela Palomo-Munoz and Undraw.co.
+23
+
+Accessible
+Purpose-built hardware taking
+Human-Wildlife Centered Design
+into account
+Advances in Technology
+1
+Do No Harm
+Hardware Advances allowing
+Citing & Co-authoring
+Collaborations
+represented parties & locations
+Repeatable
+Software Advances allowing
+No silver bullet solutions being
+more accurate data collection
+attempted or parachute science
+Equitable
+Open-Source
+1
+Solution is fully open-source
+1
+hardware, software, and
+repairs